Paradigm Shift for Future Tennis

Part 1 · pp. 5–34

RüdigerDillmann, University of Karlsruhe, Faculty of Informatics, Institute of Anthropomatics, Humanoids and Intelligence Systems Laboratories, Kaiserstr. 12, 76131 Karlsruhe, Germany

Yoshihiko Nakamura, TokyoUniversityFac. Engineering, Dept. Mechano-Informatics,7-3-1Hongo, Bukyo-ku Tokyo, 113-8656, Japan

Stefan Schaal, University of Southern California, Department Computer Science, Computational Learning & Motor Control Lab., Los Angeles, CA 90089-2905, USA

David Vernon, Department of Robotics, Brain and Cognitive Sciences, Italian Institute of Technology, Genoa, Italy

Dr. Tijana T. Ivancevic Citech Research IP Pty Ltd. QLIWW IP Pty Ltd. & Tesla Science Evolution Institute, Mawson Lakes P.O. Box 342, Burn side, 5066 Adelaide, Australia E-mail: Tijana. ivancevic@ alumni. adelaide. edu. au

Dr. Bojan Jovanovic Sports Academy Save Kovacevica3/5 21000 Novi Sad, Serbia E-mail: jovanboj@yahoo.com

Sasa Jovanovic University of Novi Sad Art Academy QLIWW IP Pty Ltd Fruskogorska 30/143 21000 Novi Sad, Serbia E-mail: 2plus2studio@gmail.com

Dr. Milk a Djukic Alfa University School of Management in Sport Palmira Toljotija 3 Novi Beograd, Serbia E-mail: milk a_djukic@yahoo. com

Natalia Djukic QLIWW IP Pty Ltd. Tesla Science Evolution Institute South Australia P.O. Box 342, Burn side, 5066 Australia E-mail: Natalia. Djukic@ quantum leap into wealthy world. com

Dr. Alex an dar Lukman Sports Academy St. Deligradska 27 11000 Belgrade, Serbia E-mail: spak@sbb. rs

Cognitive Systems Monographs ISSN 1867-4925

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a spec ific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

In a historic Wimbledon 2009fi nal, thecelebratedSwissplayerRoger Federer became the greatest men’stennisplayerofalltime, beating the popular American Andy Roddick in an epicfivesets:5-7,7-6(8-6),7-6(7-5), 3-6,16-14. It was a battle of the two top serves in the game. Roddick won thefirst set. The second set went into a tiebreak; Roddick had4 set points, yet he lost them all, and with them, the set. Federer proved successful in the next tiebreak, as well. However, Roddick came back very strong in the fourth set, broke Federer, and won it, six games to three. The American continued, starting the fifth set absolutely on fire, yet could not endure the four hour and sixteen minute marathon than followed. Federer won this battle of muscles and nerves to make tennis history.

This was Federer’s 15th Grand Slam title, the total list consisting of: Wimbledon 2003, Australian Open 2004, Wimbledon 2004, USOpen 2004, Wimbledon 2005, USOpen 2005, Australian Open 2006, Wimbledon 2006,US Open 2006, Australian Open 2007, Wimbledon 2007, USOpen 2007,US Open 2008, French Open 2009, and Wimbledon 2009. It was also Federer’s sixth Wimbledon crown(2003,2004,2005,2006,2007,2009).

Pete Sampras, holder of the previously unbeaten record of 14 Grand Slam titles and seven-time Wimbledon champion(1993,1994,1995,1997,1998, 1999,2000), waswatchingthematchfromtheRoyalBox, along with fellow tennis legends; Sweden Bjorn Borg, afive-time Wimbledon champion(1976, 1977,1978,1979,1980), and Australian Rod Laver, two-time Wimbledon championofanOpenEra(1968,1969). Inaddition, theTVcommentators werenoneotherthanAmericanJohnMcEnroeandGermanBorisBecker, both three-time Wimbledon champions(respectively,1981,1983,1984and 1985,1986,1989).

What’smore, thiswasFederer’sfourthGrandSlamfinalsvictoryover Andy Roddick. Previously, Roger beat Andy at the finals of Wimbledon 2004 (46,75,76(3),64); Wimbledon 2005(6-2,7-6(2),6-4); andUSOpen 2006 (62,46,75,61). With his latest victory, the Swiss managed to extend his overallmasteryovertheAmericanto 19-2, including 8-0 at Grand Slams. Andallthisagainstanex-number-one player in the world. How can this astonishing resulting factor, 19-2, be possible? Is it due to Federer’s forehand? His backhand? Or, perhaps his commonly underestimated serve? Could it simplybetheentireFedererpackage? And if so, then what makes up this outstanding package? Notwithstandingtheexceptionalqualitiesoftheexnumber-one, inthisbookwewillrevealthesecretweaponofRogerFederer. And much more: we will reveal the blueprint of a future tennis champion.

What is a paradigm shift? This beautiful term invented by Thomas Kuhn in 1962 in his book“The Structure of Sci ent ificRevolution”, implies on revolutionary changes in hard science theories. Today, popular and accepted in many areas paradigm shift depicts a dramatic change of view and perceptions. Is it time to radically change last hundred years of science developmentand come on track of unifiedfield? Mechanics-oriented paradigm of today’sscienceiscomingtothe“no through road”journey proving that we can’tbe treated as a computer. Human beings and human existence, as well as the whole creation have the matrix beyond the physical reality, closely connected witheverythingandeveryoneintheUniverse.

A century ago, Nikola Tesla – the King of electricity – said: “When science begins the study of non-physical phenomena, it will make more progress in one decade then in all the previous centuries of its existence.”Tesla is one example of scientist who had holistic and deep approach to our Universe and his achievements consequently dramatically improved the quality of our life.

This book startswith revelations that make obvious the limitations of today’stenniswhichdoesnotusetheLawsofModernBiomechanics(see http://adsabs. harvard. edu/doi/10.2478/s 11534-009-0148-z) and Neurophysiology. The second part of the book includes anew approach to the quantum mind of achampion. This paradigm shift started with the work of eminent physicists John A. Wheeler and David Bohm, and their ideas that we are active participants in the creation. Before them, Erwin Schr¨odinger, one of the fathers of quantum mechanics, mentioned the term“entanglement”as the very essential part of his wave mechanics. Today Guardian Science states: “Entanglementsoftwoormoreparticlesthathaveonceinteractedalways remain bound in a very strange, hardly understandable way even when they are far apart and their connection being independent of distance.” Very powerfulquantumbrainisthemaincharacteristicsofRogerFederer, as well as someothersportchampionslikeMichaelSchumacher.

1 The Federer Phenomenon and His Secret Weapon. . . . . . . 3 1.1 The Best Serve Ever: It’sBiomechanicsand Neuro-physiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Secret behind Federer’sServe. . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Biomechanics of Federer’sShots. . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 The Mental Secret behind Federer’sWinners . . . . . . . . . . . . . . 8

2 The Law of Muscular Structure and Function. . . . . . . . . . . . 11 2.1 Busting the Myth of“Muscle Memory”. . . . . . . . . . . . . . . . . . 11 2.2 Structure and Function of Skeletal Muscles. . . . . . . . . . . . . . . 11 2.3 Slow and Fast Muscles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 The Law of Functional Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1 Anatomical Description of an Ordinary Tennis Serve . . . . . . . 17 3.2 Anatomical Description of an Ordinary Tennis Forehand. . . 18 3.3 Anatomical Description of an Ordinary Backhand. . . . . . . . . 18

4 The Law of Energy Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1 The Sources of Energy for Muscular Work. . . . . . . . . . . . . . . . 21 4.2 Maximal Oxygen Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 The Problem of Lactic Acid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 PhysiologicalEfficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.5 A Proper Warm-Up for Tennis. . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.6 Nutrition and Hydration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 The Law of Basic Biomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.1 Mechanics of Topspin and Backspin Shots. . . . . . . . . . . . . . . . 27 5.2 The Need for Biomechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3 The Basic Biomechanical Unit. . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.4 Degrees–of–Freedom in Human Joints . . . . . . . . . . . . . . . . . . . . 30 5.5 Minimum Dynamics Necessary for Tennis Biomechanics. . . . 30 5.6 TheDifferencebetweenMomentandMomentum. . . . . . . . . . 32 5.7 Dynamical Balance of Tennis Players. . . . . . . . . . . . . . . . . . . . 32

6 The Law of Neuro-motor Control . . . . . . . . . . . . . . . . . . . . . . . . 35 6.1 Control of Tennis Movements by the Brain. . . . . . . . . . . . . . . 35 6.2 The‘Missing Link’between Brain and Muscles . . . . . . . . . . . . 37 6.3 TheReflexArc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.4 The Role of SpinalReflexesinMovement . . . . . . . . . . . . . . . . . 39

7 The Law of Cognition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7.1 The Role of Sports Psychology. . . . . . . . . . . . . . . . . . . . . . . . . . 41 7.2 Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7.3 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.4 Genetically Coded Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.5 Fight-or-Flight Reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.6 The Importance of Creative Visualization in Motor Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

8 The Law of High Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8.1 The Current Tennis Performance Criteria . . . . . . . . . . . . . . . . . 49 8.2 The Main Purpose of Sports Science. . . . . . . . . . . . . . . . . . . . . 49

9 The Law of Athleticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9.1 Fast Legs of a Tennis Player. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9.2 Jumping Ability of Tennis Players. . . . . . . . . . . . . . . . . . . . . . . 51 9.3 More Dynamic Performance of Traditional Strength Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 9.4 A General Example of True Athleticism. . . . . . . . . . . . . . . . . . 53 9.5 TheEfficiencyTechnique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

10 The Law of Muscular Sling shots. . . . . . . . . . . . . . . . . . . . . . . . . 57 10.1 Busting the Myth of Loops and Swings in Tennis Strokes . . . 57 10.2 The Muscular Slingshot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 10.3 The Stretch–Reflex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 10.4 The Plyometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.5 The Role of Flexibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.6 How to Prevent Injuries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

11 The Law of Whip–Like Movements. . . . . . . . . . . . . . . . . . . . . . 67 11.1 Biomechanical Kinetic Chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 11.2 Biomechanics of the Power High-Forehand . . . . . . . . . . . . . . . . 68

12 The Newton-Euler Biomechanics Law. . . . . . . . . . . . . . . . . . . . 71

13 The Law of Superb Weapons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 13.1 Roddick’sServePhenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . 73 13.2 The Best Strength Exercise for the Serve. . . . . . . . . . . . . . . . . 74 13.3 The Best Speed Exercise for the Serve. . . . . . . . . . . . . . . . . . . 74 13.4 Common Technical Misconceptions about Tennis Forehand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13.5 A Complex Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 13.6 The Main Characteristics of a Good Return. . . . . . . . . . . . . . 75

14 The Law of Blitz–Chess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

15 The Law of ArtificialIntelligence. . . . . . . . . . . . . . . . . . . . . . . . 81

16 The Law of Mental Training. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 16.1 The Best Preparation for Mental Speed . . . . . . . . . . . . . . . . . . . 87 16.2 The MostEffectiveVisualizationExerciseforTennis. . . . . . . 87 16.3 The Common Problems with a Player’sMentalGame . . . . . . 87 16.4 Focus on the Balland Still Be Aware of the Opponent. . . . . 88 16.5 The Mental ProfileofaFutureTennisChampion. . . . . . . . . . 88 16.6 MasteringtheEffectiveVisualization. . . . . . . . . . . . . . . . . . . . . 90 16.7 Be Happy and Think Clearly All the Time. . . . . . . . . . . . . . . . 91 16.8 Realistic Goals versus a Wish to Bea Champion. . . . . . . . . . 91

17 The Quantum Intention Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 17.1 Quantum Brain and Mind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 17.1.1 Classical Neurodynamics. . . . . . . . . . . . . . . . . . . . . . . . . 93 17.1.2 Computation at the Edge of Chaos. . . . . . . . . . . . . . . . 95 17.1.3 Connectionism, Control Theory and Brain Theory . . . 98 17.1.4 Neocortical Bio physics. . . . . . . . . . . . . . . . . . . . . . . . . . . 99 17.1.5 Quantum Neurodynamics. . . . . . . . . . . . . . . . . . . . . . . . 103 17.1.6Bi-stable Perception and Consciousness. . . . . . . . . . . . 105 17.2 The Mental Quantum Leap INTENTION⇛ACTION⇛MANIFESTATION. . . . . . . . . 108

18 The Model for a Future Tennis Champion. . . . . . . . . . . . . . . 111 18.1 Superb Tennis Weapons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 18.2 Mind of a Champion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 18.2.1 Nine Essential Characteristic of a Champion’s Brain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

19 Tennis Evolution: An Intensive 12 Week Programme. . . . 115 19.1 Beginners Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 19.2 Intermediate Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 19.3 Advanced Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

20 Modern Tennis Physiology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 20.1 Muscular System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 20.1.1 Muscular Histology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 20.1.2 Classical Theories of Muscular Contraction. . . . . . . . . 122 20.1.3 The Equivalent Muscular Actu at or. . . . . . . . . . . . . . . . 130 20.1.4 Bio chemistry of Muscular Contraction. . . . . . . . . . . . . 130 20.2 Muscular Energy Systems and Their Flows. . . . . . . . . . . . . . . 132 20.3 Musculo–Skeletal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 20.3.1 Biodynamical Simulations. . . . . . . . . . . . . . . . . . . . . . . . 137 20.4 StretchReflexandMotorServo. . . . . . . . . . . . . . . . . . . . . . . . . 141 20.5 Human Brain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 20.5.1 Basics of Brain Physiology. . . . . . . . . . . . . . . . . . . . . . . . 151 20.5.2 Neural Field Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 20.5.3 Behavioral and Motivational Dynamics . . . . . . . . . . . . . 171 20.6 Neural Synchronization-REF. . . . . . . . . . . . . . . . . . . . . . . . . . 172 20.6.1 Oscillatory Phase Neurodynamics. . . . . . . . . . . . . . . . . 173 20.6.2 Kuramoto Synchronization Model. . . . . . . . . . . . . . . . . 173 20.7 Cerebellar Movement Control. . . . . . . . . . . . . . . . . . . . . . . . . . . 173 20.8 Brain Topologyversus Small–World Topology . . . . . . . . . . . . . 179

21 Modern Tennis Physics: Nonlinear and Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 21.1 Basics of Nonlinear Dynamics and Chaos Theory. . . . . . . . . . 183 21.1.1 Poincar´e’s Qualitative Dynamics, Topology and Chaos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 21.1.2 Smale’sHorseshoe: Chaos of Stretching and Folding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 21.1.3 Lorenz’Weather Prediction and Chaos . . . . . . . . . . . . . 199 21.1.4 Feigenbaum’sConstantandUniversality. . . . . . . . . . . 202 21.1.5 May’sPopulationModellingandChaos. . . . . . . . . . . . 204 21.1.6H´enon’s2 DMapandItsStrangeAttractor. . . . . . . . . 207 21.1.7 Nonlinear Neurodynamics. . . . . . . . . . . . . . . . . . . . . . . . 210 21.2 Basics of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 21.2.1 History of Quantum Mechanics. . . . . . . . . . . . . . . . . . . . 217 21.2.2 Double–Slit Experiment and the Riemann Sphere. . . 220 21.2.3 Feynman Path Integral in Quantum Mechanics . . . . . . 225

22 Modern Tennis Biomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 22.1 Jet Methods in Time–Dependent Lagrangian Biomechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 22.1.1 ConfigurationManifoldforAutonomous Biomechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

22.1.2 Biomechanical Bundle, Sections and Connections . . . . 231 22.1.3 Biomechanical Jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 22.1.4 Lagrangian Time–Dependent Biomechanics. . . . . . . . . 235 22.1.5 Local Muscle–Joint Mechanics. . . . . . . . . . . . . . . . . . . . 236 22.1.6 Time–Dependent Geometry of Biomechanics . . . . . . . . 237 22.2 Jet Spaces in Modern Hamilton i an Biomechanics. . . . . . . . . . 238 22.2.1 The Covariant Force Law . . . . . . . . . . . . . . . . . . . . . . . . . 240 22.2.2 Autonomous Hamilton i an Biomechanics . . . . . . . . . . . . 243 22.2.3 Time–Dependent Hamilton i an Biomechanics. . . . . . . . 245 22.2.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 22.3 Cerebellum Motion Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 22.3.1 Cerebellum as an Adaptive Path–Integral Comparator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

23 Modern Tennis Psychology: Nonlinear and Quantum. . . . 259 23.1 Cognition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 23.1.1 Intelligence and Factor Analysis . . . . . . . . . . . . . . . . . . . 259 23.1.2 Problem Solving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 23.2 Situation Awareness and Decision Making. . . . . . . . . . . . . . . . 315 23.2.1 Simple, IntuitiveSA–Model . . . . . . . . . . . . . . . . . . . . . . . 316 23.2.2 High–Dimensional and Time–VaryingSA–Model. . . . 316 23.2.3 NonlinearAnalysisoftheSA–Model. . . . . . . . . . . . . . . 317 23.3 Quantum Brain and Cognition. . . . . . . . . . . . . . . . . . . . . . . . . . 318 23.3.1 Bio chemistry of Microtubules. . . . . . . . . . . . . . . . . . . . . 318 23.3.2 KinkSolitonModelofMT–Dynamics. . . . . . . . . . . . . . 319 23.3.3 Macro–and Microscopic Neurodynamical Self–similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 23.3.4 Dissipative Quantum Brain Model . . . . . . . . . . . . . . . . . 328 23.4 Nonlinear Quantum Psychodynamics. . . . . . . . . . . . . . . . . . . . 333 23.4.1 Classical versus Quantum Probability . . . . . . . . . . . . . . 336 23.4.2 The Life Space Foam. . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 23.4.3 Geometric Chaos and Topological Phase Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 23.4.4 Joint Action of Several Agents. . . . . . . . . . . . . . . . . . . . 352

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

The momentous five-setter in a historic Wimbledon 2009 final, eventually ended when Federer broke Roddick’sserveforthefirst time all day(he himself was broken in the fourth set), with Roddick missing on a forehand. Federer had a total of 107 winners and38 unforced errors(whichisaW/Efactorof 107/38=2.8). Roddick, on the other hand, had 74 winners and 33 unforced errors (a W/E factor of 2.2). Hence, there ald iffere nce in these statistics is the incredible number of Federer’swinners. How does he make so many winners?

Firstly, the cheapest way toachieve a winner is by an ace. Federer served 50 aces, comparedtoRoddick’s 27 aces(whichisanA-factor of 50/27=1.85), to win this historic duel. On paper, his serve was more powerful than the most powerful serve in the game. But how can this be? Did Roddick serve poorly? No, Andy’s serve was unbroken, repeatedly around 220 km/h and twice at 230 km/h (143m/h), until the 77th–and last–game. Yet he only had 27 aces, while Federer, with a serve that was mostly under 200km/h, hit an astounding 50 aces. So, does this mean that the speed of the ball is not important for aces? No, the ball-speed is actually the most important factor for an ace; but it is not the only important factor. Is it topspin then? Unlikely, as Federer and Roddick have about the same amount of topspin on both theirfirst and second serves. What else is there? Well, this is the key question to ask, not just for this particular match, buttoexplainthewholeFedererphenomenon. Before answering this question, let’sbiomechanicallyanalyzethefas test serve in the game.

From Roddick’s serve (see Figure 1.1), we can see that, as is properly advised by many coaches, at a certain point the racquet-head is parallel to Roddick’sback. However, while most coaches assume(and hence teach their players)that the back should be upright at this instant, in Roddick’s case, it is not. At this point, his torso is already leaning forward, as well as twisted around on the vertical

Fig.1.1. The best serve ever of Andy Roddick: its biomechanical chain, whip-like movement and stretch-reflexinaction.

body axis. This ‘little’ difference from the common (static) serve concept already shows the two most important physical(dynamical)factorsofRoddick’sserve:

1. A whip-like movement, the main biomechanical factor(proximal-to-distal kinetic chain, in which there is always a time-lag between two consecutive joints:firstlegs, then torso, then shoulder, then elbow, then wrist); and 2. The stretch-reflexinaction, themainneuro-physiological factor(stretching the leg-extensors, torso, shoulder, elbow and wrist muscles prior to their ballistic contractions).

From both biomechanical and neuro-physiological point of view, Roddick’s serve is by far the best ever in the tennis game. It is the only true whip-like serving action(the model of a proper serving movement)andtheonlytruestretchreflex-based serve in tennis.

In modern handball, volleyball, American football, rugby and water-polo, it is common knowledge that a high-performance throw(or smash)necessarily resembles the whip-like movement of an elite javelin throw. A biomechanical term for this whip-like movement is the kinetic chain: the sequentialflowof energy and momentum from bigger segments to smaller ones. Tennis requires sequencedactivationofmusclesandmovementofbonesandjointstoachieve the motions, positions, and velocities seen in a player. Never rotate the“hips and shoulder together”, as you may often hear coaches advising while teaching the forehand. Instead, alwaysfirst rotate the hips, then the shoulders, then the elbow, then the wrist. It is harder to hit the“sweet-spot”on a tennis racquet with this“loose”movement, but the result is unbelievable. All elite players do these shots – which are not in textbooks or coaching manuals – for their winners. Kinetic energy and momentum, as well as muscular power, are developed from the legs, hips and trunk muscles and transferred to the arm muscles. This allows the energy, momentumandpowertobetransferredefficientlytothehands, moving the racquet-head with maximum speed towards the ball. Furthermore, to achieve the highly-efficient whip-like technique of tennis serves, forehands and backhands, a proper sequencing of muscle stretch-reflex based actions must take place.

Andy Roddick holds the record for the world’s fastest tennis serve: 246 km/h (153 m/h). His serve isa proper biomechanical chain, a full whip-like movement and the stretch-reflexinaction. The stretch-reflexcausesastretchedmuscle to contract stronger and at the same time inhibits the antagonist muscle from contracting (that is, slowing the movement). Because this is an involuntary reflex response, the rate of contraction is sign ifican tly(several times)faster and more powerful than a completely voluntary muscularcontraction. In fact, the faster the muscle is stretched eccentrically, the greater the force will be on the following concentric contraction.

When we train our skeletal muscles so that theyefficientlyutilizethestretchreflex(and the related stretch-shortening cycle), thenweeffectivelymakepowerfulslingshots out of them. The prototype of such a muscular slingshot is a javelin throw; in particular, the world-record throwofJanZelezny(who was at that time only 80kg!). We will talk more about Roddick’sservelater.

Sobothbiomechanicallyandneuro-physiologically, Andy Roddick has the best serve ever, clearly better than Federer’s serve. However, it is better by only about 10%(or, even less), on average. And there is a hidden factor on Federer’sside that is greater than 10%.

The explanation of the Ace-performance factor of 1.85 between Federer’s and Roddick’s serves is clearly non-physical: non-biomechanical and non-physiological. The explanation is cognitive. Roger Federer is simply the best blitz-chess player in the tennis game. This means three cognitive factors: He effectively plans his moves in advance; Heeffectivelyreads(anticipates)the opponent’smoves and acts accordingly; and Heeffectivelydisguiseshisownmoves. These three superior cognitive factors give Federer a huge mental advantage over his opponents, with the exception of his“nemesis”, Rafael Nadal–the current world number one, who appears to have similar cognitive factors(together with an extraordinary physical fitness and the huge amount of topspin on all of his shots).

In particular, thistennisblitz-chessisthekeytoFederer’saces. He plans the ball placement, anticipates Roddick’sreturnbeforeevenserving, and then disguises the motion of the serve. Intheepicalmatchat Wimbledon 2009, particularly in the tiebreaks and at the end of thefifthset, Andy could not read Roger’sserve, while Roger was reading more and more of Andy’sserves. And Federer is able to do the same in many other matches, against many other players.

Blitz-chessisthekeytomostofFederer’sin credible winners. Again, hefirst plans, then anticipates the opponent’smove, andfinally disguises his own shot. And we can always expect another winner. In the next report, we will talk about Roger’sforehand, backhand and volley winners.

Federer’sforehand, backhand and volleys(as well as his serve, which was addressed in thefirstpartoftheWimbledon 2009 Finals report)are much more dynamic than the textbook ones. They are actually quite similar to the forehand, backhand and volleys(as well as serve)of Pete Sampras. In both the cases of Roger and Pete, you will never see the two compulsory phases of classical tennis ground-strokes: “loading” and “hitting”. Instead, their most efficient winners are hit by apparently “loose” movements. They pretty much resemble the incredible 190km/h(120m/h)forehandofFrenchGaelMonfils, which was hit not by strength, but by a combination of arm speed and jointflexibility.

Even more importantly, themosteffectiveshots(of Roger and Pete, as well as Rafa Nadal)never use big loops or swings.

As this is a very common misconception, we need to elaborate on it a bit. The basic idea of textbook tennis(which means, used by a great majority of coaches and players), as well as the simplistic biomechanics of loops and swings in tennis shots(serve, forehand and backhand)are derived from the concept of asimple physical pendulum. An arm with a hand holding a racket has been seen as a more-or-less single rigid body, with no more than3 degrees of freedom(DOF). And really, if you have a robotic arm with only3 DOF, and you want to hit a ball with it, then you need to have a loop; and, even more, if you want to hit the ball hard, you need to have a big swing. That is absolutely true. In the language of modern biomechanics,“its phase space is asimple circle.”

That is how we originally go tour current loops–to play“nice tennis”; and swings – to be able to “hit hard”. In particular, you need a big swing, as you want to use all of the potential energy of the racket’sweight. Hence, as preparation for the shot(say, a forehand), you lift the racket-head as high as possible along the circle(that is, above your head). This is the common picture behind all loops and swings in tennis. Although simple to understand, it is not as easily implemented, which is where the many expensive lessons on the court come in. This system has produced thousands of young tennis players, all hitting the ball in virtually the same way, with the same distinguished tennis movements: loops and swings.

However, this picture is wrong! Our reality is much more complex than this simplistic model of a loop (circle) and a swing (potential energy along the circle), which produces only“tennis ballet”.

Firstly, the human arm has9 DOF(not includingfingers). Just the shoulder, elbow and wrist together have9 DOF. This is a redundant system, because the racket itself has only 6 DOF (three translations and three rotations). This means that there is an infinite number of possible ways to hit the ball, and we are able to choose the best way in each particular situation. Therefore, a real phase space for the human arm with a racket is much more complicated than the simplistic circle-loop model. Besides, every coach advises, quite correctly, not to use the arm only, but the wholebody. SoweactuallyhaveseveralhundredDOFatour disposal to perform each tennis stroke. Do you still believe that asimple circle and its associated loop is an appropriate model for the serve and ground-strokes of a tennis champion? Absolutely not, itissufficient only for small children!

Secondly, there is the question of utilizing the racket’s own potential energy – that is, raising the racket-head up high as preparation for the shot. This would make sense only for very weak players and very heavy rackets(in the same way that a weak player needs along swing to accelerate a massive racket). However, with a true athlete (and every elite tennis player has to be one such athlete), the current 400 gramracket is just not heavy enough to make this potential-energy contribution to the shot sign ificant. It would be sign ificant with a4 kgracket, but not with the current 400 gram one.

Put simply, loopsaretotallyuseless: it does not matter at all how you get into position for the shot. Swings are almost useless. They could be used, say, in thefifthhourofafive-set tennis match, to compensate for lack of strength, but this same fatigue would also make it hard to raise the racket-head high above your head in thefirstplace.

Inaddition, we see that simplistic mechanics does not work with the human body. Roboticsdiscoveredthisinthepastthreedecades; tennis science has yet to learn it.

The on lyefficient type of movements in tennis shots(serve, forehand and backhand), asusedbybothRogerFedererandPeteSampras, are whip-like movements, based on muscular sling shots. In other words, all the shots of Roger and Pete, justasAndyRoddick’sserve, clearly show the following two components:

1. A whip-like movement, the main biomechanical factor(proximal-to-distal kinetic chain, in which there is always a time-lag between two consecutive joints:firstlegs, then torso, then shoulder, then elbow, then wrist); and 2. The stretch-reflexinaction, themainneuro-physiological factor(stretching the leg-extensors, torso, shoulder, elbow and wrist muscles prior to their ballistic contractions).

As already stated above (in the case of Federer’s serve), Roger is simply the best blitz-chess player in the tennis game. Again, this means three cognitive factors:

1.Heeffectivelyplanshismovesinadvance; 2.He effectively reads(anticipates) the opponent’s moves and acts accordingly; and 3.Heeffectivelydisguiseshisownmoves.

These three superior cognitive factors give Federer a huge mental advantage over his opponents, withtheexceptionofRafaNadal, who appears to have similar cognitive factors.

Similarly to Federer’saces, the tennis blitz-chessisthekeytoallofFederer’s winners. He plans the ball placement, anticipates the opponent’sreturnbefore executing a shot, and then disguises the motion ofthestroke itself. He did it against Andy Roddick in theepical match at Wimbledon 2009, and he has done it in many other matches, against many other players.

Theblitz-chessisthekeytomostofFederer’sin credible winners. Again, he firstplans, then anticipates the opponent’smove, andfinally disguises his own shot. And we can always expect another winner.

The so-called complex reaction has the highest importance in every sports duel, including tennis. In general, a complex reaction has two components: smart anticipation and lightning-fast reaction. Youfirst anticipate the opponent’smovement, and then you react by making your own movement.

The basic speed of a reaction is largely genetically predetermined and cannot be sign ifican tly improved. The best means for developing it is training based on the stretch-reflex.

On the other hand, good anticipation of the opponent’sactionsistheessentialcharacteristicofamasterinanysportscompetition. The best anticipation is actually called mind reading, andthisisthediffere nce between master and disciple. Itis Federer’sgreateststrength. It can be learned, and it should be learned if you have high expectations in tennis.

Apart from getting experience from many tennis tournaments(atdifferent competition levels), the best tool for developing anticipation in young players is the blitz-chess.

Actually, the best preparation for the mental speed in tennis is the so-called lightning chess, or bullet chess. It is the faster version of a blitz chess game, whereeachsidehaslessthan3 minutes to complete all of their moves. Often, bullet chess is so fast that tactics and skill are secondary to quick moves.

1. Confusion about strategy; 2. Trying to do too many things at once; 3. Being easily distracted; 4. Too much concern about winning or losing; 5. Perfectionism;

6. Complacency; 7. Having no clear plan or goal; 8. Too much spontaneity and creativity; 9. Lack of humility; 10. Inappropriate reactions to errors; Negative self-talk(self-criticism).

We could argue that Andy Roddick still feels some of these problems, and that by resolving them, hewouldeasilybeabletowinGrandSlams. At the same time, both Roger and Rafahavealreadyresolvedtheminamosteffectiveway.

1. Set clear goals; 2. Create action steps that will take you closer to these goals (learn to visualize: try to“see”in your mind what you want toachieve); 3. Use positive self-talk(self-criticism); 4. Practice yoga breathing techniques; 5. Always put forth 100%effort; 6. Be processoriented; 7. Stay detached; Not dwell on the future or past – stay in the present moment; 8. Be non-reactive to the opinions of others; 9. Tolerate your inability to be perfect; 10. Do what it takes to have fun and smile; 11.Definew inning in such away that it includes more than just thefinalscore.

Finally, you need to focus on the ball and yet stillbeaware of the opponent and the court. This is a big problem for most players, although Roger and Rafa seem to solve it consistently and routinely. Aside from a great tolerance for competitive pressure, focus is one of the key characteristics of a champion. You really need to “fix your eyes on the ball,” as every coach advises, and yet have a full amount of the so-called situational awareness(SA), which is defined in modern psychology as“the perception of the elements in the environment within a volume of time and space, the comprehension of their meaning, and projection of their status in the near future”. Forexample, SAisakeyfactorinthetrainingoffighter pilotsandFormula1 car-racing drivers. The outcome of any complex situation criticallydependsonSA. Therefore, you need to develop:

1. A strong focus on the moving tennis ball; 2. TheabilityforstrongSA, a sdefinedabove; and 3. TheabilitytofocusonboththeballandSAatthesametime.

This is similar to learning to play a hard piece on the piano:first you learn the right-hand part, then you learn the left-hand part, and then you learn how to combine them together. Obviously, physical and mental fatigue(including other disturbances, suchas rain, wind, a noisy audience, the big overhead screen, etc) willgraduallyreducebothfocusandSA. Therefore, besides physical speed and strength, you obviously need great mental strength and endurance to be able to win.

Roger Federer has all of these mental qualities. That is why he is the greatest tennis player ever. That is the secret behind the Federer phenomenon. It is not his forehand, his backhand, or his serve. The real secret is Federer’spowerful mind.

Roger Federer is a phenomenon, no doubt about that. However, for all those talented kids who want to be the active participators in the tennis game – not just the passive observers–the most important question is: How can we go beyond Federer? Can we formulate the blueprint for thefutureFederer? In other words, Is there really such a thing as Tennis Science?

In the following Chapters, we will formulate the main laws of tennis science.

From the sci ent ificperspective, the common term“muscle memory,”so popular among coaches and players, is shear nonsense. If neural motor pathways are damaged there is no any“musclememory”left. This means that all the motor memory(containing all acquired motor skills)is in the neural system, not in the muscles. Muscles have their structure and function(of generating muscular force), they can be exercised and trained in strength, speed, endurance and flexibility, but they are still only dum beffectors(justlikeexcretoryglands). They only respond to neural command impulses. To understand the process of training motor skills one needs to know the basics of neural motor control. Within the motor control muscles are force generators, working in antagonistic pairs. Theyarecontrolledbyneuralreflexfeedbacksand/or voluntary motor inputs. If any of these neural pathways are damaged, muscles are deadflesh without any memory left(see The Law of Neuro-Motor Control).

Skeletal muscles are composed of musclefiberswrappedbyfasciclesthatalso envelopedbloodvessels. Muscles are connected to adjacent bones via elastic tendons, whicharemadeupofepimysium, perimysiumandendomysium(see Figure 2.1). Each musclefiber(or muscle cell)is made up of many myofibrils. Eachmyofibrilcontainsaseriesofcontractileunitscalledsarcomeres, made up from two types of proteinfilaments: thin act infilamentsandthickmyosin filaments(see Figure 2.2). See Appendix 1, for technical details.

The muscles are“slaves”that contract(shorten)only when their“masters”, called motor units, fire (see Figure 2.3), otherwise they relax. Therefore, muscles can only pull (contract), never push. They usually work in mutually antagonistic pairs(see The Law of Basic Biomechanics).

Fig.2.1. Structure of human skeletal muscles. Epimysiumconsistsofcollagenfibers that surround muscle tissue. Perimysiumisaconnectivetissuethatseparatesadjacent fasciculi(small bundles of musclefibers)in a skeletal muscle. Endomysiumsurrounds individualfibers. Epimysiumandperimysiumcontain blood vessels and nerves.

Fig.2.2.Basicdiagramofsarcomereshortening.

1. Contractility(shortening while generating a force); 2. Excitability(responding to neural stimuli); 3. Elasticity(recoil ing to original resting length after being stretched).

1 The force–velocity curve (Figure 2.4) shows the relationship between muscle tension (or, generated force)and the velocity of its shortening or lengthening. This fundamental muscular curve is used to analyze theeffectsofthespeed–strength training, as well as to identify musclefibretypesusedindifferentphysicalactivities.

Fig.2.3. One motor nerve/neuron and the muscle fibers that it controls via two motor units. Motor unitsfirefullyornotatall(allornoneLaw).

1. Concentric contraction, in which muscle shortens and makes positive movement–its generated force is greater than the loading force; 2. Isometric contraction, in which muscle generates force without shortening (there is no visible movement)–its generated force is equal to the loading force; 3. Eccentric contraction, in which muscle generates force, yet the negative movement occurs because the loading force is greater than the muscle– generated force.

1. When a muscle is activated and required to lift a load which is lessthan the maximum tetanic tension it can generate, the muscle begins to shorten. Contractions that permit the muscle to shorten are referred to as concentric contractions. An example of a concentric contraction in the raising of a weight during a bicep curl. In concentric contractions, the force generated by the muscle is always lessthan the muscle’smaximum. As the load that the muscle is required to lift decreases, contraction velocity increases. This occurs until the musclefinallyreaches its maximum contraction velocity. By performing aseries of constant velocity shortening contractions, a force-velocity relationship can be determined.

2. In isometric contraction the muscle is activated, but instead of being allowed to lengthen or shorten, it is held at a constant length. An example of an isometric

Fig. 2.4. Muscular force–velocity curve related to three types of muscular contractions: (a)concentric contraction(gives the positive movement),(b)isometric contraction (no movement), and(c)eccentric contraction(negative movement). The graph shows movementsperformedwithfourdifferentspeeds. The shaded area under the curve represents muscular power.

contraction would be carrying an object in front of you. The weight of the object would be pulling downward, but your hands and arms would be opposing the motion with equal force going upwards. Since your arms are neither raising or lowering, your biceps will be isometrically contracting. The force generated during an isometric contraction is wholly dependant on the length of the muscle while contracting.

3. During normal activity, muscles are often active while they are lengthening. Classic examples of this are walking, when the quadriceps(knee extensors)are active just after heel strike while the kneeflexes, or setting an object down gently(the armflexorsmustbeactivetocontrolthef all of the object). As the load on the muscle increases, itfinally reaches a point where the external force on the muscle is greater than the force that the muscle can generate. Thus even though the muscle may be fully activated, it is forced to lengthen due to the high external load. This is referred to as an eccentric contraction(please remember that contraction in this context does not necessarily imply shortening). There are two main features to note regarding eccentric contractions. First, the absolute tensions achieved are very high relative to the muscle’s maximum tetanic tension generating capacity (you can set down a much heavier object than you can lift). Second, the absolute tension is relatively independent of lengthening velocity. This suggests that skeletal muscles are very resistant to lengthening.

Yes, some muscles are more suited for explosive speed–strength type activities, while others are adapted for endurance exercises. More precisely, there are three basic types of human skeletal muscles:

• Type I-slow twitch muscles, which are dark–red in color(under electronic microscope; they are predominately used in the aerobicenergy system; they havehighconcentrationofmyoglob in, lotsofcapillariesandmitochondria.) • TypeIIA-intermediate fasttwitch, which are reddish–white in color; their contraction is moderately fast; they have moderate myoglobin concentration; they are predominately used in anaerobic glycolysis energy system. • Type IIB - ultra fast twitch, which are white in color; their contraction speed is the fastest; they have low myoglobin concentration; they are predominately usedinanaerobicalactaticenergysystem.

There is no such thing as specialized“tennis muscles”. What is crucial for the success in tennis isn euro–motor control, of which the most important is the stretch–reflex(we will talk about it later).

Classical tennis serve has three stages:1(i)the balltoss,(ii)the jump, and(iii) thefinishingsmash.

(i)In the case of a right–handed player(like Federer), the balltoss is thrown with the left arm. The feet are apart, and the balltoss is performed with the contractions of the left deltoid eus, the biceps and the palmarflexorsmuscles. This movement is done simultaneously with two other preparatory actions.

Thefirst one of these preparatory actions is raising the rightarm,“loading”. The muscles used to carry this out are the right deltoid eus, supraspinatus(a muscle going over theshoulder blade) and the biceps brachii. The second action is bending the knees, and thus preparing for the second stage of the serve(the jump). There are noflexormusclesusedtobendtheknees, for the bending of knees is accomplished by gravity alone(actually, leg–extensors are used in eccentric fashion).

(ii)Thefirst–serve jump is performed high and forward. It is achieved by instantaneous actions of all the leg extensor muscles; left and right soleus, quadriceps femoris and gluteus maximus muscles. Jumping is the second part of “loading”in the serve. At the same time as the player lift soff,the racquet is placed behind the body, ina“back–scratching”position, and the right shoulder’srotationtowardstheballbegins. This movement involves the right biceps brachii and wrist extensor muscles. While in the air, the feet naturally join together (with Federer, the feet join in the air, not on the ground).

(iii)Thefinishingsmashtakesplaceintheair, before the player returns to the ground. To end the serve, the shoulders are rotated and the ball is hit simultaneously. By then, the shoulders should have been fully rotated and the feet prepared for landing. The internal and external obliques abdominal muscles complete the shoulder rotation. Hitting the ball is performed by the latissimus

1 We will skip the notorious preparation here.

dorsi, then pectoralis major andfinallytricepsbrachiimuscles. To adda bit of spin or slice to the serve, the wrist is flicked slightly at the end, using the palmar flexors.

Standard tennis forehand, in the case of a right–handed player(like Federer), basically has two phases: (i)preparation, or “loading”, and (ii)hitting the ball.

(i)Preparation for the forehand includes two simultaneous actions. One is stepping into the right position, with the leftleg forward. The other is the first half of the loop movement, lifting the racquet above the shoulders in a curvedc-shaped movement.(This does not need to be too far back, like in Hewitt’sforehand, but can be more to the side.)This is accomplished by the right deltoid eus and biceps brachii muscles.

(ii)Hitting the ball includes four main movements. Thefirst of these four is a right hip rotation towards the ball, while the feet are still on the ground. The right gluteus maximus and medius muscles carry out this action.

Secondly, a leap into the air is necessary to be able to hit the ball from a higher body position, so as not to hit the net. This is performed by all the leg extensor muscles: left and right soleus, quadriceps femoris and gluteus maximus –working together.

Thirdly, an arm swing of the racquet, the second part of the loop action. The right pectoralis major, deltoid eus and biceps brachii muscles complete this.

Lastly, to create topspin; a slight twist of the wrist to just brush over the ball. This is done by the right palmarflexors.

Acommon right–handed player’stwo-handed backhand(e.g., Andy Roddick’s) basically has two stages:(i)preparation, or“loading”, and(ii)hitting the ball.

(i)Preparation for the double-handed backhand includes two phases. One is stepping into the right position, with the right leg forward. The other is lifting the racquet in a loop movement similar to the forehand loop, only this time the racquet tends not to go further than about the shoulder–level. This is accomplished by both the left and right deltoid eus and biceps brachii muscles.

(ii)Hitting the ball includes four main movements. The first of these four is a left hip rotation towards the ball. The left gluteus maximus and medius muscles carry out this action, helped by the left knee extension, which is performed by the quadriceps femoris. Secondly, a rotation of the left shoulder towards the ball, achieved by both the internal and external obliques abdominal muscles.

Thirdly, the arm swing of the racquet is performed by pulling of the rightarm and pushing of the left arm. The pulling action of the right arm is accomplished by the right latissimus dorsi, pectoralis major and triceps brachii muscles. The pushing action of the left arm is accomplished by the left deltoid eus, pectoralis major and biceps brachii muscles.

Lastly, to create topspin; a slight twist of both wrists. This is done by the left palmarflexorsandrightdorsalflexors.

Part 2 · pp. 35–64

1. ATP–CP(or, anaerobic1 alactatic)system, which lasts 10–15 seconds, uses stored ATP2 and creatine phosphate (CP), with no by–products. This energy source is related to speed and strength. It is essential for the serve and winning shots in tennis. 2. Glycolysis(or, anaerobic lactic)system, which lasts 15 seconds–3minutes, usesbloodglucoseandmuscularglycogentomakeATP; its by–product is lactic acid. This energy source represents anaerobic endurance. It is essential in long, exhausting tennis relays. 3. Aerobic system,3which lasts from 2–3 minutes to several hours, uses glucose, glycogen, fats, and proteins to make ATP within the aerobic energy pathway; its by–products are carbon dioxide and water. This energy source represents aerobic endurance. It is essential in the fourth andfifthsetofanyserious tennis match.

In general, the maximal oxygen consumption, orVO 2− max, is the maximum volume of oxygen (O 2) that the human body can consume by breathing air at sea

1“ Anaerobic”means without the use of oxygen, that is, none of its metabolic activity will involve 02. 2 Adenosine–Triphosphate(ATP)is the fundamental energy source for muscular work, madeinthemitochondriaofmuscularcells. It gives energy for muscular contraction according to the equation:

where ADPis Adenosine–Diphosphate and Piis inorganic phosphate. ATP is resynthesizedbothaerobically and anaerobically. 3“ Aerobic”means in the presence of oxygen(02), that is, all of its metabolic activity will involve 02.

Fig.4.1.O 2− consumption byskeletalmusclesinmitochondri as.

level during an intense whole–body exercise. This volume is expressed as a rate in millilitersperkgbodyweightper minute(ml/kg/min). AsO 2− consumption is linearly related to energy expenditure, when one measure sO 2− consumption, they are indirectly measuring an individual’s maximal capacity for aerobic work.

Every cell consume sO2 inordertoconvertfoodenergytousableATPfor cellular work (see Figure 4.1). However, it is muscle that has the greatest range in oxygen O 2− consumption (at rest, muscle uses little energy; however, contracting musclecellshavehighdemandsforATP, proportionally to the work intensity). Endurance athletes have developed a strong cardiovascular system, as well as a strongoxidativecapacityintheirskeletalmuscles. To receive thi sO2 and use it tomakeATPformuscularcontraction, our musclefibers depend on2 things: (i)anexternaldeliverysystemtobringO2 from the atmosphere to the working muscle cells, and (ii) mitochondria in the muscle cells to carry out the process of aerobic energy transfer. In other words, one needs a big and efficient heart–pump to del i verO 2− rich blood to the muscles, andonealsoneedsmitochondria–rich muscles to use theO2 and support high rates of exercise.

As the fourth or (even worse, fifth hour) of a five-set tennis match approaches its end, the main problem for both players becomes the accumulation of lactic acid. It feels like‘muscle soreness’. Every tennis player has experienced this painful fatigue. This is how it happens.

Thecarbohydratesyouconsumeconsistofseveraldifferentsugarmolecules; sucrose, fructose, glucose, etc. However, by the time the liver does it’sjob, all of these sugars are converted to glucose which can be taken up by all cells. Muscle fibers take up glucose and either use it immediately, or store it in the form of long glucose chains(polymers)called glycogen. During exercise, glycogen is broken back down to glucose which then goes through a sequence of enzymatic reactions that do not require oxygen to proceed. All of these reactions occur out in the cell fluid, or cytosol. These reactions proceed very rapidly and yield some energy for muscle work in the process. This glycogen/glucose breakdown pathway is called the anaerobic glycolysis(or, glucose breakdown without oxygen)pathway. Every single glucose molecule must go through this sequence of reactions for useful energytobewithdrawnandconvertedtoATP, theenergymoleculethatfuels muscle contraction, and all other cellular energy dependant functions.

In a single contracting muscle fiber the frequency and duration of contractions willdetermineATPdemand. ATPdemandwillbemetbybreakingdowna combination of two energy sources: fattyacidsandglucosemolecules(ignoring the small contribution of protein for now). As ATP demand increases, the rate of glucosefluxthroughglycolyticpathwayincreases. Therefore at high workloads within the singlefiber, therateofpyruvicacidproduction will be very high. If the musclefiberispackedwithlotsofmitochondria, pyruvatewilltendtobe convertedtoAcetylCoAandmoveintothemitochondria, with relatively little lactate production. Additionally, fatty acid metabolism will account for a higher percentageoftheATPneed. Fat metabolism does not produce lactate, ever! If lactate is produced from glucose breakdown, it will tend to be transported from the area of high concentration inside the muscle cell to lower concentration out of the musclefiber and into extra-cellularfluid, then into the capillaries.

Now let’s look at an entire muscle, say the quadriceps muscle group during cycling. At a low workload, glycolytic flux is low (fatty acid breakdown is relatively high at low intensities) and thepyruvate produced is primarily shuttled into the mitochondria foroxidative breakdown. Since the intensity is low, primarily slow twitch muscle fibers are active. These fibers have high mitochondrial volume. As workload increases, morefibers are recruited and already recruitedfibershave higher duty cycles (more work and less rest). Now ATP demand has increased in the previously activefibers, resulting in higher rates of pyruvic acid production. A greater proportion of this production is converted to lactic acid rather than enteringthemitochondria, duetocompetitionbetweenthetwoenzymesLDH and PDH. Meanwhile, some fast twitch motor units are starting to be recruited. This will add to the lactate produced in and transported out from the working muscle due to the lower mitochondrial volume of these fibers. The rate of lactate appearance in the bloodstream increases.

The quadriceps is just one of several muscles that are very active in cycling. With increasing intensity, increased muscle mass is called onto meet the force production requirements. All of these muscles are contributing more or less lactic acid to the extra-cellular space and blood volume, depending on theirfibertype composition, training status and activity level. However, the body is not just producing lactate, butalso consuming it. The heart, liver, kidneys, and inactive muscles are all locations where lactic acid can be taken up from the blood and eitherconvertedbacktopyruvicacidandmetabolizedinthemitochondriaor used as a building block tore-synthesize glucose(in the liver). These sites have low intracellular lactate concentration, so lactic acid is transported into these cells from the circulatory system. If the rate of uptake, or disappearance, of lactate equals the rate of production, or appearance, in the blood, then blood lactate concentration stays nearly constant. But, when the rate of lactate production exceeds the rate of uptake, lactic acid accumulates in the blood volume, then one can see the onset of blood lactate accumulation. This is the traditional “lactate threshold”(LT).

1. Exercise intensity; 2. Training status of active muscles(highermitochondrialvolumeimproves capacityforoxidativemetabolismathighglycolyticfluxrates); 3. Fiber type composition(slow twitchfibers produce less lactate at a given workload than fasttwitchfibers, independent of training status); 4. Workload distribution(a large muscle mass working at a moderate intensity will develop less lactate than asmall muscle mass working at a high intensity); and 5. Rate of blood lactate clearance(with training, bloodflowtoorganssuchas theliverandkidneysdecreaseslessatanygivenexerciseworkload, due to decreased sympathetic stimulation; this results in increased lactate removal from the circulatory system by these organs).

Recall from above that high level endurance performance depends on two factors: (i)ahighVO 2- max, and(ii)a high lactate threshold. YourVO 2- max sets the upper limit for your sustainable work potential. On the other hand, the lactate threshold tells us something about how much of the cardiovascular capacity you can take advantage of in a sustain edeffort; it is determined by skeletal muscle characteristics and training adaptations. MultiplyingVO 2- maxwithLTgives usa measure of theeffectivesizeofyourenduranceengine. Now one comes to efficiency. What do esefficiencyhavetodowithenduranceperformance?

Physiological efficiency is defined as the percentage of energy expended by the body that is converted to mechanical work:

One can measure the mechanical work performed using an ergometer, like a bicycle ergometer, or rowing machine. One can measure the energy expended by the body indirectly via its oxygen consumption at sub maximal workloads. With some basic bio chemistry, one can convert the oxygen consumption one measures duringexercisetoastandardmeasureofenergylikekJoules, or Calories. And, one can do the same for the work one measures on the ergometer. Work divided by Time equals Power. Power is measured in watts and is a measure of the intensity of work. Intensity(watts)times exercise duration(minutes)gives us total work, againmeasuredinkJoulesorCalories.

Whilst the warmup for participation in any sporting or exercise activity is accepted as being essential for minimizing injuries and improving performance, the methodsbywhichmanysportsattempttoachievethisarelessthanideal. The warmup method used by many tennis players usually includes an initial jog around thefieldorcourt, followed by 10–15 minutes of static stretching. This is then followed by a few drills, and the players then begin their training session or game. Whilst the basis behind these methods may appear to be sound applications of current training principles, acloseranalysisrevealsmajorlimitations with this method of preparing a player for tennistraining.

The main physiological reason for a warmup include: to increase core temperature(anincreaseinrectaltemperatureofaleastonetotwodegreeCelsius appears to be sufficient); to increase heart rate and blood flow to skeletal tissues, whichimprovestheefficiencyofoxygenuptakeandtransport, carbondioxide removal, and removal and breakdown of anaerobic byproducts(lactate); to increase the activation ofthe Central Nervous System (CNS; therefore increasing co-ordination, skill accuracy and reaction time); to increase the rate and force of muscle contraction and contractile mechanical efficiency (through increased muscle temperature); and to increase the suppleness of connective tissue(resulting in less incidence of musculo–tendinous injuries).

The major criticism against the“typical warm-up”is that it does not adequately prepare the athletes for the demands placed upon them in the ensuring session. Generally the initial jog is at a pace that has a minimal effect upon body temperature, and usually consists of jogging forwards, and in a straight line.

Similarly, the stretching performed is usually that of static stretching, with most stretches performed slowly and with the athletes either standing still or sitting on the ground. This method of stretching has been shown to be beneficial for the increase in limb range of motion, and aims to relax the muscles so that they are less resistant to passive stress for stretching. But this type of stretching does not prepare the muscle and connective tissue for the active contraction - relaxation process that will occur with any running, jumping or hitting movements as required in a tennis game situation.

During this stretching period(typically from 5-20 minutes), the body is very efficient in removing excess body heat, so the small increase in body temperature from the initial jog is quickly lost if the athlete does nothing but statically stretch for this time. This is even more prevalent in cold climates or cold seasons. In general, many injuries occur at the beginning of a competition due largely to an inadequate preparation for the activity. Also, inadequate warming-up can lead to lessthan optimal speed and skill levels that could result in quick scoring by the opponent or individual early in the game leading then to opponents having to catch-up placing more pressure on the player(s)involved.

The proper warm-up should be the complete physical and mental preparation for the dynamic actions to follow. The players should be able to begin the game or training session totally ready to perform at maximal intensity if required.

The initial jog is now replaced with a more dynamic series of running exercises that include regular alternation of running forwards, backwards, sideways, high knee drills, butt flicks, crossovers, bounding, jumps and progressive sprints. This component will only take 2-4 minutes depending on the climate. It is expected that the athletes are breathing quite heavily at the end of this short series of exercises.

With the stretching component, static stretching can still be included in the program, as many athletes still feel they need some static stretching to really prepare themselves. However, the dynamic stretching component is very important for the spec ificpreparationofthemusculaturetodynamicmovements. Dynamic stretching i sdefined as repetitive contractions of an agonist(prime mover)muscle to produce quick stretches of the antagonistic muscle, so any active callisthenicmovementcanbeclass ifiedasdynamicstretching(jumping, body rotations, bending, etc).

Tennis matches can last much longer than bouts in many other sports, so good nutrition and hydration are very important. Because of the amount of muscle exertion and stamina needed for tennis, the most important part of a player’s diet is complex carbohydrates. Professional players eat up to eight servings of pasta, rice and bread everyday. This is because the energy provided by carbohydrates is released slowly, compared with simple carbohydrates like sugars which provide a quick burst in energy.

Tennis players can lose up to 2.5 liters offluid an hour through perspiration, so it is important to replace thesefluidsbydrinkingwhenyou play. This means hydrating (drinking) before, during and after playing. Some drinks are better at hydrating you than others, andsomelikecaffeinateddrinksactuallydehydrate you. The best things to drink when you’re playing tennis, or any other sport, are energy drinks, water, and fruit juice. Because of their sugar content, soft drinks give a short-term energy boost for up to half an hour. Sports energy drinks have two big advantages over water. Firstly, they contain carbohydrates, which are very important for maintaining energy levels. Secondly, sports drinks replace electrolytes, suchas sodium, which are lost through perspiration.

A topspin shot is hit by sliding the racquet up and over the ball as it is struck. By dragging the racquet over the ball, the friction between the racquet’s strings and the ball is used to make the ball spin forward, towards the opponent. The shot dips down after impact and also bounces at an angle lower to the ground than a shot hit with no topspin. As a ball travels towards a player after bouncing, it has natural topspin that is caused by the friction of the tennis court. When hitting a topspin shot, the player is reversing the spin of the ball, which requires more energy.

A backspin shot is hit in the opposite manner, by sliding the racquet underneath the ball as it is struck. This causes the ball to spin towards the player who just hit it as it travels away. Generating slice, or backspin, requires only about half the rackethead speed compared to hitting topspin, because the player is not required to change the direction in which the ball is spinning. The oncoming ball bounces offthe court with topspin, spinning from top to bottom as it comes toward the player. When aplayer returns the ball with a slice shot the direction in which the ball spins around the axis of rotation is maintained. The direction of the shot changes, but the ball continues to spin from top to bottom, from the player’sperspectiveasitmovesawayfrom the player.

In tennis, one transfers the energy from their body to the ball via a tennis racket to generate speed and spin of the ball. Energy can be either potential(stored energy)or kinetic energy(energy of movement). A spec ifictypeofpotential energy is elastic energy(that is, the energy which causes, or is released by, the elastic distortion of a solid or a fluid). An example of elastic energy is the energy stored in a spring under tension. The human equivalent would be energy stored in muscles and their tendons under tension. On the other hand, kinetic energy spec ificallyreferstotheworkrequiredto accelerate the ball from a resting position to a desired velocity.

Let’sex amine how the body transfers the necessary energy to the ball in a tennis stroke. Here, one thinks of the body as aseries, or a chain, of linkages connected to one another and affectingeachotherinaspecificsequence. For example, the foot is a link, which is connected to the leg by the ankle joint, which is in turn connected to the thigh by the knee joint and soon. During the initiation of a forehand groundstroke the feet are oriented for either an open stance or close stance position. The shoulder and torso are turned approximately 45 degrees, which in turn causes a“coiling”of the abdomen and pelvis, which in turn produce a slight knee bend. With the current forehand the racket is held fairly high at about head level. In this position there exists a great deal of potential energy, both in the form of gravity with the rackethead up high and the form of elastic stored energy in the tensed muscles that are stretched in the coiled position(both internal and external abdominal obliques muscles, pectorals major, forearm muscles, hip girdle musculature, quadriceps femoris). This energy is released in sequence and there is an overlap in the sequence of linkages. As the racket starts to drop and begin an oval path(loop)the hips start to uncoil. The hips and knees begin to straighten. In sequence with the uncoiling of the hips the next event is the uncoiling of the torso and then the shoulders as the racket is brought forward to contact the ball. At the same time, the back leg is fully extended to powerfully drive the body up and forward. In fact, many professional players actually leave the ground during this point. At ball contact only medium grip pressure is required to guide and stabilize the racket. This is because the forward momentum will carry the racquet through the ball without much effort. After contact the shoulder and torso and hips naturally rotate towards the non-dominant side following the path of the racquet resulting in a stretch of the opposite side musculature which decelerates the racquet.

Naturally, all of this occurs in onefluid motion with precise timing so that maximum energy(and momentum)transfer occurs from loading to releasing. And, for maximum racket-head speed, some body segments may be slowed down to increase the speed of the racket, as in cracking a whip. Thus, we one see that not only one does need some basic biomechanics at all levels of his/her tennis maturity, but as one advances in tennis, one even needs a special biomechanics of whip–like movements, which is crucial to make every serve, forehand and backhand–efficientweapons(we will talk about this in detail later).

The basic biomechanical unit consists of a pair of mutually antagonistic muscles producing a common muscular torque, TMus, in the same joint, around the same axes. The most obvious example is the biceps–triceps pair (see Figure 5.1). Note that in the normal vertical position, the triceps downward action is supported by gravity, that is the torque due to the weight of the forearm and the hand (with the possible load in it).

Fig.5.1. Basic biomechanical unit: left–tricepstorqueTTriceps; right–biceps torque TBiceps.

Fig.5.2. Degrees-of-freedom in main human joints.

Some joints are only slightly movable, formed by two bones held together by cartilage, withoutjointcavity(e.g., an intervertebral joint in the spine consists of two vertebra sand an intervertebral disc between them). On the other hand, major joints involved in human movement, like shoulder, hip, elbow and knee, are composed of several bones separated by a joint cavity, lubricated by synovial fluid and enclosed in afibrousjointcapsule. Different joint shaved ifferentdegrees– of–freedom(DOF)of movement: hinge joints have1 DOF, gliding and saddle joints have2 DOF, while ball-and-socket joints have3 DOF(see Figure 5.2).

Briefly, dynamicsofhumanmotionisgovernedbytheNewton-Eulerequations of a rigid body motion in our3 Dspace(see Figure 5.3 depicting a tennis ball modelled as a free rigid body 1). A rigid body freely moving in space has 6 degrees of-freedom:3translations(alongtheX, Y, Z-axes)and3 rotations(around the X, Y, Z-axes). Translational motion i sdefined by3 Newton’sequations of motion of the type:

torque(T)=inertia-moment(I)×rotational-acceleration (aR).

In Figure 5.3, with each axis(denotedbyindexx, y, z)there is associated a translationalF-equation and a rotation alT-equation; vdenotesvelocity, while dot over a quantity denotes it’srate-of-change(with respect to time). Also, inertia-moment(I)=mass(m)×moment-arm squared(l 2).

1 It is obvious that this is only an approximation, as a tennis ball is not rigid, but rather a soft body. However, soft–body dynamics is much more complicated, so let’ssticktoourrigid–body approximation. Even true rigid–body dynamics is more complicated.

Fig.5.3.Newton–Eulerdynamicsofatennisballmodelledasafreerigidbody.

torque(T)=rate-of-change of angular momentum( ˙pR).

Newton’s Causality Principle states: a force is a cause of acceleration, which is a cause of velocity, which is a cause of linear motion. Similarly, a torque is a cause of angular acceleration, which is a cause of angular velocity, which is a cause of angular motion.

In biomechanics, the only active force is muscular force. In every major human joint an antagonistic pair of muscles generates a driving torque. Any kind of human motion is a result of driving torques in major human joints.

power(P)=force(F)×velocity(v).

Physiologically, it corresponds to the area under the force–velocity curve(see Figure 2.4). Muscular power is the key element of all power sports, including tennis.

I∼mr 2: mass × moment–arm2 (∼means“proportional to”) pR= IvR: moment of inertia × angular velocity, analogous to: p = mv pR= p × r: angular momentum=linear momentum × moment–arm

T=˙pR= I ˙vR: torque=derivative of angular momentum; this is analogous to: F=˙p = m˙v: force=derivative of linear momentum

T= F× r: torque=force × moment–arm

Note that in anatomical literature our moment–arm is called lever–arm.

The above Newton-Eulerdynamics(depictedinFigure 5.3) has its neural sensors. The vestibular organ in the inner ear helps to maintain equilibrium by sending the brain information about the motion(both linear and angular)and position of the head. Thevestibularorgansconsistofthreemembranoussemicircular canals (SCC), and two large sacs, the utricle and saccule. All the vestibular organs share acommon type of receptor cell, the hair cell.

TheSCCwithinthevestibularorganofeachearcontainfluidandhairreceptorcellsencasedinsideafragilemembranecalledthecupula. Thecupulais located in a widened area of each canal called the ampulla. When you move your head, thefluidintheampullalagsbehind, pushingthecupulaaverytinybit which causes the hairs to also bend a very tiny bit. The bending hairs stimulate the hair cells, which in turn trigger sensory impulses in the vestibular nerve going to the brain to“report”the movement. Hair cells are amazingly sensitive.

Forexample, acupulamovement of even athousandth of an inch is detected by the brain as a big stimulus.

TheSCCarepositionedroughlyatrightanglestooneanotherinthethree planes of space. Thus, the canals react separately and in combination to detect different types of angular head movement. They detect when one nods in an up and down motion(pitch), when one tilts their head to the side towards their shoulder(roll), and when one shakes their head“no”in aside-to-side motion (yaw). TheSCCareresponsiblefordetectinganykindofrotationalmotionin the head, thuseffectivelysensingtheEuler’sdynamics(Figure 5.3).

Two other vestibular organs are located in membranous sacs called the utricle andthesaccule. Ontheinsidewallsofboththeutricleandthesacculeisabed (amacula)of several thousand hair cells covered by smallflat piles of calcium carbon ate crystals which look likes and, imbeddedinagel-like substance. The crystalsarecalledotoliths, a word which literally means“ear stones.”In fact, theutricleandthesacculeareoftencalledtheotolithorgans.

When a person’she a dis in the normal erect position, the hair cells in the utriclelieapproximatelyinahorizontalplane. When the head is tilted to one side, the stones want to slide “downhill.” This moves the gel just enough to bend the sensory hairs. The bending hairs stimulate the hair cells, which in turn send a signal to the brain about the amount of head tilt. The stones also move if the person is accelerated forward and back, or side to side. Similarly, the hair cellsinthesacculeareorientedinsomewhatofaverticalpositionwhenthe head is erect. When a person tilts their head, or is accelerated up and down(as in an elevator), or moved forward and back, theotolithsmoveandasignalis sent to the brain. Thesignalsfromtheotolithsinthesacculeandtheutricle complement each other and give us an integrated signal about our movement. Theotolithorgansareprimarilyresponsiblefordetectinganydegreeoflinear motion of the head, thus effectively sensing the Newton’s dynamics (Figure 5.3).

All of the body’svoluntarymovementsare controlled by the brain. One of the brain areas most involved in controlling these voluntary movements is the motor cortex(see Figure 6.1).

Fig.6.1. Mainlobes(parts)of the brain, including motor cortex.

In particular, to carry out goal–directed tennis movements, your motor cortex must first receive various kinds of information from the various lobes of the brain: information about the body’s position in space, from the the par i etal lobe; about the goal to be attained and an appropriate strategy for attaining it, fromthe anterior portion of the frontal lobe; about memories of past strategies, from the temporallobe; and soon.

There is a popular“motor map”, a pictorial representation of the primary motor cortex(or, Brodman’sare a 4), called Penfield’smotorhomunculus 1( see Figure 6.2). The most striking aspect of this motor map is that the areas assigned to various body parts on the cortex are proportional not to their size, but rather to the complexity of the movements that they can perform. Hence, the areas for the hand and face are especially large compared with those for the rest of the body. This is no surprise, because the speed and dexterity of human hand and mouth movements are precisely what give us two of our most distinctly human faculties: the ability to use tools and the ability to speak. Also, stimulations appliedtotheprecentralgyrustriggerhighlylocalizedmusclecontractionson the contralateral side of the body. Because of this crossed neural control, this motor center normally controls the voluntary movements on the opposite side of the body.

Fig.6.2. Primary motor cortex and its motor map, the Penfield’shomunculus.

Planning for any tennis movement is done mainly in the forward portion of the frontal lobe of the brain(see Figure 6.3). This part of the cortex receives information about the player’scurrentpositionfromseveralotherparts. Then, like the ship’s captain, it issues its commands, to Brodman’s area 6, the premotor cortex. Area6 acts like the ship’slieutenants. It decides which set of muscles to contract to achieve the required tennis movement, then issues the corresponding orders to the primary motor cortex (Area 4). This area in turn activates spec ific muscles or groups of muscles via the motor neurons in the spinal cord.

1 Note that there is a similar“sensory map”, with a similar complexity–related distribution, called Penfield’ssensoryhomunculus.

Fig.6.3. Planning and control of movements by the brain.

Between brain (that plans the movements) and muscles (that execute the movements), the most important link is the cerebellum(see Figure 6.1). For you to perform even so simple a gesture as touching the tip of your nose, it is not enough for your brain to simply command your hand and arm muscles to contract. To make the various segments of your hand and arm deploy smoothly, youneed an internal“clock” that can precisely regulate the sequence and duration ofthe elementary movements of each of these segments. That clock is the cerebellum.

The cerebellum performs thisfinecoordinationofmovementinthefollowing way. First it receives information about the intended movement from the sensory and motor cortices. Then it sends information back to the motor cortex about the required direction, force, and duration of this movement (see Figure 6.4).It acts like an air trafficcontrollerwhogathersan unbelievable amount of information at every moment, including(to return to our original example)the position of your hand, your arm, and your nose, the speed of their movements, and the effects of potential obstacles in their path, so that your finger can achieve a “soft landing”on the tip of your nose.

Therefore, toensureefficiencyoftennismovements, i.e., that all the movements are fast, precise, and wellcoordinated, the nervous system must constantly receive sensory information from the outside world and use this information to adjust and correct the hand’strajectory. The nervous system achieves these adjustments chiefly by means of the cerebellum, which receives information about the positions in space of the joints and the body from the proprioceptors. Even for a movement as simple as picking up a glass of water, one can scarcely

Fig.6.4. The cerebellum loop for movement coordination.

Fig.6.5. Five main components of areflexarc:(1)sensory receptor,(2)sensory (afferent)neuron,(3)spinalinterneuron,(4)motor(efferent)neuron, and(5)effector organ–skeletal muscle.

imagine trying to consciously specify the sequence, force, amplitude, and speed of the contractions of every muscle concerned. Therefore, the cerebellum clearly does a very important job in tennis.

Areflexis a spinal neural feedback, the simplest functional unit of a sensory– motor control. Itsreflexarcinvolvesfivebasiccomponents(see Figure 6.5):(1) sensory receptor, (2) sensory (afferent) neuron, (3) spinal interneuron, (4) motor (efferent)neuron, and(5)effectororgan–skeletal muscle. Its purpose is the quickest possible reaction to the potential threat.

While the motor cortex plans the movements(top control level), the cerebellum makes themefficient(middle control level), at the most basic level, movement is controlled by the spinal cord alone, with no help from the brain. The neurons of the spinal cord thus take charge of thereflexmovementsaswellas the rhythmic movements involved in walking. Wewilltalkaboutreflexeslater. For the lightning–speed of the future tennis game, they are the most important neural part.

The main topics of sports psychology are:(i)motor(and general)learning;(ii) behavioral patterns(e.g.,fight-or-flight);(iii)visualization;(iv)concentration; (v)relaxation strategies;(vi)selftalk(introspective thought)strategies;(vii) arousal strategies; and(viii)stress management. We will talk about each of them in some detail in this book.

Learning is a relatively permanent change in behavior that marks an increase in knowledge, skills, or understanding thanks to recorded memories. A memory1 is the fruit of this learning process, the concrete trace of it that is left in your neural networks.

More precisely, learning is a process that lets us retain acquired information, affectivestates, and impressions that can influenceourbehavior. Learning is the main activity of the brain, inwhichthisorgancontinuouslymodifiesits own structure to bet terreflecttheexperiencesthatonehashad. Learning can also be equated wit hen coding, thefirst step in the process of memorization. Its result, memory, is the persistence both of autobiographical data and of general knowledge.

But memory is not entirely faithful. When you perceive an object, groups of neurons in differentparts of your brain process the information about its shape, color, smell, sound, and soon. Your brain then draws connections among these different groups of neurons, and these relationships constitute your perception of the object. Subsequently, whenever you want to remember the object, you must reconstruct these relationships. The parallel processing that your cortex does for this purpose, however, can alter your memory of the object.

1“ The purpose of memory is not to let us recall the past, but to let us anticipate the future. Memory is a tool for prediction”–Alain Berthoz.

Also, in your brain’s memory systems, isolated pieces of information are memorizedlesseffectivelythanthoseassociatedwithexistingknowledge. The more associations between the new information and things that you already know, the better you will learn it.

If you show a chess grand master a chessboard on which a game is in progress, he can memorize the exact positions of all the pieces in just a few seconds. But if you take the same number of pieces, distribute them at random positions on the chessboard, then ask him to memorize them, he will do no betterthan youorI. Why? Because in thefirstcase, he uses his excellent knowledge of the rules of the game to quickly eliminate any positions that are impossible, and his numerous memories of past games to draw analogies with the current situation on the board.

Forgetting is another important aspect of memorization phenomena. Forgetting lets you get rid of the tremendous amount of information that you process every day but that your brain decides it will not need in future.

Human memory is fundamentally associative. You can remember anew piece of information better if you can associate it with previously acquired knowledge that is alreadyfirmlyanchored in your memory. And the more meaningful the association is to you personally, themoreeffectivelyitwillhelpyoutoremember. Memory has three main types: sensory, short–term and long–term (see Figure 7.1).

2 Attentiveness is often said to be the tool that eng raves information into memory. 3 It is easier to learn when the subject fascinates you. Thus, motivation is a factor that enhances memory. 4 Your emotional state when an event occurs can greatly influence your memory of it. Thus, if an event is very upsetting, you will forman especially vivid memory of it. The processing of emotionally-charged events in memory involves norepinephrine, a neurotransmitter that is released in larger amounts when one is excited or tense. As Voltaire put it,“That which touches the heart is engraved in the memory.” 5 Our memory systems are thus context ual. Consequently, when you have trouble remembering a particular fact, you maybe able to retrieve it by recollect ing where you learnt it or the book from which you learnt it.

Fig.7.1. Human cognitive memory.

The sensory memory is the memory that results from our perceptions automatically and generally disappears in lessthan a second.

The short–term memory depends on the attention paid to the elements of sensory memory. Short–term memory lets you retain apiece of information for lessthan a minute and retrieve it during this time.

The working memory is a novel extension of the concept of short–term memory; it is used to perform cognitive processes(like reasoning)on the items that are temporarily stored in it. It has several components: a control system, a central processor, and a certain number of auxiliary“slave”systems.

The long–term memory includes both our memory of recent facts, which is often quite fragile, as well as our memory of older facts, which has become moreconsolidated (see Figure 7.2). It consists of three main processes that take place consecutively: encoding, storage, and retrieval (recall) of information. The

Fig.7.2. Long–term memory.

purposeofencodingistoassign a meaning to the information to be memorized. Storage can be regarded as the active process of consolidation that makes memories less vulnerable to being forgotten. Lastly, retrieval(recall)of memories, whether voluntary or not, involvesactivemechanismsthatmakeuseofencodingindexes. In this process, information is temporarily copied from long-term memory into working memory, so that it can be used there.

Retrieval of information encoded in long-term memory is traditionally divided into two categories: recall and recognition. Recall involves actively reconstructing the information, whereas recognition only requires a decision as to whether one thing among others has been encountered before. Recall is more difficult, because it requires the activation of all the neurons involved in the memory in question. In contrast, in recognition, evenif apart of an object initially activates only apart of the neural network concerned, thatmaythensufficetoactivatethe entire network.

Long–term memory can be further divided into explicit memory(which involves the subjects’ conscious recollection of things and facts) and implicit memory (from which things can be recalled automatically, without the conscious effort needed to recall things from explicit memory, see Figure 7.2). Episodic (or, auto bio graphic)memory lets you remember events that you personally experienced at a spec ifictimeandplace. Semantic memory is the system that you use to store your knowledge of the world; its content is thus abstract and relational and is associated with the meaning of verbal symbols.

Procedural memory, which is unconscious, enables people to acquire motor skills and gradually improve them. Implicit memory is also where many of our conditioned reflexes and conditioned emotional responses are stored. It can take place without the intervention of the conscious mind.

Whatever behavior one initiates, be it drinking, playing, reading, making strategic alliances, or making eyes at someone, itis always because one is subjectively feeling certain needs. A spec ificsituationleadstoaspecificneuralactivitypatterninapers on’sbrainthatinturnleadstoaspecificbehavior.

Each individual’sgenesactivateauniqueprogram for the development of that person’snervoussystem. But how this nervous system actually develops depends on each person’sin ter actions with the environment, that is on unique personal experiences. The behaviors that this person is capable of are determined by the unique activity patterns of his/her nervous system, some of which are experienced as thoughts, emotions, memories, etc. Any given behavior by this person results from the interaction between his/her neural activity at that spec ificmomentandhis/her perception of this spec ificbehavior.

Firstly, our most primitive behaviors(likeourreflexes)are concerned with the present. These are incapable of adaptation. They make us react to external or internal stimuli automatically. Neurophysiologically speaking, these behaviors represent the activation of the“reptilian”structures of the brain, bringing the hypothalamusandthebrainstemintoplay.

Secondly, our learned behaviors add our past experience to our present actions. These more sophisticated behaviors involve remembering pleasant or unpleasant sensations that one experienced in the past and the actions that caused us to experience them at the time. These behaviors represent most of the social and cultural knowledge that one acquires. In connection with these behaviors emotions arise, the awareness of the cardiovascular adjustments necessary for action. Neurophysiologicallyspeaking, these behaviors represent the activation of the“mammalian”structures of the brain, bringingthelimbicsystemintoplay.

Thirdly, our imagined behaviors respond to the present through past experience, by anticipation of the future result. They involve more elaborate planning. They call on the creative imagination, and hence on the associative cortex, to develop strategies for ensuring that our actions will be gratifying rather than painful. They represent the creative and innovative abilities of the human mind. Neurophysiologicallyspeaking, these behaviors represent the activation of the “neocortical”structures of the brain, bringing the associative areas of the cerebral cortex into play.

A human and animal behavior is a set of movements coordinated by the nervous system to preserve the structure of the organism. The basic behavior is therefore to approach or explore the resources available in the environment. When an action to acquire one of these resources is rewarded, this gratifying behavior is positively reinforced, and the strategy through which the need was sat isfiedis memorized.

The other main basic behavior is to avoid pain, and hence to avoid situations that might lead to the organism’s premature death. Fight,flight, and inhibition

Fig. 7.3. Behavior is memorized if it succeeds and discarded for an alternative strategy if it fails.

of behavior are the three possible behavioral responses to a nociceptive stimulus. Just like gratifying behaviors, the response offighting,fleeing, or inhibiting behavior inresponse to a threatening stimulus can bee i thereffectiveorineffective. The associated behavior is then memorized as either a winning or a losing strategy(see Figure 7.3).

Our environment is full of potentially gratifying objects that stimulate our approach behaviors. But it is also full of other people who also want to use these resources to ensure their own well-being. Each individual must therefore learn to decode other people’sin tentions in order to choose the attitude that will make hisorherownactionsthemosteffective.

Forexample, if you resisted somebody who was trying to take some resource away from you, and you ended up not only losing the resource butalso getting injured in the struggle, then the next time, the memory of this failure might very well cause you to simply runaway.

And if this person were someone from whom you could notflee(forexample, because you were economically dependent on them), then you would learn that the best thing to do would be to inhibit your own behavior and accept your subordinate status. This is thewayhowsocialhierarchiesare established.

The three reactions that let you avoid pain trigger hormonal and vasomotor adjustments that are controlled by the sympathetic nervous system. The activation of this system provides increased motor autonomy, mainly by increasing the oxygen a tion of the skeletal muscles. But these changes cannot last i ndefinitely, or they would interfere with the healthy functioning of those parts of the body, suchas the internal organs, that were left temporarily short of blood. Once the source of the threat has disappeared, the body’sequilibriummustthereforebe rapidly restored.

Mental imagery, or creative visualization plays animportant role in the learning of sport movements and the improvement of motor performance. The latest research is attempting toidentifythedeterminantsofefficientmentalimaging in sport. Studies have already shown that people who are better at generating mental images make faster progress in motorlearning, as do people who practice a movement physically before imagining it mentally. Researchonmental imagery has shown that the processes for actually producing a movement and for representing it mentally are identical. Various experimental approaches have been used to show, forexample, that the mental representation of an action seems to be based on the same mechanisms as the motor preparation for it.

With mental chro no me try, forexample, it has been shown that visual mental images preserve the spatial and structural characteristics of the object or scene that they represent. Forexample, it has been shown that the visual travel time between two points in a mental image of an object is proportional to the distance between these two points on the actual object.

Researchers have also found some physiological indicators whose activation resulted solely from mental imaging of actions. For example, a group of researchers measured how physical training and training by mental imaging affectedthe strength offingermuscles. Physical training increasedfingermusclestrength by 30%, but mental imaging alone still increased it by 22%. Since the subjects did not make any muscle contractions during their mental imagery training, the observed changes did not come from the peripheral motor system but actually came from the activation of circuits in the central motor system.

Here is yet another example. When subjects were asked to imagine walking or running at various speeds on a treadmill, theirheartrateandtotalventilationincreasedinproportiontothespeedimaginedinthecourseofthismental exercise, even though their oxygen consumption remained steady.

According to the theory for which Swedish neurobiologist David Ing var coined the clever term“future memory”, the par i etal cortex is capable of producing internal models of movements to be performed, prior to any processing in the premotorandmotorcortices. According to this theory, the brain is constantly simulating movements, only some of which are eventually externalized. This theory could provide a conceptual foundation for the mentaltraining done by athletes and musicians, aswellasforre-education through motor imagery.

The concept of motor imagery extends to the sensory modalities as well. When someone has any given sensory experience, the nre-imagines its later, the brain activity in the two cases will be similar in both location and intensity. Because brain activity continuously influencesthebody, and vice versa, any experience has a given effect on the body, and re–visualizing it would generate a similar pattern of brain activity with similar effects on the endocrine system, the immune system, and soon. Speaking very generally, visualization can therefore be regarded as a form of auto suggestion or self–hypnosis which, by generating emotions, may have a beneficialphysiologicaleffectonthebody.

Question: Do you think that top tennis players, like Federer, Nadal, Djokovic, and Roddick, can improve their performance with an appropriate sport–science knowledge?

Answer: Absolutely! Modern sports science can improve performance of any athlete! Very briefly, Roger Federer needs only superb tennis weapons (explained below). Rafael Nadal needs the same plus Federer’s tennis knowledge. Andy Roddick mostly needs Federer’santicipationandmentalstrength. Novak Djokovic needssuperbtennisweaponsplusFederer’stennisknowledge.

The overriding principle governing general sports performance is the attempt of an individual, or a group of individuals, to perform a given task“in the best possible way.”In this chapter we will focus on biomechanical and physiological principles of the performance optimization in the future tennis game. Forthe tennis performance criteria one can use the 10 points of the standard tennis game statistics(inbrackets are the current ranksofRogerFederer, the greatest tennis champion, on October 22nd 2007, asgivenbyAT Ptennis. com):1

• Service game:(i)number of aces(4),(ii)1stservepercentage(29),(iii)1st serve points won(6),(iv)2ndservepointswon(1),(v)service games won (3), and(vi)breakpoints saved(8). • Return of service:(vii)points won returning1 stserve(4),(viii)points won returning2 ndserve(17),(ix)breakpoints converted(36), and(x)return games won(10).

The very purpose of sports science is to provide solutions for two essential sport problems:

1. A direct–training problem: given the set of empirically proclaimed talents, develop the champion model. 2.An inverse–selection problem: given the champion model, develop the talent model.

Thus, sports science is all about training methods(directed to make a champion) and selection methods(directed tofindingtalents). In sports science all the

1 Although one can immediately see that the standard statistics does not give us the full picture ofthe current tennis, as Federer is here ranked (1) only once, while, e.g., Ivo Karlovic, currently world number 25, is here ranked(1)four times.

Part 3 · pp. 65–94

champions are represented by the champion model for a particular discipline (e.g., tennis), and all the talents are represented by the talent model for the same discipline. In statistical language, both the champion and the talent have the same factor structure–only it is fully developed incase of the champion and yet undeveloped in case of the talent. For example, Nadal, Roddick, Federer, and Djokovic had all been talents. However, so far only one of them has proved to bea real champion – Roger Federer, the man who apparently defiesall tennis statistics. Today, in my opinion, the highest chances to become future tennis championshaveNadal, Djokovic and Andy Murray.

Running fast is the direct result of the athletes stride rate and stride length. Now, the question is how do one maximizes both of these toachieve top–level performances in the sprints, or fastest runs on the court. One cannot have a maximum stride length and stride rate and be their fastest; what is needed is a maximum stride rate with an optimal stride length. Maximal stride rate is how fast one can produce one stride, or about 10of them in 20 meters. Stride rate is dependent upon a number of factors including, strength and mechanics. In order to produce greater stride rates one must be able to execute the correct stride cycle as fast as possible and with optimal length. Optimal stride length is one that allows the athlete to execute the correct stridepattern in as short a time frame as possible.

On the other hand, ground time is the largest contributor to stride rate. Itis known that almost all athletes spend approximately the same amount of time in the air during the sprint stride. Thebigdiffere nce comes in the amount of time spent on the ground. The goal of all sprinters and fast-legged tennis players should be to spend as little time on the ground as possible. In order toachieve this, they need the necessary plyometric strength(explained later)to get them through the correct cycle.

Also, during the short sprints on the court, at each leg joint the musculotendinous units absorb force by stretching(eccentric)just before they shorten (concentric)to generate the take–offforce.

Jumping ability i sdefined as leg power. Recall that the muscular power is the product of muscular force(F)and movement velocity(v). It is represented by the area under the force–velocity curve(see Figure 2.4 above). In other words, it is the ability to generate muscular force quickly.

If you look at the force–velocity curve, you will see that high levels of power occur in the mid–range of either force or velocity. If an athlete develops greater power, this, in turn, enhances his ability to generate both force(strength)and velocity(speed). This amalgam of speed and strength maybe more useful for athletic performance than strength alone.

Therefore, the leg power is the expression of the leg strength at speed. Inthe future tennis game, players at all levels will require certain levels of leg power to be successful, the higher the level, the higher the power needed.

The velocity(speed)component of the jumping ability i sdefined as the stretch–shortening cycle(to be explained later). Before one can focus on the stretch–reflexanditsderivative, stretch–shortening cycle, one needs to know how to develop the strength (force) component of the jumping ability. There are several common methods for this, including:

1. General leg–strength exercises. These exercises are necessary to develop the force component of power. Squat exercises suchas full squats, front squats, half squats, and split squats, develop the jumping musculature to a large degree. Very few, if any, successful highlevelfieldeventathleteswouldnot have good squatting ability. Even in novice athletes, some form of squatting could be performed with medicine balls or shot puts to challenge the jumping musculature. Although there is great benefitinperforminggeneralstrength exercises suchas squats and its variants, one must be careful not to use the squat as an end in itself as these exercises are slow in nature thus do not replicate the exact demands of the power events. 2. Special leg–strength exercises. These exercises attempt to convert general strength to spec ificjumpingstrength. Some examples may include Olympic lifts and their variants and jump squats. Some researchers suggest that the Olympic lifts are very similar in structure to a vertical jump. As jumping and throwing events have some component in the vertical direction the Olympic lifts are excellent exercises for these events. 3. Plyometric leg–strength exercises. These exercises are necessary to develop the velocity component of power and attempt to provide power improvement in away which is spec ificto the required technique of an athlete. Examples of such exercises would include bounding and hopping, weighted bounding and hopping inaddition to single and double legged boxjumps.

Take forexample the popular bench–press exercise. You can perform itina traditional, bodybuilding–type form, where you remain tight throughout your entire body, and throughout the entire repetition. Alternatively, you can do it ina Russian plyometric form, which is more relaxed. This later one involves a faster negative movement (eccentric contraction), and a ballistic pressing action (concentric contraction), in reality more like a push press than a bench press (this is technically termed the stretch–shortening cycle). It is clear that the traditional exercise gives you a slow strength only, while the plyometric one gives you both strength and speed, that is the dynamical power. And this dynamical power is what one really needs for future tennis.

In simple mechanical words, given that force(F)equals mass(m)×acceleration (a), and mass is basically always constant in tennis, what one really needs is a huge acceleration. This huge acceleration will produce a huge racket-head speed, which will generate the lightning–speed of the ball.

In contemporary tennis, themostprominentexampleofathleticismisRafael Nadal. Other examples include the current champion, Roger Federer, and American James Blake.

Notwithstanding the common public sympathy towards athleticism of the popular players, we need to remark that every serious 9- year-old (or older) gymnast can do a somersault on the street; or, equally, on the tennis court during the tennis game! This shows that the very idea of“athleticism”has historically been a little-bit misunderstood within the tennis community.

Let us have a look at the real athleticism of a person with almost the same body posture and weight as Federer’s. Weare naturally talking about the track and-fieldhero, Jan Zelezny, the javelin world record holder, winner of three Olympic gold medals(Barcelona, Atlanta, Sydney)andthreeWorldChampionshipsgoldmedals(Stuttgart, Gothenburg, Edmonton). He hasfiveworld javelin records, including the actual one of 98.48m(set in 1996, Germany), as well as 34 performances over 90.00 m (which is more than all other javelin throwers combined).

This guy is not “a big boy”. He was not over 2 m in height and not over 130 kg (as some“experts”might suggest, by considering his super–human throws). He didnotlookatalllikeArnoldSchwarzenegger, but rather as a younger and biggerbrotherofBruceLee. Zelezny’sbodyheightwasonly 1.86 m,(that was exactlybetweenRoddick’sand Federer’s), while his bodyweight(at the time of his world-record throw)was only 80 kg. This is the same weight as Federer and the same as 1.87 mtallNovakDjokovic. At the same time, he is5 kgless than Nadal, or8 kglessthanRoddick, or8 kglessthanTommyHaas, even on ekg less then the 1.82 mtallFernandoGonzalez. That is to say, except for Federer and Djokovic, all these current elite tennis players are simply too heavy. Therefore, you do not need to be heavy and bulky tobe able to hit hard! Tennis is not shot–put! The mass of the tennis racket is half the mass of the javelin. Therefore, commonsense would say that the best javelin thrower of all times had to be heavier than top tennis players. Yet, at the time of his fantastic world record(as well as his 34 throws over 90.00 m), Zeleznylookedprettyskinny. What he had, though, were muscles made of steel and lightning–fastreflexes. That is, Zelezny’sbodywasanefficientslingshot!

In other words, a man with a body very similar to the current champion, Roger Federer, was able to throw an 800 gjavelin (purposely reconstructed that it cannotflyfar), almost 100 meters. Could you imagine the speed of a tennis serveorforehandwithmusclessimilartoZelezny’s. With muscles trained(for years) in utilizing the stretch–reflex. Zelezny would not only break the strings, if he attempted the tennis serve, but would also brake the frame of the“modern” tennis racket. A totally new racket technology would be required for players with muscles similar to Zelezny’s. This example shows the real athleticism: after years of proper stretch–reflex training, the body becomes a big slingshot, composed of a number of small sling shots: legs, hips, torso, shoulders, elbow and wrist–all working as a kinetic chain, in a dynamical power–sequence.

Now, we do not suggest that a tennis player will ever need to be able to throw a javelin close to 100 meters, like Zelezny. No, a 70 m throw of the current (reconstructed)javelin, quite achievable for junior javelin throwers, would give a 300+km/hservewithatechniquesimilartoRoddick’s, as well as 240+km/h forehand and a 200+km/h(single–handed)backhand with a technique similar to Federer’s.

What we are suggesting is that the future tennischampionwillnecesarily havesomethingsimilartoZelezny’sbodyheightandweight(which was, atthe time of his record throws, quitesimilartoNovakDjokovicbodyheightand weight), but more importantly, Zelezny-like slingshot muscles. It is just a matter of proper stretch–reflexbasedmuscularconditioning. This conditioning would give the future tennis champion the ability to serve consistently over 300 km/h,1 forehand consistently over 240 km/hand backhand consistently over 200 km/h. Just imagine the current top tennis players playing against someone with these three superb tennis weapons. Would they be able to win asingle game?

Basically, for dealing with any sport–science issue, one has two possible approaches: sport physiology and sport biomechanics.

Sport and exercise physiology gives us a valid description of the energy systems involved in any kind of cyclic sport activity(like running, cycling, swimming, rowing.. . )of either short–, middle–or long–distance.

On the other hand, the current popular sci ent ificdisciplinedesignedfor analyzing sport techniques is biomechanics. Anatomical biomechanics describes various techniques using mostly methods from functional anatomy, while Newton–Eulerdynamics(see Appendix 2: The Newton-Euler Law in Tennis Biomechanics) provides their mathematical modelling (and associated computer simulations), forthepurposeofansweringtheWhat–If questions(by varying

1 You don’thavetobe9 feet tall nor does your combined height including your extended arm, racquet, and jump height need to be 9 feet or more. This is because both gravity and aerodynamic drag act on the ball during itsflight. Gravity accelerates the ball downwards while drag(air resistance)creates a retarding force slowing the ball’sforwardmotion. This causes a curved trajectory.

the athlete’spar a meters). Biomechanics can confiden tly do two things:(i)give us all Newtonian mechanical principles as guidance for development of any sport technique, and(ii)roughly describe the technique of a current champion(using the concept of bio-kinetic chains).

Both sport physiology and biomechanics are legitimate sci ent ifictoolsfor understanding standard sport activities and/or human movements. However, neither of them can give us the most precious answers to the crucial question: Whatisthemostefficient technique for any particular human movement, both cyclic and non-cyclic? When it comes to understanding, prediction and control ofthemostefficient sport movements as well as training methods, both sport physiology and biomechanics fail.

Forexample, neither of them canhelp us understand the fastest ever tennis serveofAmericanAndyRoddick(153 m/h), or the fastest tennis forehand of French Gael Monfils(120 m/h). These two guys are neither the tallest(which would give them a huge leverage), nor the strongest(which would give them a huge force), nor the fastest(which would give them a huge speed–arguably, Rafael Nadal is both stronger and faster then both of them)in the game. They are record holders for the serve and the forehand, and it wouldn’thurteven Federer to have a stronger serve and forehand – he would not need to go through so many tie–breaks.

To answer this, in our opinion, the most sign ificant question in elite sport is the question that explains the secret technique of champions and to do this one needs a completely differentscientificap proach. What is the secret behind Roddick’sserveandMonfils’forehand? The secret is the stretch–reflex. Moreover, the stretch–reflexist he unique secret behind all highly–efficient movements in sport, including all athletic throws, jumps, sprints, all weight lifts, fast gymnastic movements and soon.

For his fastest serve, Roddick used the stretch–reflexin(the prime mover muscles of)all major joints: right shoulder, right elbow, right wrist, both hips, both knees and both ankles. For his record forehand, Monfilsusedthestretch– reflexinthesamejoints. Thetwotechniqueslookcompletelydifferent, but the neuro-physiological basis is the same; the stretch–reflexutilizedin(the prime mover muscles of)the same joints.

The underlying similarity between these two apparently different sport movements also teaches us another important lesson: the“monkey see, monkey do” approach of copying others’technique(without even an attempt to understand what’sreallyhappening), used by almost all coaches and athletes, sometimes even sport scientists–is blind. It shows only trivial and superficialsimilarities, which are good enough only for kindergarten sport, without any ability to understand, predictor control the movement. Surely, sport science should be able to do betterthan just“monkey see, monkey do.”It should be able to provide the means for understanding, predicting, controlling and developing the most efficienthumanmovements.

The stretch–reflexisthemostefficientfeedback-control mechanism in the human body. Therefore, any human movement that uses it simultaneously in several major joints–is immediately high lyefficient.

The stretch–reflex recipe reads: make efficient sling shots out of all your major muscles: quadriceps, gluteus, soleus, pectoralis, deltoideus, biceps, triceps, palmarflexors, and all abdominal muscles. Make your major jointsflexibleandall your major muscles at the same time strong, fast and elastic, so that you can safely stretch andfirethemlikeslingshots. Train them so that you make them efficientslingshots. That is, base your strength and speed training on utilizing the stretch–reflexofallmajormuscles, previously conditioned by theflexibilitytraining. At the same time, base your sport technique(e.g., serve, forehand, backhand, etc)on the stretch–reflex, understanding that your muscles truly are natural sling shots ready tofire. Therefore, you need to develop your body full of sling shots, and your tennis weapons(e.g., serve, forehand, backhand, etc) need to be based on those sling shots, so that in areal game situation you can efficientlyfirethem. That is all! It is as simple as that!

You can easily see that popular plyometrics training is apart of this new science, but only a minor part. The concept of plyometrics training lacks the underpinning knowledge of the stretch–reflex, so naturally it proposes only a very limited number of plyometric exercises. Once one fully understands the applied stretch–reflex, one can design any number of plyometric exercises for any part of the body. One just needs to make his/her body full of sling shots, like Bruce Lee did three and a half decades ago, and like all athletics jumpers, throwers, sprinters and weightlifters do today.

The basic idea of coaches and players, as well as the simplistic biomechanics of loops and swings in tennis shots(serve, forehand and backhand)are derived from the concept of asimple physical pendulum. An arm with a hand holding a racket has been seen as a more-or-less single rigid body, with no more than 3 degrees of freedom(DOF). And really, if you have a robotic arm with only3 DOF, and you want to hit a ball with it, then you need to have a loop; and, even more, if you want to hit the ball hard, you need to have a big swing. That is absolutely true. In a language of modern biomechanics,“its phase space is a simple circle.”1That is how we originally go tour current loops–to play“nice tennis”, and swings–to be able to“hit hard”. In particular, you need a big swing as you want to use all the potential energy of the racket’sweight, so, as a preparation for the shot(say, forehand), you lift the racket–head as high as possible along the circle(that is, above your head). This is the common picture behind all loops and swings in tennis. Although simple to understand, it is not as easily implemented, which is where the many expensive lessons on the court come in. This system has produced thousands of young tennis players, hitting the ball in virtually the same way, with the same distinguished tennis movements: loops and swings.

However, this picture is wrong! Our reality is much more complex than this simplistic model of a loop (circle) and a swing (potential energy along the circle), which produces only“tennis ballet”.

1 For technical details on modern geometrical biomechanics with hundreds of degrees of freedom, see one of my advanced sci ent ificbooks: Human–like Biomechanics(Springer), Natural Bio dynamics(World Sci ent ific), Geometrical Dynamics of Complex Systems(Springer), High–Dimensional Chaotic and Attractor Systems (Springer), Neuro–Fuzzy Associative Machinery for Comprehensive Brain and Cognition Modelling(Springer), Appl i edD ifferentialGeometry(World Sci ent ific).

Firstly, the human arm has9 DOF(not includingfingers). Just the shoulder, elbow and wrist together have9 DOF. This is a redundant system, because the racket itself has only6 DOF(three translations and three rotations).2This means that there are infinite number of possible ways to hit the ball, and one can choose the best way in that situation. Therefore, areal“phase space”for the human arm with a racket is much more complicated than the simplistic circle–loop model. Besides, every coach advises, quite correctly, not to use the arm only, but the wholebody. SooneactuallyhasseveralhundredDOFat his/her disposal to perform a tennis stroke.3Do you still believe that asimple circle and its associated loop is an appropriate model for the serve and ground-strokes?

Secondly, the question of utilizing the racket’sownpotentialenergy, that is, rising a racket–head up high as a preparation for the shot. This would make sense only for very weak players and very heavy rackets(in the same way as a weak player needs along swing to accelerate the massive racket). Otherwise, incase of an athlete, and every elite tennis player needs to be an athlete, the current 400 gram racket is simply not heavy enough to make this potential– energy contribution to the shot sign ificant. It would be sign ificant with a4 kg racket, but not with the current 400 gram one.

Put simply, loops are totally useless: it does not matter at all how you get into the position for the shot. Swings are almost useless. They could be used say in thefifthhourofafive–set tennis match, to compensate for the lack of strength in shots, but the same fatigue would make it hard to raise the racket–head high above your head.

Also, one can see that simplistic (bio)mechanics does not work with the human body. Robotic shas already learned this lesson in the last three decades. Tennis has yet to learn it.

2 Common robotic–tennis arm would have exactly6 DOF–to match the racquet’s own 6 DOF; it wouldn’t have any excessive DOF; because of this 6–6 correspondence, a robotic arm would perform every racquet movement using the exactly prescribed trajectories. However, human arm(and human body as a whole)is a highly redundant system; thismechanicalredundancyallowseverytennisplayertoexecuteany movement in an infinite number of ways and to choose the one which is optimal with respect to racket-head speed, or spin, orenergyefficiency. 3In the wooden era, the player really needed the long,flowing swing to accelerate the old, heavy racket up to hitting speed, unless he was very strong and fast. Remember, muscular force needs to overcome inertial one, or force=mass times acceleration, and acceleration is the cause of the racket–head speed. Therefore, if one has a massive racket and a weak muscular force, one needs along movement to gradually accelerate and eventually get some racket–head speed. Inaddition, this gradual acceleration gave the player much more control of the rackethead and allowed the player to hit the ball at approximately the same location on the racket face each time it was swung, because the“sweetspot”was small.

Fig. 10.1. Important points on a tennis racket. The common “sweet spot” is technically called the center of percussion(the point along the racquet’slengthwhereanimpact produces no impulse reaction at the axis of rotation). Just below it is the vibration node. Note that the center of mass is below the racket–head.

The stretch–reflexcausesastretchedmuscleto contract stronger and at the same time inhibits the antagonist muscle from contracting(that is, slowing the movement). Becausethisisaninvoluntaryreflexresponsetherateofcontraction is sign ifican tly(several times)faster and more powerful than a completely voluntary muscular contraction. In fact, the faster the muscle is stretched eccentrically, the greater the force will be on the following concentric contraction (see Appendix 1, for technical details).

Closely related to the muscular stretch–reflexist he stretch–shortening cycle, which occurs when elastic loading, through an eccentric muscularcontraction, is immediately followed by an explosive concentric muscularcontraction. The tension developed in the musculo-tendinous junction by the eccentric loading

4 With the whip–likemovementitismoredifficult to hit the ball at exactly the same location on the racket–head each time. However, due to the characteristics of the modern racket and the heavy topspin strokes used, the resulting balltrajectory is much less sensitive to the exact location of the ball impact on the strings. New rackets appear to have larger“sweet spots”, technically called the centers of percussion, see Figure 10.1. 5 General experience from all other hitting, kicking and throwing sports or games.

of the muscle causes it to act in a similar manner to a rubber band. When this stored energy is released, it helps to increase the strength of the following concentric contraction. These neuromuscular considerations have huge ram ificat ions for both the composition and training of the tennis serve, forehand and backhand.

When one trains his/herskeletalmusclessothattheyefficientlyutilizeboth the stretch–reflexandtherelatedstretch–shortening cycle, the no neeffectively makes powerful sling shots out of them.

The prototype of such a muscular slingshot is the javelin throw.

The stretch–reflex(or, myotaticreflex)is the secret behind both speed and strength in sport. Itisthequickestreflex(see Figure 6.5) in the human body, in which thereflexarcisaclosed loop: both the receptor and theeffectorarein the same skeletal muscle(see Figure 10.2).

When a person walks and accidentally steps into a hole, the stretch–reflex of the stepping knee extensor(quadriceps muscle)is activated, generating additional force for knee extension, to re–establish the lost balance. What happens is that a muscle spindle (a length sensor within the quadriceps muscle) is stretched, causing a reflex arc to fire, generating a higher force than the voluntary muscular contraction alone. If the muscle is sharply stretched prior to its contraction, it generates a stronger force than without previous stretching. That is, the muscle behaves like a nonlinear spring(or,“elastic muscular component”). As an anatomical consequence, the attached body segment moves faster.

Fig.10.2.Reflexarcofamyotaticstretch–reflex.

The stretch–reflexist he basis for all fast cyclic movements(suchas sprinting)as well as for any combination of fast individual movements. It is also the basis forefficient weightlifting and athletic throws. In other words, speed and strength might haved ifferentmuscularcharacteristics, but they have the same neural support: the stretch–reflex. Finally, the stretch–reflexist he only proper physiological basis behind the popular plyometrics muscular training.

Biomechanically speaking, the stretch–reflex(or, if you insist, the stretch– shortening cycle) involves a sharp eccentric contraction immediately followed by a strong reflex concentric contraction. There is the stretch–reflex in action in all, mostlycyclic, fast movements in full joint amplitude(e.g., in sprinting, one has a maximal knee flexion immediately followed by a sharp knee extension, followed by a maximal kneeflexion. . . ). Whenever one of the two antagonistic muscles is stretched, either by his antagonist, or by gravity, it responds by generating a stronger force than it would be able to generate voluntarily without the previous stretching. As an anatomical result, one also has the faster movement of the attached body segment(a calf in the sprinting case).

The modern–day method for developing the jumping ability called “plyometrics” is actually an almost half a century old Russian explosive strength method called the“shock method,”firstlyproposedin 1959 by Professor Yuri Verkhoshansky (see bibliography at the end of this book). Plyometrics is generally used for improving the explosive strength and the muscular reactive capacity. It is a type of exercise that utilizes a rapid eccentric movement, followed by a short amortization phase, and then followed by an explosive concentric movement, which enables the synergistic muscles to engage in the (myotatic) stretch–reflex during the stretch–shortening cycle. Plyometric exercises use explosive movements to develop muscular power, the ability to generate a large amount of force quickly. Plyometric training acts on both the musculotendinous and neurological levels to increase an athlete’spoweroutputwithoutnecessarilyincreasingtheirmaximumstrengthoutput. Plyometrics are used to increase the speed or force of muscular contractions, often with goals of increasing the height of a jump or speed of a punch or throw.

Recall that for a muscle to cause movement, it must shorten; this is known as a concentric contraction. There is a maximum amount of force with which a certain muscle can concentrically contract. However, if the muscle is lengthened (eccentric contraction) just prior to the contraction, it will produce greater force through the storage of elastic energy. This effect requires that the transition time between eccentric contraction and concentric contraction(amortization phase) be very short. This energy dissipates rapidly, so the following concentric contraction must follow the eccentric stretch before this occurs. The process is frequently referred to as the stretch–shortening cycle, and is one of the underlying mechanisms of plyometric training.

Inaddition to the elastic–recoil of the musculo–tendon ous system there is a neurological component. The stretch–shortening cycle affectsthesensoryresponseofthemusclespindles(stretch–reflex)and golgi tendon organs(GTO). It is believed that during plyometric exercise, theexcitatorythresholdofthe GTO’sisincreased, meaning they become less likely to send signals to limit force production when the muscle has increased tension. This facilitates greater contraction force than normal strength or power exercise, and thus greater training ability.

The muscle spindles are involved in the stretch–reflexandaretriggeredby rapid lengthening of the muscle as well as absolute length. At the end of the rapid eccentric contraction, the muscle has reached a great length at a high velocity. This may cause the muscle spindle to enact a powerful stretch–reflex, further enhancing the power of the following concentric contraction. The muscle spindle’ssensitivitytovelocityisan other reason why the amortization phase mustbebriefforaplyometriceffect.

A longer term neurological component involves training the muscles to contract more quickly and powerfully by altering the timing andfiring rates of the motor units. During a normal contraction, motor units peak in a de-synchronized fashion until tetanus is reached. Plyometric training conditions the neurons to contractwithasinglepowerfulsurgeratherthanseveraldisorganizedcontractions. The result is a stronger, faster contraction allowing a heavy load (suchas the body)to be moved quickly and forcefully.

Some a nimalsalsotakeadvantageofthiseffect; one is the kangaroo. Ifa kangaroo needed to use 100% new energy to contract its leg muscles everytime it jumped, it would not be able tojump very far consistently. However, because of the muscles’ability to store energy from its previous jump like a spring, the kangaroo only needs to use a fraction of the total energy in the jump.

In everyday sport, plyometrics commonly refers to jumps and similar movements that involve eccentric (stretching) muscular contraction, immediately and rapidly followed by a concentric(shortening)contraction, that is the stretch– shortening cycle. The phase between these two contractions is referred to as the amortization phase. Energy stored during the eccentric phase is partially recovered during the concentric phase. In order to best use this stored energy the eccentric phase must be rapidly followed by the concentric.

One plyometric exercise that is very useful for tennis serve, involves catching and tossing a medicineball to an assistant while the exerc i ser lies on their back. The triceps and chest muscles work both while they are lengthening (catch phase) and while contracting(toss phase).

As another example, atypical long jumper may spend as little as 0.1 seconds on the ground at take-off.A great deal of muscular force must be generated in this brief period of time. Through the correct use of plyometric exercises, this rate of force development can be enhanced. Note that highvolume plyometric workouts will not enhance speed development. Therefore, by using plyometric jumping exercises one is trying to accomplish the following:(i)to shorten the time spent in the amortization phase, and (ii)to decrease the time spent on the ground yet generate maximum force.

Conventional wisdom dictates that in order to begin plyometric training there are prerequisite strength levels which are necessary. To begin to incorporate plyometric training in a program the prime concern is strength in the stabilizing muscles in order to prevent injury. The next concern after stabilization strength is eccentric strength. Eccentric strength is the limiting factor especially in more complex highvolume and highintensity plyometric training. Without adequate levels of eccentric strength, rapid switching from eccentric to concentric work becomesveryinefficient.

Proper execution of the exercises must be continually stressed regardless of the proficiency level. For the beginner, it is especially important to establish a sound technical base upon which to build the higher intensity work. Jumping is a constant interchange between force production and force reduction leading to a summation of forces utilizing all three joints of the lower body: the hip, knee, and ankle. The timing and coordination of all limb segments will yield a positive groundreaction force which results in a high rate of force production.

A key element in the execution of a proper technique is the landing. The shock of landing is not absorbed exclusively with the foot, rather it is a combination of the ankle, knee, and hip joints working together to absorb the shock of landing and then to transfer that force. The proper utilization of all three joints will allow the body to use the elasticity of the muscles to absorb the force of landing and then utilize that force in the subsequent movement.

Thus, plyometric exercises promote high movement speed, fasttwitchfibre recruitment and elastic tendon energy release. They always involve:(i)an eccentric contraction;(ii)a brief amortization phase(with no change in muscle length); and(iii)a short concentric contraction delivering maximum force in a short period of time.

The capacity of muscles, tendons, ligaments and fascia tostretch, the range of motion in the joints and the ability of the muscles tocontractandcoordinatealldefinehowonemoves. Unlike strength, speed and other motor abilities, flexibilitydevelopsand determinesefficientfluidmovement. Heavy training and competition schedules place great stresses on our capacity for movement. One requires a systematic approach to training for full recovery both mentally and physically. The implementation of a properflexibilityprogram is imperative for this recovery, but more importantly for an increase in performance.

Flexibility training should be integrated on a continual basis within a yearly training plan. When practiced regularly it provides immediate relief from fatigue and muscle soreness. This is very important in accommodating increases in training volume and intensity as the year progress. Additionally, the accumulation offlexibilitytrainingattainedbythe athlete over an extended period of time will allow for an increased capacity to maintain these gains with less work.

Stretching should never be forced, but should be done with special care. It is important that the athlete/player focuses on the muscle group that is being stretched. This fosters greater body awareness, an overlooked attribute of achampion. Flexibility has to be both sequential and rhythmical in order to accommodatetheeffectsofhightrainingloadsthat have placed a greater load ontheCNS. The constant stimulation of the nerve cells, whose high working capacity cannot be maintained for long, affectsmuscularactionandultimately athletic performance. When the period of competition and training has very high fuel consumption, fatigue sets in. Since blood glucose is depleted from the system, the CNS becomes fatigued. Properly applied stretching techniques promote enhanced blood circulation within the athlete. With this increase in circulation, there is removal of metabolic waste (lactic acid)products, as well as an increase in the transport of oxygen and nutrient to the muscle and tendon regions.

A properly designed flexibility program does not cause injury to the tissues but aids in their recovery and regeneration. Its greatest influence is at the myo-tendon junction, the transition zone between the muscle and tendon. The function of the tendon is to transmit the mechanical impulses that derive from muscular contraction to the joints. Themyo-tendon junction must adapt each time to the functional needs of the musculoskeletal system. This region also helps in cushioning abrupt and violent movement. Most injuries occurring in this region are a result of micro-tears, referred to as micro-injuries. Micro-tears result in the development of scar tissue. As time progress and these injuries go untreated, a common result is chronic pain. Associated with chronic pain at the physical level are muscle imbalances and compensation shifts. If an athlete does not adhere to a properflexibilityprogramtheymayslowly develop into an involuntary contraction machine (due to a stretch-reflex overload).6Their higher muscle tone, a result of greater tightness, affectsthenervoussystem. A possible outcome of this neural fatigue maybe muscle atrophy, as well as poor muscle coordination. This affectsthedevelopmentofpower(that is, both strength and speed).

Firstly, itis very important for an athlete to be both physically and mentally fit for the competition and/or training. Besides, an athlete should be able to identify

6 This is an example ofpossiblemisuseofthestretchreflexifmusclesarenotflexible enough.

the presence of a minor injury and distinguish it from fatigue. Here is a list of simple ways by which you can identify injury:(i)pain is sustained steadily and does not subside;(ii)an extent of tightness can be felt that restricts full–range motion;(iii)you feel light–headed or nauseous;(iv)what initially appeared to be a minor injury does not heal promptly.

Highly motivated athletes always strive to push beyond their limits, both in training and in the competitions. Unfortunately, if an athlete does not take sufficientrestwhenneeded, their muscles can actually be injured, rather than become stronger. Such injuries include: cramps (muscles become excessively tight in contractions), contusion(internal bleeding, swelling, pain and stiffnessare caused by a serious bruise), sprains (a ligament has been overstretched and torn) and strains(a muscle or tendon attachment has been over-stretched or torn).

Forexample, when lifting weights in weight–training, the body’snaturalresponse to the heavy weights is have microscopic tears in the muscles’connective tissues. This explains the soreness that one would usually feel after each weight– training exercise. It is essential for an athletetorest after weight–training. During the duration which the athlete rests, the microscopic tears are repaired. In the repair process, muscles become larger and stronger. However, if an athlete fails to obtain adequate rest after weight–training, the repair process maybe delayed or a true injury may result. This is because if the athlete continues to strenuously use the muscles without allowing rest for repair, the microscopic tears may become larger tears.

Biomechanical term for the whip–like movement is the kinetic chain: the sequentialflowof energy and momentum from bigger segments to smaller ones. Tennis requires sequenced activation of muscles and movement of bones and joints to achieve the motions, positions, and velocities seen in a player. This sequencing is known as the kinetic chain. Kinetic energy and momentum, as well as muscular power, are developed from the legs, hips and trunk muscles and transferred to the arm muscles. This allows the energy, momentum and power to be transferred efficiently to the hands, moving the racquet-head with maximum speed to the ball.

More precisely, toachieve the highly–efficient technique of the tennis serve, forehand and backhand, a proper sequencing of muscle stretch–reflexbasedactionsmusttakeplace. Two movement strategies are critical in this respect: (i)proximal-to-distal firing patterns, and(ii)active acceleration–deceleration of body segments.

Research evidence suggests that a proximal-to-distal firing pattern is the most effectiveforincreasingtheracket-head speed in the serve, forehand and backhand. In such a sequencing pattern, the stronger more heavily muscled proximal (close to the torso)joints should become activated before the weaker but faster distal joints. Thisfiringpatternhasproventhemostefficient due to the fact that it takes advantage of each joints’linear and angular momentum generating characteristics. It suggests that power(that is, both speed and strength combined)for the serve, forehand and backhand is primarily generated with:(i)leg extension,(ii)hip rotation, and(iii)trunk rotation andflexion–before the actual arm action. The actions of these proximal joints account for more than 50% of the total forces in the serve, forehand and backhand.

Asecondcharacteristicofefficient movement coordination in the serve, forehand and backhand is consecutive acceleration and deceleration of the main body segments. When done well this permits the player toachieve racket-head speeds far greater than they would if they did not use an optimal acceleration deceleration coordination pattern. The mechanism for this benefit is the transfer of both linear and angular momentum. This movement strategy aids in the transfer of momentum from the lower extremity to the upper, and from the upper extremity to the racket-head.

These two kinetic-chain patterns combined, generate a whip–like motion: when the upper leg and trunk musculature are thefirst to contract, greater separation is developed between the shoulders and hips which results in a whip effect as the hips are decelerated and the shoulders accelerate as they uncoil and the shot is released.

As already said, each muscular contraction ina chain can be either voluntary or reflex. By now, we should already know that the stretch–reflex based contraction is several time smo reefficient(that is, both faster and stronger)then the voluntary one.

Power High-Forehand (PHF), with the aim of generating the maximal possible racket-head speed, should be executed as follows(see Figures 11.1 and 11.2).

Let’sassumethatyoustartfromthebase-line(as usual). You play your normal game, but as soon as you anticipate1 a high ball coming at a waist/chest level(or higher)while passing over the service-line, you don’twa it behind the base-line for the ball to come to you, but immediately sprint as fast as you can towards the service-line and a little-bit left from the ball, while at the same time leaving the racket-head behind your right shoulder (preparing for the shot). When you get close to the ball, you make sure that you stop about half-a-meter left from the ball’s direction, with the left foot forward. This automatically forms the left closed stance (as if you want to execute the right-hand karate gyaku-zuki reverse punch), with the racket-head far behind your right shoulder. Then you are ready to intercept the ball and execute the winning shot.

Once you are in the proper left stance(as explained above), thePHF-shot is properly executed as a full biomechanical chain, based on the sequence of stretchreflexes in all included muscular actuators, one by one as a cascade. It is represented by the following sequence of individual joint movements:

1 Here, we promote the intercept ive tennis tactics, as a superior alternative to the current static tennis in which both players most of the time just wait behind the baseline for the ball to bounce from the ground and come to them. In our Power interceptivetennis, the players do not wait for the ball to bounce, but rather intercept as soonas the ball passes the serviceline and hit it in the air. This sign ifican tly shortens the time for opponent to react. If executed properly, this shot has the same effectasthesmash. However, while an opportunity for a smash usually happens only about once in a tennis match, if you develop proper intercept ive skills you will realise that most of the balls coming over the net can be intercepted. This is tactical basis for the future tennis game. Proper interception is heavily based on the anticipation of the ball coming(both direction and height while passing over the service-line). Besides, you can also intercept low balls, using your ordinary forehand and backhand as an approaching shot; you just need to be ready to hit the ball in the air(not waiting for it to bounce).

Fig.11.1. PHF–training 1:a“3o’clock”shadow execution behind the serviceline.

1. Sharp right knee extension(with the right foot/heelfirmlyontheground), while stretching the right hip muscles(gluteus medius and maximus); 2. Sharp right hip forward rotation, while leaving the right shoulder back and thus stretching the right waist muscles(external and internal abdominal muscles); 3. Sharp torso rotation to the right, while leaving the arm back and thus stretching the pectoral and deltoid muscles; 4. Sharp right-elbow movement forward, while leaving the racket-head back, and thus stretching the right triceps; 5. Sharp right-elbow extension, while still leaving the racket-head back, and thus stretching the right wristflexormuscles; and 6. Sharp right-wrist flexion, with simultaneous pronation to generate some topspin for ball control.

This, in a nutshell, ishowthePHF-shot should be executed.

Fig.11.2. PHF–training 2:a“2o’clock”execution–intercepting the high ball behind the serviceline.

Now, howdoyouknowthatyouarereallyexecutingtheproperPHF-shot? Thefirst criterion is this: it cannot be done slowly. It can be performed moderately fast or extremely fast, more tensed or more relaxed, but never slowly. This is thefirst criterion for any stretch-reflexbasedmovement.

Here is a simple familiar example: teaching long jump to small kids. You have a short run-up and then you jump. You can run slowly, but you simply cannot jump slowly. Why? Because a single-legged jump (from running) is based on the stretchreflexinalllegextensormuscles(soleus, quadriceps and gluteus). No stretchreflex, no jump–as simple as that.

It is the same with all naturally fast movements, including our PHF: it can be done in a more tense dora more relaxed way, but it cannot be executed slowly. The very nature of all stretch-reflexbasedmovementsisitsinherentspeed(as well as strength). It is not meant for slow and/or weak movements.

Modern biomechanics is rotational (see Figure 12.1) and can be formulated using Newton-Euler, Lagrangian or Hamilton i an formal ism(see Appendix 2).

Fig.12.1. Development of human biomechanics: Left–old translational approach; middle–humanoidrobotics; and right–new rotational approach.

Here we give a brief on the Newton-Euler dynamics of the three-dimensional motion of a player’s hand holding a tennis-racket, developed as a Mat he matic aT M code – ready for computer simulation. For more technical details, see [II 06a, II 06c].

We start by defining the three local Cartesian translation coordinates, sagittal, frontal and vertical, or{x 1, x 2, x 3} = {X, Y, Z}, placed at the center-of-mass of the racket, as well as the corresponding three rotation(orientation)coordinates, {θ1, θ2, θ3}={roll, pitch, yaw}, aroundtheCartesianones. In the traditional outputformofMathematicav.7, this reads:

Vars = {x1, x2, x3, θ1, θ2, θ3}

Next, byintroducingaracketmassm=m 1=m 2=m3 and three grip/ time-dependent inertia moments {J 1(t), J 2(t), J 3(t)}around the Cartesian axes, theinertialdefinitionsofbothlinearandangularmomentaaregivenin Mat he matic a 7by: ⎛

J1(t) (θ1)′′(t) J2(t) (θ2)′′(t) J3(t) (θ3)′′(t)

Now, if one introduces time-dependent muscular forces {F 1(t), F 2(t), F 3(t)}and torques,{T 1(t), T 2(t), T 3(t)}, onecanderivethethreeNewtoniandifferential equations of the translational racket motion as:

(p1)′(t) = F1(t) −m3v3(t)ω2(t) + m2v2(t)ω3(t) (p2)′(t) = F2(t) −m3v3(t)ω1(t) −m1v1(t)ω3(t) (p3)′(t) = F3(t) −m2v2(t)ω1(t) + m1v1(t)ω2(t)

(π1)′(t) = (J2(t) −J3(t)) ω2(t)ω3(t) + (m2 −m3) v2(t)v3(t) + T1(t) (π2)′(t) = (J3(t) −J1(t)) ω1(t)ω3(t) + (m3 −m1) v1(t)v3(t) + T2(t) (π3)′(t) = (J1(t) −J2(t)) ω1(t)ω2(t) + (m1 −m2) v1(t)v2(t) + T3(t)

(12.2) Thecombinedsystemofdifferentialequations(12.1)–(12.2)representsthetotalNewton-Euler dynamics of the three-dimensional motion of a player’shand holding a tennis-racket. If one can measure/provide all introduced parameters together with sample initial racket position and orientation(dependent on the initial body-configuration), one can readily simulate this system on any computer.

Eacheffectivetennisshot, be it a serve, a forehand or a backhand, is a whip– like movement performed by a complex coordination of all the body’ssegments working to place the racquet in the correct position at the right time and apply the maximal summed force to the tennis ball.

As we already emphasized several times, the best power(strength+speed) exercise for both serve and forehand (and even for single-handed backhand) isa javelin throw. Not only that, but all speed and strength exercises practiced by elite javelin throwers are perfectly suited for future champion tennis players.

In short, a superb serve, forehand and backhand are whip–like movements, each composed as a cascade of stretch–reflexes in all major joints, starting from the feet and ending with the hitting hand.

Recall that former world number one, Andy Roddick, holds the record for the world’sfastesttennisserve:153 m/h(or,246 km/h)firedatQueen’sClub, UK, in 2004. When hefirstmetPatrickMcEnroe, his Davis Cup coach, he said: “Whatever you do, don’tsayanythingtomeaboutmyserve. If I think about it, I’min trouble.”Why? Becauseitisallreflex, more precisely stretch–reflex. If you think about something that is per for me dreflexively, you simply mess it up. Therefore, it is crucial that theeliteplayerdevelopsafullyreflex–based technique. This will generate the highestpossible racket–head speed of an elite athlete and thus maximize their performance/efficiency.1

However, coaches and sports scientists should analyze the most efficient movements to be able to teach the model techniques. Forexample, Professor Bruce Elliott from the University of Western Australia, has extrapolated the contributions of the body segments to racket-head speed using3 Dvideo–and computer

1 Assuming that the player is already capable of consistently getting the serve in the square, keep their serve deep, able to serve to the opponent’s backhand, body and/or forehand at will, a ndeffectivelyusesliceand/or topspin kick.

analysis.“These contributions vary from person to person,”Elliott says,“but the data shows the clear importance of the trunk, shoulder internal rotation and wristflexionintheswingtoimpact.”

Apart from the javelin throw, the best strength exercise for the serve is the axe chop/sledgehammer. Its purpose is to simultaneously develop strength, speed and flexibility in the shoulder girdle, pectoral and upper back musculature. This exercise requires a sledgehammer or lumberaxe, approximately of the mass and handle–length of the tennis racket, as well as a stable knee–height hitting surface suchas a tractor tire, mound of dirt, or a large log. The athlete stands in front of a knee–height hitting surface. After a proper warm-up, the axe/hammer is brought over the head and swung violently down onto the hitting surface. The key for this exercise is to let the weight of the axe or hammer pull the arms back so that a stretch is felt through the shoulders and upper back. This will initiate a stretch–reflexcontraction, while developing strength, speed andflexibilityfor throwing. It is important that:(i)the athlete initiate the movement with the wholebodyrather than just the arm, which will create a whip–likeeffecton the axe or hammer; and(ii)to imitate the serve movement as close as possible.

Other strength exercises include various overhead throws, like medicineball throw and weighted ball throw. In particular, catching and tossing a medicine ball to an assistant while the exerc i ser lies on their back.

The best speed exercise for the serve is simulating the serve movement (without the ball)with a badminton racket instead of a tennis racket. This should be performed in a double series:10repetitions of a shadow serve with a tennis racket, then 10 repetitions with a badminton racket, thena2 minute pause; in such a way to make 100 of each, performed with a maximal speed (after a proper warm-up).

The wooden racket era, characterized by the heavy rackets and the weaker players, was dominated by the following5 classicalpostulatesforthepropertechniqueforthetennisforehand:

1. You must have a loop in the stroke, either big or small, but the bigger the better; 2. If you want to“hit hard,”you need to have a huge swing; 3. You need to rotate hips and shoulders together, that is simultaneously;

4. With a series of little steps you need to put yourself into the proper position for theeffectiveforehand; and 5. The wrist must befixedtotallyrigid.

The well–known result of this 5- point approach to the forehand technique was two–fold:(a)nice and slow“ballet on the court”and(b)tennis elbow injury. Today, with the light-weight metal rackets and stronger players, the 5th postulate has been dropped: one can see now plenty of wrist slaps.

However, the other4 postulatesarestillassumedvalid. Well, from the perspective of hig hefficiency/maximum performance in tennis of the future–all four postulates are wrong. They are all pure “cosmetics” without any substance.

To dispel this“myth of a proper tennis forehand,”in this chapter we have given a biomechanical description of a whip–like tennis forehand movement, composed as aseries of stretch–reflexe sin all major joints.

A complex reaction has the highest importance in every sport’sduel, including tennis. In general, a complex reaction has two components: smart anticipation and lightning–fast reaction. Youfirst anticipate the opponent’smovementand then you react by making your own movement.

Good anticipation of the opponent’sactionsistheessentialcharacteristic of a master in any sports duel. The best anticipation is actually called“mind reading”. It is the differencebetween master and disciple. It can be learned, and it should be learned if you have high expectations from tennis. Of the current top tennis players, Federer, Nadal and Djokovic have the best anticipation– and coincidentally they are world number one, two and three. Before them, Sampras and Agassi had the best anticipation, and soon. Apart from getting the experience from many tennis tournaments(atdifferentcompetitionlevels), the best tool for developing anticipation would be the bullet/blitz chess (see The Law of Blitz–Chess in Tennis).

On the other hand, the basic speed of reaction is largely genetically predetermined and cannot be sign ifican tly improved. The best means for its developing is both technique and training based on the stretch–reflex.

For the current first serve in the range of about 200 km/h−220 km/h, the available time budget for the return player is approximately 0.7 s on the slowest clay courts, approximately 0.6 s on faster hard courts, and approximately 0.5 s on the fastest lawn courts. These are average numbers, with about 10% deviations due tod ifferences in the conditions of ballflight and individual returning strategies (e.g., the position of the return player on the court etc.).

The whole return can be roughly divided into three phases:(i)anticipating the ball, (ii) movement regulation and reprogramming, and (iii) hitting the ball.

During the time of movement regulation the players adapt their racket movement to the demands of the approaching ball. The sequencing of maximum segment velocities(i.e.,firsthips, then shoulder, then elbow, andfinallywrist), which is acommon feature of both the ground-strokes and serve, cannot be clearly seen during the return. This is due to: (a) the unpredictable nature of the return and the high time pressure under which players are placed; and(b)the very high demand of precision on the return. Currently, on the 1st serve return, precision– orientation is dominant, while on the2 ndservereturn, ball–speed generation is dominant(the racket–head speed atimpact is sign ifican tly lower during1st serve returns). Also, the ability to reprogram an incorrect anticipation/decision (e.g., shifting from backhand to forehand return)within a fraction of a second is a very important factor on the return.

Therefore, it is highly recommended to:(i)train anticipatory abilities;(ii) train returns by varying the time pressure imposed on the players in order to improve their time management; and(iii)offert asks for reprogramming.

Like in any other sport game, the important part of the tennis game is tactics. Current tennis tactics will be slightly simplified in the future tennis game, as, due to the highly–increased speed of the ball, long rallies will rarely exist. Therefore, this“tennis chess”will consist of up to 3–4 movements at anyone time.

Question: Assuming that both players are right-handed players, could you show us the optimal serve tactics?

A 1. If you serve from the right side of the baseline centermark, then Figure 14.1 shows the optimal serve tactics.

Fig.14.1. Right–handed player targeting“T”, serving to the right–handed player’s backhand.

A 2. If you serve from the leftside of the centermark, then Figure 14.2 shows the optimal serve tactics.

Fig.14.2. Right–handed player targeting wide, serving to the left–handed player’s backhand.

Question: Assuming that the right–handed player is serving and left– handed player receiving, could you show us the optimal serve tactics?

A 1. If you serve from the right side of the centermark, then Figure 14.3 shows the optimal serve tactics.

Fig.14.3. Right–handed player targeting wide, serving to the right–handed player’s backhand.

A 2. If you serve from the leftside of the centermark, then Figure 14.4 shows the optimal serve tactics.

Fig.14.4. Right–handed player targeting“T”, serving to the left–handed player’s backhand.

Part 4 · pp. 95–124

In this chapter we formulate a fuzzy–logic,1attack(AT)and counter–attack (CA)model for the tennis game.

1 Fuzzy logic is a powerful problem–solving methodology with a myriad of applications in embedded control and information processing(e.g., traincontrollers, aircraft auto pilots, air conditioning, control of nuclear reactors, etc). Fuzzy provides aremarkablysimplewaytodrawdefiniteconclusionsfromvague, ambiguous or imprecise information. Inasense, fuzzy logic resembles human decisionmaking with its ability to work from approximate data andfindprecisesolutions. Fuzzy logicusuallyworksasasetoffuzzyIF–THENrules. Forexample, incase of a heater controller with two inputs(temperature and humidity)and one output(fan speed)one has the following set of fuzzy rules:

IFtemperatureIScoldANDhumidityIShighTHENfan-spdIShigh IFtemperatureIScoolANDhumidityIShighTHENfan-spdISmedium IFtemperatureISwarmANDhumidityIShighTHENfan-spdISlow IFtemperatureIShotANDhumidityIShighTHENfan-spdISzero

IFtemperatureIScoldANDhumidityISmedTHENfan-spdISmedium IFtemperatureIScoolANDhumidityISmedTHENfan-spdISlow IFtemperatureISwarmANDhumidityISmedTHENfan-spdISzero IFtemperatureIShotANDhumidityISmedTHENfan-spdISzero

IFtemperatureIScoldANDhumidityISlowTHENfan-spdISmedium IFtemperatureIScoolANDhumidityISlowTHENfan-spdISlow IFtemperatureISwarmANDhumidityISlowTHENfan-spdISzero IFtemperatureIShotANDhumidityISlowTHENfan-spdISzero.

where the two sets, Odim=2∋omandSRdim=3∋srn, contain the temporal fuzzy variables {om= om(t)} and {srn= srn(t)}, respectively opponent–related (target information)and serve–related, partitionedbyoverlappingGaussians, μ(z) =

O OP P ONENT −IN: o 1= Opp. Posit. Left. Right : (center, medium, wide), o 2= Opp. Antcp. Left. Rght : (run Center, stay, run Wide), sr 1= 1. Serve. Speed : (low, medium, high) sr 2= 2. Serve. Spin : (low, medium, high) sr 3= 3. Serve. Placement : (center, medium, wide)

O: OP P ONENT −IN o1= Opp.Posit.Left.Right sr1= 1.Serve.Speed sr2= 2.Serve.Spin sr3= 3.Serve.Place

B. Attack–Maneuver: Serve–Volley. The generic advanced AT–dynamics is givenbyacompositionofFAMfunctors wherethenewn−category, RVdim=2∋rvp, contains the opponent–anticipation driven volley–maneuver, expressed by fuzzy variables {rvp= rvp(t)}, partitioned byoverlappingGaussiansandgivenby:

RV RUN−V OLLEY: rv 1= RV. For: (base Line, center, net Close) rv 2= RV.L.R. : (left, center, right)

O: OP P ONENT −IN o1 =Opp.Posit.L.R.

o2 =Opp.Anticip.L.R.

sr1 =1. Serve. Speed sr2 =2. Serve. Spin sr3 =3. Serve. Place where then−categories Bdim=5∋bK, Rdim=3∋rJ , Sdim=4∋sk, contain the fuzzy variables{bK=bK(t)},{rJ=rJ (t)}and{sk=sk(t)}, respectively defining the ball inputs, our player’srunningmaneuverandhisshot–response, K.e., b 1= Dist.L.R. b 2= Dist.F.B. b 3= Dist.V ert b 4= Speed b 5= Spin r 1= Run.L.R. r 2= Run.F.B. r 3= Run.V ert s 1= Backhand s 2= Forehand s 3= V olley s 4= Smash

Here, theexistenceofefficient weapons within the S SHOT −OUTarsenal–space, namelysk(t) : s 1= Backhand, s 2= Forehand, s 3= V oleyands 4= Smash, is assumed.

The universes of discourse for the fuzzy variables {bK(t)}, {rJ (t)} and {sk(t)}, partitionedbyoverlappingGaussians, aredefined respectively as:

b 1= Dist.L.R. : (very Left, left, center, right, very Right), b 2= Dist.F.B. : (base Line, center, net Close), b 3= Dist.V ert : (low, medium, high), b 4= Speed : (low, medium, high), b 5= Spin : (highT opSpin, lowT opSpin, flat, low Back Spin, high Back Spin).

r 1= Run.L.R. : (very Left, left, center, right, very Right), r 2= Run.F.B. : (close Front, front, center, back, far Back), r 3= Run.V ert : (squat, normal, jump).

s1= Backhand : (low, medium, high), s2= Forehand : (low, medium, high), s3= V olley: (backhand, block, forehand), s4= Smash : (low, medium, high).

B. Advanced Return. The advanced CA–dynamics includes both the information about the opponent and (either conscious or subconscious) decision making. This generic CA–scenario is formulated as the following composition + fusion of FAMfunctors:

spectively containing information about the opponent as a target, as well as our own aiming decision processes. In the case of advanced tennis return, this reads:

wherethetwoadditionaln−categories, Odim=4∋omandDdim=5∋dl, contain the fuzzy variables{om=om(t)}and{dl=dl(t)}, respectivelydefining the opponent–related target information and the aim–related decision processes, bothpartitionedbyoverlappingGaussiansanddefined as:

o 1= Opp. Posit.L.R. : (left, center, right), o 2= Opp. Posit.F.B. : (net Close, center, base Line), o 3= Opp. Anticip.L.R. : (run Left, stay, run Right), o 4= Opp. Anticip.F.B. : (run Net, stay, run Base).

d 1= Aim.L.R. : (left, center, right), d 2= Aim.F.B. : (net Close, center, base Line), d 3= Aim.V ert : (low, medium, high), d 4= Aim. Speed : (low, medium, high), d 5= Aim. Spin : (highT opSpin, lowT opSpin, no Spin, low Back Spin, high Back Spin).

b1= Dist.L.R. b2= Dist.F.B. b3= Dist.V ert b4= Speed b5= Spin r1= Run.L.R. r2= Run.F.B. r3= Run.V ert d1= Aim.L.R. d2= Aim.F.B. d3= Aim.V ert d4= Aim.Speed d5= Aim.Spin

O: OP P ONENT −IN o1= Opp.Posit.L.R. o2= Opp.Posit.F.B. o3= Opp.Anticip.L.R. o4= Opp.Anticip.F.B.

s1= Backhand s2= Forehand s3= V olley s4= Smash

The best preparation for the mental speed in tennis is the so–called lightning chess, or bullet chess. It is the faster version of the blitz chess game, where each sidehaslessthan3 minutes to complete all of their moves. Often bullet chess is so fast that tactics and skill are secondary to quick moves. Under United States Chess Federation(USCF)rules, bulletgamesareconsideredblitz. Every year, an over-the-boardlightningchesstournamentisheldinApeldoorninthe Netherlands. The time control is2 minutes per player per game. It is the only official championship in bullet chess. The winner can claim the title Open Dutch Champion in Lightning Chess.

1. You need to decide what level in tennis you want toachieve(e.g.,“world number one,” or“in the top 10,” etc.). This needs to be something that you believe that you can do and that you deserve. 2. Imagine yourself as a winner at that level. 3. Visualize that your everyfirst serve results in an ace. 4. Visualize that you canefficientlyreturneveryserve. 5. Visualize that your every ground-stroke is a winner. 6. Visualize that you are full of energy. 7. Visualize the enjoyment of all your aces and winners.

winning and losing;(v)perfectionism;(vi)complacency;(vii)having no plan or clear goal;(viii)too much spontaneity and creativity;(ix)lack of humility;(x) inappropriate reaction to errors; and(xi)negative selftalk(self–criticism).

To improve your mental game you need to: (i) set clear goals; (ii) create action steps that will take you closer to these goals(learn to visualize: try to“see”in your mind what you want to achieve); (iii)use positive self–talk (self–criticism); (iv) practice yoga breathing techniques; (v) always put forth 100% effort; (vi) be processoriented;(vii)stay detached;(viii)no future tripping, no past tripping –stayin the present;(ix)be non–reactive to the opinions of others;(x)tolerate your inability to be perfect;(xi)do what it takes to have fun and smile;(xii) definew inning in away that includes more than just thefinalscore.

Yes, this is a great problem for most players, andapparentlyRogerFederer solves it consistently and routinely. Aside from a great tolerance to competitive pressure, focus is one of the key characteristics of achampion. You really need to“fixyoureyesontheball,”as every coach advises, and yet to have a full amount of the so–called situation awareness(SA), which i sdefined in modern psychologybyM.R. Endsley(1995)as“the perception of the elements in the environment within a volume of time and space, the comprehension of their meaning, and projection of their status in the near future”(see Appendix). For example, SAisakeyfactorinthetrainingoffighterpilotsandFormula1 racing drivers. TheoutcomeofanycomplexsituationcriticallydependsonSA. Therefore, you need to develop:(i)strong focus on the moving tennis ball;(ii) abilityforstrongSA, a sdefinedabove; and(iii)to have both focus on the ball andSAatthesametime. This is similar to learning to play a hard piece on a piano:first you learn the right–hand part, then you learn the left–hand part, then you learn how to combine them together. Clearly, both physical and mental fatigue(combined with other disturbances, likee.g., rain, wind, noisyaudience, big overhead screen, etc.)willgraduallyreducebothfocusandSA. Therefore, besides physical speed and strength, you obviously need mental strength and endurance, to be able to win.

The future tennis champion will bear are combination of talent, hard work and the right mental profile. Even in today’s tennis, often thedifferencebetween the good and elite players is their mental qualities. In this respect, most important are the following three psychological characteristics:(i)confidence,(ii)anxiety, and(iii)motivation.

The whole thing is never to get negative about yourself. Sure, it’s possible that the other guy you’re playing is tough, and that he may have beaten you the last time you played, and okay, maybe you haven’t been playing all that well yourself. But the minute you start thinking about these things you’redead. I go out to every match convinced that I’mgoingto win. That is all there is to it.

In general, elite athletes tend to have very high levels of confidenceandfeel that these high levels are needed for the performances that they are looking for. About 90% of all elite athletes have a very highlevel of self-confidence. Confidenceisusuallyaresultofan athlete anticipating success in their upcoming event. An athlete’santicipatedoutcomeisthegreatestindicatorofconfidence. This expectation for success can be based on an athlete’sconfidenceinthemselves, emotional readiness, physical ability, knowledge of the opponent, goals, strategies, physical condition, or on the coach. To reach the very pinnacle of sport, an athlete must have high confidence in their abilities; and getting to that elitelevel and all the preceding successes that it took to get to that level must surely build the confidencelevelsofanathlete.

(ii)The link between anxiety and poor performance in sport has been known for along time. Stories abound of athletes or teams that performed poorly because they underestimated their opponent(below optimum anxiety levels)or worried themselves out of the game(above optimum anxiety levels). Dealing with anxiety successfully is an important characteristic of a future tennis champion. The ability to cope with pressure and anxiety will be an integral part of a champion.

One of the earliest models that attempted to explain the relationship between arousal/anxiety and performance was the so–called inverted–U hypothesis, which stated that as arousal increased, performance would increase as well; but if arousal became too great performance would deteriorate. In other words, as stress began to build an individual still felt confident in their ability to control it and performance would improve. However, once a stressor became so great that the individual started to doubt their ability to cope with it, performance would decline.

An individualistic approach was added to this hypothesis when the concept of individualized zones of optimal functioning, or IZOFs, was developed. According to this theory, each individual has an optimal level of pre-performance anxiety. If the athlete is in this“zone,” peak performances will be the result. However, if anxiety levels are too high or too low, the athlete will not see optimum results. IZOFscanbedeterminedbyrepeatedlymeasuringanxietyandperformanceor through athlete’sre call of anxiety levels prior to peak performances.

Depending on the individual, a nxietylevelscanhaveavarietyofeffectson athletic performance. Today it is well-known that anxiety can be reduced through mental imagery, relaxation, and cognitive intervention. These methods not only aim at reducing stress and anxiety levels butalso aim to improve confidence levels. ThegoalistohelptheathleteenterhisorherIZOF.

(iii)To become achampion in any sport requires many hours, days and years of training. Often this training is rigorous, painful, or exhausting. However, the athletes who have reached the pinnacle of their sport have more than likely put in their time to get to achieve that high level of success. To do this, these athletes must have something that motivates them to continually push their bodies, and comeback from whatever struggles or setbacks they may experience along the way. This motivation may come intrinsically or extrinsic ally. Intrinsic motivation is an athletes’personaldrivetoachievetheirgoal. This maybe setting a school record, winning a race, or defeating a particular opponent. Extrinsic motivation is the resulting motivation from an outside source suchas parents, coaches, or teammates.

There are many players who have the talent to succeed but very few who have the motivational drive to do what it takes to become achampion. In light of this, it appears that intrinsic motivation maybe the greater determinant of achieving success in sports. Toachieve at an elitelevel in sport, an athlete must have the motivation to train hard on a daily basis and to overcome any obstacles or setbacks that they might face in reaching or maintaining that level of performance.

Overall, itseems that the following traits would be common among elite athletes: extreme self-confidence, low performance anxiety, and high motivation. Thesethreethingsareverycloselyrelatedandwouldseemtoformacyclic pattern, positively influencingoneanother.

In the 1980s and 90s, Dr. Denis Waitley implemented what he called “Visual Motor Rehearsal”(VMH)intotheU.S. Olympics program. He and his researchers had found that when an athlete competed in an event only in their mind, the same nervous reaction in the body occurred as when they did their event in real life. This was another proof: Themindcannottellthediffere nce between an actual,“real–life” event and a vividly imagined one.1In this way, theVMHhas shown that by merely thinking and visualizing their event, athletes can enhance their performance.2

How to apply the VMH to tennis? Before a tennis match, sit back, relax, close your eyes and play the game in your mind. DotheVMHeverytimeyouhave animportant match. ThemoreyoupracticethemoreeffectiveVMHwillbefor

1 See “The Psychology of Winning,”a famous tape program for self–improvement, by Dr. Denis Waitley. 2 More technically speaking, relationships between sensory and motor events can be learned, although the events may have no prior association. Forexample, although mostdriversknowhowtorespondtoatrafficsignal, the sensory–motor rules governing these relationships require learning, because no intrinsic association exists between a trafficsignalcolororitsspatialpositi on and the appropriate movements to modify the speed of an automobile. These arbitrary cue–response associations become learned through experience, commonly by trial and error.

you. Do the same with learning anew technique:first watch the chosen ideal(a champion) and then visualize the same movement as performed by yourself. The more you visualize the less you need to practice physically.

The brain is very happy when you are focused on what you love doing. The more you focus on what you truly love and desire, the volume gets turned down inthosepartsofthelimbicsystemwherethedestructiveemotionsoffear, anger, depression and anxiety are controlled. This allows you to think more clearly. Youalsoturnupthevolumeinotherpartsofthelimbicsystem that generate positive emotions. When this happens, you get a release of dopamine, endorphins, and a variety of stress-reducing hormones and neurotransmitters, which enhances clear thinking. The more you focus on what you truly love, the healthier you are likely to be, and the more youfeelthepositiveeffectsofthosestress-reducingneuro-chemicals in your body and mind. You can have a decrease in negative emotions and an increase in positive emotions when you align yourself with what you believe is most important to you.

When one focuses on the big questions, the really big questions, one is challenging their brains to think outside the box, and this causes the structure of our neurons to change, particularly in our frontal lobes, that part of the brain that controls logic, reason, language, consciousness, and compassion. Newaxons grow, reaching out tonewdendrites to communicate in ways that our brains have never done before. When contemplating the big questions one uses their frontal lobes to alter the function of other parts of their brains.

In this section we will introduce modern concepts of quantum brain and mind, mostly following a modern trend set-upinthejournalNeuroQuantology. For example, according to some of the recent papers from this journal(see[Per 03]), ‘brain is classical’(i.e., governed by classical bio physics), while‘mind is quantum’,1and human consciousness is generated by‘neuro-quantization’, which takes place inside brain’smicrotubules(see, e.g.[Pen 89, Pen 94]).

+ wijfi(vi), (i, j= 1, ..., N), (17.1) where vj= vj(t) represent the activation potentials in the jth neuron, Cjand Rj denote input capacitances and leakage res i stances, synaptic weights wijrepresent conduct an ces, Ijrepresent the total currents flowing toward the input nodes, and thefunctionsfiaresigmoidal.

Geometrically, equation (17.1) defines a smooth autonomous vector–field X(t) inNDneurodynamicalphase–spacemanifoldM, and its(numerical)solution forthegiveninitialpotentialsvj(0)definestheautonomousneurodynamical phase–flowΦ(t) : vj(0) →vj(t)onM.

In AI parlance, equation (17.1) represents a generalization of three well–known recurrentNNmodels(see[Hay98,Kos92,II07b]):

1 From Delight we all came into existence; in Delight we grow and play our respective roles; at the end of our journey’sclosewereturntoDelight–Upanishads.

(i)continuous Hopfieldmodel[Hop 84], (ii)GrossbergART–family cognitive system[CG 83], and (iii)Hecht–Nielsen counter–propagation network[Hec 87, Hec 90].

Physiologically, equation(17.1)isbasedontheNobel–awarded Hodgkin– Huxley equation of the neural action potential(forthesinglesquidgiantaxon membrane) as a function of the conduct an ces g of sodium, potassium and leakage [HH 52, Hod 64]:

wherebrackettermsrepresenttheelectromotiveforcesactingontheions.

+ Tijui, (i, j= 1, ..., N), (17.2) whereuiareoutputfunctionsfromprocessingelements, and Tijistheinverse of the resist ors connection–matrix becomes equation(17.1)ifweputTij=wij andui= fi[vj(t)].

where A, B, C, Dare positive constants(A is dimensionally conduct a nce),0≤ ε<<1is the fast–variable factor(dimensionallycapacitance), and I+ jare excitatoryandinhibitoryinputstothejthprocessingunit, respectively.

˙vj= −aj(vj)[bj(vj) −fk(vk)mjk], (j= 1, ..., N), (17.3) aj= 1/Cj, bj= vj/Rj−Ij, fj(vj) = uj, and constant mij= mji= Tij, the system(17.3)reducestotheHopfieldcircuit model(17.2).

The Hecht–Nielsen counter–propagation network is governed by the activation equation[Hec 87, Hec 90]:

where A, Bare positive constants and Ijareinputvaluesforeachprocessing unit.

Provided some simple conditions are sat isfied, namely, say symmetry of weights wij= wij, non–negativity of activations vjand monotonicity of transfer functionsfj, the system(17.1)isgloballyasymptoticallystable(in the sense of Liapunovenergyfunctions). Thefixed–points(stable states)of the system correspond to the fundamental memories to be stored, so it works as content– addressable memory(AM). The initial state of the system(17.1)lies inside the bas in of attraction of itsfixed–points, so that its initial state is related to appropriate memory vector. Various variations on this basic model are reported in the literature[Hay 98, Kos 92], and more general form of the vector–fieldcanbe given, preserving the above stability conditions.

Depending on the connectivity, the so–called recurrent neural networks(see [II 07b]), consisting of simple computational units, canshowverydifferenttypes of dynamics, ranging from totally ordered(linear–like behavior)to chaotic. Using the the mean–fieldtheoryapproachwithevolvingHammingdistance, Bertschinger and Natschl¨ager analyzed how the type of dynamics(ordered or chaotic) exhibited by randomly connected networks of threshold gates driven by a time-varying input signal depended on the parameters describing the distribution of the connectivity matrix(see [II 08c]). In particular, the authors calculated the critical boundary in parameters pace where the transition from ordered to chaotic dynamics takes place. Employing a recently developed framework for analyzing real-time computations, they showed that only near the critical boundary could such networks perform complex computations on time series. This result strongly supports conjecture sthatdynamicalsystemsthatare capable of doing complex computational tasks should operate near the edge of chaos, that is, the transition from ordered to chaotic dynamics(see[II 08b]).

In particular, the authors pointed out the following:(i)Dynamics near the critical line area general property of input–drivendynamicalsystemsthatsupportcomplexreal-time computations; and(ii)Such systems can be used by trainingasimplereadoutfunctiontoapproximateanyarbitrarycomplexfilter. This suggests that to exhibit sophisticated information processing capabilities, an adaptive system should stay close to critical dynamics. Since the dynamics is also influencedbythestatisticsofthein put signal, these results indicate a new role for plasticity rules: stabilize the dynamics at the critical line. Such rules would then implement what could be called dynamicalhomeostasis, which could be achieved in an unsupervised learning manner.

Now, if critical dynamics at the edge of chaos of classical adaptive systems is essentially nonlinear, then what can we say about modern adaptive quantum systems? While quantum mechanics is based on super position principle(see Chapter 3), and thus is essentially linear, adaptation introduces critical nonlinearity. In this way, we come to the nonlinear deformation/extension of the linear Schr¨odingerequation.

An impetus to hypothesize a quantum brain model comes from the brain’s necessitytounifytheneuronalresponseintoasinglepercept. Anatomical, neurophysiological andneuropsychological evidence, as well as brain imaging using fMRIandPETscans, show that separate functional maps exist in the biological brain to code separate features suchas direction of motion, location, color and orientation. How does the brain compute all this data to have a coherent perception?

To provide a partial answer to the above question, a quantum neural network (QNN) (see Figure 17.1) has been proposed (see [II 08c] and references therein), in which a collective response of a neuronal lattice is modeled using the Schr¨odinger equation:

2Δψ(x, t) + V (x)ψ(x, t), (17.4) where ψ(x, t) is the wave function, or probability amplitude, associated with the quantum–mechanical system at a space-time point(x, t)andΔis the standard Laplacian operator. It is shown that an external stimulus reaches each neuron in a lattice with a probability amplitude function ψi. Such a hypothesis would suggest that the carrier of the stimulus performs quantum computation. The collective response of all the neurons is given by the quantum super position equation, ciψi= ciψi.

TheQNNhypothesissuggeststhatthetimeevolution of the collective response ψ=ψ(x, t)is described by equation(17.4). Aneuronal lattice sets up a spatial potentialfieldV (x). A quantum process described by a quantum stateψ whichmediatesthecollectiveresponseofaneuronallatticeevolvesinthespatialpotentialfieldaccordingtoequation(17.4). Thus the‘classical brain’sets up as patio-temporal potentialfieldV (x, t)and the‘quantum brain’is excited by this potentialfieldtoprovideacollectiveresponseψ=ψ(x, t)(see[II 08c] and references there in).

Fig.17.1. Quantum neural network has three blocks: quantumstochasticfilterbased onthenonlinearSchr¨odingerequation, Kalman–likepredictorandneuralmotorservo, correspondingtotheKalman–like correct or.

theplantdefinedbythelinearSchr¨odingerequation(17.4). Informally, QNNis representedbyadaptiveNLS, thatisNLSwith‘synapticweights’wire placing quantum super position constant sci. Formally, itistheSchr¨odingerequation (17.4)withaddedcubicnonlinearityandsynapticweights.

Consider a classical per cept ron[II 07b], i.e., the system with nin put channels x 1, . . . , xnandoneoutputchannely, given by wheref(·)is the per ceptronactivationfunctionandwj are the weights tuning during learning process.

1. The weight swjareinitializedtosmallrandom numbers. 2. A pattern vector(x 1, . . . , xn)is presented to the per cept ron and the output ygeneratedaccordingtotherule(17.5) 3. The weights are updated according to the rule wj(t + 1) = wj(t) + η(d −y)xj, (17.6) wheretisdiscreettime, dis the desired output provided for training and 0 < η< 1 is the step size.

It will be hardly possible to construct an exact analog of the nonlinear activation functionf, like sigmoid and other functions of common use in neural networks, but we will show that the leaning rule of the type (17.6) is possible for a quantum system too.

Consider a quantum system with nin puts|x 1⟩, . . . , |xn⟩of the form(17.8), and the output|y⟩derived by the rule whereˆwjbecome 2 ×2 matrices acting on the basis (|0⟩, |1⟩), combined of phase shifterseiθand beam split ters, andpossiblylightattenuators,ˆFis an unknown operator that can be implemented by the network of quantum gates.

Also, quantum computation on optical modes using only beam splitters, phase shifters, photonsourcesandphotodetectorsispossible. Following this concept, wecanassumetheexistenceofaqub it[II 08c] where|α|2+ |β|2=1, where the states|0⟩and|1⟩are understood a sdifferent polarizationstatesoflight.

where|d⟩is the desired output.

For smallη(0 < η< 1/n)and normalized input states⟨xj|xj⟩= 1 the result of iteration converges to the desired state |d⟩. The whole network can be then composed from the primitive elements using the standard rules of ANN architecture. For more technical details, see[II 08c].

Let us start with the classical connection ist brain theory 2( see[Arb 98] fora general overview). A recent paper [Roy 08] proposes a engineering new theory for the supposed internal mechanisms of the brain function. It postulates that there are controllers in the brain and that there are parts of the brain that control other parts. Thus, the new theory refutes the standard connectionist theory that there are no separate controller sin the brain for higher level functions and that all control is ‘local and distributed’ at the level of the cells. Connectionist algorithms themselves are used to prove this theory. The author claims that there is evidence in the neuroscience literature to support this theory. Thus, this paper proposes a control the or etic approach for understanding how the brain works and learns. That means that control the or etic principles should be applicable to developing systems similar to the brain.

Having established the connection between connection ism and the control theory, itisasmallsteptoarguethatifconnectionistmodelsarebrain-like, then, given that they use executive cont rollers, the internal mechanisms of the brain must also use executive cont rollers. Hence, it can be generalized and claimed that there are parts of the brain that control other parts. A control the or etic approach to understanding how the brain works and learns could overcome many

2 As already mentioned, connection is mis an approach in thefieldsofartificialintelligence, cognitive psychology/cognitive science, neuroscience and philosophy of mind, that models mental or behavioral phenomena astheemergentprocessesofinterconnectednetworksofsimpleunits. There are many forms of connection ism, but the mostcommonformsuseANNs.

of the limitations of connection ism and lead to the development of more powerful models that can properly replicate the functions of the brain, suchas autonomous learning without external intervention. None of the supervised connectionist algorithms can learn without human intervention that requires setting various learning parameters by a trial-and-error process. Thus, it is impossible to build autonomous robots (software or hardware), ones that can learn on their own, withtheseconnectionistalgorithms. The new proposed control the or etic paradigm, however, should allow thefield to freely explore other means of learning in neural networks[Roy 08].

BasicbrainphysiologywillbepresentedinPartII. In this section we focus on its latest and most important part, calledneocortex3 performs a large variety of complex computations in real time. Itisconjectured that these computations are carried out by a network of cortical micro-circuits, where each micro-circuit is a rather stereotypical circuit of neurons within a cortical column. A characteristic property of these circuits and networks is an abundance of feedback

3 The neocortex(Latin for ‘new bark’ or ‘new rind’) is a part of the brain of mammals. It is the outer layer of the cerebral hemispheres, and made up of six layers, labeled ItoVI(withVIbeingtheinnermostandIbeingtheoutermost). The neo cortex is part of the cerebral cortex(alongwiththearchicortexandpaleocortex, which are cortical parts of the limbic system). It is involved in higher functions such as sensory perception, generation of motor commands, spatial reasoning, conscious thought and, in humans, language.

Theneocortexconsistsofthegreymatter, or neuronal cell bodies and unmyelinatedfibers, surrounding the deeper white matter(myelinatedaxons)inthecerebrum. Whereastheneocortexissmoothinrodentsandothersmallmammals, it has deep grooves(sulci)and wrinkles(gyr i)in primatesandotherlargermammals. These folds increase the surface area of the neocortex considerably without taking up too much more volume. Thishasallowedprimatesandespeciallyhumanstoevolve new functional areas of neo cortex that are responsible for enhanced cognitive skills such as working memory, speech, and language. The neocortex contains two primary types of neurons, excitatorypyramidalneurons(80%ofneocorticalneurons)and inhibitoryinterneurons(20%). Theneuronsoftheneocortexarealsoarrangedin verticalstructurescalledneocorticalcolumns.

The neo cortex is divided into frontal, par i etal, temporal, andoccipitallobes, which per for mdifferentfunctions. Forexample, theoccipitallobecontainstheprimaryvisualcortex, and the temporal lobe contains the primary auditory cortex. Further subdivisions or areas of neocortexare responsible for more spec ific cognitive processes. In humans, the frontal lobe contains areas devoted to abilities that are enhanced in or unique to our species, suchascomplexlanguageprocessinglocalizedtotheventrolateralprefrontalcortex(Broca’sare a)and social and emotional processing localized to the orb i to-frontal cortex.

The female human neocortexcontainsapproximately 19 billionneurons while the malehumanneocortexhas 23 billion on average[PG 97]. Additionally, the female neocortexhasmorewhitematter, whilethemaleneocortexcontainsmoregrey matter. Theimplicationsofsuchdifferences are not fully known.

connections. But the computational function of these feedback connections is largely unknown. Two lines of research have been engaged to solve this problem. In one approach, which one might call the constructive approach, one builds hypothetical circuits of neurons and shows that(under some conditions on the response behavior of its neurons and synapses) such circuits can perform spec ific computations. In another research strategy, which one might call the analytical approach, one startswith data-based models for actual cortical micro-circuits, and analyzes which computational operations such “given” circuits can perform under the assumption that a learning process assigns suitable values to some of their parameters(e.g., synapticefficaciesofreadoutneurons). An underlying assumption of the analytical approach is that complex recurrent circuits, suchas cortical micro-circuits, cannot be fully understood in terms of the usually considered properties of their components. Rather, system-level approaches that directly address the dynamics of the resulting recurrent neural circuits are needed to complement the bottom-up analysis[MJS 07]. This line of research started with the i dent ificat i on and investigation of so-called can on ical micro circuits[DKM 95]. Several issues related to cortical micro-circuits have also been addressed in the work of S. Grossberg and collaborators from Boston (see [Gro 03] and the references therein). Subsequently it was shown that quite complex real– time computations on spike trains can be carried out by such‘given’models for cortical micro-circuits(see[DM 04] for a review). A fundamental limitation of this approach was that only those computations could be modeled that can be carried out with a fading memory, more precisely only those computations that require integration of information over a time-span of 200 ms to 300 ms (its maximal length depends on the amount of noise in the circuit and the complexity of the input spike trains). In particular, computational tasks that require a representation of elapsed time between salient sensory events or motor actions, or an internal representation of expected rewards, working memory, accumulation of sensory evidence for decision making, the updating and holding of analog variables suchas forexample the desired eye position, anddifferentialprocessingofsensoryinputstreamsaccordingtoattentionalorotherinternalstatesof the neural system could not be modeled in this way. Previous work on concrete examples of artificial neural networks and cortical micro-circuit models had already indicated that these shortcomings of the model might arise only if one assumes that learning affectsexclusivelythesynapsesofreadoutneuronsthat project the results of computations to other circuits or areas, without giving feedback into the circuit from which they extract information. This scenario is unrealistic from a biological perspective, sincepyramidalneuronsinthecortex typicallyhaveinadditiontotheirlongprojectingaxonalargenumberofaxon collateralsthatprovidefeedbacktothelocalcircuit[MJS 07]. Also, abundant feedback connections exist on the network level bet we end ifferentbrainareas. For statistical mechanics of neo cortical interactions (using path integral–related methods), see[Ing 82, Ing 97, Ing 98].

pyramidal apical dendrites finish in a tuft-like branch ing in lamina I. It has been accepted(see, e.g.[Bec 08]) that the apicalbundlesofdendritesarethe basic anatomical units of the neo cortex. They are observed in all areas of the cortex that have beeninvestigatedinallmammals, including humans. It has been proposed that these bundles are the cortical units for reception, which would give them a preeminent role. Sincetheyarecomposedessentiallyofdendrites, thenamedendronwasadoptedbyJohnEccles 4[ Ecc 90, Ecc 94]. Figure 17.2(b) illustratesatypicalspinesynapsethatmakesanintimatecontactwithanapicaldendriteofapyramidalcell. The inner surface of about on confronting the synapticcleftdformsthepresynapticvesiculargrid(PVG). A nerve impulse propagating into about oncausesaprocesscalledexocytos is. A nerve impulse evokesatmostasingleexocytosisfromaPVG[Bec 08]. Exocytosisisthebasic unitary activity of the cerebral cortex. Each all-or-nothing exocytosisof synaptic transmittersubstanceresultsinabriefexcitatorypostsynapticdepolarization (EPSP). Summationbyelectrotonictransmissionofmanyhundredsofthese mill i-EPSPsisrequiredforanEPSPlargeenough(1020mV)to generate the dischargeofanimpulsebyapyramidalcell. This impulse will travel along its axon tomakeeffectiveexcitation atitsmanysynapses. This is the conventional macro-operationofapyramidalcelloftheneocortex, and it can be satisfactorily described by conventional neuroscience, even in the most complex design of neural network theory and neural group selection(see, e.g.[Sze 78]).

Experimental analysis of transmitter release by spine synapses of hippocampal pyramidalcellshasrevealedaremarkablylowexocytosisprobabilityperexcitatoryimpulse. This means that there must exist an activation barrier against opening of an ion channel in the PVG. Activation can either occur purely stochasticallybythermalfluctuations, or by stimulation of a trigger process. Recently, F. Beck has proposed a two–statequantumtriggerwhichisrealizedbyquasiparticletunneling.5This is motivated bythepredominantroleofexocytos is as thesynapticregulatorofcortical activity which is certainly not completely at random[Bec 08]. Thequasi-particle assumption allows the treatment of the

4 Sir John Carew Eccles(January 27,1903– May 2,1997) wasanAustralianneurophysiologistwhowonthe 1963 Nobel Prize in Physiology or Medicine for his work onthesynapse. He shared the prize together with Alan Lloyd Hodgkin and Andrew Fielding Huxley, whoformulatedthecelebratedHodgkin–Huxleyneuralmodel. 5 The so-calledquantumtunnelingeffectisananoscopicphenomenonin which a quantum particle violates the principles of classical mechanics by penetrating a potentialbarrierorimpedancehigherthanthekineticenergyoftheparticle.A barrier, in terms of quantum tunnel ing, may be a form of energy state analogous to a ‘hill’ or incline in classical mechanics, which classically suggests that passage through or over such a barrier would be impossible without sufficient energy. On the quantum scale, objects exhibit wave-like behavior; in quantum theory, quanta moving against a potential energy‘hill’can be described by their wave-function, which represents the probability amplitude offindingthatparticleinacertain location at either side ofthe‘hill’. If this function describes the particle as being on the other side of the ‘hill’, then there is the probability that it has moved through, rather than over it, and has thus‘tunneled’.

Fig. 17.2. Schematic of neocortical bio physics(adapted from [Bec 08]): (a) 3D construct by[Sze 78] showing cortical neurons of various types; therearetwopyramidalcellsin laminaVandthreeinlaminaIII, one being shown in detail in a column to the right, and two in lamina II. (b) Detailed structure of a spine (sp) synapse on a dendrite (den); st, axon terminating in the synaptic bout on or presynaptic terminal (Pre); sv, synaptic vesicles; c, presynapticvesiculargrid(PVGintext); d, synapticcleft; e, postsynaptic membrane; a, spine apparatus; b, spines talk; m, mitochondrion[Gra 82].

complicated molecular transition a saneffectiveone–body problem whose solution follows from thetimedependentSchr¨odinger equation (in normal units, see PartII)

2Δψ(q, t) + V (q)ψ(q, t).

Here it is assumed that the activated state of the presynaptic cell lasts for a finite time period[t 0, t 1] only before i tre combines. Recombinationoftheactivated state defines the measurement process, leading to von Neumann’s collapse of the tunnel ing state[Bec 08].

ψ(q, t 1) = ψleft(q, t 1) + ψright(q, t 1), which constitutes the two amplitudes for the alternative results of the same process: which, after collapse, determine: either exocytosis has happened (ψright), orexocytosishasnothappened(ψright)(inhibition). State collapse transforms this into the probabilities[Bec 08]:

|ψright(q, t1)|2dq.

All theories that hold the neuron as the functional basis of consciousness must bridge a gap between a property of conscious experience and a fundamental tenet of neuronal processing. While evidence suggests that the brain subdivides perceptual processing into modality(e.g., the visual, the tactile)and sub modality (e.g., color, temperature), our perceptions themselves area unifiedexperience [Mas 04]. If the anatomic substrate of perceptual modality is functionally and spatiallydiscreteneuronalsubpopulations, how is information ultimately synthesize dtocreatethemanifestonenessof experience? The cognitive binding problem6 is a central question in the study of consciousness: how does the brain synthesize its modal and submodal processing systems to generate a unity of conscious experience? Binding is thought to occur at virtually all levels of perceptual (and motor)processing, and is thought to be a crucial event for consciousness itself[CK 90, Cri 94]. Although a relatively recent question in neuroscience, the binding problem may have made itsfirstappearanceinImmanuelKant’sCritique of Pure Reason. Kant’s principle of a transcendental unity of apperception7

6 The term cognitive binding problem is attributed toC. vonderMahlsburg[Mal 81]. 7 Apperception(from Latin ad + percipere= to perceive) has the following meanings: (i)inepistemology, itistheintrospectiveorreflectiveapprehensionbythemindof its own inner states;(ii)in psychology, it is the process by which new experience is assimilatedtoandtransformedbytheresiduumofpastexperienceofanindividualtoformanewwhole(in short, it is to perceive new experience in relation to past experience); (iii) in philosophy, Kant distinguished empirical apperception from transcendent a lap perception(thefirst is‘the consciousness of the concrete actual self with its changing states’, the so-called‘inner sense’; the second is‘the pure, original, unchangeable consciousness which is the necessary condition of experience as such and the ultimate foundation of the synthetic unity of experience’.

There can be in us no modes of knowledge, no connection or unity of consciousness of one mode of consciousness with another, without that unity of consciousness which precedes all data of intuitions, and by relation to which representation of objects is alone possible...For the mind couldneverthinkitsidentityinthemanifoldnessofitsrepresentations, and indeed think this identity a priori, if it did not have before its eyes the identity of its act, whereby it subordinates all synthesis of apprehension(which is empirical)to a transcendental unity, thereby rendering possible their interconnection according to a priori rules

There have been various solutions proposed for the binding problem, which can be summarized as binding by convergence(information is bound by higher order neurons that collect various responses andfireas a binding unit when the full set of inputs converge), binding by assembly(binding occurs as a result of self–organizing Hebbiancellassemblies), and binding by synchrony(binding results from the synchronized firing of neuronal sub-populations that are spatially discrete)[Mas 04].

The discussion of distributed domains of reality and consciousness leads us intoarecentviewofthebrainthatdoesnotregardtheneuronasaunitof cognitive perception, but rather regards the brain as one indivisible entity. This view is based on the emergingfieldofquantumneurodynamics[Mas 04]. The brain, insteadofbeingaNewtonianobjectobeyingclassicallaws, is posi ted to beamacroscopicquantumobjectobeyingthesamelawsfoundatthePlanck scale[Pen 94]. The large–scale quantum coherence would render the brain a Bose–Einsteincondensate, where properties of quantum wave-functions hold at amacroscopiclevel. These properties would include non-local ity and quantum super position. Such large- scale coherence can be found in superconductivity and superfluidity, where the environment is dis entangled from the quantum events. Superconductivityandsuperfluidityoccurattemperaturesjustaboveabsolute zero, where it i sdifficult for the environment to interfere with the quantum coherence. H. Fr¨olich, however, predicted that quantum coherence could also occur in the temperature of a biologic environment (e.g. the brain) with high metabolic energyandextremedielectricpropertiesofthematerialinvolved[FK 83]. It has been posited by S. Hameroffthat the cytoskeletal elements of microtubulescould be just such a material[Ham 87]. Microtubulesarecomposedoftubulindimers, which can exist in multiple con formations based ond i electric properties, leading to dip ole oscillations that could be transmitted through the length of the tubule. ThesesuperconductingFr¨olichwavescouldbetransmittedthroughout the proteinnetworkandgapjunctions: fromcytoskeletontomembraneprotein to extracellular matrix to adjacent membraneproteintocytoskeleton, and so on. It is of interest that the supporting glial cells, the cellular majority in the brain, would contribute to this network, recall i ngC. Golgi’sconceptionofthe syncytium. It is currently a point of theoretical contention whether quantum coherenceinthebrainwoulddecoherefromawaveformcollapseofsuperposed states(termed objective reduction)[HP 96], or from the‘warm, wet, and noisy’ environment of the brain[Teg 00]. It has been suggested that the hollow core of microtubules, ordered water, andactingelationmaybuffert he quantum events from the environment. Magnetic resonance imaging of the brain by quantum coherence(albeit an induced arti fact of the procedure)has also been suggested as proof of principle[HHT 02].

Hameroffhassuggestedthatquantumcomputationwithinthemicrotubules could be involved in consciousness, andproposestheactivityofgeneralanesthetics as support for this claim [HW 83]. He has argued that the similar activity of diverse chemical structures within the class of anesthetic scan be explained byacommonactionofbindingtohydrophobicdomainsandmodulatingdipole momentsinthetubulincomponentsofmicrotubules. This is supported by the factthatanestheticsinhibitprokaryoticmotility, afunctionmediatedbymicrotubules. ItisofinterestthatrecentstudiesusingquantitativeEEGsuggest that anestheticsofvariouspharmaco logic properties may act by interrupting cognitive binding.

Recall from Gestalt psychology(see, e.g.[HHB 01]) thatbi–stable perception arises whenever a stimulus can be thought in two different alternatives ways. The essential points are here on the terms ‘be aware of ambiguity’ and ‘manifest state of only one per cept at time’. Forexample consider the ambiguous Gestalt picture called Rubin face8 given in Figure 17.3. A quantum–mechanical counterpart

8 Rubin’s vase, also known as the figure–groundvase) is a famous set of cognitive optical illusions developed around 1915 by the Danish Gestalt psychologist Edgar Rubin. His fundamental principle of cognitive optical illusions reads: When twofieldshave acommon border, and one is seen asfigure and the other as ground, the immediate perceptualexperienceischaracterizedbyashapingeffectwhichemergesfromthe common border of thefieldsandwhichoperatesonlyononefieldoroperatesmore strongly on one than on the other (see[HHB 01]). The illusion generally presents the viewer with a mental choice of two interpretations, each of which is valid. Often, the viewer sees only one of them, and only realizes the second, valid, interpretation after sometime or prompting. When they attempt to simultaneously see the second andfirstinterpretations, they suddenly cannot see thefirstinterpretationanymore, and no matter how they try, they simply cannot encompass both interpretations simultaneously–oneoccludestheother. The illusions are useful because they are an excellent and intuitive demonstration of thefigure–ground distinction the brain makes during visual perception. Rubin’sfigure–ground distinction, since it involved higher–level cognitive pattern matching, in which the overall picture determines its mental interpretation, ratherthantheneteffectoftheindividualpieces, influencedtheGestaltpsychologists, who discovered many similar illusions themselves. Normally the brain class ifiesimagesbywhatsurroundswhat, establishing depth and relationships. If something surrounds another thing, the surrounded object is seen asfigure, and the presumably further away(and hence background)object is the ground, and vice versa. This makes sense, since if apiece of fruit is lying on the ground, one would want to pay attention to the‘figure’and not the‘ground’. However, when the contours are not so unequal, ambiguity starts to creep into the previously simple inequality, and the brain must begin‘shaping’what it sees; it can of such mind processes was proposed in [Con 08]. In this quantum model of mental states, the state of the potential consciousness is represented by a quantum state–vector in Hilbertspace. If we in dic a tee.g., a 2D case with potential states |1⟩and|2⟩, the potential state|ψ⟩ofbi–stable perception is given by

Here a and bre present the respective probability amplitudes of the two stable perception states, so that|a|2gives the probability that the state of consciousness, represented by the percept |1⟩will be finally actual i zed or manifested during perception; conversely, |b|2represents the probability that state(percept)|2⟩of consciousness will be actual i zed or manifested during perception. Closely–related is bin ocular rivalry model of[Man 07], with a wave-function between perceptual events describing a state of potential consciousness and wave-function collapse when perceptual event is manifested.

Fig.17.3. Rubinface: ambiguousfigureinducingtwodifferentrepresentations, vase or faces. Thesubjectisawarethatdifferentrepresentationsarepossible, but he can perceive only one of the two possible percepts at any one time. The bi–stable perception is described as:|ψ⟩=a vase|1⟩+ bfase|2⟩. In manifest or actual state of consciousness the subject will select or the vase or the faces[Con 08].

be shown that this shaping overrides and is at a higher level than feature recognition processes that pull together the face and the vaseimages–one can think of the lower levels putting together distinct regions of the picture(each region of which makes sense in isolation), but when the brain tries to make sense of it as a whole, contradictionsensue, and patterns must be discarded.

On the other hand, consciousness, which has traditionally been based on Descartes’statement“Cogitoergosum,”refers to our capacity of asking questions about the so-called reality(see[Pop 08]). It concerns thefieldof‘potential information’instead of the so-called‘actual information’.9In order to describe its contents, a measure of potential information was suggested in terms of the capacity of actualities in the possible contents. This definit i on includes a variety of relevant consequences, i.e., the appearance of a memory, the question of time down to relationofmolecularbiophotons.10Consciousness works for evolutionary purposes which is not a matter of fact but a process. The improvement and the optimization of consciousness is at the same time the most important healing power of life. In other words, the communication system of biological systems is basedoncavityresonator waves and light guides. This example shows the identity of mitotic figures with the force pattern of cavity reson at or waves within the cells. These forces regulate the migration of the molecules in such away that no error takes place. At the same time such a pattern represents in the average one coherentbiophoton. They are the origin of consciousness.

1. Self confidenceaccordingtothereferencepointofDescartes:“Iam,”resulting from the transformation of the physical existence of the body into ‘doubts’and back to the control, and soon. The end of this process is the adjustment of actual and potential information. 2. I dent ificat i on(awareness)according to the transformation of the actual information of the physical existence of matter or radiation into‘doubts’and back to the control, and so on. This leads after adjusting actual and potential information to the confirmationof“Youare”or“Itis.” 3. Prediction: Repeated transformations of the actual information ofthephysicalexistenceofdifferent objects into the possibilityfieldofthepotential information under adjustment of the potential information of the memory which provide boundary conditions in the control processes. The transformations are again finished as soon as there is an equality of actual and potential information. The result is a statement:“It will be.”

9 Thesimplestsystemwhereactualandpotentialinformationaremutuallytransformedarecavityresonators. The potential information(in bit)is identical to the Q−value of there son at or. Theactualinformationreflectsthedistributionofthe codedsequenceontheemittedwave. The actual information cannot exceed there the potential one. It corresponds to the maximum number of switch–on/offprocesses of a wave that i sreflectedNtimeswithintheresonator. 10 Biophotons (BPHs) are weak photons within or emitted from living organisms. BPH emissionoriginatesfromade-localized coherent electromagneticfieldwithinliving organisms and is regulated by thefield. Based on experimental results concerning Pois son and sub-Pois son distributions of photo count statistics, thecoherentpropertiesofBPHsandtheirfunctionsincellcommunicationaredescribedin[Cha 08]. FunctionsofBPHrolesarediscussedinsomeimportantprocesses, includingDNA replication, transcript i on, protein synthesis, and cell signalling, and in the processes of oxidativephosphorylationandpho to synthesis.

4. Memory and Inspiration. Repeated transformation and re-transformation of potential information of the past and the presence. After adjustment of these probability distributions the statement will follow:“It could have been,”or “It could become.”

17.2 The Mental Quantum Leap: INT ENT ION⇛ACT ION⇛MANIF EST AT ION

The Mental Quantum Leap represents the Mental–to–Physical transition leap: INT ENT ION⇛ACT ION⇛MANIFEST AT ION, whichisformallydefinedbytheFeynmanpathintegral:

On the left-hand side of this quantum-fieldpath-integralthereisDirac’str an sition amplitude:⟨Manifestation|Action|Intention⟩, whose absolute square |⟨Manifestation|Action|Intention⟩|2is the overall probability for the transition leap: INT ENT ION⇛ACT ION⇛MANIFEST AT ION. In other words, theabsolutesquareP= |⟨Manifestation|Action|Intention⟩|2is the transition probability Pof occurring thefinalstateofPhysicalManifestationgiventhe initial state of Mental Intention, via the action-at-a distance within a quantum bio-electro-magneticfield.

On the right-hand side of(23.73)isthecelebratedFeynmanpathintegral of modern quantumfieldtheory. In the words of Stephen Hawk ing, it is the ‘wavefunctionoftheUniverse’. Technically, it is the complex-valued integral over the bio-electro-magnetic quantumfieldΦ.The Dirac’ssymbol

 Σdenotes the Lebesgueintegration (withthefieldmeasure D[Φ])over the continuousfield spectrum and summationΣover the spectrum of discrete quantum particles.

Finally, the key term in thefieldpathintegral(23.73)is the physical-field actionS[Φ], which is dimensionally: Effort = Energy × T ime. Therefore:

phys i calEffort, weshallgetthedesiredPhysicalManifestation.

Now, let’selaboratealittle-bit on this creative quantum–leap process. Historically, thefirstquantumparticle-fieldtheorywasdevelopedin 1949 by John A. Wheeler and RichardP. Feynmanastheirfamousaction–at–a–distanceelectrodynamics(which is explained in‘plain English’in Feynman’sNobellecture). Here, wegiveamentalinterpretationtotherelativisticaction–at–a–distance fieldtheory(occurringwithintheEinsteinspace-timemanifold), based on two (or more)thought–particles, moving with light-like speed, each in in its own time. The key to our quantum leap is the Mental Focus, which we define as the synchronization of thoughts.

Formally(using Einstein’ssummationconvention)the action–at–a–distance S[Φ] = S[x, y; ti, tj]reads:

2 , (17.11) Intention ≤ti, tj, t ≤Manifestation.

Here˙xi(ti)=dxi/dti and˙yj(tj)=dyj/dtjarethespace-time velocities of two thought-particlesxi(ti)andyj(tj), moving in their own times(ti)and(tj), respectively. Thefirst term in(17.11)represents the mental potential energy of Intention, formalized as an interaction between the two elementary particles. It is a double integral over a delta function of the square of interval I2 between the two thought–particles; thus, interaction occurs only when this interval, representing the cognitive distance between the two thoughts, vanishes; that is, when mental synchronization, or mental focus, occurs. Therefore, the subsequent physical action isefficient only after the mental focus. It is given by the second term in(17.11), which represents the physical kinetic energy:

generated bytheRiemannianmetrictensorgijof the Einstein space-timemanifold(see[II 06c, II 07e]).

Part 5 · pp. 125–154

The tennis champion of the future is Roger Federer’shypotheticalyounger brother, whoknowsandcandoeverythingthatRogerknowsandcando. In addition, he is both physically and mentally stronger and faster. Because of this addition, hewillgenerallybemuchmoreefficient in the future tennis game, which will be much faster than today’s tennis, due to improved racquet technology and players’improved psychophysical abilities.

The superb tennis weapons of the future champion will demonstrate their general efficiencyinallgamesituations:

1.An aggressive 300+km/h serve, targeting either the “T”–spot, or the weaker side of the opponent; the secondserve is the same as thefirst 300+km/h serve; 2. An aggressive, passing return, based on the‘mind/body–reading’anticipation and lightning–fast reaction(based on the stretch–reflex) 3.An aggressive, penetrating 240+km/h forehand and single–handed 200+km/hbackhand(or, a two-handed 220+km/hbackhand), hit consistently from anybody configurationandfromanypositiononthecourt; 4. Inaddition, a serve–volley game can add-up to the general aggressive attitude.

A quick and accurate decision–making is crucial: both for the serve and for the aggressive return, as well as for the most appropriate shotselection in any other game situation. It is based onefficientlyreadingtheballand accurately predicting the opponent’snextmove. In two words, this is what we call the mental speed. It will be trained(for years)by means of blitz and lightning–chess.

match/tournament. In two words, this is what we call the mental strength. It will be trained(for years)by means of visualization exercises.

High ability of learning on the spot means adjusting/changing both technique and tactics in the most appropriate way according to the situation on the court. (Forexample, if playing against a baseline player–being able to come to the net a ndefficientlyfinishthepointwiththewinningvolley.)

• Natural talent; • Commitment(to regular and consistent training/competition); • Passion(desire to train, compete and win); • Self–determination(responsibility for their performance); • Self–belief(I can do it, self–confidenceintheirowncapacity); • Planned approach(clear goals; Federer himself admits that“scheduling has been a very important factor in my success – it helps me to heal from injuries and mentally getaway”); • Quick adaptation(to the changing situation on the court); • Mind–body balance; • Competitive toughness(like Federer, Nadal, Henin); and • Perseverance(persistence);

1. Acceleration(fast sprinting up to 15 m, from any starting body configuration); 2. Agility(quick changing of directions in both sprinting and jumping); and 3. Athleticism, which has two components: (i)ability to throw the current(reconstructed)officialjavelin 60 m;1and (ii)ability to perform a somersault on a court.2

Optimal body height(posture)for the future male tennis champion will be around 1.85–1.87m,(like Federer, Nadal, Roddick, Djokovic), while his optimal body–weight will be 80 kg(like Federer and Djokovic). Optimal age for a male champion will be 22–27(+/-2) years old.

If we would like to make a hypothetical future champion out of the current players, thenitwouldbeacombinationofRogerFedererwithAndyRoddick’s power–serve, Rafael Nadal’sagilityandGaelMonfils’power–forehand.

1 This means doing several years of parallel javelin training during the teenage years; this would give a 300+km/hserve. 2 We do not suggest that the future tennis players will actually have to perform somersaults on the court. Rather, any gymnastics–trained kid that can perform a somersault on the street with a safe landing on his feet obviously have all the abilities commonly covered by the umbrella of athleticism: superb jump take–off, superb coordination and superb balance. Every elite tennis player can have this true athleticism if in his youth he had several years of parallel gymnastics training.

At the beginning we claimed that Roger Federer has the brain of the sports champion. What does it really mean? In this section we will answer to this question, usingasanexampleMichaelSchumacher,3the greatest driver in the car-racing history. How can someone win 91 Grand Prix races in a single lifetime? WhatisthesecretbehindthegreatestF1 legend?

Michael Schumacher was one of the brightest stars in sport history. He had the ultimatebrainofaChampion. There are nine essential characteristics of the brainofaChampionineverycomplex, speed-characterized sport. These areas follows:

1. Willpower ThefirstmarkofaWorldChampioninanysportistheindomitable will power. This ability must be extremely well developed as it was in the extremely rare case of seven-time World Champion, Michael Schumacher.

2. Focus/Concentration From the indomitable will power follows the sharpest focus. This is an essential ability of every Champion, and is a capability which allows one to fully concentrate on asingle thing for several hours, as if nothing else exists. Schumihadthesharpestfocusinthe history ofF1 racing.

3. Planning Schumacher’s character made him an extremely thorough planner of every race. This was clear in his 19/94/95 seasons, while driving for Be net ton. This was even more powerfully demonstrated in his“golden period,”20/00/04, withhisunbeatableFerrariteam. They collectively planned and carefully executed every single detail before and during the race.

4. Risk Taking Certain amount of risk-taking is necessary for high achievements in anyfield. Forexample, gymnastic rules award 0.20 of a mark for high risk, which means that a perfectly executed exercise is not enough for a perfect mark. High risk is also required. Schumiwaswell-known for his highlevel of risk-taking(whichevengothimdisqualifiedfromaChampionshipin 1997, aftercollisionwithhisrivalJacquesVilleneuve, who after that won the Championship).

3 German car-racingdriverMichaelSchumacher, popularlycalledSchumi(or, Schuey), was a records even-time Formula One World Champion(1994,1995,2000, 2001,2002,2003,2004) with a record of 91 Grand Prix races won. He has been officiallydeclared“statistically the greatest driver”inthehistoryofFormulaOne (or, F 1) racing. Besides winning drivers’Championships and race victories, Schumi alsoholdsmanyotherrecordsinF1 racing, including: fastestlaps, pole positions, points scored(1,369)and most races won in asingle season.

5. Situation Awareness The modern buzzword,“situation awareness”, has been defined (by M. Endsley) as “The perception of the elements in the environment within time and space, the comprehension of their meaning, and projection of their status in the near future.”It is necessary for optimal decision-making. Situation awareness is one of the most important conditions for safety in everyday car driving; even more so, inF1 racing. This was one of Michael Schumacher’s superior abilities as the greatest driver ever.

6. Decision-Making Decision-making is the“sine-qua-non” for winning in any complex game/situation. In the past, it was evident in soccer, whenasingle movementofPeleorMaradonadecidedthewholematchandultimatelythe world Championship. Today, similar decision-making capability can be seen in tennis in the most important winning shots from Roger Federer. The same ability wasalsooneofSchumi’swinningqualities, namely the ability to decide when and where to push the car to its maximum, as it is impossible for the car to be pushed to its maximum all the time.

7. Anticipation of Opponents’Moves One of most dominant characteristics of a master in any competitive field is their ability to read the mind of an opponent. Like Roger Federer in tennis, Schumacher also had this crucial ability to sense, predict, or anticipate when and were his rivals will attack. This enabled him to prepare and execute an effectivecounter-attack which helped him on more than one occasion.

8. Reflexes In all speed-dominated sports, the Champion must have lightningfast reflexes. As F1 is probably the fastest sport, clearly Schumi was no exception to this rule. From his early youth in cart-driving, totheendofhisChampion career, he had lightning-fastreflexes. When combined with his superior anticipation of his opponents’moves, thisgavehimthefastestandmosteffective complex reaction.

9. Complete Situation Control As a result of the above 8 characteristics, follows complete situation control. Schumi had the ability to control the whole race, from start tofinish. This was most clearly demonstrated during his“golden period,” 200004, whenhewonmoreracesandChampionshipsthananyotherdriverin F1 history.

In summary, Michael Schumacher had the strongest willpower, the most thorough planning, the sharpest focus/concentration, the highest risk-taking, the most complete situation awareness, the quickest and most appropriate decision making, the most accurate anticipation, andthefastestreflexes. As a result, he had the highest rate of complete situation-control(see Appendix for technical details), which enabled him to win 91F1 Grand-Prix races and seven World Championships.

Here we present an intensive 12– week programme for quick development of the basic tennis skills.

Every lesson lasts 60 minutes.

Every lesson startswith a5 min warm-up(skipping or various types of running).

Every lessonfinisheswitha 5- min cool-down(stretching). Athletics on Thursdaysincludessprints, long jumps, and discus throws. Itis followed by javelin throws.

Gymnastics on Fridaysincludes: monkey bars, forward/backward rolls and assisted somersaults.

50 min 5Km running on Sundays means completion of 5km running under 50 minutes.

Interval swimming on Sundays includes 12.5 msprintswithstrongbreathing for recovery.

20- Min. Chess on Thursday sand Saturdays means 20 minutesofBlitzChess (or Bullet Chess)with a clock,30sec per move.

Every lesson lasts 60 minutes.

Every lesson startswith a5 min warm-up(skipping or various types of running).

Every lessonfinisheswitha 5- min cool-down(stretching). 20- Min. Chess on Thursdays means 20 minutes of Blitz Chess (or Bullet Chess) with a clock,15sec per move.

Fig.19.1.12Week Programme–Beginners Group.

Every lesson lasts 60 minutes.

Every lesson startswith a5 min warm-up(skipping or various types of running).

Every lessonfinisheswitha 5- min cool-down(stretching).

Fig.19.2.12Week Programme–Intermediate Group.

While performing serve, forehand and backhand drills, focus on a stretch-reflex based whipping movements(kinetic chains).

Once a week perform measuring the ball speed in serve, forehand and backhand using a police radar-gun.

20- Min. Chess on Thursdays means 20 minutes of Blitz Chess (or Bullet Chess) with a clock,15sec per move.

Fig.19.3.12Week Programme–Intermediate Group.

Human skeletal and face muscles, accounting for more than 40% of the body weight in man, consist of bundles of elongated, cylindric cells called muscle fibers, 50to 200μ in diameter and often many centimeters long. Bundles of muscle fibers, eachcalledfasciculus, are surrounded by a connective tissue covering, theendomysium(see, e.g.,[Mou 80, Mar 98]).

A muscle consists ofanumberoffasciculiencasedina thick outer layer of connective tissue, the perimysium. At both ends of a muscle the connective tissue me lds into a tendon by which the muscle is attached to the face or bony skeleton. In some muscles(fusiform), the musclefibers run the whole length of muscle between the tendons, which format opposite ends. In most muscles(pennate), one of the tendons penetrates through the center of the muscle; musclefibers run at an angle to the axis of the whole muscle from the central tendon to the perimysium.

Like other cells, muscle cells are surrounded by a cell membrane, thesarcolemma. Myofibrils, the contractile elements, are numerous parallel, lengthwise threads1to3 in diameter thatfillmostofthemusclefiber. Thecrossstriations, seen in the skeletal and face muscles with electron microscope, are located in the myofibrils. Squeezed between the myofibrils and the sarcolemma is a small amount of cytoplasm, thesarcoplasm, in whicharesuspendedmultiplenucle i, numerousmitochondria, lysosomes, lipiddroplets, glycogengranules, and other intracellularinclusions. Thesarcoplasmcontainsglycogen, glycolyticenzymes, nucleotides, creatine phosphate, aminoacids, andpeptides.

Sarcoplasmalsocontainsawell–developedendoplasmicreticulum, which in muscleiscalledsarcoplasmicreticulum. Thesarcoplasmicreticulumformsan extensive hollow membranous system within the cytoplasm surrounding the myofibrils. Periodically, there are branch ing in vag i nations of thesarcolemma called T tubulesor transverse tubules. The sarcoplasmic reticulum bulges out on either side of the T tubules to form large lateral cisternae. The T tubule and two sets of lateralcisternaeconstituteatriad. The triad splay animportant role in muscle excitation-contraction coupling(by release ofC a++ions).

Two types of musclefibersarefoundinhumanskeletalandfacemuscles: red and white muscle fibers, being his to chemically and functionally distinctive. Many muscles are mixed, containing both types offibers, which can be distinguished by various his to chemical stains. In addition to muscle cells and fibroblasts in the connective tissue, a whole muscle containsfatcellsandhistiocytes.

Each muscle fiber contains numerous contractile elements -myofibrils(1−3 μ in diameter)which are biological machines that utilize chemical energy from metabolism of food in the form of adenosine triphosphate, AT Phydrolysisto produce mechanical work. An understanding of contractility and muscle function requires, thus, both his to–mechanical and bio–energetic insight.

Contractile machinery unit of the myofibril, sarcomere(1.5 −3.5 μlong; on electronmicroscopeitisseenasboundedbytwoZlines, with Hzonein the middleoftheAband)is constituted of a great number of longitudinal protein filaments of two kinds: thick, myosinfilaments(about 120˙ A in diameter and about 1.8 μlong; they are located in the center of the sar come rearranged in a hexagon a lar ray about 450˙ A apart)and thin, act infilaments(about 80˙ Ain diameter and about 1.0 μlong; they are anchored into the transversefilaments forming the Zline) (see Figure 20.1). Each myosin filament is surrounded by six act infilaments. Eachmyosinfilament has two heads and two projections from opposite sides at about 143˙ A intervals along its length.

Essential for the contraction process are cross bridges(see Figure 20.1). They extend from myosin filaments to touch one of the adjacent act in filaments. Each thin filament receives cross bridges from the three adjacent thick filaments. Duringshorteningthetwosetsofinterdigitatingfilaments slide with respect to each other, cross andfinallyoverlapeachother. Thisprocessofmuscleshorteninginvolvingprogressiveinterdigitationofthetwosetsofproteinfilaments represents the slidingfilament mechanism, discovered and mathematically formulated as a microscopic theory of muscularcontraction in 1954–57 by A.F. Huxley[HN 54, Hux 57].

According to Huxley, the myo sin heads and cross bridges are elastic elements with a mechanism for attaching themselves transiently to spec ific sites on the thin filaments. The following cyclic events take place during muscularcontraction:

1. The cross bridges extend from myo sinfilaments and attach themselves to spec ific sites on act in filaments. The probability that attachment will occur is f(x), where x is the instantaneous distance between the equilibrium position (0)and the maximum distanceforattachmenthalong the myofibrillar axis. 2. The cross bridgesdetachwithprobabilityg(x).

Fig.20.1. Cellular structure of the voluntary(skeletal)human muscle:(a)Muscular fiberswiththeircross–sections;(b)Sarcomerewithoverlappingmyofilaments.

If we let Nequalthedensityofcross bridges and nthe fraction of cross bridges that are attached, then nNequals the density of attached cross bridges. Huxley’s rate equation for cross–bridge attachment–detachment, i.e. the slidingfilament model of muscularcontraction is now given by:

function of the interfilamentar overlap, and its maximum is about the middle of the shortening, where the a cto–myosin overlap is maximal. This is the so–called parabolic length–tension curve of muscularcontraction.

The dynamic force–velocity relation of muscular contraction isfirstlydiscovered in 1938, by A.V. Hill [Hil 38], in his thermodynamic studies of muscular work, and put into the basis of macroscopicmuscle–load dynamics. Hill’s famous hyperbolic force–velocity curve has the equation:

and says that the muscle force is greatest in isometric conditions(without motion), while the velocity of shortening is maximal without external load; in other words, muscle is either‘strong’or‘fast’, but no both. Constants a a ndb correspond respectively to the energy dissipated during the contraction and the velocity of the mechano–chemical processes.

Hill showed that energy change in muscle during contraction can be described by the following thermodynamic relation:

where Uisthetotalenergychangeassociatedwithcontraction, A is the activation heat(i.e., the heat production associated with the activation of the contractile elements), Wis the mechanical work performed by the muscle by lifting a load,αΔxistheshorteningheat, and Mis the maintenance heat of contraction.

The activation heat begins and is almost completely liberated before any tension is developed, i.e. it is predominantly connectedwiththeexcitation– contraction coupling process, and corresponds in time to the latency relaxation of muscle. It is associated with the internal work required to transform the contractile elements from the resting to the active state. Part of the activation heat probably is associated with a change in the elastic properties of muscle, but about two thirds of it is associated with the release of Ca++ions from the triads, itsbindingbytroponinandthesubsequentrearrangementofthethinfilament proteins. The activation heat is greatest for thefirst twitch after a period rest and becomes smaller with succeeding twitches.

The maintenance heat begins at about the time tension begins and can be divided into two parts: the labile maintenance heat and the stable maintenance heat. For isometric contractions at shorter than rest length, both the labile and the stable heats diminish. For stretched muscle, the labile heat is approximately constant, whereas the stable heat diminishes with stretching and is roughly proportional to the degree of interfilamentaroverlap. The stable heat has quite different values in functionally different muscles; it is law when the muscle maintains tens i onefficientlyandviceversa.

The shortening heat is proportional mainly to the distance of shortening and does not depend greatly on the load, the speed of shortening, or the amount of work performed. SincemechanicalworkisW=PΔx, substituting this in the above thermodynamic relation(20.4)gives the heat equation:

From the analogy of the term(P+α)in the heat equation(20.5)and the term (P + a) in the force-velocity equation (20.3), Hill was able to show a rough equivalence between the coefficient of the shortening heat α and the force-velocity constant a. The shortening heat is greatest for thefirst twitch after a period of rest and is less for subsequent twitches.

Last, note should be made of ther moe last i cheat. Generally speaking, resting muscle has rubber like thermoelastic properties, whereas actively contracting musclehasspringlikethermoelasticproperties. During the development of tension the change inelastic properties is accompanied by an absorption of heat by the muscle. As tension falls during relaxation, an equivalent amount of heat is released by the muscle owing to its elastic properties. The various kinds of muscle heat must be corrected for the ther moe last i cheat. However, for a complete cycle of contraction and relaxation, the net heat produced by thermoelastic mechanisms is zero.

In the same seminal paper[Hil 38], Hill also proposed a three–elementrheologicalmodeloftheskeletalmuscle–tendon complex(see Figure 20.2). In this model the length–tension property of muscle is represented by an active contractile element(CE)in parallel with a passive elastic element. Total isometric muscle force is assumed to be the sum of muscle force when it is inactive(passive)and when it is maximally excited(active). The muscle is in series with tendon, which is represented by a nonlinear spring. Pennationangle(α)is the angle between tendon and musclefibers. Tendons lack length is the length of tendon at which force initially develops during tendon stretch. The model was

Fig.20.2. Hill’smo del of the skeletal muscle–tendon complex.

scaled to represent each muscle by specifying the muscle’speak force, optimal fiberlength, tendons lack length, and pen nation angle based on data collected in anatomical experiments.

Hill’smuscle–tendon model has been widely applied in biomechanical musculo–skeletal modelling.

Dynamics of human skeletal and face muscles is in the most sophisticated form described in the series of papers of Hatze (see [Hat 78]). His muscle–control model involvesexcitationdynamicsofneuro–muscular inputs(motor units)andcontractiondynamicsbasedonHuxley’ssliding–filament theory of muscle contraction. In brief, Hatze’smyocyberneticscanbedividedintoexcitationdynamics and contraction dynamics. The excitation dynamics of a single muscle fibre stimulated by trains of normalized nerve impulsesα(t)is represented by the system where ρ∗(ξ) is normalized Cadensityfunction, k(ξ) is filamentary–overlap function, h(˙xi)is velocity–dependence function; c 1, . . . , c 6, d 1, d 2,VN, VT , q0 aredefinedconstants; VT β(t)is action potential as appearing in the interior of the T –system of the fibre, while γ(t) denotes the free Ca–i on concentration in the interfilamentaryspace; the variableδqexpresses thestretchpotentiationinducedbyanelongationofthetetanizedfibre.

˙ξ=a 1[1/a2 arcs in h a3 ln( q∗k(ξ) where a 1, . . . , a 4, b 1, b2 aredefinedconstants, fSE/¯fis the normalized force across the series elastic element, b1k 1(ξ) is the passive sar com ere tension, and q∗is the active state.

The celebrated Hodgkin–HuxleyHH–neuron model is described by the nonlinear coupled differentialequations for the four variables, Vforthemembranepotential, andm, handnforthegatingvariablesofNaandKchannels, and it is given by[HH 52, Hod 64]

HerethereversalpotentialsofNa, KchannelsandleakageareVNa=50mV, VK=−77mVandVL=−54.5mV; the maximum values of corresponding conductivitiesaregNa=120 mS/cm 2,gK=36mS/cm2 andgL=0.3mS/cm 2; thecapacityofthemembraneisC=1μF/cm 2. The external, input current is given by δ(t −tin).

In equation (20.9), tin is the nth firing time of the spike–train inputs, gsynand Vc denotetheconductanceandthereversalpotential, respectively, ofthesynapse, τ sis the time constant relevant to the synapse conduction, and α(t) is the alpha function given by α(t) = (t/τ s) e−t/τ sΘ(t).

whereΘ(t)istheHeavisidefunction. TheHHmodelwasoriginallyproposed to accountforthepropertyofsquidgiantaxons[HH 52, Hod 64] and it has been generalizedwithmodificationsofionconductances. TheHH–type models have beenwidelyadoptedforastudyonactivitiesoftransducerneuronssuchas motorandthalamusrelayneurons, which transform the amplitude–modulated input to spike–train out puts.

Hodgkin–Huxley theory of neural action potential was adapted by Noble [Nob 62] as a model of muscular action potential. Noble model has the same form as the HH–neuron model (20.8), with changed the values of constants, so that the whole signal is about 10 times slower. Noble’smodelwaslatermodifiedbyHatze’s muscularexcitationdynamics(20.6)and complemented by his contraction dynamics(20.7).

Now, tosimplifyHatze’smyiocybernetics, and yet to retain all the necessary excitation–contraction dynamics, as well as to establish the neuro–muscular inter–connection, we propose herein approach of recurrent diffus i on physics. The EFS–responsemappingFofaskeletalorfacemuscle, i.e., the response of the musclesystemMtotheefferentfunctionalstimulationfromtheneuralnetworksystemN–can be stated in the form of the force generator time behavior, F: R →Homt(N, M), where: tdenotesstimulationtime, NandM correspondtotheleftR–moduliofneuralandmuscularsystems. The mapping Fcanbeconsideredasaneffectofafifth–order transmission cascade (F 1→F 2→F 3→F 4→F 5), whereFi (i=1, . . . , 5) represent neural action potential, synapticpotential, muscular action potential, excitation–contraction coupling and muscle tension generating, respectively(see[II 06a, II 06b]).

According to[Nob 62, Hak 93, Hak 02], all transmission components of the system(F 1→F 2→F 3→F 4→F 5), whereFi (i = 1, . . . , 5) can be considered as being some kind of diffus i on processes, forming thefifth–order transmission flux cascade.

Mapping F(for all included motor units in the particular muscle contraction) can be described by fifthorderrecurrent, distributedparameterdiffus i on system [II 06a, II 06b]

∂z2 −Jk(Vk), withboundaryconditionatz= 0,

Vk(0, t)=V0 sin(2πft)=S(t), (k= 1, . . . , 5).

ThesingleelementF 4,(k= 1, . . . , 5) behavior is now given by

Vk(z, t)=V0 exp(−zk/m) sin(2πf(t −zk/n)),

For muscle–mechanical purpose, thepresenteddistributedmapFcanbefirst mathematicallyapproximatedwiththecorrespondinglumpedparameterRkCk electric circuit (where the second circuit represents the Ecclesmodelofsynaptic activation(see[Ecc 64, EIS 67, Ecc 86]) and the last one corresponds to the low passfilterrepresentingthecontractionprocessitself), atx = tendon

(bkzk−1 −zk), (k= 1, . . . , 5), whereTk=RkCkaretimecharacteristicsof the circuits in cascade, andbkare corresponding input gains(conversion factors).

The single muscle behavior in the lumped approximation form is given by the recurrent sum of its transient and weighting terms(with time legsτk) zk(t)=bkzk−1(1 −exp(−t/Tk))+zk exp(−(t −τ k)/Tk).

where a(having dimension of force)corresponds to energy dissipated during the contraction, b(having dimension of velocity)isthephosphagenicenergy transducingrate, whileccorrespondstothesecond derivative of the stress– strain curve of the series viscoelastic element[Wil 56] of the muscular actu at or (assumed in exponential three–parameterform).

jgi(xi)+Si, (i, j= 1, . . . , n),

∂z2 −Jk(Vk), (k= 1, . . . , 5);

(bkzk−1 −zk), (k= 1, . . . , 5), (20.11) zk(0)=0, z0= S(t), z5=F(t). (20.12)

Equations(20.10,20.12)constitute a3n–dimensional phase–space(forn=5or k= i) being a hiper–cube≡neuro–muscular controlspace. The feedback control F−1ofthemappingFisperformedbymuscularautogeneticmotorservo.

It is now well–known(see[Hou 79, HBB 96]) thatvoluntarycontractionforceF of a skeletal or face muscle system M is reflexly excited (positive reflex feedback +F−1byresponsesofitsspindlereceptorstostretchandisreflexlyinhibited (negativereflexfeedback−F−1byresponsesofitsGolgitendonorganstocontraction. Stretch and unload in greflexesaremediatedbycombinedactionsof several auto genetic neural pathways.

James Houk’sterm‘auto genetic’means that the stimulus excites receptors located in the same face or body muscle that is the target of thereflex response. The most important of these muscle receptors are the primary and secondary endings in muscle–spindles, sensitive to length change – positive length feedback +F−1, and the Golgi tendon organs, sensitive to contractile force - negative force feedback−F−1.

The gain Gofthelengthfeedback+F−1can be expressed as the positional stiffness(theratioG≈S=dF/dxoftheforceF–change to the length x−change)ofthemusclesystemM. The greater the stiffnessS, the less will the muscle be disturbed by a change in load and the more reliable will be the performanceofthemusclesystemMinexecutingcontrolledchangesinlength +F−1.

The auto genetic circuits (+F−1) and (−F−1) appear to function as servoregulatoryloopsthatconveycontinuouslygradedamountsofexcitationandinhibitiontothelarge(alpha)skeletomotorneurons. Small(gamma)fusimotor neuronsinnervatethecontractilepolesofmusclespindlesandfunctiontomodulatespindle–receptor discharge.

Asingleskeletalmuscle,(e.g., the triceps brachii muscle, see Figure 20.3),is attached at its origin to a large area of bone (the humerus in case of the triceps). At its other end, the insertion, it tapers into a glistening white tendon which, (in case of the triceps is attached to the ulna). As the triceps contracts, the insertion is pulled toward the origin and the arm is straightened or extended at the elbow. Thus the triceps is an extensor. Because skeletal muscle exerts force only when it contracts, a second muscle–aflexor–is needed toflexorbendthejoint (e.g., the biceps brachii muscle is theflexoroftheforearm). Together, they(the biceps and triceps)makeup an antagonistic pair of muscles, which we will call forming the equivalent muscular actu at or. Similar pairs, i.e., equivalent muscular actuators, working antagonistic ally across other joints, provide for almost all the movement of the skeleton. The equivalent muscular actu at or has the role of ‘driver’in bio dynamics. It generates the equivalent muscular torque, which is the primary cause of human–like motion[II 06a, II 06b].

Fig.20.3. An antagonistic pair of human skeletal muscles, oneflexorandtheother extensor(in the case of the forearm, biceps brachii and triceps brachii, respectively), forming the equivalent muscular actu at or–the primary cause of the human–like motion.

The immediate energy source for contraction inhuman muscles is adenosine triphosphateATP(see[Mou 80, Mar 98]). Muscle contains about2 μmoleATP/ gram wet weight. The myo sin head istheonlysiteofthemajorATPhydrolysis inactive muscle. AttheconcentrationsofATP, ADP(adenosinediphosphate), andPi(inorganicphosphate)presentinthesarcoplasm, ATPhydrolysisyields about 11.5 kcal/mole. About 0.3 μmole of ATP/gram muscle ishydrolyzed by a single muscle twitch. TheATPhydrolysisoverallscheme:

represents actually the complex six–step chain–reaction{ki},(i=1, . . . , 5), which can be summarized as follows:

1. MyosinreactsrapidlywithATPtoformacomplex; the myo sin is from its resting(low–energy)form converted to an energy–rich form. 2. While complexed to the myosin, ATP is hydrolyzed to ADP and Pi. Reaction 2 is much more rapid than reaction 1.( This step is extremely temperature sensitive). 3. In reaction 3, whiletheADPandPiarestillattachedtothemyosin, the latter is converted to a low-energy form. This step is slow, rate limiting in the sequence of reactions, and insensitive to temperature changes. 4. Reactions4 and5 are rapid.

where Mismiosininlow–energy form, M∗ismiosininenergy–rich form, and symbol↔actually represents a pair of reversible reactions{ki, ki−1},(i= 1, . . . , 5). Muscles also contain about 20 μmoleCP/gram(creatine phosphate). CreatinephosphatecanphosphorylateADPtoformATPinareversiblereaction catalyzedbytheenzymecreatinekinase.

Musclecontainslargeamountsofcreatinekinase; it amounts to more than 25 percentofthesolublecytoplasmicprotein. AssoonasATPishydrolase, the ADP formed is very rapidly rephosphorylated by CP and the ATP is regenerated. ThusCPformsareservoirofenergy–rich phosphate bonds to quickly replenish thesarcoplasmicATP.

Ultimately, ATPis produced by glycolysis and respiration [Mou 80].In glycolysis (the so called Embden–Meyerhoffpathway), glucose is degraded to pyruvate, or to lactic acid in the absence ofO 2, yielding2 molesAT P/moleglucosemetabolized. Intracellular glycogen granules provide avery readily available source of glucose. Muscles normally contain9to16gm/kgglycogenor, for a well–fed man of average height and weight, the total glycogen stores in muscle amount to 300 −500 gram, with another 55 −90 gram in the liver. Glycogen breakdown in muscle begins immediately on stimulation, and the amount of muscle glycogen depleted is proportional to the mechanical work done. Glycogenishydrolyzed by the enzymephosphorylasetoglucose–1–phosphate, which then enters the glycolytic pathway.

Red musclefibers respond to a stimulus with a relatively slow twitch(maximum shortening velocity about 17mm/sec)and therefore are also called slow fibers, whereas white musclefibersreact to a stimulus with a rapid twitch (maximum shortening velocity about 42mm/sec)and therefore are also called fast fibers. Red muscle has a more extensive blood supply than white muscle. Red musclefibers are able to sustain activity for long periods of time whereas white muscle fibers characteristically produce short bursts of great tension followed by the rapid onset of fatigue.

WholeredandwhitemusclesdifferinATPaseactivity, and, indeed, the pur ifiedcontractileproteinmyosinextractedfromredandwhitemusclediffers in ATPaseactivity, afindingassociatedwithdifferentmyosinlightchains. White muscleandwhitemuscleactomyosin show the greater ATPaseactivity. The innervation of red andwhite muscled iffers, and, indeed, whether a given muscle is red or white results from trophic influencesofthemotornerve.

Slow musclefibersaregenerallythinnerandpossessmanysarcosomes (mitochondria)containing large amounts of respiratory enzymes, as well as copious quantities of the O 2− carrying protein myoglobinin the sarcoplasm and man lipid droplets. Thenumeroussarcosomes andhighlevelofmyoglobin gives lowfibers their red color. Fast (or white)muscle fibers, on the other hand, are generally of largerdiameterandcontainlargeamountsofphosphorylaseandglycolyticenzymes and large deposits of glycogen. Slow muscles derive energy predominantly from respiration, whereas in fast muscle fibers, glycolysis and lactate production are more prominent.

Human energy production spans the range of human movements from those requiring large bursts of energy over short periods of time(as in sprint running)–to those activities requiring small but sustained energy production(as in marathon running). Even within the same activity, the energy requirements change from one moment to the next(see any textbook on exercise physiology).

Adenosine triphosphate(AT P)is the immediately usable form of chemical energy for muscular activity. It is stored in most cells, particularly muscle cells. Other forms of chemical energy, suchas that available from the foods we eat, must be transferred into the AT Pform before they can be utilized by the muscle cells.

ThechemicalstructureofAT Piscomplicated, but for our purposes it can besimplified, sayingthatAT Pconsistsofalargecomplexofmoleculescalled adenosine and three simpler components called phosphate groups. The last two phosphate groups represent‘highenergybonds’. In other words, they store a highlevel of potential chemical energy. When the terminal phosphate bond is broken, energy is released, enabling the cell to perform work. The kind of work performed by the cell depends on the cell type. Forexample, mechanical work (contraction)is performed by muscle cells(skeletal, smooth and heart muscle), nerve conduction by nerve cells, secret i on by secretory cells (e.g., endocrine cells), and soon. All‘biological’work performed by any cell requires the immediate energyderivedfromthebreakdownofAT P.

Since energy is released whenAT Pisbroken down, it is not too surprising that energy is required to rebuild or resynthesize AT P. The building blocks for AT P synthesis are the by-products of its breakdown, adenosinediphosphate(ADP) and inorganic phosphate(Pi). Therefore, there are two essential formulas, the direct: AT P→ADP+ Pi + energy, and the inverse one: ADP+ Pi→AT P. The energy for AT Pre synthesis comes from three different series of chemical reactions that take place within the body. Two of the three depend upon the food we eat, whereas the other depends upon a chemical compound called phosphocreatine. The energy released from anyone of these series reactions is coupled withtheenergyneedsofthereactionthatresynthesizesAT P. In other words, the separate reactions are functionally linked together in such away that the energy released by the one is always used by the other. Biochemists refer to these functional links as coupled reactions, and it has been shown that such coupling isthefundamentalprincipleinvolvedinthemetabolicproductionofAT P.

PCi san abbreviation for phosphocreatine, another one of those‘energy–rich’ phosphatecompoundscloselyrelatedtoAT P. Forexample, PC, likeAT P, is stored in muscle cells, and when it is broken down(i.e., when its phosphate group is removed), a large amount of energy is released. The released energy, of course, is coupled to the energy requirement necessary for the resynthesis of AT P. In other words, as rapidly asAT Pisbroken down during muscular work, it is continuously reformed from ADPand Piby the energy liberated during the breakdown of the stored PC: ADP+ Pi→AT P. ForeverymoleofPCbroken down, onemoleofAT Pisresynthesized.

ThetotalmuscularstoresofbothAT PandPC(collectively referred to as phosphagens)are very small–only about 0.3 mole in females and 0.6 mole in males. Thus, the amount of energy obtainable through this system is limited. In fact, ifyou were to run 100 mas fast as you could, thephosphagen stores in the working muscles would probably be empty by the end of the sprint. However, theusefulnessoftheAT P−PCsystemliesintherapidavail ability of energy rather than in the quantity. Forexample, activities suchas sprinting, jumping, swinging and other similar skills requiring only a few seconds to complete are all depend entuponthestoredphosphagensfor their primary energy source.

This system is also known as‘anaerobic glycolysis’.‘Glycolysis’refers to the breakdown of sugar; ‘anaerobic’ means without oxygen. In this system, the breakdown of sugar (a carbohydrate, one of the food stuffs)supplies the necessary energy from which AT Pis manufactured. When the sugar is only partially broken down, one of the end products is lactic acid (hence the name lactic acid system).

When lactic acid accumulates in the muscles and blood reaches very high levels, temporary muscular fatigue results. This is a very definite limitation, and is the main cause of the‘early’fatigue. Another limitation of the lactic acid systemthatrelatestoitsanaerobicqualityisthatonlyafewmolesofAT P canberesynthesizedfromthebreak down of sugar as compared to the yield possible when oxygen is present. Forexample, onlythreemolesofAT Pcanbe manufactured from the anaerobic breakdown of 180 grofglycogen(glycogen is the storage from glucose or sugar in the muscle). As we will soon see, the aerobic breakdown of 180gr of glycogen results in enough energy to resynthesize 39 moles ofAT P!

The lactic acid system, liketheAT P−PCsystem, is extremely important to us, primarilybecauseittooprovidesforarapidsupplyofAT Penergy. For example, exercises that are performed at maximum rates for between1 min and 3 min, suchas sprinting 400 mand800m, depend heavily upon the lactic acid systemforATPenergy. Also, in some performances, suchas running 1500 mor a mile, the lactic acid system is used predominantly for the‘kick’at the end of the race.

In the presence of oxygen, the complete breakdown of 180gr of glycogen to carbon dioxide(CO 2) and water(H2O)yields enough energy to manufacture 39 moles ofAT P. This series of reactions, like the anaerobic series, takes place within the muscle cell, but is confined to specialized sub cellular compartments called mitochondria. Mitochondriaareslipper–shaped cell bodies often referred to as the ‘powerhouse’ of the cell because they are the seat of the aerobic manufacture ofAT Penergy. Musclecellsarerichwithmitochondria.

The aerobic breakdown of carbohydrates, fats, and even proteins provides energyforAT Pre synthesis: ADP+ Pi→AT P. SinceabundantAT Pcanbe manufactured without yielding fatiguing by–product, the aerobic system is most suited for endurance activities. Thecarbondioxidethatisproduceddiffuses freely from the muscle cell into the blood and is carried to the lung, where it is exhaled. The water that is formed is useful within the cell itself, since the largest constituentofthecellis, in fact, water.

Another feature of the aerobic system that should be noticed is that concerned with the type of food stuffrequired for breakdown. Not only glycogen but fats and proteins as well can be aerobically broken down to carbon dioxide and water, withenergyreleasedforAT Psynthesis. Forexample, the breakdown of 256gr off at will yield 130 moles ofAT P. During exercise, both glycogen and fats, but not protein, areimportantsourcesofAT P–yielding energy.

The amount of oxygen one needs to consume from the environment in order to synthesize one mole of AT Pis approximately 3.5l if glycogen is the food fuel and about 4.0 lwith fat. At rest, most of us consume between 0.2 land 0.3 lof oxygen per minute. In other words, amoleofAT Pisaerobicallymanufacturedevery 12 min to20 min under normal resting conditions. During maximal exercise, a moleofAT Pcanbeaerobicallysuppliedtotheworkingmuscleseveryminute by most of us. For the highly trained endurance athlete, more than 1.5 moles of AT Pcanbeaerobicallysynthesized and supplied to the muscles every minute duringmaximaleffort.

In summary, then, the aerobic system is capable of utilizing both fats and glycogen for resynthesizing large amounts of AT Pwithout simultaneously generating fatiguing by–products. Therefore, the aerobic system is particularly suited formanufacturingAT Pduringprolonged, endurance–type activities. Forexample, during marathon running, approximately 150 moles of AT Pare required over 2.5h the race takes. Such a large, sustained output of AT Penergy is possible only because early fatigue can be avoided and large amounts of food(glycogen and fats)and oxygen are readily available.

1. bythestoredphosphagens, AT PandPC; 2. by the lactic acid system(anaerobic glycolysis); and 3. by the oxygen, or aerobic, system. And ability of each system to supply the majorportionoftheAT Prequiredinanygivenactivityis related to the spec ifickindofactivityperformed.

Forexample, in short–term, high–intensity types of activities, suchas the 100m dash, mostoftheAT Pissuppliedbythereadilyavailablephosphagensystem. In contrast, longer–term, lower intensity types of activities, suchas the 42.2km−marathon, are supported almost entirely by oxygen, or aerobic, system.

In the middle range of activities are those tasks that rely heavily on the lacticacidsystemforAT Pproduction–e.g., the 400 mand800 mdashes. Also in the middle are those activities that require a blend of both anaerobic and aerobic metabolism–e.g., the 1500 mandmileruns. In these latter activities, theanaerobicsystemssupplythemajorportionofAT Pduringthesprintat both the start andfinishoftherace, with the aerobic system pre dominating during the middle or steady–state period of the run.

Just from these examples it can be seen that what is at work in physical activities is an energy continuum, acontinuumrelatingthewayinwhichAT P is made available and the type of physical activity performed.

control. In case of biomechanics, one has the passive damping contribution to the joint torques, Ti(t, qi, pi), whichhasthebasicstabilizingeffecttothecomplex human movement. Thiseffectcanbedescribedby(q, p)-quadratic form of the Rayleigh–Vander Pol’sdissipationfunction, where a i and bidenote dissipation parameters. Its partial derivatives ∂R/∂p give risetoviscousforcesinthejointswhicharelinearinpiandquadraticinqi. It isbasedontheunforcedVanderPol’soscillator

˙x 2– with the velocity term˙xreplaced by our momentumtermp 2.

which reduces to the gradient system in case H= 0 (as well as to the conservative system inc a seR = 0).

Muscular dynamics describes the intern alex citation and contraction dynamics ofequivalentmuscularactuators, anatomically represented by resulting action of antagonistic muscle-pairsforeachuniaxialjoint. We attempt here into describe the equivalent muscular dynamics in the simplest possible way (for example, Hatzeused 51 non line ard ifferentialequationstoderivehis, arguably most elaborate, myocyberneticmodel), and yettoincludethemainexcitation and contraction relations.

The active muscular contribution to the joint torques, Ti(t, qi, pi), should describe the intern alex citation and contraction dynamics of equivalent muscular actuators, anatomically represented by resulting action of antagonistic muscle pairs per each active degree-of-freedom.

i denote the maximal isometric muscular torques applied at i-thjoint, whileτidenotethetimecharacteristicsofparticularmuscularactuators. This is a rotational-jointformofthesolutionoftheWilkie’smuscularactive-state element equation, wherex=x(t)represents the active state of the muscle,βdenotes the element gain, A corresponds to the maximum tension the element can develop, and S= S(r)is the‘desired’ active state as a function of motor unit stimulus rate r.

where a i(having dimension of torque)a ndb i(having dimension of momentum) denotetherotationalHill’spar a meters, corresponding to the energy dissipated during the contraction and the phosphagenic energy conversion rate, respectively.

i , (i = 1, . . . , N). (20.17)

Herewepresentthebasicsyntaxfornumericalsolutionofordinarydifferential equations(ODEsforshort)in Sci lab.

1 Sci lab is a free simulation(for non-commercial use)package, developedbyConsortium Sci lab(DIGITEO, INRIA, ENPC). It is similartoMatlabT M. Current version is 5.1.1. See http://www. sci lab. org/.

ThestandardfunctionforsolvingexplicitODE-systems is"ode", defined by: dy/dt=f(t, y). It can be called using several formats.

where parameters in the bracket shave the following meaning: y 0= real vector or matrix of initial conditions; t 0= realscalardenotinginitialtime; t=realtime-axis vector for the solution; f=external(function or character string or list); y=matrix of solution vectors:

wheresquarebrackets[...]denoteoptions: "rtol"and"a tol"are real constants or real vectorsofthesamesizeasy; jac=external jacob i an; w, iw=real vectors.

The"type"is one of the following strings of characters: [adams, stiff, rk, rkf, fix, discrete, roots]. It automatically selects between non-stiff"adams" predictor-correct or method and"stiff"Backward Differentiation Formula(BDF)method. Ituses non-stiff method initially and dynamically monitors data in order to decide which method to use. "rk": Runge-Kuttamethodoforder 4(RK 4); "rkf": adaptive Runge-Kutta-Fehlbergmethodoforder 4 and 5( RKF 45); this is the fastest algorithm for non-stiff and mildly stiff problems. "fix": this is"rkf"with simple interface (only"rtol"and"a tol"parameters); this is the simplest method to try. "discrete": performs discrete-time simulation, e.g.:

y=ode("discrete", y 0,k 0, kvect, f); "roots": ODE-solverwithroot-finding capabilities.

where: root=ODEsolverwithroot-finding capabilities; ng=integer; g=external(function or character string or list); k 0= integer(initial time). kvect: integervector;

dy/dt=f(t, y). It is an external i.e. a function with specified syntax, orthenameofaFortransubroutineora C-function(string)with specified calling sequence or a list. If fis a Sci lab function, its syntax must be: ydot=f(t, y); wheretisarealtimescalar, yisarealstatevector, and ydotisarealvectorequaldy/dt.

ydot=f(t, y, u 1,u 2,...,un); This syntax allows to use parameters as the arguments ofrealf. Thefunctionfcanreturnapxqmatrix instead of a vector.

Optional input parameters can be given to control the error of the solution:"rtol"and"a tol"are thresholds for relative and absolute estimated errors. The estimated error ony(i)is:

rtol(i)*abs(y(i))+a tol(i) and integration is carried out as far as this error is small for all components of the state. If rto land/or a to lisa constant, rtol(i)and/or a tol(i)are set to this constant value. Defaultvaluesforrtolandatol are:

Part 6 · pp. 155–184

rtol=1.d-5and a tol=1.d-7 for most solvers and rtol=1.d-3and a tol=1.d-4for"rfk"and"fix". For stiff problems, itisbettertogivetheJacobian oftheRHSfunctionastheoptionalargument"jac". It is an external. If"jac"is a function, the syntax should be:

J=jac(t, y), wheretisarealtimescalarandyarealstate-vector. The resulting matrix Jmustevaluatetodf/dx, i.e. J(k, i)=dfk/dxi, withfk=k-thcomponentoff.

where(x 1, x 2, x 3)=(ρ, phi, theta)=Eulerangles(roll, pitch and yau), while I 1, I 2, I3 are local inertia moments(see The Newton-Euler Biomechanics Law for biomechanical details). Numerical solution of this ODE-system for the initial conditions:

IC: x1(0) = 0,x2(0) = 1,x3(0) = 1 isgiveninFigure20.4.

//Euler’srigidbodyequations //Consider the following system: //dx 1=x 2*x 3;dx 2=-x 3*x 1;dx 3=-0.51*x 1*x2 //withIC: [x 1(0);x 2(0);x 3(0)]=[0;1;1].

//inertia moments: I 1=1; I 2=1; I 3=0.51;

Fig.20.4. Sci lab simulation of the basic Eulerrigidbodyequations.

//numerical solution and plot: scf(1); clf;//open and clear new figure window sol=ode(x 0,t 0,t, Euler Body);//numerical solution plot(t’, sol’); //basicplot

The myotatic stretch–reflex, also known as the monosynaptic reflex,2is the simplest reflex known: it involves only two neurons (one sensory, or afferent and one motor, orefferent), withonesynapsebetweenthem. Therefore, it depends only onthemonosynapticconnection between primary afferentfibersfrommuscle spindlesandmotorneuronsinnervatingthesamemuscle.

The muscle spindle(see Figure 20.5) contains specialized elements that sense muscle length and velocity of length change. The spindle is innervated by large myelinated afferent fibers known as type Ia afferent fibers. The cell bodies of these neuronsareclusterednearthespinalcordinthedorsalrootganglia. They are

2 Itiscalledamonosynapticreflexbecauseitdependsonlyonthesimpleconnection between primary afferentfibersfrommusclespindlesandmotorneuronsinnervatingthesamemuscle. Inotherspinalreflexessuchasthoseproducedbycutaneous stimuli, oneormoreinterneuronsmaybeinterposedbetweentheprimaryafferent fibersandthemotorneurons.

an example of a bipolar cell: one branch of the cell’s axon goes out to the muscle and the other runs into the spinal cord. InthespinalcordtheIaafferentfibers makemonosynapticexcitatoryconnectionstoalphamotorneuronsinnervating the same muscle from which they arise and motor neuronsinnervatingsynergisticmuscles. Theyalsoinhibitsmotorneuronscontrollingantagonisticmuscles through an inhibitory inter neuron. Half the neurons in the brain are inhibitory. Theyreleaseneurotransmittersthathyperpolarizethemembranepotentialof thepostsynapticcell, thus reducing the likelihood offiring. The muscle spindle issensitivetostretchsothatwhenthemuscleisstretchedtheIaafferentfibers increase thefiring rate. This leads to the contraction of the same muscle and its synergists and relaxation of the antagonist muscle. Thereflextherefore tends to counteract the stretch, enhancing the spring like properties of the muscles.

Fig.20.5. Schematic of a muscular length–sensor, the muscle spindle.

A physician tapping thereflexhammeragainst the patellar tendon traditionally elicits the special case of myotatic stretch–reflex, called the knee jerk reflex. The tap of thereflex hammer causes stimulation of the muscle spindles afferents and Golgitendonorgans, by altering the stretch and length of the muscle. The afferentneuronsynapsesintheanteriorhornofthespinalcordwithanipsilateralefferentneuron. Theefferentneuronleadstoaneuromuscularjunction with the respect i veeffectormusclethequadricepsfemor is. The stretch–reflex also plays a central role in the maintenance of balance.

The intensity of the stretch–reflexismodulatedbyexcitatoryandinhibitory supraspinal input. Damage to supraspinal input, affecting descending input, precipitatesabnormallyhighgainofthestretch–reflex. Theeffectisanincreasein musclestiffness, alsoreferredashypertonus.

About three/four decades ago, James Houk[Hou 67, Hou 78, Hou 79] pointed out that stretch and unload in greflexesweremediatedbycombinedactionsof several auto genetic neural pathways. In this context,“auto genetic”(or, autogenic)means that the stimulus excites receptors located in the same muscle that is the target of thereflexresponse. The most important of these muscle receptors are the primary and secondary endings in muscle spindles, sensitive to length change, andtheGolgitendonorgans, sensitive to contractile force. The autogeneticcircuitsappeartofunctionasservo-regulatoryloopsthatconvey

Fig.20.6.Houk’sautogeneticreflexes.

continuouslygradedamountsofexcitationandinhibitiontothelarge(alpha) skeletomotorneurons. Small(gamma)fusimotorneuronsinnervatethecontractile poles of muscle spindles and function to modulate spindle–receptor discharge. Houk’sterm“motorservo”has been used to refer to this entire control system, summarizedbytheblockdiagraminFigure 20.6.

Prior toastudybyMatthews, it was widely assumed that secondary endings belong to the mixed population of “flexor reflex afferents,” so called because their activation provokes theflexorreflexpattern–excitationofflexormotoneurons andinhibitionofextensormotoneurons. Matthews’results indicated that some category of muscle stretch receptor other than the primary ending provides important excitation to extensor muscles, and he argued forcefully that it must be the secondary ending.

The primary and secondary muscle spindle afferentfibers both arise from a specialized structure within the muscle, the muscle spindle, a fusiform structure 4–7 mmlongand 80–200μ in diameter. The spindles are located deep within the muscle mass, scattered widely through the muscle body, and attached to the tendon, theendomysiumortheperimysium, so as to be in parallel with the extrafusalorregularmusclefibers. Although spindles are scattered widely in muscles, they are not found throughout. Muscle spindle(see Figure 20.5) containstwotypesofintrafusalmusclefibers(in trafusalmeansinsidethefusiform spindle): the nuclear bagfibers and the nuclear chainfibers. The nuclear bag fibers are thicker and longer than the nuclear chain fibers, and they receive their name from the accumulation oftheirnucleiintheexpandedbag-like equatorial region-the nuclear bag. The nuclear chain fibers have no equatorial bulge; rather theirnucleiarelinedupintheequatorialregion-the nuclear chain. Atypical spindle contains two nuclear bagfibers and 4-5 nuclear chainfibers.

The pathways from primary and secondary endings are treated commonly by Houkin Figure 20.6, since both receptors are sensitive to muscle length and bothprovokereflexexcitation. However, primary endings show an additional sensitivity to the dynamic phase of length change, called dynamic responsiveness, and they also show a much–enhanced sensitivity to small changes in muscle length.

The motor servo comprises three closed circuits (Figure 20.6), two neural feedback pathways, and one circuit representing the mechanical interaction between a muscle and its load. One of the feedback pathways, that from spindle receptors, conveys information concerning muscle length, and it follows that this loop will act to keep muscle length constant. The other feedback pathway, that from tendon organs, conveys information concerning muscle force, and it acts to keep force constant.

In general, it is physically impossible to maintain both muscle length and force constant when external loads vary; in this situation the action of the two feedbackloopswillopposeeachother. Forexample, an increased load force will lengthen the muscle and cause muscular force to increase as the muscle is stretched out on its length-tension curve. The increased length will lead to excitationofmotoneurons, whereas the increased force will lead to inhibition. It follows that the net regulatory action conveyedbyskeletomotoroutput will depend on some relationship between force change and length change and on the strength of the feedback from muscle spindles and tendon organs. Asimple mathematicalderivationdemonstratesthatthechangeinskeletomotoroutput, theerrorsignalofthemotorservo, Shouldbeproportionaltothedifference between a regulated stiffness and the actual stiffness provided by the mechanical properties of the muscle, where stiffnesshas the units of force change divided by length change. The regulated stiffness is determined by the ratio of the gain of length to force feedback.

It follows that the combination of spindle receptor and tendon organ feedback will tend to maintain the stiffness of the neuromuscular apparatus at some regulated level. If this level is high, due to a high gain of length feedback and alow gain of force feedback, one could simply forget about force feedback and treat muscle length as the regulated variable of the system. However, if the regulated level of stiffness is intermediate in value, i.e. notappreciablydifferent from the average stiffnessarisingfrommusclemechanicalpropertiesintheabsenceofreflexactions, one would conclude that stiffness, or its inverse, compliance, is the regulatedpropertyofthemotorservo.

In this way, theautogeneticreflexmotorservo provides the local, reflexfeedbackloopsforindividualmuscularcontractions. A voluntary contraction force Fofhumanskeletalmuscleisreflexlyexcited(positive feedback+F−1)by the responsesofitsspindlereceptorstostretchandisreflexlyinhibited(negative feedback−F−1)bytheresponsesofitsGolgitendonorganstocontraction. Stretch and unload in greflexesaremediatedbycombinedactionsofseveralautogeneticneuralpathways, formingthemotorservo.

In other words, branches of the afferent fibers also synapse with with interneurons that inhibit motor neurons controlling the antagonistic muscles – reciprocal inhibition. Consequently, the stretch stimulus causes the antagonists to relax so that they cannot resists the shortening of the stretched muscle caused by the mainreflexarc. Similarly,firingoftheGolgitendonreceptorscausesinhibition of the muscle contracting too strong and simultaneous reciprocal activation of its antagonist.

Recall that human brain is the most complicated mechanism in the Universe. Roughly, it has its own physics and physiology. A closer look at the brain points toaratherrecursivelyhierarchicalstructureandaveryelaborateorganization (see[II 06b]for details). An average brain weighs about 1.3kg, and it is made of:∼77%water,∼10%protein,∼10%fat,∼1%carbohydrates,∼0.01% DNA/RNA. The largest part of the human brain, thecerebrum, is found on the top and is divided down the middle into left and right cerebral hemispheres, and front and back into frontal lobe, parietallobe, temporallobe, and occipital lobe. Further down, and at the back lies a rather smaller, spherical portion of the brain, the cerebellum, and deep inside lie a number of complicated structures likethethalamus, hypothalamus, hippocampus, etc.

Boththecerebrum and the cerebellum have comparatively thin outer surface layers of grey matter and larger inner regions of white matter. The grey regions constitute what is known as the cerebralcortexandthecerebellar cortex. It is in the grey matter where various kinds of computational tasks seem to be performed, while the white matter consists of long nervefibers(axons)carrying signals from one part of the brain to another. However, despite of its amazing computational abilities, brain is not a computer, at least not a‘Von Neumann computer’[Neu 58], but rather a huge, hierarchical, neural network. It is the cerebral cortex that is central to the higher brain functions, speech, thought, complex movement patterns, etc. On the other hand, the cerebellum seems to be more of an ‘auto mat on’. It has to do more with precise coordination and control of the body, and with skills that have become our‘second nature’. Cerebellum actions seem almost to take place by themselves, without thinking about them. They are very similar to the unconscious reflex actions, e.g., reaction to pinching, which may not be mediated by the brain, but by the upper part of the spinal column.

Various regions of the cerebral cortex are associated with very spec ificfunctions. The visual cortex, a region in the occipital lobe at the back of the brain, is responsible for the reception and interpretation of vision. The auditory cortex, in the temporal lobe, deals mainly with analysis of sound, while the olfactory cortex, in the frontal lobe, deals with smell. The soma to sensory cortex, just behind thedivisionbetweenfrontalandparietallobes, has to do with the sensations of touch. There is a very spec ificmappingbetweenthevarious parts of the surface of the body and the regions of the soma to sensory cortex. Inaddition, justin frontofthedivisionbetweenthefrontalandparietallobes, in the frontal lobe, there is the motor cortex. The motor cortex activates the movement ofdifferent parts of the body and, again here, there is a very spec ificmappingbetweenthe various muscles of the body and the regions of the motor cortex. All the above mentioned regions of the cerebral cortex are referred to as primary, since they are the one most directly concerned with the input and output of the brain. Near to these primary regions are the secondary sensory regions of the cerebral cortex, where information is processed, while in the secondary motor regions, conceived plans of motion get translated into spec ificdirectionsforactualmusclemovement by the primary motor cortex. The most abstract and sophisticated activity of the brain is carried out in the remaining regions of the cerebral cortex, the association cortex.

The basic building blocks of the brain are the nerve cells or neurons. Among about 200 typesofdifferent basic types of human cells, the neuron is one of the most specialized, exotic and remarkably versatile cell. The neuron is highly unusual in three respects: its variation in shape, its electrochemicalfunction, and its connectivity, i.e., its ability to link up with other neurons in networks. Let us start with a few elements of neuron micro anatomy(see[II 06b]). There is a centralstarlikebulb, called the soma, which contains the nucleus of the cell.A long nervefibre, known as the axon, stretches out from one end of the soma. Its length, in humans, canreachuptofewcm. The function of an axon is to transmit the neuron’s output signal, in which it acts like a coaxial cable. The axon has the ability of multiple bifurcation, branch ing out into many smaller nerve fibers, and the very end of which thereisalwaysasynapticknob. At the other end of the soma and often springing offin all directions from it, are the tree–like dendrites, along which input data are carried into the soma. The whole nerve cell, as basic unit, has a cell membrane surrounding soma, axon, synoptic knobs, anddendrites. Signals pass from oneneurontoanotheratjunctions known as synapses, whereasynapticknobofoneneuronisattachedtoanotherneuron’s soma or dendrites. There is very narrow gap, of afewnm, between thesynaptic knobandthesoma/dendritetowherethesynapticcleftisattached. The signal from one neuron to another has to propagate across this gap.

A nervefibreisacylindricaltubecontainingamixedsolutionofNaCland KCl, mainly the second, so there areNa+, K+, andCl−ionswithinthetube. OutsidethetubethesametypeofionsarepresentbutwithmoreNa+than K+. In there stingstatethereisanexcessofCl−overNa+andK+inside the tube, giving it a negative charge, while it has positive charge outside.A nerve signal is a region of charge reversal traveling along thefibre. At its head, sodium gates open to allow the sodium toflowinwardsandatitstailpotassiumgatesopentoallowpotassiumtoflowoutwards. Then, metabolicpumps act to restore order and establish the resting state, preparing the nervefibre for another signal. There is no major material(ion)transport that produces the signal, just in and out local movements of ions, across the cell membranes, i.e., asmallandlocaldepolarizationof the cell. Eventually, the nerve signal reachestheattachedsynapticknob, at the very end of the nervefibre, and triggers it to emit chemical substances, known as neurotransmitters. It is these substances that travel across the synaptic cleft to another neuron’s soma or dendrite. It should be stressed that the signal here is not electrical, but a chemical one. What really is happening is that when the nerve signal reaches the synaptic knob, thelocaldepolarizationcauselittlebagsimmersedinthevesiculargrid, the vesicles containing molecules of the neurotransmitter chemical(e.g., acetylcholine)to release their contents from theneuronintothesynapticcleft, the phenomenon of exocytosis. These molecules then diffuse across thecleft to interact with receptor proteins on receiving neurons. On receiving a neurotransmitter molecule, the receptor protein opens a gate that causes alocaldepolarization of the receiver neuron.

It depends on the nature ofthesynapticknobandofthespecificsynaptic junction, if the next neuron would be encouraged tofire, i.e., to start anew signal along its own axon, or it would be discouraged todoso. In the former case we are talkingaboutexcitatorysynapses, while in the latter case about inhibitorysynapses. At any given moment, one has to add up theeffectofall excitatorysynapsesandsubtracttheeffectofalltheinhibitoryones. If the net effectcorrespondstoapositiveelectricalpotentialdiffere nce between the inside and the outside of the neuron under consideration, and ifitis bigger than a critical value, then the neuronfires, otherwise it stays mute.

1. The neural axon is an all or none state. In the all state a signal, called a spike or action potential(AP), propagates indicating that the summation performedinthesomaproducedanamplitudeoftheorderoftensofmV. In the none state there is no signal traveling in the axon, only the resting potential (∼−70mV). It is essential to notice that the presence of a traveling signal in the axon, blocks the possibility of transmission of a second signal. 2. The nerve signal, upon arriving at the ending of the axon, triggers the emission of neurotransmitter sinthesynapticcleft, which in turn cause the receptors to open up and allow the penetration of ionic current into the post synapticneuron. Theefficacyofthesynapseisaparameterspecifiedbythe amount of penetrating currentperpresynapticspike. 3. The post synaptic potential (PSP) diffuses toward the soma, where all inputs in a short period, fromallthepresynapticneuronsconnectedtothepostsynapticaresummedup. TheamplitudeofindividualPSP’sisabout1mV, thus quite a number of inputs is required to reach the‘firing’threshold, of tensofmV. Otherwise the postsynapticneuronremainsinthe resting or none state.

The cycle–time of a neuron, i.e., the time from the emission of a spike in the presynaptic neuron to the emission of a spike in the postsynaptic neuron is of the order of 1 −2 ms. There is also some recovery time for the neuron, after itfired, of about 1 −2ms, independently of how large the amplitude of thedepolarizing potential would be. This period is called the absolute refractory period of the neuron. Clearly, it sets an upper bound on the spike frequency of 500−1000/ sec.

In the types of neurons that we will be interested in, the spike frequency is considerably lower than the above upper bound, typically in the range of 100/ sec, or even smaller in some areas, at about 50/ sec. It should be noticed that this ratherexoticneuralcommunicationmechanismworksveryefficientlyanditis employed universally, bothbyvertebratesandinvertebrates. The vertebra tes have gone even further in perfection, by protecting their nervefibers by an insulating coating of myelin, a white fatty substance, which incidentally gives the white matter of the brain, discussed above, its color. Because of this insulation, the nerve signals may travel undisturbed at about 120m/sec(see[II 06b]).

A very important anatomical fact is that each neuron receives some 104 synapticinputsfromtheaxonsofotherneurons, usuallyoneinputperpresynaptic neuron, and that each branch ing neural axon forms about the same number (∼104)ofsynapticcontactsonother, postsynapticneurons. Acloser look at our cortex then would expose a mosaic-typestructureofassembliesofafew thousand densely connected neurons. These assemblies are taken to be the basic cortical processing modules, and their size is about 1(mm)2. The neural connectivity gets much sparser as we move to larger scales and with much less feedback, allowing thus for autonomous local collective, parallel processing and more serial and integrative processing of local collective outcomes. Taking into account that there are about 1011 nerve cells in the brain(about7 × 1010 inthecerebrum and3 × 1010 in the cerebellum), we are talking about 1015 synapses.

A very useful set of phenom eno logical rules has been put forward by Hebb [Heb 49], the Hebbrules, concerning the underlying mechanism of brain plasticity. According to Hebb, asynapsebetweenneuron1 and neuron2 would be strengthened whenever the firing of neuron 1 is followed by the firing of neuron 2, and weakened whenever it is not. It seems that brain plasticity is a fundamental property of the activity of the brain.

Many mathematical models have been proposed to try to simulate learning process, based upon the close resemblance of the dynamics of neural communication to computers and implementing, one way or another, the essence of the Hebb rules. These models are known as neural networks. They are closely related toadaptiveKalmanfiltering(see[II 06c]).

Let us try to construct a neural network model for a set of Ninterconnected neurons(seee.g.,[Kos 92,II 07b]). Theactivityoftheneuronsisusuallyparameterizedby Nfunctionsσi(t), (i=1, 2, . . ., N), andthesynapticstrength, representing thesynapticefficacy, byN × Nfunctionsjik(t). The total stimulus of the network on a given neuron(i)is assumed to be given simply by the sum of the stimuli coming from each neuron

Si(t) = jik(t)σk(t), (summationoverk) where we have i dent ifiedt he individual stimuli with the product of thesynaptic strength jikwith the activity σkof the neuron producing the individual stimulus. Thedynamicequationsfortheneuronaresupposedtobe, in the simplest case with Fa nonlinear function of its arguments. The dynamic equations controlling the time evolution ofthesynapticstrengthsjik(t)are much more involved and only partially understood, andusuallyitisassumedthatthej−dynamics is suchthatitproducesthesynapticcouplings. The simplest version of a neural network model is the Hopfieldmodel[Hop 82]. In this model the neuron activities are conveniently and conventionally taken to be switch–like, namely±1, and thetimetisalsoaninteger–valued quantity. This all(+1)or none(−1)neural activityσi is based on the neurophysiol ogy. The choice±1is more natural the usual‘binary’one(bi=1or 0), from a physicist’spointofviewcorresponding to a two–state system, like the fundamental elementsoftheferromagnet, i.e., theelectronswiththeirspinsup(+)or(−).

N ) = sign(Si(t + i/N)), (20.20) which provides a rather explicit form for(20.19). If, assuggestedbytheHebb rules, thejmatrixissymmetric(jik=jki), the Hopfielddynamics[Hop 82] corresponds to a sequential algorithm for looking for the minimum of the Hamilton i an

H= −Si(t)σi(t) = −jikσi(t)σk(t).

The Hopfield model, at this stage, is very similar to the dynamics of a statistical mechanics Is ing–type, or, more generally a spin–glass, model(see[II 07b]. This mapping of the Hopfield model to a spin–glass model is highly advantageous because we have now a just ificat i on for using the statistical mechanics language of phase transitions, like critical points or attractors, etc, to describe neural dynamics and thus brain dynamics. This simplified Hopfield model has many attractors, correspondingtomanydifferent equilibrium or ordered states, endemic in spin– glass models, and an unavoidable prerequisite for successful storage, in the brain, ofmanydifferent patterns of activities. In the neural network framework, itis believed that an internal representation(i.e., a pattern of neural activities)is associated with each object or category that we are capable of recognizing and remembering. According to neurophysiol ogy, it is also believed that an object is memorizedbysuitablychangingthesynapticstrengths. The so–called associative memory is generated in this scheme as follows [II 06b]: An external stimulus, suitably involved, produces synapticstrengthssuchthataspec ificlearnedpattern σi(0) = Piis ‘printed’ in such a way that the neuron activities σi(t) ∼Pi(II learning), meaning that theσi will remain for all times close toP i, corresponding to a stable attractor point(IIIcodedbrain). Furthermore, if are plication signal is applied, pushing the neurons toσivaluespartiallydifferentfromPi, the neurons should evolve toward thePi. In other words, the memory is able to retrieve the information on the whole object, from the knowledge of apart of it, or even in the presence of wrong information(I Vrecallprocess). Clearly, ifthe externalstimulusisverydifferentfromanypreexistingσi= Pipattern, it may eithercreateanewpattern, i.e., createanewattractorpoint, or it may reach a chaotic, random behavior(Iuncodedbrain).

Despite the remarkable progress that has been made during the last few years in understanding brain function using the neural network paradigm, it is fair to say that neural networks are rather artificial and a very long way from providing a realistic model of brain function. It seems likely that the mechanisms controlling the changes in synapticconnectionsaremuchmore complicated and involved thantheonesconsideredinNN, as utilizing cytoskeletalrestructuringofthe sub–synapticregions. Brain plasticity seems to play an essential, central role in the workings of the brain! Furthermore, the‘binding problem, i.e., how to bind together all the neurons firing to different features of the same object or category, especially when more than one object is perceived during asingle conscious perceptual moment, seems to remain unanswered. In this way, we have come along way since the times of the‘grandmotherneuron’, whereasinglebrain location was invoked for self observation and control, i dent ifiedwiththepineal glandsbyDescartes[II 06b].

Ithasbeenlongsuggestedthatdifferentgroupsofneurons, responding to acommon object/category,firesynchronously, implying temporal correlations [SG 95]. If true, such correlatedfiring of neurons may help us in resolving the binding problem[Cri 94]. Actually, brainwaves recorded from the scalp, i.e., theEEGs, suggest the existence of some sort of rhythms, e.g., the‘α−rhythms’ of a frequency of 10Hz. More recently, oscillations were clearly observed in the visual cortex. Rapid oscillations, aboveEEGfrequenciesintherangeof 35to75Hz, called the‘γ−oscillations’or the‘40Hzoscillations’, have been detected in the cat’svisualcortex[SG 95]. Furthermore, it has been shown that these oscillatoryresponsescanbecome synchronized in a stimulus–dependent manner. Studies of auditory–evoked responses in humans have shown inhibition of the 40 Hz coherence with loss of consciousness due to the in duc tion of general anesthesia. These striking results have prompted Crick and Koch to suggest that this synchronized firing on, or near, the beat of a ‘γ−oscillation’ (in the 35–75 Hz range) might be the neural correlate of visual awareness[Cri 94]. Such a behavior would be, of course, a very special case of a much more general framework where coherentfiring of widely–distributed, non–local groups of neurons, inthe ‘beats’ofx−oscillation(of spec ificfrequencyranges), bind them together in a mental representation, expressing the oneness of consciousness or unitarysense of self. While this is a bold suggestion[Cri 94], it is should be stressed that in a physicist’slanguageitcorrespondstoaphenomenologicalexplanation, not providing the underlying physical mechanism, based on neuron dynamics, that triggers the synchronized neuronfiring. On the other hand, the Crick–Koch binding hypothesis[CK 90, Cri 94] is very suggestive(see Figure 20.7) and in compliance with the central bio dynamic adjunct i on[II 06b] coordination = sensory⊣motor: brain ⇆body.

Fig.20.7. Fiber connections between cortical regions participating in the perception– action cycle, reflecting again our sensory–motor adjunct i on. Empty rhomboids stand for intermediate areas or sub areas of the labelled regions. Notice that there are connections between the twohierarchiesatseverallevels, not just at the toplevel.

Fig.20.8.A 1–to–many relation: Thalamus⇒Cortex in the human brain(with permission fro mE. Izhikevich).

network[Izh 99b], consistingofnquasi–periodiccorticaloscillatorsX 1, ..., Xn forcedbythethalamicinputX 0( see Figure 20.8)

Fig.20.9. Atypicalneuron, containingalloftheusualcellorganelles. However, itis highly specialized for the conduct a nce of nerve impulse.

the spinal cord and in the peripheral nerves and ganglia. Neurons consist of four major parts, including the dendrites(shorter projections), which are responsible for receiving stimuli; the axon(longer projection), which sends the nerve impulse away from the cell; the cell body, which is the site of metabolic activity in the cell; and the axon term in als, which connect neurons to other neurons, or neurons to other body structures. Each neuron can have several hundredaxon term in als that attach to another neuron multiple times, attach to multiple neurons, or both. Some types of neurons, suchas Purkinjecells, have over 1000 dendrites. The body of a neuron, fromwhichtheaxonanddendritesproject, is called the soma and holds the nucleus of the cell. The nucleus typically occupies most of thevolumeofthesomaandismuchlargerindiameterthantheaxonanddendrites, which typically are only about a micro meter thick or less. Neuronsjoin to one another andtoothercellsthroughsynapses.

Adefiningfeatureofneuronsistheirabilitytobecome‘electrically excited’, that is, to undergo an action potential–and to convey thi sex citation rapidly along their axons as an impulse. The narrow cross section of axons and dendrites lessens the metabolic expense of conducting action potentials, although fatter axons convey the impulses more rapidly, generally speaking.

Manyneuronshaveinsulatingsheathsofmyelinaroundtheiraxons, which enable their action potentials to travel faster than in unmyelinated axons of the same diameter. Formed by glial cells, the myelin she a thing normally runs along the axon in sections about1 mmlong, punctuated by unsheathed nodes of Ranvier. Neuronsandgliamakeupthetwochiefcelltypesofthenervous system.

An action potential that arrives at its term in us in one neuron may provoke an action potential in another through release of neurotransmitter molecules across thesynapticgap.

Fig.20.10. Three structural classes of human neurons:(a)multipolar,(b)bipolar, and(c)unipolar.

1. Themultipolarneurons, the majority of neurons in the body, in particular in the central nervous system. 2. Thebipolarneurons, sensory neurons found in the special senses. 3. Theunipolarneurons, sensory neurons located in dorsal root ganglia.

Figure 20.11 depictsageneralmodelofaconvergentcircuit, showingtwoneuronsconvergingononeneuron. This allows one neuron or neuronal pool to receive input from multiple sources. For example, the neuronal pool in the brain that regulates rhythm of breathing receives input from other areas of the brain, baroreceptors, chemoreceptors, and stretch receptors in the lungs.

Fig.20.11. Aconvergentneuralcircuit: nerve impulses arriving at the same neuron.

Fig.20.12. Organization of the human nervous system.

Glia are specialized cells of the nervous system whose main function is to ‘glue’ neuronstogether. Specializedgliacalled Schwanncellssecretemyelinsheaths around particularly longaxons. Gliaof the various types greatly outnumber the actualneurons.

The human nervous system consists of the central and peripheral parts(see Figure 20.12). The central nervous system(CNS)refers to the core nervous system, which consists of the brain and spinal cord(as well as spinal nerves). The peripheral nervous system(PNS)consists of the nerves and neurons that reside or extend outside the central nervous system–to serve the limbs and organs, forexample. The peripheral nervous system is further divided into the soma to– motoricnervoussystemandtheautonomicnervoussystem(see Figure 20.13).

TheCNS is further divided into two parts: the brain and the spinal cord. The average adult human brain weighs 1.3to 1.4kg(approximately3 pounds). The

Fig.20.13. Basic divisions of the human nervous system.

brain contains about 100 billion nerve cells(neurons)andtrillionsof‘support cells’calledglia. Further divisions of the human brain are depicted in Figure 20.14. The spinal cord is about 43 cmlonginadultwomenand45 cmlongin adult men and weighs about 35-40 grams. The vertebral column, the collection of bones (back bone) that houses the spinal cord, is about 70 cm long. Therefore, the spinal cord is much shorter than the vertebral column.

Fig.20.14. Basic divisions of the human Brain.

ThePNS is further divided into two major parts: the somatic nervous system and the autonomic nervous system.

The somatic nervous system consists of peripheral nervefibers that send sensory information to the central nervous system and motor nervefibersthat project to skeletal muscle.

The autonomic nervous system (ANS)controls smooth muscles of theviscera (internal organs) and glands. In most situations, we are unaware of the workings of the ANS because it functions in an involuntary, reflexive manner. For example, we do not notice when blood vessels change size or when our heartbeats faster. TheANSismostimportantintwosituations:

1.In emergencies that cause stress and require us to ‘fight’ or take ‘flight’, and 2. In non–emergencies that allow us to‘rest’and‘digest’.

Fig.20.15. Basic functions of the sympathetic nervous system.

1. The sympathetic nervous system(see Figure 20.15), 2. The parasympathetic nervous system(see Figure 20.16), and 3. The ent eric nervous system, which is a mesh work of nerve fibersthat innervatetheviscera(gastrointestinaltract, pancreas, gallbladder).

Fig.20.16. Basic functions of the parasympathetic nervous system.

• motor(efferent)neurons–carryinformationawayfromtheCNS(for muscle control). 2.• cranialneurons–connect the brain with the periphery, and • spinalneurons–connect the spinal cord with the periphery. 3.• somaticneurons–connect the skin or muscle with the central nervous system, and • visceralneurons–connect the internal organs with the central nervous system.

1.• IntheCNS, collectionsofneuronsarecallednuclei. • InthePNS, collections of neurons are called ganglia. 2.• IntheCNS, collectionsofaxonsarecalledtracts. • InthePNS, collections of axons are called nerves.

Cerebral Cortex. The word‘cortex’comesfromtheLatinwordfor‘bark’(of a tree). This is because the cortex is a sheet of tissue that makes up the outer layer of the brain. The thickness of the cerebral cortex varies from2to6mm. The right and left sides of the cerebral cortex are connected by a thick band of nerve fibers called the ‘corpus callosum’. In higher mammals such as humans, the cerebral cortex looks like it has many bumps and grooves. A bump or bulge on the cortex is called a gyr us (the plural of the word gyr us is ‘gyr i’) and a groove is called a sulcus (theplural of the word sulcus is ‘sulci’). Lower mammals like rats and mice have very few gyr i and sulci. The main cortical functions are: thought, voluntary movement, language, reasoning, and perception.

Cerebellum. The word‘cerebellum’comesfromtheLatinwordfor‘little brain’. The cerebellum is located behind the brain stem. In some ways, the cerebellum is similar to the cerebral cortex: the cerebellum is divided into hemispheres and has a cortex that surrounds these hemispheres. Its main functions are: movement, balance, and posture.

Brain Stem. The brain stem is a general term for the area of the brain between thethalamus and spinal cord. Structures within the brain stem include theme dull a, pons, tectum, reticularformationandtegmentum. Some of these areas are responsible for the most basic functions of life such as breathing, heart rate and blood pressure. Its main functions are: breathing, heart rate, and blood pressure.

Hypothalamus. Thehypothalamusiscomposedofseveraldifferentareas an dislocated at the base of the brain. Although it is the size of only a pea (about 1/300 of the total brain weight), thehypothalamusisresponsiblefor some very important functions. Oneimportantfunctionofthehypothalamus is the control of body temperature. Thehypothalamusactslikea‘thermostat’ by sensing changes in body temperature and then sending signals to adjust the temperature. Forexample, if we are too hot, thehypothalamusdetectsthisand then sends a signal to expand the capillaries in your skin. This causes blood to be cooledfaster. Thehypothalamusalsocontrolsthepituitary. Its main functions are: body temperature, emotions, hunger, thirst, sexual instinct, and circadian rhythms. Thehypothalamusis‘the boss’oftheANS.

Thalamus. Thethalamusreceivessensoryinformationandrelaysthisinformation to the cerebral cortex. The cerebral cortex also sends information tothe thalamuswhichthentransmitsthisinformationtootherareasofthebrainand spinal cord. Its main functions are: sensory processing and movement.

Limbic System. Thelimbicsystem (or the limb i care as) is a group of structures that includes the amygdala, the hippocampus, mammillary bodies and cingulategyrus. These areas are important for controlling the emotional response to a given situation. The hippo campus is also important for memory. Its main function is emotions.

Hippocampus. Thehippocampusisonepartofthelimbicsystem that is important for memory and learning. Its main functions are: learning and memory.

Basal Ganglia. The basal ganglia area group of structures, including the globuspallidus, caudatenucleus, subthalamicnucleus, putamenandsubstantianigra, that are important in coordinating movement. Its main function is movement.

Mid brain. The mid brain includes structures such as the superior and inferior colliculiandrednucleus. There are several other areas also in the mid brain. Its main functions are: vision, audition, eye movement(see Figure 20.17), and body movement.

Fig.20.17. Opticalchiasma: the point of cross–over for optical nerves. By means of it, information presented to either left or right visual half–fieldisprojectedtothe contralateral occipital areas in the visual cortex. For more details, seee.g.[II 06b].

A nerve is an enclosed, cable–like bundle of nerve fibers or axons, which includes thegliathatensheathetheaxonsinmyelin(see[Mar 98,II 06b]).

Nerves are part of the peripheral nervous system.Afferentnervesconvey sensory signals to the brain and spinal cord, forexample from skin or organs, whileefferentnerves conduct stimulatory signals from the motor neurons of the brain and spinal cord to the muscles and glands.

These signals, sometimes called nerve impulses, are also known as action potentials: Rapidly traveling electrical waves, which begin typically in the cell body ofaneuronandpropagaterapidlydowntheaxontoitstiporterminus’.

Nerves may contain fibersthat all serve the same purpose; for example motor nerves, the axons of which all terminate on musclefibers and stimulate contraction. Or they be mixed nerves.

An axon, or ‘nerve fibre’, is a long slender projection of a nerve cell or neuron, whichconductselectricalimpulsesawayfromtheneuron’scellbodyorsoma. Axons are ineffect the primary transmission lines of the nervous system, and as bundlestheyhelpmakeupnerves. Theaxonsofmanyneuronsaresheathedin myelin.

On the other hand, adendriteisaslender, typicallybranchedprojectionof a nerve cell or neuron, which conducts the electrical stimulation received from other cells through synapsestothebodyorsomaofthe cell from which it projects.

Many dendrites convey this stimulation passively, meaning without action potentials and without activation of voltage–gated i on channels. Insuchdendrites the voltage change that results from stimulationatasynapsemayextend both towards and away from the soma. In other dendrites, though an action potential may not arise, nevertheless voltage–gated channels help to propagate excitatory synapticstimulation. Thispropagationisefficientonlytowardthesomadueto an uneven distribution ofchannelsalongsuchdendrites.

The structure and branch ing of a neuron’sdendritesstronglyinfluenceshow it integrates the input from many others, particularly those that input only weakly(moreatsynapse). This integration is in aspects‘temporal’–involving the summation of stimuli that arrive in rapid succession–as well as‘spatial’ –entailingtheaggregationofexcitatoryandinhibitoryinputsfromseparate branches or‘arbors’.

Spinal nerves take their origins from the spinal cord. They control the functions of the rest of the body. In humans, there are 31 pairs of spinal nerves: 8 cervical,12 thoracic,5 lumbar,5 sacral and1 coccygeal.

As the traveling signals of nerves and as the localized changes that contract muscle cells, action potentials are an essential feature of animal life. They set the pace of thought and action, constrainthesizesofevolvinganatomiesand enable centralized control and coordination of organs and tissues(see[Mar 98]).

When a biological cell or patch of membrane undergoes an action potential, the polarityofthetransmembranevoltageswingsrapidlyfromnegativetopositive and back. Within anyone cell, consecutive action potentials typically are indistinguishable. Also bet we end ifferent cells the amplitudes of the voltage swings tend to be roughly the same. But the speed and simplicity of action potentials vary sign ifican tly between cells, in particular bet we end ifferentcelltypes.

Minimally, an action potential involves a depolarization, a re–polarizationand finallyahyperpolarization(or‘under shoot’). In specialized muscle cells of the heart, such as the pacemaker cells, a ‘plateau phase’ of intermediate voltage may precedere–polarization.

The trans membrane voltage changes that take place during an action potential result from changes in the per me ability of the membrane to spec ificions, the internal and external concentrations of which are in imbalance. In the axon fibers of nerves, depolarization results from the inward rush of sodium ions, while re–polarizationandhyperpolarizationarisefromanoutwardrushofpotassium ions. Calcium ions make up most or all of thedepolarizing currents at an axon’s pre–synapticterminus, in musclecellsandinsomedendrites.

The imbalance of ions that makes possible not only action potentials but the resting cell potential arises through the work of pumps, in particular the sodium–potassiumexchanger.

Changesinmembranepermeabilityandtheonsetandcessationofioniccurrentsreflecttheopeningandclosingof‘voltage–gated’ i on channels, which provide portals through the membrane for ions. Residing in and spanning the membrane, these enzymes sense and respond to changes in transmembrane potential.

Action potentials are triggered by an initial depolarization to the point of threshold. This threshold potential varies but generally is about 15 millivoltsabove the resting potential of the cell. Typically action potential initiation occurs at asynapse, but may occur anywhere along the axon. In his discovery of‘animal electricity’, L. Gal van i elicited an action potential through contact of hisscalpel with the motor nerve of a frog he was dissecting, causing one of its legs to kick as in life.

In the fine fibers of simple (or ‘unmyelinated’) axons, action potentials propagate as waves, which travel at speeds up to 120 meters per second.

The propagation speed of these ‘impulses’ is faster in fatter fibers than in thin ones, other things being equal. In their Nobel Prize winning work uncovering the wave nature and ionicmechanism of action potentials, AlanL. Hodgkin and

AndrewF. Huxleyperformed their celebrated experiments on the ‘giant fibre’ of Atlantic squid [HH 52]. Responsible for initiating flight, this axon is fat enough to be seen without a microscope(100to 1000 times larger than is typical). This is assumed to reflect an adaptation for speed. Indeed, the velocity of nerve impulses in thesefibers is among the fastest in nature.

Manyneuronshave insulating sheaths of myelin surroundingtheiraxons, which enable action potentials to travel faster than in unmyelinated axons of the same diameter. The myelin she a thing normally runs along the axon in sections about 1 mmlong, punctuated by unsheathed‘nodesofRanvier’. Because the salty cytoplasm of the axon is electrically conductive, and because the myelin inhibits charge leakage through the membrane, depolarization at one nodeissufficient to elevate the voltage at a neighboring node to the threshold for action potential initiation. Thus in myelinated axons, action potentials do not propagate as waves, butrecuratsuccessivenodesandineffecthopalong the axon. Thismodeofpropagationisknownassaltatoryconduction. Saltatory conduction is faster than smooth conduction. Some typical action potential velocities areas follows:

The disease called multiple sclerosis(MS)is due to a breakdown of myelin she a thing, and degrades muscle control by destroying axons’ability to conduct action potentials.

Depolarization andre–polarization together are complete in about two milliseconds, while under shoots can last hundreds of milliseconds, depending on the cell. Inneurons, the exact length of the roughly two–millisecond delay i nre– polarization can have a strong effect on the amount of neurotransmitter released at a synapse. The duration of the hyperpolarization determines a nerve’s ‘refractory period’ (how long until it may conduct another action potential) and hence the frequency at which it willfireundercontinuousstimulation. Both of these properties are subject to biological regulation, primarily (among the mechanisms discovered so far)acting onion channels selective for potassium.

A cell capable of undergoing an action potential is said to be excitable.

Fig.20.18.Neuronformingasynapse.

Figure 20.18). Synapsesdefinet he circuits in which the neurons of the central nervous system inter connect. They are thus crucial to the biological computations that underlie perception and thought. They also provide the means through which the nervous system connects to and controls the other systems of the body (see[II 06b]).

Anatomy and Structure. Ataclassicalsynapse, a mushroom-shaped bud projectsfromeachoftwocellsandthecapsofthesebudspressflat against one another(see Figure 20.19). At this interface, the membranes of the two cells flankeach other across a slender gap, the narrowness of which enables signaling molecules known as neurotransmitters to pass rapidly from one cell to the other bydiffus i on. Thisgapissometimescalledthesynapticcleft.

Synapsesareasymmetricbothin structure and in how they operate. Only the so–called pre–synaptic neuron secretes the neurotransmitter, which binds to receptorsfacingintothesynapse from the post–synapticcell. Thepre–synaptic nerveterminalgenerallybudsfromthetipofanaxon, while the post–synaptic targetsurfacetypicallyappearsonadendrite, a cell body or another part of a cell.

The release of neurotransmitter is triggered by the arrival of a nerve impulse (or action potential)and occurs through an unusually rapid process of cellular secret i on: Withinthepre–synapticnerveterminal, vesicles containing neurotransmitters it‘docked’andreadyatthesynapticmembrane. The arriving action potential produces an influxofcalciumionsthroughvoltage–dependent, calcium–selective i on channels, at which point the vesicles fuse with the membrane and release their contents to the outside. Receptors on the opposite side

Fig.20.19.Structureofachemicalsynapse.

ofthesynapticgapbind neurotransmitter molecules and respond by opening nearby ion channels in the post–synaptic cell membrane, causingions to rush in or out and changing the local trans membrane potential of the cell. The result is excitatory, in thecaseofdepolarizingcurrents, or inhibitory in the case of hyper polarizing currents. Whether a synapse is excitatory or inhibitory depends on what type(s) of ion channel conduct the post–synaptic current, which in turn is a function of the type of receptors and neurotransmitter employed at thesynapse.

Excitatory synapses in the brain show several forms of synapticplasticity, including long–term potent i a tion(LTP)and long–term depression(LTD), which areinitiatedbyincreasesinintracellularCa 2+ that are generated through NMDA(N-methyl-D-aspartate)receptors or voltage–sensitiveCa 2+ channels. LTPdependsonthecoordinatedregulationofanensembleofenzymes, including Ca 2+/ calmodulin–dependent protein kinase II, adenylyl cyclase 1 and 8, and calc in eur in, allofwhicharestimulatedbycalmodulin, aCa 2+– binding protein.

The amount of current, or more strictly the change in transmembrane potential, depends on the ‘strength’ of the synapse, which is subject to biological regulation. One regulatory mechanism involves the simple coincidence of action potentials inthesynapticallylinkedcells. Because the coincidence of sensory stimuli(the sound of a bell and the smell of meat, forexample, intheexperimentsbyNobel Laureate IvanP. Pavlov)can give rise to associative learning or conditioning, neuroscientistshavehypothesizedthatsynapticstrengtheningthroughcoincident activity in two neurons might underlie learning and memory. This is known astheHebbiantheory[Heb 49]. It is related to Pavlov’sconditional–reflexlearning: it is learning that takes place when we come to associate two stimuli in the environment. One of these stimuli triggers areflexiveresponse. The second stimulus is originally neutral with respect to that response, but after it has been paired with thefirststimulus, it comes to trigger the response in its own right.

Technically, synaptictransmissionhappensintransmitter–activated i on channels. Activation of a presynaptic neuron results in a release of neurotransmitters into the synaptic cleft. The transmitter molecules diffuse to the other side of the cleftand activatereceptorsthatarelocatedinthepostsynapticmembrane. So– calledionotropicreceptorshaveadirectinfluenceonthestateofanassociated ionchannelwhereasmetabotropic receptors control the state of the i on channel by means of a biochemical cascade of g−proteins and second messengers. In any case the activation of the receptor results in the opening of certain i on channels and, thus, inanexcitatoryorinhibitorypostsynapticcurrent(EPSCorIPSC, respectively). The transmitter–activated i on channels can be described as an explicitly time–dependent conductivity gsyn(t) that will open whenever a presynaptic spike arrives. The current that passes through these channels depends, as usual, on thedifferenceofitsreversal potential Esyn and the actual value of the membrane potential,

Isyn(t) = gsyn(t)(u −Esyn).

The parameter Esyn and the function gsyn(t) can be used to characterize different typesofsynapse. Typically, a super position ofexponentialsisusedforgsyn(t). Forinhibitorysynapses Esynequalsthereversalpotentialofpotassiumions (about-75mV), whereasforexcitatorysynapsesEsyn≈0.

Theeffectoffastinhibitoryneuronsin the central nervous system of higher vertebratesisalmostexclusivelyconveyedbyaneuro–transmitter calledγ− aminobutyricacid, orGABAforshort. Inadditiontomanydifferent types of inhibitoryinterneurons, cerebellarPurkinjecellsformaprominentexampleof projectingneuronsthatuseGABAastheirneuro–transmitter. Theseneurons synapseontoneuronsinthedeepcerebellarnuclei(DCN)and are particularly importantforanunderstandingofcerebellarfunction.

The parameters thatdescribetheconductivityof transmitter–activated i on channels at a certain synapsearechosen so as to mimic the time course and the amplitude of experimentally observed spontaneous postsynapticcurrents. For example, the conduct a nce¯gsyn(t) of inhibitory synapses in DCN neurons can be describedbyasimpleexponentialdecaywithatimeconstantofτ=5msand an amplitude of¯gsyn= 40pS, where t(f)denotes the arrival time of a presynaptic action potential. The reversal potential is given by that of potassium ions, viz. Esyn= −75mV(see[II 06b]).

alsoGABAergic, but their post synapticcurrentismadeupoftwodifferent components. There is a fast component, thatdecayswithatimeconstantof about5ms, and there is a component that is ten times slower. The underlying postsynapticconductanceisthus of the form

Θ(t −t(f)).

Now, most of excitatorysynapsesintheverteb rate central nervous system rely on glutamate as their neurotransmitter. Thepostsynapticreceptors, however, canhaveverydifferent pharmacological properties and of tend ifferent types of glutamatereceptorsarepresentinasinglesynapse. These receptors can be class ified by certain amino acids that may beselective agonists. Usually, NMDA (N–methyl–D–aspartate) and non–NMDA receptors are distinguished. The most prominent among then on–NMDAreceptorsareAMPA–receptors. I on channels controlledbyAMPA–receptorsarecharacterizedbyafastresponsetopresynapticspikesandaquicklydecayingpostsynapticcurrent. NMDA–receptor controlled channels are sign ifican tly slower and have additional interesting properties that are due to a voltage–dependent blocking by magnesium ions (see[GMK 94]).

Excitatorysynapsesincerebellargranulecells, forexample, contain two different types of glutamate receptors, viz. AMPA–andNMDA–receptors. The time course of thepostsynapticconductivity caused by an activation of AMPA– receptorsattimet = t(f)can be described as follows, with rise timeτ rise=0.09ms, decaytimeτdecay=1.5ms, and maximum conduct a nce¯gAMP A=720pS. ThenumericalconstantN=1.273normalizes the maximum of the braced term to unity(see[GMK 94]).

NMDA–receptor controlled channels exhibit a sign ifican tly richer repertoire of dynamic behavior because their state is not only controlled by the presence or absence of their agonist, butalso by the membrane potential. The voltage dependence itself arises from the blocking of the channel by acommon extracellular i on, Mg 2+. UnlessMg 2+ is removed from the extracellular medium, the channels remain closed at the resting potential even in the presence of NMDA. If the membrane is depolarized beyond −50mV, then the Mg 2+− block is removed, the channel opens, and, in contrast to AMPA–controlled channels, stays open for 10–100 milliseconds. Asimpleansatzthataccountsforth is additional voltage dependenceofNMDA–controlledchannelsincerebellargranulecellsis

,τ rise=3ms,τ decay=40ms, N=1.358, ¯gNMDA= 1.2nS,α = 0.062mV−1,β= 3.57mM, and the extracellular magnesium concentration o= 1.2mM(see[GMK 94]). Finally, Though NMDA–controlled ion channels are permeable to sodium and potassiumions, their per me ability toC a 2+ is evenfiveortentimeslarger. Calciumionsareknown to play an important role in intracellular signaling and are probably also involved in long–termmodificationsofsynapticefficacy. Calcium influxthroughNMDA–controlled i on channels, however, is bound tothecoincidenceofpresynaptic(NMDAreleasefrompresynapticsites)andpostsynaptic (removaloftheMg 2+− block)activity. Hence, NMDA–receptors operate as a kindofamolecularcoincidencedetectorsastheyarerequiredforabiochemical implementationofHebb’slearningrule[Heb 49].

ThebasisofallCNSactivity, as well as the simplest example of our sensory– motor adjunct i on, is thereflex(sensory–motor)action, RA. It occurs at all neural organizational levels. We are aware of some reflex acts, while others occur without our knowledge.

In particular, thespinalreflexact i on i sdefined as a composition of neural pathways, RA=EN◦CN◦AN, whereENistheefferentneuron, A Nisthe afferentneuronandCN=CN 1, ..., CNnisthechainofnconnectorneurons (n=0for the simplest, stretch, reflex, n≥1forallotherreflexes). In other words, we have the following loop-diagram:

inwhichRecisthereceptor(for a complex–type receptor as eye, see Figure 20.17), Effistheeffector(e.g., muscle), PSCistheposterior (or, dorsal) horn of the spinal cord, and ASCistheanterior (or, ventral) horn of the spinal cord. InthiswaydefinedmapRA: Rec→Effisthesimplest, one–to–one relation betweenonereceptorneuronandoneeffectorneuron(e.g., patellarreflex, see Figure 20.20).

Now, inthemajorityofhumanreflexarcsachainCNofmanyconnector neurons is found. There maybe link–ups with various levels of the brain and spinal cord. Every receptorneuron ispotentiallylinkedintheCNSwithalarge numberofeffectororgansalloverthebody, i.e., themapRA: Rec→Effis one–to–many. Similarly, everyeffectorneuronispotentiallyincommunication with receptors all over the body, i.e., the map RA : Rec →Effis many–to–one.

Fig.20.20. Schematic of asimple knee–jerkreflex. Hammer strikes knee, generating sensory impulse to spinal cord. Primaryneuronmakes(monosynaptic, excitatory) synapsewithanteriorhorn(motor)cell, whoseaxontravelsviaventralroottoquadricepsmuscle, which contracts, raising foot. Hamstring(lower)muscle is simultaneously inhibited, viaaninternuncialneuron.

link–upwithcentersathigherandlowerlevelsoftheCNS. In this way, higher centers in the brain canmodifyreflexactswhichoccurthrough the spinal cord. These centers can send suppressing or facilitating impulses along their pathways to the cells in the spinal cord[II 06b].

Through such‘functional’link–ups, neuronsindifferentpartsoftheCNS, when active, can influenceeach other. This makes it possible for Pavlov’s cond it i one dreflexestobeestablished. Suchreflexesformthebasisofalltraining, so that it become sdifficulttosaywherereflex(or involuntary)behavior ends and purely voluntary behavior begins.

In particular, the control of voluntary movements is extremely complex. Many different systems across numerous brain areas need to work together to ensure proper motor control. Our understanding of the nervous system decreases as we moveuptohigherCNSstructures.

The brain is the supervisory center of the nervous system consisting of grey matter(superficialpartscalledcortexanddeepbrainnuclei)andwhitematter (deep parts except the brain nuclei). It controls and coordinates behavior, homeostasis 3(i.e., negative feedback control of the body functions such as heartbeat,

3 Homeostasis is the property of an open system to regulation its internal environment so as to maintain a stable state of structure and functions, by means of multiple dynamic equilibrium controlled by interrelated regulation mechanisms. The term was coined in 1932 byW. Cannon from two Greekwords[homeo–man]and[stasis– stationary]. Homeostasis is one of the fundamental characteristics of living things. It is the maintenance of the internal environment with intolerable limits. All sorts of factors affectthesuitabilityofourbodyfluidstosustainlife; these include properties like temperature, salinity, acidity(carbondioxide), and the concentrations blood pressure, fluid balance, and body temperature) and mental functions (such as cognition, emotion, memory and learning)(see[Mar 98,II 06b]).

The vertebratebraincanbesubdivided into:(i)medullaoblongata(or, brain stem);(ii)myelencephalon, divided into: ponsandcerebellum;(iii)mesencephalon(or, mid brain);(iv)diencephalon; and(v)telencephalon(cerebrum).

Sometimes a gross division into three major parts is used: hindbrain(including medullaoblongataandmyelencephalon), mid brain(mesencephalon)and fore brain(includingdiencephalonandtelencephalon). The cerebrumandthecerebellumconsis teach of two hemispheres. Thecorpuscallosumconnectsthetwo hemispheresofthecerebrum.

The cerebrumandthecerebellumconsis teach of two hemispheres. Thecorpuscallosumconnectsthetwohemispheresofthecerebrum. The cerebellum is acauliflower–shaped section of the brain(see Figure 20.21). It is located in the hindbrain, at the bottom rear of the head, directlybehindthepons. The cerebellum is a complex computer mostly dedicated to the intricacies of managing walking and balance. Damagetothecerebellumleavesthesuffererwithagait thatappearsdrunkenandisdifficult to control.

The spinal cord is the extension of the central nervous system that is enclosed in and protected by the vertebral column. It consists of nerve cells and their connections(axonsanddendrites), withbothgraymatterandwhitematter, with the former surrounded by the latter.

Cranialnervesarenerveswhich start directly from the brain stem instead of the spinal cord, and mainly control the functions of the anatomic structures of the head. Inhumananatomy, there are exactly 12 pairs of them:(I)olfactory nerve, (II) optic nerve, (III) oculomotor nerve, (IV) Trochlear nerve, (V) Trigeminalnerve,(VI)Abducensnerve,(VII)Facial nerve,(VIII)Vestibulocochlear nerve(sometimes called the auditory nerve),(IX)Glossopharyngealnerve,(X) Vagus nerve, (XI) Accessory nerve (sometimes called the spinal accessory nerve), and(XII)Hypoglossalnerve.

The optic nerve consists mainly of axons extending from the ganglionic cells of the eye’s retina. Theaxons terminate in the lateral geniculate nucleus, pulvinar, of nutrients and wastes(urea, glucose, various i on, oxygen). Since these properties affectthechemicalreactionsthatkeepbodiesalive, there are built-in physiological mechanisms to maintain them at desirable levels. This control is achieved with various organs and glands in the body. Forexample[Mar 98,II 06b]: Thehypothalamus monitors water content, carbon dioxide concentration, and blood temperature, sendingnerveimpulsestothepituitaryglandandskin. Thepituitaryglandsynthesizes ADH (anti–diuretic hormone) to control water content in the body. The muscles can shivertoproduceheatifthebodytemperatureistoolow. Warm–blooded animals (homeotherms)have additional mechanisms of maintaining their internal temperature through homeostasis. Thepancreasproducesinsulintocontrolblood–sugar concentration. The lungs take in oxygen and give out carbon dioxide. Thekidneys remove urea and adjust i on and water concentrations. More realistic is dynamical homeostasis, orhomeokinesis, whichformsthebasisoftheAnochin’stheoryof functional systems andsuperiorcolliculus, all of which belong to the primary visual center. From the later algeniculatebodyandthepulvinarfibers pass to the visual cortex.

In particular, the optic nerve contains roughly one million nervefibers. This number is low compared to the roughly 130 million receptors in the retina, and implies that substantial pre–processing takes place in the retina before the signals are sent to the brain through the optic nerve.

Inmostvertebratesthemesencephalonisthehighestintegrationcenterin the brain, whereas in mammals this role has been adopted by the telencephalon. Thereforethecerebrumisthelargestsectionofthemammalianbrainandits surface has many deepfissures(sulci)and grooves(gyr i), giving an excessively wrinkled appearance to the brain.

The human brain can be subdivided into several distinct regions: The cerebral hemispheres form the largest part of the brain, occupying the anterior and middle cranial fossae in the skull and extending backwards over the tentoriumcerebelli. They are made up of the cerebral cortex, the basal ganglia, tractsofsynapticconnections, andtheventriclescontainingCSF.

The diencephalonincludes the thalamus, hypothalamus, epithalamus and subthalamus, and forms the central core of the brain. It is surrounded by the cerebral hemispheres.

Themidbrainislocatedatthejunctionofthemiddleandposteriorcranial fossae.

Theponssitsintheanteriorpartoftheposteriorcranialfossa;thefibres within the structure connect one cerebral hemisphere with its opposite cerebellar hemisphere.

Themedullaoblongataiscontinuouswiththespinalcord, and is responsible for automatic control of the respiratory and cardiovascular systems.

Thecerebellumoverliestheponsandmedulla, extendingbeneaththetentoriumcerebelliandoccupyingmostoftheposteriorcranialfossa. It is mainly concerned with motor functions that regulate muscle tone, coordination, and posture.

The frontal lobe is concerned with higher intellectual functions, suchas abstract thought and reason, speech(Broca’sare a in the left hemisphere only), olfaction, and emotion. Voluntary movement is controlled in the pre central gyr us (the primary motor area, see Figure 20.22).

Theparietallobeisdedicatedtosensoryawareness, particularly in the post central gyr us(the primary sensory area, see Figure 20.22). It is also associated with abstract reasoning, language interpretation and formation of a mental egocentric map of the surrounding area.

Theoccipitallobeisresponsibleforinterpretationandprocessingofvisual stimuli from the optic nerves, and association of these stimuli with other nervous inputs and memories.

The temporal lobe is concerned with emotional developmentand formation, and also contains the auditory area responsible for processing and discrimination

Part 7 · pp. 185–214

Fig.20.21. The human cerebral hemispheres.

Fig.20.22. Penfield’s‘Homunculus’, showing the primary soma to sensory and motor areas of the human brain.

of sound. It is also the area thought to be responsible for the formation and processing of memories.

The dynamical system from which the temporal evolution of neural activation fieldsisgenerated is constrained by the postulate that localized peaks of activation are stable objects, or, in mathematical terms,fixedpointattractors. Sucha fielddynamicshasthegenericform(see[Ama 77,II 08c]):

τ ˙u(x, t) = −u(x, t) + resting level + input + interaction, (20.21) whereu(x, t)is the activationfielddefinedoverthemetricdimensionxand timet. Thefirstthreetermsdefineaninputdrivenregime, in which attractor solutions have the form u(x, t) = resting level + input.

The rate of relaxation is determined by the time scale parameterτ.Theinteractionstabilizeslocalizedpeaksofactivationagainstdecaybylocalexcitatory interaction and again std iffus i on by global inhibitory interaction. InS. Amari’s formulation, the conceptual model(20.21)is spec ified as a continuous model for neural activity in cortical structures[Ama 77], dx′w(x −x′)σ(u(x′, t)), (20.22) where h < 0 is a constant resting level, S(x, t) is spatially and tempo rally variable input function, w(x)isaninteractionkernelandσ(u)isasigmoidalnonlinear threshold function. The interaction term collects input from all thosefieldsites x′atwhichactivationissufficientlylarge. Theinteractionkerneldeterminesif inputs from those sites are positive, driving up activation(excitatory), or negative, driving down activation (inhibitory). Excitatory input from nearby location and inhibitory input from all field locations generic ally stabilizes localized peaks of activation. For this class of dynamics, detailed analytical results provide a framework for the inverse dynamics task facing the mode ler, determiningadynamicalsystemthathastheappropriateattractorsolutions.

The so-calledneuralattractordynamics(NAD)isbasedonadiscretizationfor singleneuronsofAmari’sneuralfieldequation(20.22). The so–called discrete Amar i equation describes the temporal evolution of the activity of all single neurons considering positive and negative contributions from external input and internal neural interactions. Since only activated neurons can have an impact on other neurons, the neural attractor dynamics is nonlinear, a ndeffectsofbistability and hysteresis can be used for low-level memory and neural competition.

h, the constant negative resting level of neural activation; σ(.),asigmoidal function, which maps the value of neural activity onto[0, 1], given by i , the adequate stimulus provided by sensory input of a certain duration; ui, activity of behavioral neuron i, i.e., activity of behavior i; cmot, a constant for weighting the motivational contribution, cmot< |h|; αbeh exc, i, all excitatory contribution of motivation neurons connected to neuron i; αmot inh, i, all inhibitory contribution of motivation neurons connected to neuron i. In this framework, a nonlinear neural dynamical and control system generates the temporal evolution of behavioral variables, such that desired behaviors are fixed-pointattractorsolutionswhileun-desiredbehaviorsarerepellers.

This kind of attractor&repellerdynamicsprovides the basis for understanding cognition, both natural and artificial.

Recent studies have provided new evidence for the physiological relevance of oscillator ysynchronizationinhumanmotor and cognitive functions. Synchronization in the beta frequency band seems to have a particularly important role in long–range communication. Electrophysiologicalrecordingsofbasalganglia– thalamocorticalcircuitsinhealthymonkeys, amonkeymodelofParkinson’s diseaseandpatientswithParkinson’sdiseasehaveprovidednewinsight sin to the functional roles of oscillations and oscillatory synchronization in normal and disturbed motor behavior. Specifically, enhanced beta and reduced gamma oscillations are associated with the poverty and slowness of movement that is characteristic of Parkinson’s disease. In addition, tremor seems to arise from abnormal synchronization of oscillations in several cortical and sub cortical brain areas. Chronic high–frequency deep brain stimulation, which can be delivered through electrodes that have been implanted in spec ificbasalgangliatargetstructures, greatly improves motor symptoms in patients with Parkinson’s disease, probably through desynchronizing effects. Pathological changes in long–range synchronization are also evident in other movement disorders, as well as in neuropsychiatric diseases.

Incoupledoscillatoryneuronalsystems, under suitable conditions, the original dynamics can be reduced theoretically to a simpler phase dynamics. The state of theithneuronaloscillatorysystemcanbethencharacterizedbyasinglephase variable ϕi representing the timing of the neuronal firings. The typical dynamics of oscillator neural networks are described by the Kuramoto model, consisting of Nequally weighted, all–to–all, phase–coupled limit–cycle oscillators, where each oscillator has its own natural frequency ωi drawn from a prescribed distribution function:

N Jij sin(ϕj−ϕi + βij). (20.24)

Here, Jijand βijare parameters representing theeffect of the interaction, while K≥0is the coupling strength. For simplicity, one can assume that all natural frequenciesωi are equal to somefixedvalueω0.We can then eliminateω0by applying the transformationϕi→ϕi+ ω0t. Using the complex representation Wi=exp(iϕi)andCij=Jij exp(iβij)in(20.24), it is easily found that all neurons relax toward their stable equilibrium states, in which the relation Wi= hi/|hi|(hi= CijWj)is sat isfied.

i=1ωi/Ncan beset to0 without loss of generality; this corresponds to observing dynamics in theco–rotating frame at frequencyΩ.

It is known that as Kisincreased from0 abovesomecriticalvalueKc, more andmoreoscillatorsstarttogetsynchronized(or phase–locked)until all the oscillatorsgetfullysynchronizedatanothercriticalvalueofKtp. In the choice ofΩ=0,the fully synchronized state corresponds to an exact steady state of the‘detailed’,fine–scaleproblemintheco–rotating frame.

Such synchronization dynamics can be conveniently summarized by considering the fraction of the synchronized(phase–locked)oscillators, and conventionally described by a complex–valuedorderparameter, reiψ= 1

Neiϕj, where the radiusrmeasuresthephasecoherence, andψis the average phase angle.

When someone compares learning a new skill to learning how to ride a bike they implythatoncemastered, the task seems imbedded in our brain forever. Well, imbedded in the cerebellum to be exact. This brain structure is the commander of coordinated movement and possibly even some forms of cognitive learning. Damage to this arealeadstomotorormovementdifficulties.

Apart of a human brain that is devoted to the sensory-motor control of human movement, that is motor coordination and learning, as well as equilibrium and posture, is the cerebellum (which in Latin means “little brain”). It performs integration of sensory perception and motor output. Many neural pathways link the cerebellum with the motor cortex, which sends information to the muscles causing them to move, and the spi no–cerebellar tract, which provides proprioception, or feedback on the position of the body in space. The cerebellum integrates these pathways, using the constant feedback on body position to fine–tune motor movements.

The human cerebellum has 7–14 million Purkinjecells. Each receives about 200,000 synapses, mostontodendriticsplines. Granulecellaxonsformtheparallelfibers. Theymakeexcitatorysynapsesonto Purkinjecelldendrites. Each parallelfibresynapsesonabout 200 Purkinjecells. Theycreateastripofexcitationalongthecerebellarfolia.

Mossyfibersare one of two main sources of input to the cerebellar cortex (see Figure 20.23). Amossyfibreisanaxonterminalthatendsinalarge, bulbous swelling. Thesemossyfibersenterthegranulecelllayerandsynapseonthe dendrites of granule cells; in fact thegranule cells reach out with little‘claws’ to grasp the term in als. The granule cells then send their axons up to the molecular layer, where they end in a T and run parallel to the surface. For this reason these axons are called parallelfibers. The parallel fiberssynapseon thehugedendritic arraysofthePurkinjecells. However, the individual parallelfibers are not a strongdrivetothePurkinjecells. The Purkinjecelldendritesfanoutwithin a plane, like the splayedfingersofonehand. If one were to turn a Purkinje cell to the side, it would have almost no width at all. The parallelfibers run perpendicular to the Purkinje cells, so that they only make contact once as they passthroughthedendrites.

Unless firing in bursts, parallel fibre EPSPs do not fire Purkinje cells. Parallel fibers provide excitation to all of the Purkinje cells they encounter. Thus, granule cellactivityresultsinastripofactivatedPurkinjecells.

Mossyfibers arise from the spinal cord and brain stem. Theysynapseonto granulecellsanddeepcerebellarnuclei. The Purkinjecellmakesaninhibitory synapse(GABA)tothedeepnuclei. Mossyfibreinputgoestobothcerebellar cortex and deepnuclei. When the Purkinjecellfires, it inhibits output from the deepnuclei.

The climbingfibrearises from the inferior olive. It makes about 300 excitatory synapsesontoonePurkinjecell. This powerful input canfirethePurkinjecell.

The parallelfibresynapsesareplastic—that is, they can be modifiedby experience. When parallelfibre activity and climbingfibreactivityconverge onthesamePurkinjecell, the parallelfibresynapsesbecomeweaker(EP- SPsaresmaller). This is called long-term depression. Weakened parallelfibre synapses result in less Purkinje cell activity and less inhibition to the deep nuclei,

Fig.20.23. Stereotypic always throughout the cerebellum.

resultinginfacilitateddeepnucleioutput.Consequently,themossyfibrecollateralscontrolthedeepnuclei.

The basket cell is activated by parallelfibers afferents. It makes inhibitory synapses onto Purkinje cells. It provides lateral inhibition to Purkinje cells. Basket cells inhibit Purkinjecellslateraltotheactivebeam.

Golgicellsreceive input from parallel fibers, mossy fibers, and climbing fibers. They inhibit granule cells. Golgi cells provide feedback inhibition to granule cells as well as feedforward inhibitiontogranulecells. Golgi cells create a brief burst ofgranulecellactivity.

Although each parallelfibretoucheseachPurkinjecellonlyonce, the thousands of parallelfibersworkingtogethercandrivethePurkinjecellstofirelike mad.

The second main type of input to the folium is the climbing fibre. The climbing fibers go straight tothe Purkinje cell layer and snake up the Purkinje dendrites, likeivyclimbingatrellis. Each climbingfibreassociateswithonlyonePurkinje cell, but when the climbing fibre fires, it provokes a large response in the Purkinje cell.

The Purkinje cell compares and processes the varying inputs it gets, and finally sends its own axons out through the white matteranddowntothedeepnuclei. Although the inhibitory Purkinjecellsarethemainoutputofthecerebellar cortex, the output from the cerebellum asawholecomesfromthedeepnuclei. Thethreedeepnucleiareresponsibleforsendingexcitatoryoutputbackto the thalamus, as well as to postural and vestibular centers.

Thereareafewothercelltypesincerebellar cortex, which can all be lumped into the category of inhibitory inter neuron. The Golgi cell is found among the granule cells. The stellateand basket cells live in the molecular layer. The basket cell(right)dropsaxonbranchesdownintothePurkinjecelllayerwherethe branches wraparound the cell bodies like baskets.

The cerebellum operates in 3’s: there are 3 highways leading in and out of the cerebellum, there are3 main inputs, and there are3 main out puts from3 deep nuclei. They are:

1. Theinferiorcerebellarpeduncle(rest i form body)contains the dorsal spinocerebellar tract (DSCT)fibers. Thesefibersarise from cells intheipsilateralClarke’scolumninthespinalcord(C 8–L 3). Thispedunclecontains thecuneo–cerebellartract(CCT)fibers. Thesefibersarisefromtheipsilateralaccessorycuneatenucleus. Thelargestcomponentoftheinferiorcerebellarpeduncleconsistsoftheolivo–cerebellartract(OCT)fibers. These fibersarise from the contralateral inferior olive. Finally, vestibulo–cerebellar tract(VCT)fibers arise from cells in both the vestibular ganglion and the vestibularnucleiandpassintheinferiorcerebellarpeduncletoreachthe cerebellum. 2. Themiddlecerebellarpeduncle(brachiumpontis)containsthepontocerebellartract(PCT)fibers. Thesefibersarisefromthecontralateralpontine grey. 3. Thesuperiorcerebellarpeduncle(brachiumconjunctivum)is the primary efferent (out of the cerebellum) peduncle of the cerebellum. It contains fibers that arise fromseveraldeepcerebellarnucle i. Thesefibers pass ips i laterally for a while and then cross at the level oftheinferiorcolliculustoform the decussationofthesuperiorcerebellarpeduncle. Thesefibersthencontinue ips i laterally to terminate in the red nucleus(‘ruber–duber’)and the motor nucleiofthethalamus(VA, VL).

The 3 inputs are: mossy fibersfrom the spinocerebellarpathways, climbing fibers from the inferiorolive, and more mossy fibers from thepons, which are carrying information from cerebral cortex(see Figure 20.24). Themossyfibers from the spinal cord have come up ips i laterally, so they do not need to cross. Thefibers coming down from cerebral cortex, however, do need to cross (as the cerebrum is concerned with the opposite side of the body, unlike the cerebellum). These fibers synapseinthepons(hence the huge block offibersinthecerebralpeduncles labelled‘cortico–pontine’), cross, andenterthecerebellumasmossyfibers.

The 3 deep nuclei are the fastigial, interposed, and dentate nuclei. The fastigial nucleus is primarily concerned with balance, and sends information mainly to vestibular and reticular nuclei. The den tate and interposed nucleiare concerned more with voluntary movement, and send axons mainly to thalamus and the red nucleus.

The main function of the cerebellum as a motor controller is depicted in Figure 22.9. A coordinated movement is easy to recognize, but we know little about how it is achieved. In search of the neural basis of coordination, a model of spinocerebellarinteractionswasrecentlypresentedinwhichthestructure-functional organizing principle is a division of the cerebellum into discrete micro–complexes.

Fig.20.24. Inputs and out puts of the cerebellum.

Each micro–complex is the recipient of a spec ificmotorerrorsignal-that is, a signal that conveys information about an inappropriate movement. Thesesignalsareencodedbyspinalreflexcircuitsandconveyedtothecerebellarcortex through climbingfibreafferents. This organization reveals salient features of cerebellar information processing, butalso highlights the importance of systems level analysis for a fuller understanding of the neural mechanisms that underlie behavior.

Fig.20.25. The cerebellum as a motor controller.

The cerebellum is crucial for the coordination of movement. The authors presented a modelofthecerebellarparavermis, a region concerned with the control of voluntary limb movements through its inter connections with the spinal cord. Theyparticularlyfocusedontheolivo-cerebellarclimbingfibresystem.

Climbing fibres are proposed to convey motor error signals (signals that convey information about inappropriate movements)related to elementary limb movements that result from the contraction of single muscles. Theactualencoding of motor error signals is suggested to depend on sensorimotor trans formations carried out by spinal modules that mediatenociceptivewithdrawalreflexes.

The termination of the climbingfibresysteminthecerebellarcortexsubdividestheparavermisintodistinctmicrozones. Functionallysimilarbutspatiallyseparatemicrozonesconvergeontoacommongroupofcerebellarnuclear neurons. The processing units formed as a consequence are termed‘multizonal micro-complexes’(MZMCs), and are each related to a spec ificspinalreflex module.

ThedistributednatureofmicrozonesthatbelongtoagivenMZMCisproposed to enable similar climbing fibre inputs to integrate with mossy fibre inputs thatarisefromdifferentsources. Anatomical results consistent with this notion have been obtained.

Within an individual MZMC, the skin receptive fields of climbing fibres, mossy fibresandcerebellarcorticalinhibitoryinterneuronsappeartobesimilar. This indicates that the inhibitory receptive fields of Purkinje cells within a particular MZMCresultfromtheactivationofinhibitoryinterneuronsbylocalgranule cells.

On the other hand, the parallelfibre–mediatedexcitatoryreceptivefieldsof the Purkinje cells in the same MZMC differ from all of the other receptive fields, butaresimilartothoseofmossyfibresinanotherMZMC. This indicates that theexcitatoryinputtoPurkinjecellsinagivenMZMCoriginatesinnon–local granulecellsandismediatedoversomedistancebyparallelfibres.

TheoutputfromindividualMZMCsofteninvolvestwoorthreesegmentsof the ipsilateral limb, indicative of control of multi–jointmusclesynergies. The distal–most muscle in this synergy seems to have a roughly antagonistic action to the muscle associated with the climbingfibreinputtotheMZMC.

The cerebellar paravermis system could provide the control of both single– and multi–joint movements. Agonist-antagonist activity associated with single–joint movementsmightbecontrolledwithinaparticularMZMC, whereascoordinationacrossmultiplejointsmightbegovernedbyinteractionsbetweenMZMCs, mediated by parallelfibres.

Two main theories address the function of the cerebellum, both dealing with motor coordination. One claims that the cerebellum functions as a regul at or of the“timing of movements.”This has emerged from studies of patients whose timed movements are disrupted.

The second,“Tensor Network Theory”provides a mathematical model of transformation of sensory(covariant)space-time coordinates into motor(contravariant)coordinatesbycerebellarneuronalnetworks.

Studies of motorlearning in the vestibulo–ocularreflexandeye-blink conditioning demonstrate that the timing and amplitude of learned movements are encoded by the cerebellum. Manysynapticplasticitymechanismshavebeen found throughout the cerebellum. The Marr–Alb us model mostly attributes motor learning to a single plasticity mechanism: the long-term depression of parallel fibersynapses. The Tensor Network Theory of sensory–motortransformations by the cerebellum has also been experimentally supported.

Now, from a physiological point of view, a neural–networks complexity has beenstudiedasaninterplaybetweendynamicsandtopologyofbrainnetworks [LBM 05]. This approach is of interest from several points of view: the brain and its structural features can be seen as a prototype of a physical system capable of highly complex and adaptable patterns in connectivity, selectively improved through evolution; architectural organization of brain cortex is one of the key features of how brain system evolves, adapts itself to the experience, and to possible injuries. Brain activity can be modeled as a dynamical process acting on a network; each vertex of the structure represents an elementary component, such as brain areas, groups of neurons or individual cells. Recall that a complexity measure, has been introduced (see [LBM 05] and reference therein) with the purpose to get a sensible measure of two important features of the brain activity: segregation and integration. The former is a measure of the relative statistical independence of small sub sets; the latter is the measure of statistical deviation from independence of large sub sets. Complexity is based on the values of the Shannonentropycalculatedoverthedynamicsofthedifferentsizedsubgraphs of the whole network. It is sensitive both to the statistical properties of the dynamics and to the connectivity. It has been shown by means of genetic algorithms that the graphs showing high values of complexity are characterized by being both segregated and integrated; the complexity is low when the system is either completely independent (segregated), or completely dependent (integrated) [LBM 05].

In general this approach has been developed within the framework of small– world networks, comprising World Wide Web structure etc. which have received much attention from researchers in various disciplines, since they were introduced by Watts and Strogatz[WS 98] as models of real networks that lie somewhere between being random and being regular.

Watts and Strogatzintroducedasimplemodelfortuningcollectionsofcoupleddynamicalsystemsbetweenthetwoextremesofrandomandregularnetworks. In this model, connections between nodes in a regular array are randomly rewiredwithaprobabilityp, suchthatp=0means the network is regularly connected, whilep=1results in a random connection of nodes. For a range of intermediate value sofpbetweenthesetwoextremes, the network retains a property of regular networks(a large clustering coefficient)and also acquires a property of random networks (a short characteristic path length between nodes).

Many examples of such small worlds, both natural and human–made, have been discussed [Str 01]. For example, a model of a social network is given in Figure 20.26, where(a)denotes people(dots)belonging to groups(ellipses), which in turn belong to groups of groups and so on. The largest group corresponds to the entire community. As we godown in this hierarchical organization, each group represents a set of people with increasing social affinity. In the example, there a rel=3hierarchical levels, eachrepresentingasubdivisioninb=3smaller groups, and thelowestgroupsarecomposedofg= 11 people, on average. This defines a social hierarchy. The distance between the highlighted individuals i and jin this hierarchy is 3.(b)Each hierarchy can be represented as a tree like structure. Differenthierarchiesarecorrelated, in the sense that distances that are short along one of them are more likely to be short along the others as well. ThefigureshowsanexamplewithH= 2 hierarchies, where highlighted in thesecondhierarchyarethosepeoplebelongingtogroupAinthefirst one.(c) Pairs of people at shorter social distances are more likely to be linked by social ties, which can represent either friendship or acquaintances hip ties(we do not distinguished them here because the ones that are relevant for the problem in question may depend on the social context). Thefigureshows, for a person in the network, the distribution of acquaintances at social distanceD=1, 2, and 3, where Distheminimumoverthedistancesalongallthehierarchies. Not surprisingly, therehasbeenmuchinterestinthesynchronizationofdynamicalsystemsconnectedinasmall–world geometry[NML 03]. Generic ally, such studies have shown that the presence of small–world connections make it easier for a network to synchronize, an effect generally attributed to the reduced path length between the linked systems. This has also been found to be true for the special case in which the dynamics of each oscillator is described by a Kuramotomodel[HCK 02a, HCK 02b].

Small–world networks are characterized by two numbers: the average path length L and the clustering coefficient C. L, which measures efficiency of communication or passage time between nodes, i sdefined as being the average number of links in the shortest path between a pair of nodes in the network. Crepresents the degree of local order, and i sdefined as being the probability that two nodes connected to acommon node are also connected to each other.

Many real networks are sparse in the sense that the number of links in the network is much less than N(N −1)/2, the number of all possible (bidirectional) links. Ononeh and, randomsparsenetworkshaveshortaveragepathlength (i.e., L ∼log N), but they are poorly clustered (i.e., C≪1). On the other hand, regularsparsenetworksaretypicallyhighlyclustered, but Lis comparable to N.(All–to–allnetworkshaveC=1andL=1, sotheyaremostefficient, but most expensive in the sense that they have allN(N −1)/2possible connections andsotheyaredenseratherthansparse.)The small–world network models have advantages of both random and regular sparse networks: they have small Lforfastcommunicationbetweennodes, and they have large C, ensuringsufficientredundancyforhighfaulttolerance. The models of small–world networks a reconstructed from a regular lattice by adding a relatively small number of

Fig.20.26. Model of a social network(modified and adapted from [MNL 03] – see text for explanation.

shortcuts at random, wherealinkbetweentwonodesuandviscalledashortcutiftheshortestpathlengthbetweenuandvintheabsenceofthelinkis more than two[Wat 99]. The regularity of the underlying lattice ensures high clustering, whiletheshortcutsreducethesizeofL.

Most work has focused on average properties of such models over different realizationsofrandomshortcutconfigurations. However, adifferent point of view is necessary when a network is to be designed to optimize its performance with a restricted number of long–range connections. Forexample, a transportation networkshouldbedesignedtohavethesmallestLpossible, so as to maximize the ability of the network to transport peopleefficiently, while keeping a reasonable cost of building the network. The same can be said about communication networks forefficient exchange of information between nodes.

Most random choices of shortcuts result in a suboptimal configuration, since they do not have any special structures or organizations. On the contrary, many

Fig.20.27. Examples of shortcut configurationwith(a)asingle center and(b)two centers.

real networks have highly structured configurationsofshortcuts. Forexample, in long–range transportation networks, the airline connections between major cities which can be regarded as shortcuts, are far from being random, but they areorganizedaroundhubs.Efficient travel involves ground transportation to a nearest airport, then flights through a hub to an airport closest to the destination, and ground transportation again at the end.

It has been shown in[NML 02] thattheaveragepathlengthLofasmall– world network with a fixed number of shortcuts attains its minimum value when there exists a ‘center’ node, from which all shortcuts are connected to uniformly distributed nodes in the network (see Figure 20.27(a)). If a small–world network has several‘centers’and its sub network of shortcuts is connected, then Lis almost as small as the minimum value(see Figure 20.27(b)).

Forexample, asmall–world geometry has been explored in a family of1D networks of Nintegrate–and–fire neurons arranged on a ring [AT 06]. To simplify the formal ism, the geometry of a neuron is reduced to a point, and the time evolutionofthemembranepotentialViofcelli, i=1, . . . , Nduringaninterspikeinterval follows the equation

[Isyn(t) + Iinh(t) + Ii(t)], (20.26) where τ mis the time constant of the neural–cell membrane, Rmits passive resistance, and Vres the resting potential. The total current entering cell i consists of a synaptic current Isyn i (t) due to the spiking of all other cells in the network that connect to cell i, aninhibitorycurrentIinh(t), which depends on the network activity, and a small external current Ii(t), which represents all other inputs into cell i.

Nonlinear dynamics depicts an irregular and unpredictable time evolution of both simple and complex deterministic dynamical systems, characterized by nonlinear coupling of its variables. Given an initial condition, the dynamic equation determines the dynamic process, i.e., every step in the evolution. However, the initial condition, when magn ified, reveals a cluster of values within a certain error bound. For a regular dynamic system, processes issuing from the cluster arebundledtogether, and the bundle constitutes a predictable process with an error bound similar to that of the initial condition. In a unpredictable nonlinear dynamicalsystem, processes issuing from the cluster diverge from each other exponentially, and after a while the error becomes so large that the dynamic equation losses its predictive power(see Figure 21.1).

Fig.21.1.Regular(or,linear)v.s.unpredictablenonlineardynamicalprocess.

Forexample, in a pinball game, any two trajectories that start out very close to each other separate exponentially with time, and in a finite (and in practice, a very small) number of bounces their separation δx(t) attains the magnitude of L, the characteristic linear extent of the whole system. This property of sensitivity toinitialconditionscanbequantifiedas whereλ,the mean rate of separation of trajectories of the system, is called the Lyapunovexponent. For anyfiniteaccuracy|δx(0)| = δxoftheinitialdata, the dynamics is predictable only up to afiniteLyapunovtime despite the deterministic and infallible simple laws that rule the pinball motion.

When a physicist says that a certain system“exhibits completely unpredictable nonlinear dynamics”, they mean that the system obeys deterministic laws of evolution, but that the outcome is highly sensitive to small uncertainties in the spec ificat i on of the initial state. Ifa deterministic system is locally unstable(positive Lyapunovexponent)and globally mixing(positive entropy), itis said to be completely unpredictable(see[Str 94,II 07a]).

In the remainder of this section we will outline the main historical pillars of nonlinear dynamics and chaos theory.

Chaos theory really started with Henry Jules Poincar´e, the last mathematical universal ist, thefatherofbothdynamicalsystemsandtopology(which he considered to be the two sides of the same coin). Together with the four dynamics giants mentioned above, Poincar´ehasbeenconsideredasoneofthegreat sci ent ificgeniusesofalltime.1

1 Recall that Henri Poincar´e (April 29, 1854– July 17, 1912), was one of France’s greatest mathematicians and theoretical physicists, and a philosopher of science. Poincar´e is often described as the last‘universalist’(after Gauss), capable of understanding and contributing in virtually all parts of mathematics. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestialmechanics. He was responsible for formulating the Poincar´e conjecture, one of the most famous problems in mathematics. In his research on the three-body problem, Poincar´ebecamethefirst person to discover a deterministic chaotic system. Besides, Poincar´eintroducedthemodern principle of relativity and was thefirsttopresenttheLorentztransformationsin their modern symmetrical form(Poincar´egroup). Poincar´ediscoveredtheremainingrelativisticvelocitytransformationsandrecordedtheminalettertoLorentzin 1905. Thus he got perfect in variance of all of Maxwell’sequations, thefinal step in the discovery of the theory of special relativity. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics,

Poincar´e conjecturedandproved anumberoftheorems. Two of them related to chaotic dynamics are:

( ˙x,˙y) = (f(x, y), g(x, y)), wherefandgarecontinuousdifferentiablefunctionsofxandy. LetSbe a closed bounded subset of the2 Dphase–spaceofFthatdoesnotcontain astationarypointofFandletCbeatrajectoryofFthatneverleavesS. Then Cis either a limit–cycleorCconvergestoalimit–cycle. The Poincar´e– Bendixson theorem limits the types of longterm behavior that can be exhibited by continuous planar dynamical systems. One important implication is that a 2 Dcontinuousdynamicalsystemcannotgiverisetoastrangeattractor. Ifa strangeattractorCdidexistinsuchasystem, then it could been closed in a closed and bounded subset of the phase–space. By making this subset small enough, any nearby stationary points could be excluded. But then the Poincar´e– Bendixsontheoremsays that Cisnotastrangeattractoratall–it is either a limit–cycle or it converge sto a limit–cycle. The Poincar´e–Bendixson theorem says that chaotic behavior can only arise in continuous dynamical systems whose phase–space has3 or more dimensions. However, this restriction does not apply to discrete dynamical systems, where chaotic behavior can arise in two or even one–dimensional.

2. The Poincar´e–Hopfindextheoremsays: Let Mbeacompact different i able manifold and vbeavector–field on Mwithisolated zeroes. If Mhasboundary, the nweinsistthatvbepointinginthe outward normal direction along the boundary. Then we have the formula where the sum is over all the isolated zeroes ofv andχ(M)istheEulercharacteristicofM. systems.

mathematical physics, and celestial mechanics. He was responsible for formulating the Poincar´econjecture, one of the most famous problems in mathematics. In his research on the three-body problem, Poincar´ebecamethefirst person to discover a deterministic chaotic system. Besides, Poincar´eintroducedthemodernprinciple of relativity and was thefirsttopresenttheLorentztransformationsintheirmodern symmetrical form(Poincar´egroup). Poincar´ediscoveredtheremainingrelativistic velocitytransformationsandrecordedtheminalettertoLorentzin 1905. Thus he gotperfectinvarianceofallofMaxwell’sequations, thefinal step in the discovery of the theory of special relativity.

Poincar´ehadtheoppositephilosophicalviewsofBertrandRussellandGottlob Frege, who believed that mathematics were a branch of logic. Poincar´estrongly disagreed, claiming that intuition was the life of mathematics. Poincar´egivesan interesting point of view in his book‘Science and Hypothesis’:“For a superficial observer, sci ent ifictruthisbeyondthepossibilityofdoubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rule.”

Fig.21.2. The 2– body, problemsolvedbyNewton(left), and the 3– body problem, firstattackedbyPoincar´e, and still the point of active research(right).

In 1887, in honor of his 60 thbirthday, OscarII, King of Swedenoffered a prize to the person who could answer the question“Is the Solar system stable?”Poincar´ewontheprizewithhisfamouswork on the 3– body problem. He consideredtheSun, Earth and Moon or biting in a plane under their mutual gravitational attractions(see Figure 21.2). Like the pendulum, this system has some unstable solutions. Introducing a Poincar´esection, he saw that homoclinic tanglesmustoccur. These would then give rise to chaos and unpredictability.

Recall that trying to predict the motion of the Moonhaspreoccupiedastronomerssinceantiquity. Accurate understanding of its motion was important for determining the longitude of ships while traversing open seas. The Rudolphine Tables of Johannes Keplerhadbeenagreatimprovementoverprevioustables, and Keplerwasjustlyproudofhisachievements. Bernoulli used Newton’swork onmechanicstoderivetheellipticorbitsofKeplerandsetanexampleofhow equations of motion could be solved by integrating. But the motion of the Moon isnotwellapproximatedbyanellipsewiththeEarthatafocus; at least the effectsoftheSunhavetobetakenintoaccountifonewantstoreproduce thedatatheclassicalGreeksalreadypossessed. To do that one has to consider the motion of three bodies: the Moon, the Earth, andtheSun. When the planetsarereplacedbypointparticlesofarbitrarymasses, the problem to be solved is known as the 3– body problem. The 3– body problem was also a model to another concern in a str on omy. In the Newtonian model of the Solar system it is possible for one of the planets to go from an elliptic orbit around the Sun to an orbit that escaped its domain or that plunged right into it. Knowing if any of the planetswoulddosobecametheproblemofthestabilityoftheSolarsystem.A planet would not meet this terrible end if Solar system consisted of two celestial bodies, but whether such fate could befall in the 3– body case remained unclear.

After many failed attempts to solve the 3– body problem, natural philosophers started to suspect that it was impossible to integrate. The usual technique for integrating problems was tofindtheconservedquantities, quantities that do notchangewithtimeandallowonetorelatethemomentaandpositionsdifferenttimes. Thefirst sign on the impossibility of integrating the 3– body problem camefromaresultofBurnsthatshowedthattherewerenoconservedquantitiesthatwerepolynomialinthemomentaandpositions. Burns’result did not preclude the possibility of more complicated conserved quantities. This problem wassettledbyPoincar´e and Sundmanintwoverydifferentways.

In an attempt to promote the journal Acta Mat he matic a, Gustaf Mittag– Lefflergot the permission of the King Oscar II of Sweden and Norway to establish a mathematical competition. Severalquestionswereposed(although the king would have preferred only one), and the prize of 2500 kronerwouldgotothe best submission. One of the questions was formulated by the‘father of modern analysis’, Karl Weierstrass:

“Given a system of arbitrary mass points that attract each other according to Newton’slaws, under the assumption that no two points ever collide, try tofindarepresentationofthe coordinates of each point as aseries in a variable that is some known function of time and for all of whosevaluestheseriesconvergesuniformly. This problem, whose solution would considerably extend our understandingoftheSolarsystem...”

Poincar´e’ssubmissionwontheprize. He showed that conserved quantities that were analytic in the momenta and positions could not exist. To show that he introduced methods that were very geometrical in spirit: the importance of phase flow, the role of periodic or bits and their cross sections, the homo clinic points (see[CAM 05]).2

Poincar´epointedoutthattheproblemwasnotcorrectlyposed, and proved that a complete solution to it could not be found. His work was so impressive that in 1888 the jury recognized its value by awarding him the prize. He found that the evolution of such a system is often chaotic in the sense that a small perturbation in the initial state, suchas a slight change in one body’sinitialposition, might leadtoaradicallydifferentlaterstate. If the slight change is not detect able by our measuring instruments, then we will not be able to predict whichfinal state will occur. One of the judges, thedistinguishedKarlWeierstrass, said, “This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that itspublicationwillinaugurateanewerainthehistoryofcelestialmechanics.” Weierstrass did not know how accurate he was. In Poincar´e’s paper, he described new mathematical ideas suchas homo clinic points. The memoir was about to

2 The interesting thing about Poincar´e’sworkwasthatitdidnotsolvetheproblem posed. He did notfindafunctionthatwouldgivethe coordinates as a function of time for all times. He did not show that it was impossible either, but rather that it couldnotbedonewiththeBernoullitechniqueoffindingaconservedquantityand trying to integrate. Integration would seem unlikely from Poincar´e’sprize–winning memoir, but it was accomplished by the Finnish–born Swedish mathematician Sundman, who showed that to integrate the 3– body problem one had to confront the 2– body collisions. He did that by making them go away through a trick known as regularizationofthecollisionmanifold. The trick is not to expand the coordinates as a function oft i met, but rather as a function of3t. To solve the problem for alltimesheusedaconformalmapintoastrip. This allowed Sundmantoobtaina series expansion for the coordinates valid for all times, solving the problem that was proposed by Weirstrass in the King Oscar II’s competition. Though Sundman’s work deserves better credit than it gets, itdidnotliveuptoWeirstrass’sexpectations, and the series solution did not‘considerablyextendourunderstandingoftheSolar system.’The work that followed from Poincar´edid.

bepublishedinActaMathematicawhenanerror was found by the editor. This errorinfactledtofurtherdiscoveriesbyPoincar´e, which are now considered to be the beginning of chaos theory. The memoir was published later in 1890. Poincar´e’sresearchintoorbitsaboutLagrangepointsandlow-energy transfers was not utilized for more than a century afterwards.

In 1889 Poincar´e proved that for the restricted three body problem no integrals existapartfromtheJacobian. In 1890 Poincar´eprovedhisfamousrecurrence theorem, namely that in any small region of phase–space trajectories exist which pass through the region infinitelyoften. Poincar´epublished3 volumes of‘Les m´ethods nouvelledelam´ecanique celeste’between 1892 and 1899. He discussed convergence and uniform convergence of the series solutions discussed by earlier mathematicians and proved themnottobeuniformlyconvergent. The stability proofsofLagrangeandLaplacebecameinconclusiveafterthisresult.

Poincar´eintroducedfurthertopological methods in 1912 for the theory of stability of or bits in the 3– body problem. It fact Poincar´eessentiallyinvented topology in his attempt to answer stability questions in the three body problem. He conjectured that there are infinitely many periodic solutions of the restricted problem, theconjecturebeinglaterprovedbyGeorgeBirkhoff.The stability oftheorbitsinthethreebodyproblemwasalsoinvestigatedbyLevi–Civita, Birkhoffand others(see[II 06c]for technical details).

To examine chaos, Poincar´eusedtheideaofasection, today called the Poincar´esection, which cuts across the or bits in phase–space. While the original dynamical system alwaysflows in continuous time, onthe Poincar´esection we can observe discrete–time steps. More precisely, the original phase–space flow(see[II 06c])is replaced by a nite rated map, which reduces the dimension of the phase–space by one(see Figure 21.3). Later, toshowwhataPoincar´e section would look like, H´enondevisedasimple2D–map, which is today calledtheH´enonmap: xnew=1 −ax 2+ by, ynew=x, with parameters

Fig.21.3. The2 DPoincar´esection, reducing the3 Dphase–space, using the ite rated map: xnew= F(x, y), ynew= G(x, y).

a=1.4, b=0.3. Given any starting point, this map generates a sequence of points settling onto a chaotic attractor.

As an inheritance of Poincar´ework, thechaosoftheSolarsystemhasbeen recently used for the SOHO project,3to minimize the fuel consumption need for the space flights. Namely, in a rotating frame, a spacecraft can remain stationary at5 Lagrange’spoints(see Figure 21.4).

Fig. 21.4. The Lagrange’spoints(L 1, ...L 5) used for the space flights. Points L 1, L 2, L3 onthe Sun–Earth axis are unstable. TheSOHOspacecraftusedahaloorbitaround L1 toobservetheSun. ThetriangularpointsL4 andL5 are often stable. A Japanese rescuemissionusedachaoticEarth–Moon trajectory.

Poincar´ehadtwoproteg´esinthedevelopmentofchaostheoryinthenew world: GeorgeD. Birkhoff(see [Bir 15, Bir 27, Bir 17] for the Birkhoffcurve shorteningflow), and Stephen Smale.

3 TheSOHOprojectisbeingcarriedoutjointlybyESA(European Space Agency) andNASA(USNationalAeronauticsandSpaceAdministration), as a cooperative effortbetweenthetwoagenciesintheframeworkoftheSolarTerrestrialScience Program(STSP)comprisingSOHOandCLUSTER, andtheInternationalSolar– Terrestrial Physics Program(ISTP), with Geotail(ISAS–Japan), Wind, and Polar. SOHO was launched on December 2, 1995. The SOHO spacecraft was built in Europe by an industry team led by Matra, and instruments were provided by European and American scientists. There are nine European Principal Investigators(PI’s)and three American ones. Large engineering teams and more than 200co–investigators frommanyinstitutionssupportthePI’sin the development of the instruments and in the preparation of their operations and data analysis. NASAisresponsiblefor the launch and mission operations. Large radio dishes around the world which form NASA’sDeepSpaceNetworkareusedtotrackthespacecraftbeyondtheEarth’s orbit. Mission control is based at Goddard Space Flight Center in Maryland.

In some detail, thetheoremsofJohnvonNeumannandGeorgeBirkhoffon theergodichypothesis(see footnote6 above)were published in 1912 and 1913. This line of enquirydeveloped in two directions. One direction took an abstract approachandconsidereddynamicalsystemsastransformationsofmeasurable spaces into themselves. Could we classify these transformation sina meaningful way? This lead AndreyN. Kolmogorov to the introduction of the fundamental concept of entropy for dynamical systems. With entropy as a dynamical invariant it became possible to classify a set of abstract dynamical systems known as the Bernoulli systems.

The other line that developed from the ergodichypothesiswasintryingto findmechanicalsystemsthatareergodic. Anergodicsystemcouldnothave stable or bits, as these would breaker god i city. So, in 1898 JacquesS. Hadamard published a paper on billiards, where he showed that the motion of balls on surfaces of constant negative curvature is everywhere unstable. Thisdynamical systemwastoproveveryusefulanditwastakenupbyBirkhoff.

Marston Morsein 1923 showed that it was possible to enumerate the or bits of a ball on a surface of constant negative curvature.4He did this by introducing a symbolic code to each orbit and showed that the number of possible codes grew exponentially with the length of the code. WithcontributionsbyE. Art in, G. Hedlund, and Heinz Hop fit was eventually proven that the motion of a ball on a surface of constant negative curvature was ergodic. The importance of this result escaped most physicists, one exception being N.M. Krylov, who understood that aphysicalbilliardwasadynamicalsystemonasurfaceofnegativecurvature, but with the curvature concentrated along the lines of collision. Sinai, who was the first to show that a physical billiard can be ergodic, knew Krylov’s work well.

On the other hand, the work of Lord Rayleigh also received vigorous development. It prompted many experiments and sometheoreticaldevelopmentbyB. Vander Pol, G. Duffing, andD. Hayashi. They found other systems in which the nonlinear oscillator played a role and class ified the possible motions of these systems. This concrete ness of experiments, and the possibility of analysis was too much of temptation forM.L. CartwrightandJ.E. Little wood, who set out to prove that many of the structures conjecture dbytheexperimentalistsand theoretical physicists did indeed follow from the equations of motion.

Also, G. Birkhoffhad found a ‘remarkable curve’ in a 2D map; it appeared to be non–different i able and it would be nice to see if a smooth flow could generate

4 Recall from[II 06c]that ind ifferentialtopology, the techniques of Morsetheorygive a very direct way of analyzing the to pologyofamanifoldbystudyingdifferent i able functions on that manifold. According to the basic insights of Marston Morse, a differentiablefunctiononamanifoldwill, in atypical case, reflectthetopology quite directly. Morse theory allows one tofindtheso–calledCW–structures and handledecompositionsonmanifoldsandtoobtainsubstantialinformationabout theirhomology. Before Morse, Arthur Cayley and James Clerk Maxwell developed someoftheideasofMorsetheoryinthecontextoftopography. Morseoriginally appliedhistheorytogeodesics(critical points of the energy functional on paths). These techniques were later used by Raoul Bottinhisproofofthecelebrated Bott periodicitytheorem.

such a curve. The work of Cart wright and LittlewoodleadtotheworkofN. Levinson, which in turn provided the basis for the horseshoe construction of Steve Smale.

In Russia, AleksandrM. Lyapunovparalleledthemethodsof Poincar´eand initiatedthestrongRussiandynamicalsystemsschool.A. Andronov5 carried on with the study of nonlinear oscillator sand in 1937 introduced together with LevS. Pontryagin6 the notion of coarsesystems. They were formal i zing the understanding garnered fromthestudyofnonlinearoscillators, the understanding that many of the details onhowtheseoscillatorsworkdonot affecttheoverall picture of the phase–space: there will still be limit cycles if one changes the dissipation or spring force function by a little bit. And changing the system alittle bit has the great advantage of eliminating exceptional cases in the mathematical analysis. Coarse systems were the concept that caught Smale’sat ten tion and enticedhimtostudydynamicalsystems(see[CAM 05]).

The path traversed fromergodicitytoentropyisalittlemoreconfusing. The generalcharacterofentropywasunderstoodbyNorbertWiener,7who seemed to have spoken to ClaudeE. Shannon.8In 1948 Shannon published his results on information theory, where he discusses the entropy of the shift transformation.

In Russia, AndreyN. Kolmogorov went far beyond and suggested a definit i on of the metric entropy of an area preserving transformation in order to classify Bernoulli shifts. The suggest ionwastakenbyhisstudentYa.G. Sinai and the results published in 1959.In 1967D.V. Anosov9 and Sinai applied the notion of entropy to the study of dynamical systems. It was in the context of studying theentropyassociatedtoadynamicalsystemthatSinaiintroducedMarkov partitions(in 1968), which allow one to relate dynamical systems and statistical mechanics; this has been a very fruitful relationship. It adds measure notions tothetopologicalframeworklaiddowninSmale’sdynamicalsystemspaper. Markov partitions divide the phase–space of the dynamical system into nice little

5 Recall that both the Andronov–Hopfbifurcation and a crater on the Moon are named after Aleksandr Andronov. 6 The father of modern optimal control theory(see[II 06c]) 7 The father of cybernetics 8 The father of information theory 9 Recall that the Anosovmaponamanifold Mis a certain type of mapping, fromM to itself, with rather clearly marked local directions of ‘expansion’and ‘contraction’. More precisely:

• IfadifferentiablemapfonMhasahyperbolicstructureonthetangentbundle, thenitiscalledanAnosovmap. Examples include the Bernoulli map, and Arnold cat map. • If the Anosov map is a diffeomorphism, then it is called an Anosovdiffeomorphism. Anosovproved that Anosovdiffeomorphismsarestructurallystable. • Ifaflowonamanifoldsplitsthetangentbundleintothreeinvariantsubbundles, with one sub bundle that is exponentially contracting, and one that is exponentially expanding, and a third, non–expanding, non–contracting1 Dsub–bundle, then the flowiscalledanAnosovflow.

boxes that map into each other. Each box is labelled by a code and the dynamics on the phase–space maps the codes around, inducing a symbolic dynamics. From the number of boxes needed toc overall the space, Sinaiwasabletodefinet he notion of entropy of a dynamical system. However, the relations with statistical mechanicsbecameexplicitintheworkofDavidRuelle.10Ruelleunderstood thatthetopologyoftheorbitscouldbespecifiedbyasymboliccode, and that one could associate an‘energy’to each orbit. The energies could be formally combined in apart it i on function(see[II 06c])to generate the invariant measure of the system.

Thefirstdeliberate, coordinated attempt to understand how global system’s behaviormightdifferfromitslocalbehavior, camefromtopologistSteveSmale from the University of California at Berkeley. A young physicist, making a small talk, asked what Smale was working on. The answer stunned him:“Oscillators.” It was absurd. Oscillators(pendulums, springs, or electric circuits)where the sort of problem that a physicist finished offearly in his training. They were easy. Why would a great mathematician be studying elementary physics? However, Sma lew as looking at nonlinear oscillators, chaotic oscillators –and seingthings that physicists had learned noto see[Gle 87].

Smale’s 1966 Fields Medal honored a famous piece of work in high– dimensionaltopology, proving Poincar´econjectureforalldimensionsgreater than 4; he later generalized the ideas in a 107 page paper that established the H–cobordism theorem(this seminal result provides algebraic algebraic topological criteria for establishing that higher–dimensionalmanifoldsarediffeomorphic).

10 David Ruelleisamathematicalphysicistworkingonstatisticalphysicsanddynamicalsystems. Together with Floris Takens, he coined the term strange attractor, and founded a modern theory of turbulence. Namely, in a seminal paper [RT 71] they argued that, as a function of an external parameter, the route to chaos in afluid flowis a transition sequence leading from stationary(S) to single periodic (P), double periodic(QP 2), triple periodic(QP 3) and, possibly, quadruplyperiodic(QP 4) motions, before theflowbecomeschaotic(C). 11 In the fall of1961 Steven Sma lew as invited to KievwherehemetV.I. Arnol’d, (one of the fathers of modern geometrical mechanics[II 06c]), D.V. Anosov, Sinai, and Novikov. He lectured there, andspentalotoftimewithAnosov. He suggested aseriesofconjectures, mostofwhichAnosovprovedwithinayear. It was Anosov who showed that there are dynamical systems for which all points(as opposed to an on wandering set)admitthehyperbolicstructure, and it was in honor of this resultthatSmalenamedthemAxiom–A systems. In Kiev Smalefoundareceptive audience that had been thinking about these problems. Smale’sresultcatalyzed their thoughts and initiated a chain of developments that persisted into the 1970’s.

Fig.21.5. The Smalehorseshoemapconsistsofasequenceofoperationsontheunit square. First, stretch in they−direction by more than a factor of two, then squeeze (compress)inthex−direction by more than a factor of two. Finally, fold the resulting rectangle andfititbackontothesquare, overlapping at the top and bottom, and not quite reaching the ends to the left and right(and with a gap in the middle), as illustrated in the diagram. The shape of the stretched and folded map gives the horseshoe map its name. Note that it is vital to the construction process for the map to overlap and leave the middle and vertical edges of the initial unit square uncovered.

sign ificantresearchindynamicalsystemsandchaostheory.12Smalealsooutlinedamathematical research program carried out by many others. Smaleis alsoknownforinjectingMorsetheoryintomathematicaleconomics, as well as recent explorations of various theories of computation. In 1998 he compiled a list of 18 problems in mathematics to be solved in the 21st century. This list was compiledinthespiritofHilbert’sfamouslistofproblemsproduced in 1900.In fact, Smale’slistincludessomeoftheoriginalHilbertproblems. Smale’sproblems include the Jacob i an conjecture and the Riemann hypothesis, both of which arestillunsolved.

The Smale horseshoe map(see Figure 21.5) is any member of a class of chaotic maps of the square into itself. Thistopologicaltransformation provided a basis for understanding the chaotic properties of dynamical systems. Its basis are simple: A space is stretched in one direction, squeezed in another, and then folded. When the process is repeated, it produces something like a many–layered

12 In his landmark1967 Bulletin survey article entitled‘Differentiabledynamicalsystems’[Sma 67], Smalepresentedhisprogramforhyperbolicdynamicalsystems and stability, complete with a superb collection of problems. The major theorem of the paper was theΩ−Stability Theorem: theglobalfoliationofinvariantsetsofthe map into dis joint stable and unstable parts, whoseproofwasatourdeforceinthe new dynamicalmethods. Some other important ideas of this paper are the existence of a horseshoe and enumeration and ordering of all its or bits, as well as the use of zetafunctionstostudydynamicalsystems. The emphasis of the paper is on the global properties of the dynamical system, onhowtounderstandthetopologyof the or bits. Smale’saccounttakesusfromalocaldifferentialequation(in the form of vectorfields)to the globaltopologicaldescriptionin terms of horse shoes.

Fig.21.6. The Smalehorseshoemapf, defined by stretching, folding and squeezing of the system’sphase–space.

pastrydough, in which a pair of points that end upclose together may have begun far apart, while two initially nearby points can end completely far apart.13

The horseshoe map was introduced by Smalewhilestudyingthebehaviorof theorbitsoftherelaxationVanderPoloscillator. The action of the map is definedgeometricallybysquishingthesquare, then stretching the result into a long strip, andfinally folding the strip into the shape of a horseshoe.

Most points eventually leave the squareundertheactionofthemapf. They go to the side caps where they will, under iteration, converge toafixed–point in one of the caps. The points that remain in the square under repeated iteration form a fractal set and are part of the invariant set of the map f(see Figure 21.6).

The stretching, folding and squeezing of the horseshoe map are the essential elements that must be present in any chaotic system. In the horseshoe map the squeezing and stretching are uniform. They compensate each other so that the area of the square does not change. The folding is done neatly, so that the or bits that remain forever in the square can be simply described.

Repeating this generates the horseshoe attractor. If one looks at a cross section ofthefinalstructure,itisseentocorrespondtoaCantorset.

The Smalehorseshoemapisthesetofbasictopologicaloperationsforconstructinganattractorconsistofstretching(which gives sensitivity to initial conditions) and folding (which gives the attraction). Since trajectories in phase– space cannot cross, the repeated stretching and folding operations result in an objectofgreattopologicalcomplexity. For any horseshoe map we have:

• There is an infinitenumberofperiodicorbits; • Periodic or bits of arbitrarily long period exist; • The number or periodic or bits grows exponentially with the period; and

13 Originally, Smalehadhopedtoexplainalldynamicalsystemsintermsofstretching and squeezing– with no folding, at least no folding that would drastically undermine a system’s stability. But folding turned out to be necessary, and folding allowed sharp changes in dynamical behavior[Gle 87].

• Close to any point of thefractalinvariantsetthereisa point of a periodic orbit.

More precisely, thehorseshoemapfisadiffeomorphismdefined from a region Soft he plane into itself. The region Sisasquarecappedbytwosemi–disks. The action of fis defined through the composition of three geometrically defined transformations. First the square is contracted along the vertical direction by a factor a<1/2. The caps are contracted so as to remain semi-disks attached to the resulting rectangle. Contracting by a factor smaller than one half assures that there will be a gap between the branches of the horseshoe. Next the rectangle is stretched by a factor of 1/a; the caps remain unchanged. Finally the resulting strip is folded into a horseshoe–shape and placed back in toS.

The interesting part of the dynamics is the image of the square into itself. Oncethatpartisdefined, the map can be extended to adiffeomorphismby definingitsactiononthecaps. The caps are made to contract and eventually map inside one of the caps (the left one in thefigure). The extension of fto the caps adds afixed–point to then on–wandering set of the map. To keep the class of horseshoe maps simple, the curved region of the horseshoe should not map back into the square.

The horseshoe map is one–to–one (1–1, or injection): any point in the domain has a unique image, even though not all points of the domain are the image of a point. The inverse of the horseshoe map, denotedbyf−1, cannot have as its domain the entire region S, instead it must be restricted to the image of Sun der f, that is, the domain off−1isf(S).

By folding the contracted and stretched square ind ifferentways, other types of horseshoe maps are possible(see Figure 21.7). The contracted square cannot overlap itself to assure that it remains 1–1. When the action on the square is extended to a diffeomorphism, the extension cannot always be done on the plane. Forexample, the map on the right needs to be extended to adiffeomorphismof the sphere by using a‘cap’thatwrapsaroundtheequator.

The horseshoe map is an Axiom Adiffeomorphismthatservesasamodel for the general behavior at a transverse homo clinic point, where the stable and unstablemanifoldsofaperiodicpointintersect.

The horseshoe map was designed by Smale to reproduce the chaotic dynamics ofaflowintheneighborhoodofagivenperiodicorbit. The neighborhood is chosentobeasmalldiskperpendiculartotheorbit. As the system evolves,

Fig.21.7. Other types of horseshoe maps can be made by folding the contracted and stretched square ind ifferentways.

pointsinthisdiskremainclosetothegivenperiodicorbit, tracing out or bits that eventually intersect the disk once again. Otherorbitsdiverge.

Thebehaviorofalltheorbitsinthediskcanbedeterminedbyconsidering what happens to the disk. The intersection of the disk with the given periodic orbit comes back to itself every period of the orbit and so do points in its neighborhood. When this neighborhood returns, its shape is transformed. Among the points back inside the disk are some points that will leave the disk neighborhood and others that will continue to return. The set of points that never leaves the neighborhood of the givenperiodicorbitformafractal.

A symbolic name can be given to all the or bits that remain in the neighborhood. Theinitialneighborhooddiskcanbedividedintoasmallnumberof regions. Knowing the sequence in which the orbit visits these regions allows the orbit to be pinpointed exactly. The visit a tion sequence of the or bits provide the so–called symbolic dynamics 14.

It is possible to describe the behavior of all initial conditions of the horseshoe map. Aninitialpointu 0=x, ygetsmappedintothepointu 1=f(u 0). Its iterateisthepointu 2=f(u 1)=f 2(u 0), and repeated iteration generates the orbitu 0, u 1, u 2, ...Under repeated iteration of the horseshoe map, most or bits end up at thefixed–point in the left cap. This is because the horseshoe maps the left cap into itself by an affinetransformation, which has exactly onefixed– point. Anyorbitthatlandsontheleftcapneverleavesitandconvergesto the fixed–point in the left cap under iteration. Points in the right cap get mapped into the left cap on the next iteration, and most points in the square get mapped into the caps. Under iteration, most points will be part of or bits that converge tothefixed–point in the left cap, but some points of the square never leave.

Under forward iterations of the horseshoe map, the original square gets mapped into aseries of horizontal strips. The points in these horizontal strips come from vertical strips in the original square. LetS0 be the original square, mapitforwardntimes, and consider only the points that fallback into the squareS 0, whichisasetofhorizontalstripesHn= fn(S 0) ∩S 0. The points in the horizontal stripe scamefromtheverticalstripesVn= f−n(Hn), which are the horizontal strips Hnmappedbackwards ntimes. That is, a point in Vnwill, underniterationsofthehorseshoemap, endupinthesetHnofverticalstrips (see Figure 21.8).

Now, if a point is to remain i ndefinitelyinthesquare, then it must belong to an invariant setΛ that maps to itself. Whether this set is empty or not has to be

14 Symbolic dynamics is the practice of modelling a dynamical system by a space consisting of infinite sequences of abstract symbols, each sequence corresponding to a state of the system, and a shift operator corresponding to the dynamics. Symbolic dynamics originated as a method to study general dynamical systems, now though, its techniques and ideas have found sign ificant applications in data storage and transmission, linear algebra, the motions of the planets and many other areas. The distinct feature in symbolic dynamics is that time is measured in discrete intervals. So at each time interval the system is in a particular state. Each state is associated with a symbol and the evolution of the system is described by an infinitesequence of symbols(see text below).

Fig.21.8. Iteratedhorseshoemap: pre–images of the square region.

Fig.21.9. Intersections that converge to the invariant setΛ.

determined. The vertical strips V1 map into the horizontal strips H 1, but not all pointsofV1 map back in toV 1. Only the points in the intersection ofV1 andH1 maybe long toΛ,as can be checked by following points outside the intersection for one more iteration. The intersection of the horizontal and vertical stripes, Hn∩Vn, aresquaresthatconvergeinthelimitn→∞to the invariant setΛ (see Figure 21.9).

The structure of invariant setΛcan be better understood by introducing a system of labels for all the intersections, namely a symbolic dynamics. The intersection Hn ∩Vnis contained in V 1. So any point that is in Λ under iteration mustlandintheleftverticalstripAofV 1, orontherightverticalstripB. The lowerhorizontalstripofH1 istheimageofAandtheupperhorizontalstripis theimageofB, soH 1= f(A) ∩f(B). The strips Aand Bcanbeusedtolabel the four squares in the intersection ofV1 andH 1( see Figure 21.10)as:

The set ΛB•A consist of points from strip A that were in strip Bin the previous iteration. A dot is used to separate the region the point of an orbit is in from the region the point came from.

Fig.21.10. The basic domains of the horseshoe map in symbolic dynamics.

Thisnotationcanbeextendedtohigheriteratesofthehorseshoemap.The verticalstripscanbenamedaccordingtothesequenceofvisitstostripAor stripB.Forexample,thesetABB⊂V3consistsofthepointsfromAthatwill alllandinBinoneiterationandremaininBintheiterationafterthat:

Working backwards from that trajectory determines a small region, the set ABB, withinV3.

All the points inΛAB•BBareinBandwillcontinuetobeinBforatleastone more iteration. TheirprevioustrajectorybeforelandinginBBwasAfollowed byB.

Anyone of the intersectionsΛP •Fofahorizontalstripwithaverticalstrip, where Pand Fare sequences of As and Bs, is an affine transformation of a small regioninV 1. If Phasksymbolsinit, an diff−k(ΛP •F )andΛP •Fintersect, then the region ΛP •Fwill have a fixed–point. This happens when the sequence P isthesameasF. Forexample,ΛABAB•ABAB⊂V4 ∩H4 has at least onefixed– point. This point is also the same as thefixed–point inΛAB•AB. By including more and more A Bsin the Pand Fpart of the label of intersection, the area of the intersection can be made as small as needed. It converge sto a point that is part of a periodic orbit of the horseshoe map. The periodic orbit can be labelled by the simplest sequence of As and Bs that labels one of the regions the periodic orbitvisits. For every sequence of As and Bsthereisaperiodicorbit.

The Smale horseshoe map is the same topological structure as the homoclinic tangle. Todynamicallyintroducehomoclinictangles, let us consider a classical engineering problem of escape from a potential well. Namely, if we have a motion, x=x(t), ofadampedparticleinawellwithpotentialenergy V=x 2/2 − x 3/3( see Figure 21.11) excited by a periodic driving force, Fcos(wt)(withthe periodT=2π/w), we are dealing with a nonlinear dynamical system given by [TS 01]

Fig.21.11. Motion of a damped particle in a potential well, driven by a periodic force Fcos(wt),. Up: potential (x−V )−plot, with V= x 2/2−x 3/3; down: the corresponding phase (x−˙x)−portrait, showing the safe bas in of attraction– ifthe driving is switched off(F= 0).

Now, if the driving is switched off, i.e., F= 0, we have an autonomous 2D–system with the phase–portrait(and the safe bas in of attraction)giveninFigure 21.11 (below). The grey area of escape starts over the hill top to infinity. Once we start driving, the system (21.1)becomes 3– dimensional, with its3 Dphase–space. We need to see the bas in in a stroboscopic section(see Figure 21.12). The hill–top solution still has an inset and and outset. As the driving increases, the inset and outset get tangled. They intersect one another an infinite number of times. The boundaryofthesafebasinbecomesfractal. As the driving increases even more, the so–called fractal–fingerscreatedbythehomoclinictangling, makeasudden incursionintothesafebasin. At that point, the integrity of the in–well motions is lost[TS 01].

Recall that an attractor is a set of system’sstates(i.e., points in the system’s phase–space), invariant under the dynamics, towards which neighboring states inagivenbasinofattractionasymptoticallyapproach in the course of dynamic evolution.15Anattractorisdefined as the smallest unit which cannot be itself decomposedintotwoormoreattractorswithdistinctbasinsofattraction. This restrictionisnecessarysinceadynamicalsystemmayhavemultipleattractors, each with its own bas in of attraction.

Conservative systems do not have attract ors, since the motion is periodic. Fordissipativedynamicalsystems, however, volumes shrink exponentially, so attractorshave0 volumeinnDphase–space.

15 A bas in of attraction is a set of points in the system’sphase–space, such that initial conditions chosen in this set dynamically evolve to a particular attractor.

Part 8 · pp. 215–244

Fig.21.12. Dynamicsofahomoclinictangle. The hill–top solution of a damped particle in a potential well driven by a periodic force. As the driving increases, the inset and outset get tangled.

In particular, a stablefixed–pointsurroundedbyadissipativeregionisan attractorknownasamapsink.16Regularattractors(corresponding to0 Lyapunov exponents) act as limit cycles, in which trajectories circle around a limiting trajectorywhichtheyasymptoticallyapproach, but never reach. The so–called strangeattractors 17 a rebounded regions of phase–space(correspondingtopositiveLyapunov characteristic exponents) having zero measure in the embedding phase–spaceandafractaldimension. Trajectories within a strange attractor appear to skip around randomly.

In 1963, Ed Lorenz from MIT was trying to improve weather forecasting. Using a primitive computer of those days, he discovered thefirstchaoticattractor. Lorenz used three Cartesian variables, (x, y, z), to define atmospheric convection. Changing in time, these variables gave him a trajectory in a(Euclidean)3D– space. From all starts, trajectories settle onto a chaotic, or strange attractor.18

16A map sink is a stable fixed–point of a map which, in a dissipative dynamical system, is an attractor. 17 Astrangeattractorisanattractingsetthathaszeromeasureintheembedding phase–spaceandhasfractaldimension. Trajectories within a strange attractor appear to skip around randomly. 18 Edward Lorenz isaprofessorofmeteorologyatMIT who wrote the first clear paper on deterministic chaos. The paper was called‘Deterministic Non periodic Flow’and itwaspublishedintheJournalofAtmosphericSciencesin 1963. Before that, in 1960, Lorenz began a project to simulate weather patterns on a computer system calledtheRoyalMcBee. Lacking much memory, the computer was unable to create complex patterns, butitwasabletoshowtheinteractionbetweenmajormeteorologicaleventssuchastornados, hurricanes, easter lies and we ster lies. A variety of factors was represented by a number, and Lorenzcouldusecomputerprintoutsto analyze the results. After watching his systems develop on the computer, Lorenz began to see patterns emerge, and was able to predict with some degree of accuracy what would happen next. While carrying out an experiment, Lorenz made an accidental discovery. He had completed a run, and wanted to recreate the pattern.

More precisely, Lorenz reduced the Navier–Stokesequationsforconvective B´enardfluidflowin to threefirstordercouplednonlineardifferentialequations and demonstrated with these the idea of sensitive dependence upon initial conditions and chaos(see[Lor 63, Spa 82]).

Fig.21.13.B´enardcells, showing atypical vortex of a rolling air, with a warm air rising in a ring and a cool air descending in the center(left). Asimple model of the B´enardcellsprovided by the celebrated‘Lorenz–butterfly’(or,‘Lorenz–mask’)strange attractor(right).

Using a printout, Lorenz entered some variables into the computer and expected the simulation to proceed the same as it had before. To his surprise, the pattern began to diverge from the previous run, and after a few‘months’of simulated time, thepatternwascompletelydifferent. Lorenzeventuallydiscoveredwhyseemingly identicalvariablescouldproducesuchdifferentresults. When Lorenz entered the numbers to recreate the scenario, the printout provided him with numbers to the thousandth position(such as 0.617). However, the computer’sinternalmemoryheld numbers up to the millionth position (such as 0.617395); these numbers were used to create the scenario for the initial run. This small deviation resulted in a completely divergent weather pattern in just a few months. This discovery creates the groundwork of chaos theory: In a system, small deviations can result in large changes. This concept is now known as a butterflyeffect.

Lorenz definit i on of chaos is: “The property that characterizes a dynamical system in which most or bits exhibit sensitive dependence.”Dynamicalsystems(like the weather)are allaround us. They have recurrent behavior(it is always hotter in summer than winter)butareverydifficult top in down and predict apart from the very shortterm.‘What will the weather be tomorrow?’–can be anticipated, but ‘What will the weather be in a months time?’is an impossible question to answer.

Lorenz showed that with a set of simple differential equations seemingly very complex turbulent behavior could be created that would previously have been considered as random. He further showed that accurate longer range forecasts in any chaotic system were impossible, thereby overturning the previous orthodoxy. It had been believed that the more equations you add to describe a system, the more accurate will be the eventual forecast.

wherex, yandzaredynamicalvariables, constituting the3 Dphase–space of the Lorenzsystem; anda, band care the parameters of the system. Originally, Lorenz used this model to describe the unpredictable behavior of the weather, where x is the rate of convective overturning (convection is the process by which heat is transferred by a movingfluid), yisthehorizontaltemperature overturning, andzistheverticaltemperatureoverturning; the parameters are: a ≡P−proportional tothe Prandtlnumber(ratio of thefluid viscosity of a substance to its thermal conductivity, usually set at 10), b ≡R−proportional to the Rayleigh number(differe nce in temperature between the top and bottom of the system, usually set at 28), andc≡K−a number proportional to the physical proportions of the region under consideration(width to height ratio of the box which holds the system, usually set at 8/3). The Lorenzsystem(21.2)has the properties:

1. Symmetry:(x, y, z) →(−x, −y, z)for all values of the parameters, and 2. Thez−axis(x=y=0)is invariant(i.e., all trajectories that start on it also end on it).

Nowadays it is well–knownthattheLorenzmodelisaparadigmforlow– dimensionalchaosindynamicalsystemsinsynergeticsandthismodelorits modificat ions are widely investigated in connection with modelling purposes in meteorology, hydrodynamics, laser physics, superconductivity, electronics, oil industry, chemical and biological kinetics, etc.

The3 Dphase–portraitoftheLorenzsystem(21.13)shows the celebrated ‘Lorenzmask’, aspecialtypeoffractalattractor(see Figure 21.13). It depicts the famous‘butterflyeffect’,(i.e., sensitive dependence on initial conditions) –thepopularideainmeteorologythat‘theflapp ing of a butterfly’swings in Brazil can setoffatornadoinTexas’(i.e., atinydifferenceisamplifieduntil twooutcomesaretotallydifferent), so that the longterm behavior becomes impossible to predict(e.g., longterm weather forecasting). The Lorenzmask has the following characteristics:

1. Trajectory does not intersect itself in three dimensions; 2. Trajectory is not periodic or transient; 3. General form of the shape does not depend on initial conditions; and 4. Exact sequence of loops is very sensitive to the initial conditions.

Mitchell Jay Feigenbaum(born December 19,1944; Philadelphia, USA)isa mathematicalphysicistwhosepioneeringstudiesinchaostheoryledtothediscoveryoftheFeigenbaumconstant.

In 1964 hebegangraduatestudiesattheMIT. Enroll ing to study electrical engineering, he changed to physics and was awarded a doctorate in 1970 fora thesis on dispersion relations under Francis Low. After short positions at Cornell

University and Virginia Polytechnic Institute, he was offered a longer–term post at Los Alamos National Laboratory to study turbulence. Although the group wasultimatelyunabletounraveltheintractabletheoryofturbulentfluids, his research led him to study chaotic maps.

Manymathematicalmapsinvolvingasinglelinearparameterexhibitapparentlyrandombehaviorknownaschaoswhentheparameterliesinacertain range. As the parameter is increased towards this region, the map undergoes bifurcations at precise values of the parameter. At first there is one stable point, the nbifurcatingtooscillatebetween two points, the nbifurcatingagaintooscillate between four points and soon. In 1975 Feigenbaum, usingtheHP-65 computer he was given, discoveredthattheratioofthediffere nce between the values at which such successive period–doublingbifurcations(calledtheFeigenbaumcascade)occur tends to a constant of around 4.6692. He was then able to provide a mathematical proof of the fact, and showed that the same behavior and the same constant would occur in a wide class of mathematical functions prior to the onset of chaos. For thefirst time this universal result enabled mathematicians to take theirfirst huge step to unravelling the apparently in tract able ‘random’behavior of chaotic systems. This‘ratio of convergence’is now known astheFeigenbaumconstant.

More precisely, the Feigenbaum constant δ is a universal constant for functions approaching chaos via successive period doubling bifurcations. It was discovered by Feigenbaumin 1975, while studying thefixed–points of the ite rated function f(x)=1 −μ|x|r, and characterizes the geometric approach of the bifurcation parameter to its limiting value (see Figure 21.14) as the parameter μ is increased forfixedx[Fei 79].

The Logisticmapisawellknownexampleofthemaps that Feigenbaum studiedinhisfamousUniversalitypaper[Fei 78].

withr=2several hundred times for aseries of discrete but closely spaced values of μ,discarding thefirst hundred or so points before the iteration has settled down to itsfixed–points, and then plotting the points remaining. The plot on the right more directly shows the cycle may be constructed by plotting function fn(x)−x as a function ofμ,showingtheresultingcurvesforn = 1, 2, 4. Simulated in Mat he matic aT M.

In 1986 Feigenbaumwasawarded the Wolf Prize in Physics. He has been Toyota Professor at Rockefeller University since 1986.

For details on Feigenbaumuniversality, see[Gle 87].

˙x = births −deaths + migration, (21.3)

19 The Rev. Thomas Robert Malthus, FRS(February,1766–December 23,1834), was an Englishdemographerandpoliticaleconomistbestknownforhispessimistic but highly influential views. Malthus’s views were largely developed in reaction to the optimistic views of his father, Daniel Malthus and his associates, notably Jean-Jacques Rousseau and William Godwin. Malthus’ses say was also inresponse to the views ofthe Marquisde Condorcet. In An Essay on the Principle of Population,first published in 1798, Mal thus made the famous prediction that population would outrun food supply, leading to a decrease in food per person: “The power of population is so superior to the power of the earth to produce subsistence forman, that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction; and often finish the dreadful work themselves. But should they fail in this warofextermination, sicklyseasons, epidemics, pestilence, and plague, advance in terrific array, and sweep offtheir thousands and tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world.”This Principle of Population was based on the idea that population if unchecked increases at a geometric rate, whereas the food supply grow satan arithmetic rate. Only natural causes(eg. accidents and old age), misery(war, pestilence, andaboveallfamine), moral restraint and vice(whichforMalthusincludedinfanticide, murder, contraception and homosexuality)could check excessive population growth. Thus, Malthusregarded his Principle of Population as an explanation of the past and the present situation of humanity, as well as a prediction of our future. The eight major points regarding evolution found in his 1798 Essay are:(i)Population level is severely limited by subsistence.(ii)When the means of subsistence increases, population increases.(iii) Population pressures stimulate increases in productivity. (iv) Increases in productivity stimulates further population growth.(v)Since this productivity can never keep up with the potential of population growth for long, there must be strong checks on population to keep it inline with carrying capacity.(vi)It is through individual cost/benefitdecisionsregardingsex, work, and children that population and production are expanded or contracted.(vii)Positive checks will come into operation as population exceeds subsistence level.(viii)The nature of these checks will have sign ificanteffectontherestofthesocioculturalsystem.

whereb, darepositiveconstantsandx 0=x(0)is the initial population. Thus, accordingtotheMalthusmodel(21.4), ifb > d, the population grows exponentially, whileifb<d, it dies out. Clearly, thisapproachisfairlyoversimplified and apparently fairly unrealistic. (However, if we consider the past and predicted growth estimates for the total world population from the 1900, we see that it has actually grown exponentially.)

This simple example shows that it i sdifficult to make long–term predictions (or, even relatively short–term ones), unlessweknowsufficientfactstoincorporateinthemodeltomakeitareliablepredictor. In the long run, clearly, there must be some adjustment to such exponential growth. Fran¸coisVerhulst [Ver 838, Ver 845]20 proposed that a self–limiting process should operate when a population becomes too large. He proposed the so–called log is tic growth population model, wherer, Karepositiveconstants. Inthe Verhulstlogisticmodel(21.5), the constant Kisthecarryingcapacityoftheenvironment(usually determined by the available sustaining resources), while the per capita birth rate rx(1−x/K) is dependentonx. There are two steady states(where˙x = 0) for(21.5):(i)x = 0 (unstable, since linearization about it gives˙x ≈rx); and (ii) x = K(stable, since linearizationaboutitgives d dt(x −K)≈−r(x −K), solimt→∞x=K). The carryingcapacityKdeterminesthesizeofthestablesteadystatepopulation, while risa measure of the rate at which it is reached(i.e., the measure of the dynamics)–thus 1/ risa representative time scale of the response of the model to any change in the population. The solution of(21.5)is

Evolution i sts John Maynard Smith and Ronald Fisher were both critical of Malthus’theory, thoughitwasFisherwhoreferredtothegrowthrater(used in log is tic equation)astheMalthusianparameter. Fisher referred to“...arelicofcreationistphilosophy...” in observing the fecundity of nature and deducing (as Darwin did)that this therefore drove natural selection. Smithdoubtedthatfaminewas the greatlevellerthatMalthusinsisteditwas. 20 Fran¸coisVerhulst(October 28,1804– February 15,1849, Brussels, Belgium)wasa mathematician and a doctor in number theory from the University of Ghent in 1825. Verhulstpublishedin 1838 the log is tic demographic model(21.5).

In the mid 20th century, ecologists realized that many species had no overlap between successive generations and so population growth happens in discrete– timestepsxt, rather than in continuous–timex(t)as suggested by the conservative law(21.3)anditsMaltus–Verhulstderivations. This leads to study discrete–timemodelsgivenbydifferenceequations, or, maps, of the form where f(xt) is some generic nonlinear function of xt. Clearly, (21.7) is a discrete– time version of(21.6). However, insteadofsolvingdifferentialequations, if we know the particular form of f(xt), it is a straightforward matter to evaluate xt+1 and subsequent generations by simple recurs i on of (21.7). The skill in modelling a spec ific population’s growth dynamics lies in determining the appropriate form off(xt)toreflectknownobservationsorfactsab out the species in question.

In 1970s, Robert May, a physicist by training, wontheCrafoordPrizefor ‘pioneering eco logical research in theoretical analysis of the dynamics of populations, communities and ecosystems’, by proposing a simple log is tic map model for the generic population growth (21.7).21May’s model of population growth is the celebrated log is tic map[May 76, May 73, May 76], where ris the Malthusianparameter that varies between 0 and 4, and the initial value of the population x 0= x(0) is restricted to be between 0 and 1. Therefore, in May’slogistic map (21.8), the generic functionf(xt)gets a spec ificquadratic form f(xt) = r xt (1 −xt).

For r< 3, the xt have a single value. For 3 < r< 3.4, the xt oscillate between two values(see bifurcation diagram 22 on Figure 21.15). Asrincreases, bifurcations occur where the number of it erates doubles. Theseperioddoublingbifurcations continuetoalimitpointatrlim=3.569944at which the period is 2∞ and the dynamics become chaotic. The r values for the first two bifurcations can be found analytically, they are r 1= 3 and r 2= 1+

6. We can label the successive values ofratwhichbifurcationsoccurasr 1,r 2,... The universal number associated withsuchperioddoublingsequencesiscalledtheFeigenbaumnumber,

21 Lord Robert May received his Ph.D. in theoretical physics from University of Sydney in 1959. He then worked at Harvard University and the University of Sydney before developing an interest in animal population dynamics and the relationship between complexity and stability in natural communities. He moved to Princeton University in 1973 and to Oxford and the Imperial College in 1988. May was able to make major advances in thefieldofpopulationbiologythrough the application of mathematics. His work played a key role in the development of theoretical ecology through the 1970 sand 1980s. He also applied these tools to the study of disease and to the study of bio–diversity. 22 A bifurcation diagram shows the possible long–term values a variable of a system can get in function of a parameter of the system.

Fig. 21.15. Bifurcation diagram for the log is tic map, simulated using Mat he matic aT M.

≈4.669.

This series of period–doubling bifurcations says that close enough to rlimthe distance between bifurcation points decreases by a factor ofδforeach bifurcation. Thecomplexfractalpatterngotinthiswayshrinksindefinitely.

MichelH´enon(born 1931 in Paris, France)is a mathematician and astronomer. He is currently at the Nice Observatory. In a str on omy, H´enoniswellknown for his contributions to stellar dynamics, mostnotablytheproblemofglobular cluster(see [Gle 87]). In late 1960s and early 1970s he was involved in dynamical evolution of star clusters, in particular theglobular clusters. He developed a numerical technique using Monte Carlo methods, to follow the dynamical evolution of a spherical star cluster much faster than the so–calledn−body methods. In mathematics, heiswellknownfortheH´enonmap, asimplediscretedynamicalsystemthatexhibitschaoticbehavior. Lately he has been involved in the restricted 3− body problem.

His celebrated H´enon map[Hen 69] is a discrete–time dynamical system that is an extension of the log is tic map(21.8) and exhibits a chaotic behavior. The map wasintroducedbyMichelH´enonasasimplifiedmodelofthePoincar´esection ofthe Lorenzsystem(21.2). This2D–map takes a point(x, y)in the plane and mapsittoanewpointdefinedbyequations

The map depends on two parameters, a a ndb, whichforthecanonicalH´enon map have values of a=1.4a ndb=0.3(see Figure 21.16). For the can on ical valuestheH´enonmapischaotic. For other values of a and bthe map maybe chaotic, intermittent, or converge to a periodic orbit. An overview of the type

Fig.21.16.H´enonstrangeattractor(see text for explanation), simulated using Dynamics SolverT M.

ofbehaviorofthemapatdifferentparametervaluesmaybeobtainedfrom its orbit(or, bifurcation)diagram(see Figure 21.17). For the can on ical map, an initial point of the plane will either approach a set of points known as the H´enonstrangeattractor, or diverge to infinity. TheH´enon attractor isafractal, smoothinonedirectionandaCantorsetinanother. Numerical estimates yield a correlation dimension of 1.42±0.02 and a Hausdorffdimension of 1.261±0.003 fortheH´enonattractor. As a dynamical system, thecanonicalH´enonmapis interesting because, unlike the log is tic map, itsorbitsdefyasimpledescription. The H´enon map maps two points into themselves: these are the invariant points. Forthecanonicalvaluesofaandb, one of these points is on the attractor: x=0.631354477...andy=0.189406343...This point is unstable. Points close to thisfixed–point and along the slope 1.924 will approach thefixed–point and points along the slope-0.156 will move away fromthefixed–point. These slopes arisefromthelinearizationsofthestablemanifoldandunstablemanifoldof thefixed–point. The unstable manifold of thefixed–point in the attractor is containedinthestrangeattractoroftheH´enonmap. TheH´enonmapdoesnot haveastrangeattractorforallvaluesoftheparametersaandb. Forexample, bykeepingbfixedat 0.3 the bifurcation diagram shows that for a=1.25the H´enon map has a stable periodic orbit as an attractor. Cvitanovic et al. [CGP 88]

Fig.21.17. BifurcationdiagramoftheH´enonstrangeattractor, simulated using Dynamics SolverT M.

showedhowthestructureoftheH´enonstrangeattractorcouldbeunderstood intermsofunstableperiodicorbitswithintheattractor.

n, yn+1=bxn, therearethreebasinsofattraction(see Figure 21.18).

n, yn+1= bxn,witha = 0.475.

where a = 0.24 is a parameter. It is an area–preserving map, and simulates the Poincar´emapof period or bits in Hamilton i an systems. Repeated random initial conditions are used in the simulation and their gray–scale color is selected at random.

n.

Fig.21.19.Phase–plot of the area–preservinggeneralizedH´enonmap, simulated using Dynamics SolverT M.

xn+1=xn+ w1 + aP 1(xn, yn) mod 1, yn+1=yn + w2 + aP 2(xn, yn) mod 1, rs)], with(r, s)=(0, 1), (1, 0), (1, 1), (1, −1), while A(i) rsw ere selected randomly in the range[0, 1].

The so–called complex nonlinearity[II 08b]plays the key role in modern computational neuroscience. All standard neural models are nonlinear and for certain values of their system parameters they can exhibit unpredictable nonlinear behavior, mostcommonlystrangeattractors. In this chapter I will brieflyreview three especially popular fields within the computational neuroscience: (i) spiking &burstingneurons, (ii)synchronization in oscillatory neural networks, and (iii) neural attractor dynamics. There are many applications of this approach, including: brain science, medical diagnosis, autonomous robotics, pattern class ificat i on, as well as nonlinear and adaptive control theory.

A neuron is said tofirea burst of spikes when itfires two or more action potentialsfollowedbyaperiodofquiescence. A burst of two spikes is called a doublet, of three spikes is called a trip let, four–quadruplet, etc. Most spiking neuronscanburstifstimulatedwithacurrentthatslowlydrivestheneuron above and below thefiring threshold. Such a current could be injected via an elect rodeorgeneratedbythesynapticin put.

Throughout this section, vdenotesthemembranepotential. All the parameters in themodelsarechosensothatvhasm Vscaleandthetimehasms scale. To compare computational cost, one assumes that each model, written as a dynamical system˙x=f(x), is implemented using the simplest,fixed–step first–order Eulermethod, with the integration time step chosen toachieve a reasonable numerical accuracy(see[IJP 08,II 07a, II 07b]for more details).

where vis the membrane potential, I is the input current, anda, b, c, andvtrsh are the parameters. When the membrane potential vreaches the threshold value vtrsh, the neuron is said tofireaspike, and vis reset toc. The I&Fneuroncan fire tonic spikes with constant frequency, and it is an integr at or. The I&F neuron canfiretonicspikeswithconstantfrequency, and it is an integr at or. It is the simplest model to implement when the integration time stepτis1ms. Because I&Fneuronhasonlyonevariable, itcannothavephasicspiking, bursting of any kind, rebound responses, threshold variability, bistabilityofattractors, or autonomous unpredictable nonlinear dynamics. Because of thefixedthreshold, the spikes do not have latencies.

The I&F model is1D, hence it cannot burst or have other properties of cortical neurons. One may think that having a second linear equation describing activation dynamics of a high–thresholdK−current, can make an improvement, e.g., endowthemodelwithspike–frequency adaptation. Indeed, eachfiringincreasestheK−activationgateviaDiracδ−function and produces an outward current that slows down the frequency of tonic spiking. This model is fast, yet still lacks many important properties of cortical spiking neurons.

tomodelthalamo–corticalneurons. Herehdescribestheinactivationof the calciumT −current, g, vh, vT , τ+andτ−are parameters describing dynamics of theT −current, and His the Heavisidestepfunction. Having this kind of a second variable creates the possibility for bursting and other interesting regimes but is already a much slower(depending on the value ofv).

˙z= I+ (b + iw)z, if Imz= atrshthenz←−z 0(z), (21.9) wherez=x + iy∈Cis a complex–valuedvariablethatdescribesoscillatory activity of the neuron. Hereb, w, andatrshareparameters, i =√ wheretherealpartxisthecurrent–like variable, while the imaginary party is the voltage–like variable. The resonate–and–fire model is simple a ndefficient. Whenthefrequencyofoscillationw= 0, it becomes an integr at or.

An alternative to the leaky I&F neuron is the quadraticI&F neuron, also known asthetheta–neuron, ortheErmentroutcanonicalmodel. It can be presented as

˙v= I+ a(v −vrest)(v −vtrsh), If v= vtrshThen v←vrest, where vrestandvtrsharetherestingand threshold values of the membrane potential. This model is can on ical in the sense that any Class1 excitable system describedbysmoothODEscanbetransformedintothisformbyacontinuous change of variables. It takes only seven operations to simulate 1 ms of the model, and this should be the model of choice when one simulates large–scale networks ofintegrators. Unlike its linear analogue, thequadraticI&Fneuronhasspike latencies, activity dependent threshold(whichisvtrshonlywhenI=0), and bistabilityofrestingandtonicspikingmodes.

whereF, G, and Haresomefunctions. This model is quite expensive to implement as a large–scalespikesimulator.

Morris and Lecar suggested a simple 2 Dmodel to describe oscillations in barnaclegiantmusclefiber. Because it has bio physically meaningful and measurable parameters, the Morris–Lecarneuronmodelbecamequite popular in computational neuroscience community. It consists of a membrane potential equation with instantaneous activation of Cacurrent and an additional equation describingsloweractivationofKcurrent, with parameters: C=20 μF/cm 2,gL=2 mmho/cm 2,VL=−50 mV , gCa= 4 mmho/cm 2, VCa=10 mV , gK=8 mmho/cm 2,VK=−70 mV , V 1=0 mV , V 2=15 mV , V 3=10 mV , V 4=10 mV ,¯λ=0.1 s−1, and applied current I(μA/cm 2). The model can exhibit various types of spiking, but could exhibit tonic bursting only when an additional equation is added, e.g., slowinactivationofCacurrent. In this case, the model becomes equivalent to the classical Hodgkin–Huxleyneuralmodel.

Wilson-Cowan model describes the dynamics of interactions between populations of excitatory E(t) and inhibitory I(t) neuronal populations. The model includes the following components:

1 + exp(aθ).

Among spiking neuron models, the oldest one, celebrated Hodgkin–Huxleymodel (or, HH–model), is expected to be the most realistic in the biological sense [HH 52, Hod 64,HH 52]. Nobel Laureates Alan Hodgkin and Andrew Huxleyperformed experiments on the giant axon of the squid and found three different types of ion current, viz., sodium, potassium, and a leak current that consists mainly of Cl− ions. Specificvoltage–dependent i on channels, one for sodium and another one for potassium, control the flow of those ions through the cell membrane. The leak current takes care of other channel types which are not described explicitly. From a biophysical point of view, action potentials are the result of currents that pass through ion channels in the cell membrane. In an extensive series of experiments on the giant axon of thesquid, Hodgkin and Huxleysucceeded to measure these currents and to describe their dynamics in the form of an analogous circuit, which in turn can be characterized by a set of nonlinear ODEs. These equations contain an empirically adequate description of the feedback system which changes the conduct an ces (or equivalently, the per me abilities) of the membrane as a function of time and potent i ald ifference.

TheHH–neuronmodelisdescribedbythenonlinearcoupledODEsforthe four variables, Vforthemembranepotential, andm, handnforthegating variables of sodium and potassium channels,

HerethereversalpotentialsofNa, anKchannelsandleakageareVNa=50 mV, VK= −77 mVandVL= −54.5mV; the maximum values of corresponding conductivitiesaregNa=120 mS/cm 2,gK=36mS/cm2 andgL=0.3mS/cm 2; thecapacityofthemembraneisC=1μF/cm 2. The external, input current is given by whereΘ(t)istheHeavisidefunction. TheHHmodelwasoriginallyproposed toaccountforthepropertyofsquidgiantaxonsandithasbeengeneralized withmodificationsofionconductances. TheHH–type models have been widely adoptedforastudyonactivitiesoftransducerneuronssuchasmotorandthalamusrelay neurons, which transform the amplitude–modulatedinputtospike– train out puts. In this section, one pays attention to data–processingneurons whichreceiveandemitthespike–train pulses.

jk denoterespectivelyconductancesofexcitatoryandinhibitorysynapsesofHebbianorgeneralizedHebbiantype.

AnotherplausiblebiologicalNN–modelisFitzHugh–Nagumomodel(FHM, [Fit 61, NAY 60]), which is actually a lower–dimensionaldescendantoftheHH– model. TheFHNequationscorrespondtoanexcitablethresholdmodelbut, as will be seen briefly, duetotheircubicnonlinearity, they exhibit the characteristic behavior of a bistablesystem. Our main objective is to show analytically the appearanceofSRinthismodelunderaperiodicexternalforcing.

where 0 < a < 1 is essentially the threshold value, b and γare positive constants and I a is the applied current. The driftfieldforthismodelisgivenby u 1(v, w)=v(a −v)(v −1) −w, u 2(v, w)=bv −γw.

γ v= 0.

bw. Phase plane analysis was us edeffectivelybyR. Fitz Hugh to understand various aspects of the HH equations and the two-variable Fitz Hugh–Nagumo model. Hisreviewalsodefines some basic mathematical terminology of nonlinear dynamics applied to neurodynamics.

Quantum mechanics was born on December 14,1900, when Max Plan ckannouncedhisfamousformulaon the black-body spectrum. He succeeded in deriving the correct shape of the black-body spectrum which now bears his name, eliminating the ultraviolet catastrophe. However, this involved an assumption so bizarre that even he distanced himself from it for many years afterwards: that energy was only emitted in certainfiniteunits, or“energyquanta”. Indefining his black-body radiation law, Max Plankintroducedtheideaofelectricoscillators in the walls of the cavity, vibrating back and forth under thermal agitation. He assumed that all possible frequencies would be present, as well as the average frequency to increase at higher temperatures as heating the walls caused the oscillators to vibrate faster and faster until thermal equilibrium was reached. Therefore, this was ultimately the thermodynamics problem and thus Planck applied Boltzmann’s statistical formulation of the Second Law of Thermodynamics, inparticularBoltzmann’sex pression for the entropy(measure of disorder)ofa collection of particles, or oscillators. He found that he had to choose energy units proportional to the oscillator frequencies, namely: e=nh f, where eis the energy, nisthenumberofoscillators(thatequallysharetheenergye), fis the frequency and his an on zero constant that would latergetPlanck’sname. This formula implies that it is not possible for an oscillator to absorb and emit energy in a continuous range—it must gain and lose energy dis continuously, in small indivisible units of the form: e = h f, which Planck called “energy quanta”. This discovery by Planck showed that at the microscopic level, energy and matter are discontinuousandforthishewontheNobelPrize.

In 1905, Albert Einsteintook Planck’sboldideaonestepfurther. Assuming that radiation could only transport energy in such units, called “photons”, he was able to explain the so-called photo electric effect, which is related to the processes used in present-day solar cells and the image sensors in digital cameras. In 1905, Albert Einsteinpublishedthreepapersinasinglevolumeofthetopphysics journal Annalender Physik(inwhichPlanckwastheEditor), as an unknown physicist, he instantly became the world-leader. Thefirst paper explained the photo elect riceffect, basedonPlanck’senergy-quantizationformula: e=hf. Einsteinshowedthatilluminatingradiationwasactuallyacollectionoflightparticles. If one assumes that the incident light consists of photons (light energy quanta)ofmagnitudehf, itispossibletoconceiveoftheejectionofelectrons (matterenergyquanta)as follows. Energyquantapenetratethesurfacelayer of the metal of the target elect rode. Their energy is partly transferred into the kineticenergyoftheelectrons: hf−P, where Pis the characteristics of the radiating metal; alsosomeelectronsareejected. Einstein came up with the photo electric formula: qV=hf−P, qis the electron charge, and Visthe threshold voltage to which the plate must be charged in order to offset the kinetic energy(and reduce the electron current to zero). This tangible discovery finally wonforEinsteinthelongexpectedNobelPrize, which he could not win for his Relativity Theory.

In 1911, Ernest Rutherfordhadconvincinglyarguedthatatomsconsisted of electron sor biting a positively charged nucleus much like a miniature solar system. This planetary atom model won for Rut her ford the Nobel Prize.23However, Rut her ford’ssolarmodelwasunstable, which does not correspond to our everyday reality which consists purely of stable atoms. If the electron sare like planets in a microscopic solar system, moving in circular or bits about the nucleus, and thus accelerating, they would radiate continuously as the classical electrodynamicswouldpredict. They would lose all their energy in a fraction of a second. Clearly, additional assumptions would be needed to complete the picture, in particular with regard to the details of atomic structure.

In 1913, Niels Bohr entered the micro-world scene. He was the grandfather of quantum physics and the founder of the mainstream Copenhagen Interpretation of quantum mechanics. Bohr made a real breakthrough in this new exciting field ofsciencebypostulatingthattheamountofangularmomentuminanatom wasquantized, whiletheelectrons were confined to a discrete set of or bits, each withadefiniteenergy. If the electron jumped from one orbit to a lower one, the energydiffere nce was sen toffin the form of a photon. If the electron was in the innermostallowedorbit, thentherewerenoorbitswithlessenergytojumpto, and therefore the atom was stable. Niels Bohr quantized the angular momentum of an electron in units of Planck’s constant h. The angular momentum is defined asL=mvr, where mis the mass of a particle, visits speed around the orbitandristheradiusoftheorbit. Constant angular momentum means that there is no torque acting on the particle. Bohr’sfirstpostulateisthequantum orbital condition: an atom can exist only in one of several special or bits with no emission of radiation, which is contrary to the expectations of classical physics. Theseorbitsarecalledstationarystatesandhavethequantizedvaluesof the orbitalangularmomentum: L=mvr=nh/(2π). And when an electron is in a stationary state, it is ready to make a quantum jump from one orbit to another. In other words, the angular momentum L of an electron cannot take any arbitrary value(as it is the case in classical mechanics)butonlycertaininteger values: L=1 ·h/(2π)inthefirstorbit(n=1); L=2 ·h/(2π)in the second

23 Note that the popular solar model is still in use in chemistry.

orbit(n=2)etc. Thisintegerniscalledtheprincipalquantumnumber. This discoverywonBohrtheNobelPrize. It was also a foundation for today’sview of the micro-world. In particular, Bohrdefined the popular“quantum jump”or “quantum leap”.

The famous wave–particledualityofmatterwasproposedbyFrenchprince Louisde Brogliein 1923 inhisPh.D. thesis: thatelectronsandotherparticles acted like standing waves. Such waves, like vibrations of a guitar string, can only occur with certain discrete (or quantized) frequencies. The idea was so new that the examining committee went outside its circle for advice on the acceptability of the thesis. Einstein gave a favorable opinion and the thesis was accepted.

In November 1925, Erwin Schr¨odinger gave a seminar on deBroglie’s work in Zurich. When he wasfinished, P. Debyesaidineffect,“You speak about waves. But where is the wave equation?”Schr¨odingerwentontoproduceandpublish his famous wave equation, the master key for so much of modern physics.

Another formulation of quantum mechanics, involving infinitematrices, was proposedbyWernerHeisenbergandhisco-workers, Max Born and Pasquale Jordan, around the same time. Next year, Paul Diracshowedthatboth Schr¨odinger’swavemechanicsandHeisenberg’smatrixmechanicswereequivalentandsubsumedbyhisownvectormechanics. All of these people won the Nobel Prize for physics. With this new powerful mathematical underpinning, quantum theory made explosive progress. Within a few years, a host of hitherto unexplained measurements had been successfully explained, including spectra of more complicated atoms and various numbers describing properties of chemical reactions. But what was this quantity, the “wave–function”, which Schr¨odinger’s equation described? This central puzzle of quantum mechanics remains a potent and controversial issue to this day. Max Born had the dramatic insight that the wave–function should be interpreted in terms of probabilities, which won him the Nobel Prize. If we measure the location of an electron, the probability of findingitinagiven region depends on the intensity of its wave–function in that region. This interpretation suggested that a fundamental randomness was built into the laws of nature. Einstein was deeply unhappy with this interpretation, and expressed his preference for a deterministic Universe with the oft-quoted remark“Ican’tbelievethatGodplaysdice”. Schr¨odingerwasalsouneasy. Wave functionscoulddescribecombinationsofdifferentstates, so-called super positions. Forexample, anelectroncouldbeinasuperpositionofseveraldifferent locations. Schr¨odingerpointedoutthatifmicroscopicobjectslikeatomscould be in strange super positions, socouldmacroscopicobjects, since they are made of atoms.

The Copenhagen Interpretation of quantum mechanics is still the dominant interpretation as taught at the majority of universities. This interpretation evolved fromdiscussionsbetweenBohrandHeisenberg(mostly as answers to Einstein’s questions) in the late 1920s, and addresses the mystery by asserting that observations, or measurements, are special. So long as a quantum system is unobserved, its wave–functionevolvesbyobeyingtheSchr¨odingerequation–a continuous and smooth “unitary” evolution, which produces the super position of all possible states of a quantum system. However, the act of observing the quantum system triggers an abrupt change in its wave function, commonly called a“collapse”: the observer sees the card in on edefiniteclassicalstate(face up or facedown) and from then onward only that portion of the wave–function survives. Nature supposedly decided which particular state to collapse into at random, withthe probabilities determined by the wave function. In this way, the Copenhagen Interpretation makes a distinction between the observer and the observed; when no one is watching, a system evolves deterministic ally according to a wave equation, but when someone is watching, the wave–function of the system “collapses” to the observed state, which is why the act of observing changes the system. Although this provided a strikingly successful calculation recipe, there was a lingering feeling that there ought to be some equation describing when and how this collapse occurred. Many physicists took this to mean that there is something fundamentally wrong with quantum mechanics, and that it would soon be replaced by some even more fundamental theory that provided such an equation.24

Light consists of quantumparticlescalledphotons, andtheFigure 21.20 shows aphotonsourcewhichweassumeemitsphotonsoneatatime. There are two slits, A a ndB, and a screen behind them. Thephotonsarriveatthescreen as individual events, where they are detected separately, just as if they were ordinaryparticles. The curious quantum behavior arise in the following way, as nicelydescribedbySirRogerPenrose. If only slit A were open and the other closed, there would be many places on the screen which the photon could reach. If we now close the slit A and open the slit B, we may again find that the photon could reach the same spot on the screen. However, if we open both slits, and if we have chosen the point on the screen carefully, we may nowfindthatthephoton cannot reach that spot, even though it could have done so if either slit alone were open. Somehow, the two possible things which the photon might do cancel each other out. This type of behavior does not take place in classical physics. Either one thing happens or another thing happens – we do not get two possible things which might happen, somehow conspiring to cancel each other out(see [II 08a]and references there in).

The way we understand the outcome of this experiment in quantum mechanics is to say that when the photon i sen route from the source to the screen, the stateofthephotonisnotthatofhavinggonethroughoneslitortheother, but is some mysterious combination of the two, weighted by complex numbers. That is, we can write the state of the photo nasa waveψ−function,25which is

24 See Max Tegmark, John Arch i bald Wheeler“100Years of Quantum Mystery,”Sci ent ificAme ric an no.284, pp.68-75,(2001). 25In the Schr¨odingerpicture, the unitary evolutionUof a quantum system is described by the Schr¨odingerequation, which provides the time rate of change ofthe quantum state or wave functionψ= ψ(t).

Fig.21.20. The two–slitexperiment, withindividualphotonsofmonochromaticlight (see text for explanation).

wherez1 andz2 are complex numbers(not both zero), while|·⟩denotes the quantumstateket–vector.

Now, in quantum mechanics, we are not so interested in the sizes of the complexnumbersz1 andz2 themselves as we are in their ratio–it is only the ratio of these numbers which has direct physical meaning(as multiplying a quantum state with an on zero complex number does not change the physical situation). The Riemann sphere is a way of representing complex numbers (plus ∞)and their ratios on a sphere on unit radius, whose equatorial plane is the complex–plane, whose center is the origin of that plane and the equator of this sphere is the unit circle in the complex–plane. We can project each point on the equatorial complex–planeontotheRiemannsphere, projecting from its south poleS, which corresponds to the point at infinity in the complex–plane. To represent a particular complex ratio, sayu=z/w(withw̸=0), we take the stereo graphic projection from the sphere onto the plane.

2 particle, suchas an electron, a proton, oraneutron, then the various combinations of their spin states can be realised geometrically on the Riemann sphere. Spin –1

2 particles can have two spin states: (i)spin–up(with the rotation vector pointing upwards), and(ii)spin–down (with the rotation vector pointing downwards). The super position of the two spin–states can be represented symbolically as

26 We are using here the standard Dirac‘bra–ket’notation for quantum states. Paul Diracwasoneoftheoutstandingphysicistsof the 20 thcentury. Among his achievements was a general formulation of quantum mechanics(having He is enberg matrix mechanicsandShr¨odingerwavemechanicsasspecialcases)andalsoitsrelativistic generalizationinvolvingthe‘Diracequation’, which he discovered, for the electron. He had an unusual ability to‘smell out’the truth, judging his equations, to a large degree, by their aesthetic qualities!

2 particle(e.g., electron, proton, neutron):|↗⟩=|↑ ⟩+q|↓⟩. The sphere is projected stereo graphically from its south pole(∞)tothe complex–plane through its equator(see text for explanation).

Different combinations of these spin states give us rotation about some other axis and, if we want to know where that axis is, we take the ratio of complex numbers u = z/w. We place this new complex number u on the Riemann sphere and the direction of u from the center is the direction of the spin axis (see Figure 21.21). More general quantum state vectors might have a form suchas[II 08a]:

wherez 1... znarecomplexnumbers(not all zero)and the state vectors |A 1⟩, ..., |An⟩might represent various possible locations for a particle(or perhaps some other property of a particle, suchas its state of spin). Even more generally, infinitesumswouldbeallowedforawaveψ−function or quantum state vector.

Now, the most basic feature of unitaryquantum evolutionU 27is that it is linear. This means that, if we have two states, say |ψ⟩and |φ⟩, and if the Schr¨odinger equation would tell us that, after some timet, the states |ψ⟩and |φ⟩would each would also hold. This would consequently apply also to linear super positions of more than two individual quantum states. Forexample, z 1|ψ⟩+z 2|φ⟩+ z 3|χ⟩ would evolve, aftertimet, toz 1|ψ′⟩+z 2|φ′⟩+ z 3|χ′⟩,if|ψ⟩, |φ⟩, and|χ⟩would each individually evolve to|ψ′⟩, |φ′⟩, and|χ′⟩, respectively. Thus, the evolution alwaysproceedsasthougheachdifferent component of a super position were oblivious to the presence of the others.

As a second experiment, consider a situation in which light impinges on a half– silvered mirror, that is a semi-transparent mirror thatreflectsjusthalfthelight (composedofastreamofphotons)falling upon it and transmits the remaining half. We might well have imagined that forastreamofphotonsimpingingon our half–silveredmirror, halfthephotonswouldbereflectedandhalfwould be transmitted. Not so! Quantum theory tells us that, instead, each individual photon, as it impinges on the minor, is separately put into a super posed state of reflection and transmission. If the photon before its encounter with the minor is in state|A⟩, then afterwards it evolves according to Utobecomeastate that can be written|B⟩+ i|C⟩, where|B⟩represents the state in which the photon is transmitted through the mirror and|C⟩thestatewherethephotonisreflected from it(see Figure 21.22). Let us write this as

The imaginary factor‘i’arises here because of a net phase shift by a quarter of a wavelength, which occurs between thereflectedandtransmittedbeamsat such a mirror.

Fig.21.22. A photon in state|A⟩impinges on a half–silveredmirroranditsstate evolvesaccordingtoUintoaasuperposition|B⟩+ i|C⟩(see text for explanation).

Fig.21.23. Mach–Zehnderinterferometer: the two parts of the photon state are brought togetherbytwofullysilveredmirrors(black), so as to encounter each other at afinalhalf–silveredmirror(white). They interfere in such away that the entire state emerges in state|F⟩, andthedetectoratGcannotreceivethephoton(see text for explanation).

doing both things at once in this strange, complex super position. To see that it cannot just be a matter of classical probability–weighted alternatives, let us take this example a little further and try to bring the two parts of the photon state, i.e., thetwophotonbeams, back together again. We can do this byfirst reflectingeachbeamwithafullysilveredmirror. Afterreflection, the photon state |B⟩would evolve according to U, into another state i|D⟩, whilst |C⟩would evolve into i|E⟩, since i 2=−1. Now, suppose that these two beams come together at a fourth mirror, whichisnowhalfsilvered(see Figure 21.23). The state|D⟩evolves into a combination|G⟩+ i|F⟩, where|G⟩represents the transmitted state and|F⟩ thereflectedone. Similarly,|E⟩evolves into|F⟩+ i|G⟩, since it is now the state |F⟩that is the transmitted state and|G⟩thereflectedone,

Our entire state i|D⟩−|E⟩is now seen (because of the linearity of U) to evolve as:

As mentioned above, the multiplying factor −2 appearing here plays no physical role, thus we see that the possibility|G⟩is not open to the photon; the two beamstogethercombinetoproducejustasinglepossibility|F⟩. This curious outcome arises because both beams are present simultaneously in the physical state of the photon, between its encounters with thefirst and last mirrors. We say that the two beams interfere with one another 28[II 08a].

Dirac’samplitude⟨qF |e−iHT|qI⟩of a quantum-mechanical system(in natural unitsℏ=c=1)with Hamilton i an Htopropagatefromapointq I to a point qFintimeTisgoverned bytheunitaryoperatore−iHT, or more completely, by thecomplexFeynmanpathintegral(in realtime):

For example, a quantum particle in a potential V (q) has the Hamilton i an operator:ˆH=ˆp 2/2m + V (ˆq), withthecorrespondingLagrangian: L =1

2m ˙q2 −V (q), so the path integral reads:

It is somewhat more rigorous to perform aso-called Wickrotationto Euclidean time, whichmeanssubstitutingt →−it and rotating the integration contour in thecomplext−plane, so that we obtain the real path integral in complex time:

knownastheEuclideanpathintegral.

0 dtL( ˙qc, qc),29whereqc(t)is therecoveredclassicalpathdeterminedbysolvingtheEuler-Lagrangianequation:(d/dt)(δL/δq) −(δL/δq) = 0, with appropriate boundary conditions.

28 Thisisapropertyofsinglephotons: each individual photon must be considered to feel out both routes that are open to it, but it remains one photon; it does not split into two photon sin the intermediate stage, but its location undergoes the strange kind of complex–number–weightedco–existence of alternatives that is characteristic of quantum theory. 29 To do an exponential integral of the form: I=

−∞dq e−(i/ℏ)f(q), we often have to resort to the following steep est-descent approximation[Zee 03]. In the limit of ℏsmall, this integral is dominated by the minimum off(q). Expanding: f(q)= f(a) +1

2f”(a)(q −a)2 + O[(q −a)3],andapplyingtheGaussianintegralrule: +∞

2 ).

Most of dynamics in contemporary human biomechanics is autonomous(see [II 06a, II 06b, II 06c, II 07e, II 07a, II 08c, II 08b]). This approach worksfinefor most individual movement simulations and predictions, in which the total human energy dissipation sare in sign ificant. However, if we analyze a 100 m-dash sprinting motion, which is in case of top athletes finished under 10 s, we can recognize a sign ificantslow-down after about 70 min all athletes–despite of their strong intention tofinis hand win the race, which is an obvious sign of the total energy dissipation. This can be seen, forexample, in a current record-braking speed-distancecurveofUsainBolt, the world-record holder with 9.69s, or in a former record-braking speed–distancecurveofCarlLewis, the former world record holder(and9 time Olympicgoldmedalist)with 9.86s(see Figure 22.1). In other words, the total mechanical energy of a sprinter cannot be conserved even for 10 s. So, if we want to develop a realistic model of intensive human motion that is longer than 7–8 s, we necessarily need to use the more advanced formal ism of time-dependent mechanics.

Similarly, if we analyze individual movements of gymnasts, we can clearly see that the highspeed of these movements is based on quickly-varying mass inertia distribution of various body segments(mostly arms and legs). Similar is the case of pirouette sin ice skating. As the total mass-inertiamatrixMij ofabiomechanicalsystemcorrespondstotheRiemannianmetrictensorgij of its configurationmanifold, wecanformulatethisproblemintermsoftimedependentRiemanniangeometry[II 06c, II 07e].

1. Human biomechanical configurationmanifoldandits(co)tangentbundles; 2. Biomechanical configuration bundle, as a time–extension of the configuration manifold;

Fig.22.1. Sprint speed–distancecurveofCarlLewisfor 100m-dash. Bottomline L: Carl Lewis’result(9.86s). ToplineM: theoretical-model curve, as an exponential solution of the linearfirst-orderdifferentialequation: Mdv/dx + Bv=E(with mass M, dampingB, input Eandoutputv), with an initial condition: v(0)=0(adapted from[IJD 09]). Also, vmdenotesmaximalspeedofthemodelcurve, while a is the initial running acceleration. TheL-curve maximum occurs at 73m, after which there is a wavy decline in speed.

3. Biomechanical jet spaces and prolongation of locomotion vector-fields developed on the configurationbundle; and 4. Time–dependent Lagrangian dynamicsusingbiomechanicaljetspaces.

Inaddition, wewillshowthatRiemanniangeometricalbasisofthisframework is defined by the Ricci flow. In particular, we will show that the exponential–like decay of total biomechanical energy (due to exhaustion of biochemical resources [IJD 09]) iscloselyrelatedtotheRicciflowontheconfigurationmanifoldof human motion.

Recall from[II 08c]that representation of an ideal humanoid–robotmotion isrigorouslydefinedintermsofrotationalconstrainedSO(3)–groups in all mainrobotjoints. Therefore, the configurationmanifoldQrobforhumanoid dynamics i sdefinedasatopologicalproductofallincludedSO(3)groups, Qrob= iSO(3)i. Consequently, thenaturalstageforautonomousLagrangian dynamicsofrobotmotionisthetangentbundleT Qrob, defined as follows. To eachn−dimensional(nD)configurationmanifoldQthereisassociatedits2nD velocity phase–spacemanifold, denotedbyT Qandcalledthetangentbundleof Q. The original smooth manifold QiscalledthebaseofT Q. There is an onto mapπ: T Q →Q, called the projection. Aboveeachpointx ∈Qthereisatangent space TxQ = π−1(x) to Q at x, which is called a fibre. The fibre TxQ ⊂T Q is the subset ofT Q, such that the total tangent bundle, T Q= disjointunionoftangentspacesTxQtoQforallpointsx∈Q. Fromdynamicalperspective, the most important quantity in the tangent bundle concept isthesmoothmapv: Q→T Q, which is an inverse to the projectionπ,i.e, π ◦v= IdQ,π(v(x)) = x. It is called the velocity vector–field. Its graph (x, v(x)) represents the cross–section of the tangent bundle T Q. This explains the dynamicaltermvelocityphase–space, giventothetangentbundleT Qofthemanifold Q. The tangent bundle is where tangent vectors live, and is itself a smooth manifold. Vector–fieldsarecross-sections of the tangent bundle. Robot’sLagrangian (energy function)isanaturalenergyfunctiononthetangentbundleT Q.1

On the other hand, human joints are more flexiblethan robot joints. Namely, every rotation in all synovial human joints is followed by the corresponding micro–translation, which occurs after the rotational amplitude is reached [II 08c]. So, representationofhumanmotionisrigorouslydefinedintermsofEuclidean SE(3)–groups of full rigid–body motion[II 06a, II 06c, II 07e]in all main human joints (see Figure 22.5). Therefore, the configuration manifold Q for human dynamics i sdefinedasatopologicalproductofallincludedconstrained SE(3)– groups,2Q= iSE(3)i. Consequently, thenaturalstageforautonomousLagrangian dynamics of human motion is the tangent bundle T Q (and forthecorrespondingautonomousHamiltoniandynamicsisthecotangentbundleT∗Q).

Therefore, the configuration manifoldQ for human musculo-skeletal dynamics i sdefinedasaCartesianproductofallincludedconstrainedSE(3)groups, Q= jSE(3)jwherejlabelstheactivejoints. The configurationmanifoldQ is coordinated bylocaljointcoordinatesxi(t), i=1, ..., n=total number of activeDOF. The corresponding joint velocities˙xi(t)live in the velocity phase spaceT Q, which is the tangent bundle of the configurationmanifoldQ.

n−manifoldQisthevectorbundleT∗Q=(TQ)∗, the(real)dual of the tangent bundle TQ. The cotangent bundle is where 1– forms live, and is itself a smooth manifold. Covector–fields(1–forms)are cross-sectionsofthecotangentbundle. Robot’s Hamiltonianisanaturalenergyfunctiononthecotangentbundle. 2 Briefly, theEuclideanSE(3)–group i sdefined as a semi direct(noncommutative) product (denoted by ▷) of3D rotations and 3D translations: SE(3) := SO(3)▷R 3. Its most important sub groups are the following(for technical details see[II 06c,

Part 9 · pp. 245–274

Fig.22.2. The configurationmanifoldQofthehumanmusculo-skeletal dynamics i sdefinedasatopologicalproductofconstrainedSE(3)groups acting in all major (synovial)human joints, Q = iSE(3)i.

where gij(x) is the material metric tensor defined by the biomechanical system’s mass-inertia matrix and dxiaredifferentialsofthelocaljoint coordinates xion Q. Besides giving the local distances betweenthepointsonthemanifoldQ, the Riemann i an metric form⟨g⟩defines the system’skineticenergy:

giving the Lagrangian equations of the conservative skeleton motion with kinetic minus-potentialenergyLagrangian L = T−V , with the corresponding geodesic form[II 06c]

2 × 2 −rotationalmatrices includedinSE(2) −group where subscripts denote partial derivatives, while Γi jkare the Chris toffel symbols of the affineLevi-CivitaconnectionofthebiomechanicalmanifoldQ.

This is the basic geometrical structure for autonomous Lagrangianbiomechanics. In the next sections will extend this basic structure to embrace the time-dependent Lagrangian biomechanics.

While standard autonomous Lag ran gian biomechanics is developed on the configurationmanifoldX, the time–dependent biomechanics necessarily includes also therealtimeaxisR, so we have an extended configurationmanifoldR × X. Slightly more generally, the fundamental geometrical structure is the so-called configurationbundleπ: X→R. Time-dependent biomechanics is thus formally developed either on the extended configurationmanifoldR × X, or on the configurationbundleπ:X→R, usingtheconceptofjets, which are based on the idea of higher–ordertangency, or higher–order contact, at some designated point (i.e., certain joint)on a biomechanical configurationmanifoldX.

In general, tangent and cotangent bundles, T Mand T∗M, of a smooth manifold M, are special cases of a more general geometrical object called fibre bundle, denotedπ: Y→X, where the wordfiber Vofamapπ:Y→Xisthepreimageπ−1(x)ofanelementx ∈X. It is a space which locally looks like a product of two spaces(similarlyasamanifoldlocallylookslikeEuclideanspace), but may possess adifferentglobalstructure. To get a visual intuition behind this fundamental geometrical concept, we can say that afibrebundleYisahomeomorphicgeneralizationofaproductspaceX× V(see Figure 22.6), whereX and Varecalledthebaseand thefibre, respectively.π: Y→Xiscalled the projection, Yx= π−1(x)denotes afibreoverapointxofthebaseX, while the mapf= π−1:X→Ydefines the cross–section, producing the graph(x, f(x)) inthebundleY(e.g., incaseofatangentbundle, f=˙xrepresentsavelocity vector–field).

Fig.22.3. A sketch of a locally trivialfibrebundleY≈X× Vasageneralization ofaproductspaceX× V ; left–main components; right–a few details(see text for explanation).

More generally, a biomechanical configuration bundle, π: Y−→X, is a locally trivialfibred(or, projection)manifoldoverthebaseX. It is endowed with an atlas offibredbundlecoordinates(xλ, yi), where(xλ)arecoordinatesofX.

All dynamical objects in time–dependent biomechanics(including vectors, tensors, differentialformsandgaugepotentials)are cross–sections of biomechanical bundles, representing generalizations of graphs of continuous functions.

An exteriordifferentialformα of order p (or, a p−formα) on a base manifold p ∧T∗X−→X[II 07e]. It has the following expression inlocalcoordinatesonX where summation is performed over all ordered collections (λ1, ..., λp).Ωp(X) is thevectorspaceofp−formsonabiomechanicalmanifoldX. In particular, the 1– formsarecalledthePfaffian forms. The contraction⌋of any vector-fieldu =uμ∂μandap−formα = αλ1...λpdxλ1∧· · · ∧dxλponabiomechanicalmanifoldXisgiveninlocalcoordinatesonXby

The LiederivativeLuα of p−form α along a vector-field u is defined by Cartan’s ‘magic’formula(see[II06c,II07e]):

Clearly, a linear connection¯Γis a special case of an affineconnectionΓ.

A pair of smooth manifoldmaps, f 1, f 2: M→N(see Figure 22.7), are said to bek−tangent(ortangentoforderk, orhaveakthordercontact)atapointx onadomainmanifoldM, denotedbyf 1∼f 2,iff f 1(x)=f 2(x) called 0 −tangent,

∂xf1(x)=∂xf2(x), called 1 −tangent, ∂xxf1(x)=∂xxf2(x), called 2 −tangent,

Inthiswaydefinedk−tangencyisanequivalencerelation.

Fig.22.4. An intuitive geometricalpicturebehindthek−jetconcept, based on the idea of a higher–ordertangency(or, higher–order contact).

A k−jet(or, a jetoforderk), denoted by jk xf, of a smooth map f: Q →Nat apointx ∈Q(see Figure 22.7), i sdefinedasanequivalenceclassofk−tangent mapsatx,

Forexample, considerasimplefunctionf: X→Y, x→y=f(x), mapping theX−axis into theY −axisinR 2. Atachosenpointx ∈Xwehave: a 0− jetisagraph:(x, f(x)); a 1− jet is a triple:(x, f(x), f′(x)); a 2− jetisaquadruple:(x, f(x), f′(x), f′′(x)), and soon, up to the or derk(wheref′(x) =df(x) xf: X→Yiscalledthek−jetmanifoldJk(X, Y ). Formally, givenabiomechanicalbundleY−→X, itsfirstorderjetmanifold J1 Ycomprisesthesetofequivalenceclassesj1 xs, x ∈X, ofsectionss : X−→Y so that sections sands′belong to the same class iff

In particular, given the biomechanical configuration bundle Q →R over the time axisR, the 1− jetspaceJ 1(R, Q)isthesetofequivalenceclassesj1 tsofsections si: R→QoftheconfigurationbundleQ→R, which are i dent ifiedbytheir valuessi(t), as well as by the values of their partial derivatives∂tsi=∂tsi(t) attimepointst∈R. The 1– jetmanifoldJ 1(R, Q)is coordinated by(t, qi,˙qi), that is by(time, coordinates and velocities)at every active human joint, so the 1– jetsarelocaljointcoordinatemaps ts : R →Q, t→(t, qi,˙qi).

TherepeatedjetmanifoldJ1J1 Yisdefinedtobethejetmanifoldofthebundle J1Y−→X. It is endowed with the adapted coordinates(xλ, yi, yi μλ.It has the local coordinates(xλ, yi, yi λ.

xsareidentifiedbytheirvaluesandthe values of theirfirstandsecondorderderivativesatthepointx ∈X.

tsof sectionssi: R →Qoftheconfigurationbundleπ: Q →R, which are i dent ified by their values si(t), as well as the values of their first and second partial derivatives,∂tsi= ∂tsi(t)and∂tts i= ∂tts i(t), respectively, attimepointst ∈R. The 2– jetmanifoldJ 2(R, Q)is coordinated by(t, qi,˙qi, ¨qi), that is by(time, coordinates, velocities and accelerations) at every active human joint, so the 2– jets are localjointcoordinatemaps:

ts : R →Q, t→(t, qi,˙qi, ¨qi).

The general form of time-dependent Lagrangianbiomechanicswithtimedependent LagrangianfunctionL(t; qi;˙qi)definedonthejetspaceX = J 1(R, Q)∼=R × T Q, with local can on ical coordinates:(t; qi;˙qi)=(time, coordinates and velocities)inactive local joints, can be formulated as[II 06c, II 07e] dtL ˙qi−Lqi= Fi (t, q,˙q) , (i = 1, ..., n), (22.3) wherethecoordinateandvelocitypartialderivativesoftheLagrangianarerespectivelydenotedbyLqiandL ˙qi.

μλ. There is 1–1 correspondence between the affineconnectionsΓon the affine jetbundleJ 1(R, Q)→Qand the linear connections Konthetangentbundle T Q →QoftheautonomousbiomechanicalmanifoldQ. This correspondence is given by the relation α.

Anyquadratic biomechanical equation (22.4) is equivalent tothegeodesic equation[II 07e] jk.

Conversely, any linear connection Kon the tangent bundle T Q →Qdefines the quadraticdynamicalequation written with respect to a given reference frame(t, qi) ≡qμ(see[II 07e]for technical details).

The right–hand side terms Fi(t, q,˙q) of (22.3) denote any type of external torques and forces, including excitation and contraction dynamics of muscular–actuators androtationaldynamicsofhybridrobotactuators, as well as(nonlinear)dissipative joint torques and forces and external stochastic perturbation torques and forces. In particular, we have[II 06a, II 06b]):

1. Synovial joint mechanics, giving thefirststabilizingeffecttotheconservative skeleton dynamics, is described by the(q,˙q)–form of the Rayleigh–Van der Pol’sdissipationfunction which are linear in˙qiandquadraticinqi.

2. Muscular mechanics, givingthedrivingtorquesandforcesFmuscle i = Fmuscle i (t, q,˙q)with(i=1, . . . , n)for human biomechanics, describestheinternalexcitationandcontractiondynamicsofequivalentmuscularactuators [Hat 78].

i denote the maximal isometric muscular torques and forces, whileτ i denote the associated time characteristics of particular muscular actuators. This relation represents a solution of the Wilkie’s muscular active–state element equation[Wil 56] whereμ = μ(t)represents the active state of the muscle,Γdenotes the element gain, A corresponds to the maximum tension the element can develop, andS= S(r)is the‘desired’active state as a function of the motor unit stimulus rate r. This is the basis for biomechanical force controller.

whereaiandbidenotetheHill’spar a meters, corresponding to the energy dissipated during the contraction and the phosphagenicenergyconversion rate, respectively, whileδijistheKronecker’sδ−tensor.

i .

Now, for the purpose of bio medical engineering and rehabilitation, human biomechanics has developed the so–called hybrid rotational actu at or. It includes, along withmuscularandviscousforces, theD.C. motor drives, asusedinrobotics [II 06a] k = ik(t) −Jk¨qk(t) −Bk ˙qk(t), with lkik(t) + Rkik(t) + Ck ˙qk(t) = uk(t), wherek=1, . . . , n, ik(t)anduk(t)denotecurrentsandvoltagesintherotors of the drives, Rk, lkand Ckareresistances, in duc tance sand capacitances inthe rotors, respectively, while Jkand Bkcorrespond to inertia moments and viscous damping soft he drives, respectively.

whereBij[q(t), t]representscontinuousstochasticdiffus i onfluctuations, and Wj(t)is anN−variable Wienerprocess(i.e., generalized Brown i an motion), with dWj(t) = Wj(t + dt) −Wj(t), (forj= 1, . . . , N).

As illustrated in the introduction, the mass-inertia matrix of human body, defining the Riemann i an metric tensor gij(q) need not be time-constant, as in case of fast gymnastic movements and pirouettes in ice skating, which are based on quick variations of inertia moments and products constituting the material metric tensorgij(q). In particular, inthegeodesicframework(22.10), the(in)stability of the biomechanical joint and center-of-mass trajectories is the(in)stability of the geodesics, and it is completely determined by the curvature properties of the underlyingmanifoldaccordingtotheJacobiequationofgeodesicdeviation [II 06c, II 07e] jkmarethecomponentsoftheRiemanncurvaturetensor. In general, thebiomechanicalmetrictensorgijisbothtimeandjointdependent, gij= gij(t, q). This time-dependent Riemann i an geometry can be formalized in terms of the Ricciflow[II 07e], the nonlinear heat–likeevolutionmetric equation:

for a time–dependent Riemannianmetricg=gij(t)onasmoothn−manifold QwiththeRiccicurvaturetensorRij. This equation roughly says that we can deform any metric on the configuration manifoldQbythenegative of its curvature; after normalization, the final state of such deformation will be a metric with constant curvature. The negative sign in(22.5)insures akind of global volume exponentialdecay,3sincetheRicciflowequation(22.5)is akind of nonlinear geometricgeneralizationofthestandardlinearheatequation

In a suitable local coordinate system, the Ricciflow equation (22.5) on a biomechanical configurationmanifoldQhasanonlinearheat–type form, as follows. Atanytimet, wecanchooselocalharmoniccoordinatessothatthecoordinate functionsarelocallydefinedharmonicfunctionsinthemetricg(t). Then the Ricciflowtakesthegeneralform[II 07e]

∂tgij=ΔQgij+ Gij(g, ∂g), where (22.6) is the Laplace–Belt ram i operator of the configuration manifold Q and Gij(g, ∂g) is a lower–order term quadratic ing and itsfirstorderpartialderivatives∂g. From the analysis of nonlinear heat PDEs, one obtains existence and uniqueness of forward–timesolutionstotheRicciflowonsometimeinterval, starting at anysmoothinitialmetricg0 onQ.

The exponentially-decaying geometrical diffus i on (22.6) is a formal description for pirouette sin ice skating and fast rotational movements in gymnastics.

Inthissectionwepresentourrecentworkonetmethodsinmoderntimedependent Hamilton i an biomechanics[Iva 10]. This work is a sequel to our previous paper on time-dependent Lagrangianbiomechanics[Iva 09]. Inthe Hamilton i an paper we have used thecovariant force law in conjunction with the for a positive function f(q). We can get some insight into its solution from the simple exponentialdecayODE,

˙q= −λq with the solution q(t) = q0e−λt, whereq=q(t)is the observe dquantitywithitsinitialvalueq0 andλis a positive decayconstant.

Fig.22.5. Biomechanical configuration manifold for a humanoid robot(a) and human (loco)motion system (b), both defined as anthropomorphic product trees of constrained motion symmetry groups, SO(3) and SE(3), respectively. Each humanoid joint consists of a pair of coupled segmentswithonlyEulerianrotationaldegreesoffreedom, defined asaconstrainedSO(3)rotational group. On the other hand, in each human(synovial) joint, besidesgrossEulerianrotationalmovements, we also have some hidden and restrictedtranslationsalongEulerian/Cartesian(X, Y, Z)−axes. Forexample, inthe knee joint, patella(kneecap)moves for about 7–10 cmfrommaximalextensionto maximalflexion). Even greater are translational amplitudes in the shoulder joint. In other words, within the realm of rigid body mechanics, a segment of a human arm or leg is not properly represented as a rigid bodyfixedatacertainpoint, but rather as a rigid body hanging on rope–like ligaments. More generally, the whole skeleton mechanically represents a system offlexiblycoupledrigidbodies, eachofthemdefined bytheEuclideangroupof(rotational+translational)motionsinR 3( see text below for explanation). This implies more complex kinematics, dynamics and control then in the case of humanoid robots(see[II 08c]for technical details).

modern geometric formalism of jet manifolds to develop general Hamilton i an approach to time-dependent human biomechanics in which total mechanical energy is not conserved.

Before we get into spec ifictechniquesof(time-dependent)Hamilton i an biomechanics, weremarkherethethreeessentialdifferences between a human (loco)motion system and a humanoid robot (or a mechanical multi-body system in general):

1. Human joints are moreflexiblethanrobotjoints(see Figure 22.5); 2. Human(or animal)muscularactuatorsaremorecomplexthenanyrobot actuators; theyhavetheirownexcitation-contraction dynamics(defined in section 22.2.3 below); 3. In a human(loco)motion system, total mechanical energy is not conserved but dissipated.

Autonomous Hamilton i an biomechanics(aswellasautonomousLagrangian biomechanics), based on the postulate of conservation of the total mechanical energy, can bederivedfromthecovariantforce law[II 06a, II 06b, II 06c, II 07e], which in‘plain English’states:

Force 1– form = Mass distribution × Acceleration vector-field, i(t, q,˙q),(i=1, ..., n)denotes any type of torques and forces acting on a human skeleton, includingexcitation and contraction dynamics of muscular–actuators[Hil 38, Wil 56, Hat 78] androtationaldynamicsofhybridrobotactuators, as well as(nonlinear)dissipative jointtorquesandforcesandexternalstochasticperturbationtorquesandforces [II 07a]. mijisthematerial(mass–inertia)metrictensor, which gives the total mass distribution of the human body, by including all segmental masses and theirindividualinertiatensors. a jis the total acceleration vector-field, including all segmental vector-fields, defined as the absolute(Bianchi)derivative˙¯viof all the segment alangularandlinearvelocitiesvi=˙xi,(i = 1, ..., n), wherenis the total number of active degrees of freedom (DOF) with local coordinates (xi).

More formally, this central Law of biomechanics represents the covariantforce functorF∗definedbythecommutativediagram:

Here, M≡Mn={xi,(i=1, ..., n)}is the biomechanical configuration n−manifold (Figure 22.2), that is the set of all active DOF of the biomechanical system under consideration (in general, human skeleton), with local coordinates (xi).

The right-hand branch of the fundamental covariant force funct or F∗: T T∗M −→T T Mdepictedin(22.8)is Lagrangian dynamicswithitsRiemanniangeometry. Toeachn−dimensional(nD)smoothmanifoldMthereisassociatedits 2nD velocity phase-spacemanifold, denoted byT Mand called the tangent bundleofM. The original configurationmanifoldMiscalledthebaseofT M. There is an onto mapπ:T M→M, called the projection. Above each point x∈MthereisatangentspaceTxM=π−1(x)to Matx, which is called a fibre. ThefibreTxM⊂T Mis the subset ofT M, such that the total tangent bundle, T M= allpointsx∈M. Fromdynamicalperspective, the most important quantity inthetangentbundleconceptisthesmoothmapv: M→T M, which is an inverse to the projectionπ,i.e,π ◦v=IdM,π(v(x))=x. It is called the velocity vector-fieldvi=˙xi.4Itsgraph(x, v(x))represents the cross–section of thetangentbundleT M. Velocity vector-fieldsarecross-sections of the tangent bundle. Biomechanical Lagrangian(that is, kinetic minus potential energy)isa natural energy function on the tangent bundle T M. Thetangent bundle is itself a smooth manifold. It has its own tangent bundle, T T M. Cross-sections of the secondtangentbundleT T Mare the acceleration vector-fields.

The left-hand branch of the fun damentalcovariantforcefunctorF∗: T T∗M −→T T Mdepictedin(22.8)is Hamiltoniandynamicswithitssymplecticgeometry. IttakesplaceinthecotangentbundleT∗Mrob, defined as follows. A dual notiontothetangentspaceTxMtoasmoothmanifoldMatapointx=(xi) withlocalisitscotangentspaceT∗ xMatthesamepointx. Similarly to the tangentbundleT M, foranysmoothnDmanifoldM, there is associated its 2 nDmomentumphase-spacemanifold, denotedbyT∗Mandcalledthecotangentbundle.T∗MisthedisjointunionofallitscotangentspacesT∗ n−manifoldMisthevectorbundleT∗M=(T M)∗, the(real)dualofthetangentbundleT M. Momentum 1– forms(or, covector-fields)piarecross-sections of the cotangent bundle. Biomechanical Hamilton i an(that is, kinetic plus potential energy) is a natural energy function on the cotangent bundle. The cotangent bundleT∗Mis itself a smooth manifold. It has its own tangent bundle, T T∗M. Cross-sections of the mixed-second bundle T T∗Mare the force 1– forms Fi=˙pi.

T M∋(xi, vi)→(xi, pi) ∈T∗M.

It is called the Legendretransformation, orfiberderivative(for details see, e.g. [II 06c, II 07e]).

The fundamental covariant force funct or F∗: T T∗M−→T T Mstates that the force 1– formFi=˙pi, definedonthemixedtangent–cotangentbundleT T∗M, causes the acceleration vector-fieldai=˙¯vi, definedonthesecondtangentbundleT T MoftheconfigurationmanifoldM. The corresponding con tra variant accelerationfunctorisdefined as its inverse map, F∗: T T M−→T T∗M.

4 This explains the dynamical term velocity phase–space, given to the tangent bundle TMofthemanifoldM.

wherejlabelstheactivejoints. The configurationmanifoldMiscoordinated bylocaljointcoordinatesxi(t), i = 1, ..., n =totalnumberofactiveDOF. The corresponding joint velocities˙xi(t) live in thevelocityphasespaceT M, which is the tangent bundle of the configurationmanifoldM.

where gij= gij(m, x) is the material metric tensor defined by the biomechanical system’smass-inertiamatrixanddxiaredifferentialsofthelocaljointcoordinatesxionM. Besides giving the local distances between the points on the manifold M, the Riemann i an metric form ⟨g⟩defines the system’s kinetic energy:

where subscripts denote partial derivatives, while Γi jkare the Chris toffel symbols of the affineLevi-CivitaconnectionofthebiomechanicalmanifoldM.

The corresponding momentum phase-spaceP =T∗Mprovidesanaturalsymplecticstructurethatcanbedefined as follows. As the biomechanical configurationspaceMisasmoothn−manifold, we can pick local coordinates{dx 1, ..., dxn}∈M. Then{dx 1, ..., dxn}definesabasisofthecotangent spaceT∗ xMasθ=pidxi, we get local coordinates {x 1, ..., xn, p 1, ..., pn} on T∗M. We can now define the can on ical symplectic form ωonP= T∗Mas:

This 2− form ωis obviously independent of the choice of coordinates {x 1, ..., xn} and independent of the base point{x 1, ..., xn, p 1, ..., pn} ∈T∗ where summation is performed over all ordered collections(λ1, ..., λp).Ωp(X)is the vectorspaceofp−formsonabiomechanicalmanifoldX. In particular, the 1– forms arecalledthePfaffian forms. 6 The can on ical 1− formθonT ∗Mis the unique 1− form with the property that, for any 1− formβwhich is a section ofT∗Mwehaveβ∗θ= θ.

If(P, ω)isa2 nDsymplecticmanifoldthenabouteach pointx ∈Pthere are local coordinates {x 1, ..., xn, p 1, ..., pn} such that ω= dpi∧dxi. These coordinates arecalledcanonicalorsymplectic. By the Darbouxtheorem,ωis constantin this local chart, i.e., dω= 0.

We develop autonomous Hamilton i an biomechanics on the configuration biomechanical manifold Min three steps, following the standard symplecticgeometry prescription(see[II 06a, II 06c, II 07e, II 08b]):

Step A. Findasymplecticmomentumphase–space(P, ω).

Asymplecticstructureonasmoothmanifold Mis an on degenerate closed 2− formωonM, i.e., foreachx∈M,ω(x)is non degenerate, anddω=0. ThecotangentbundleP=T∗Mis our momentum phase–space. On Pthere isanondegeneratesymplectic 2− formωi sdefined in local joint coordinates xi, pi∈U, Uopen in P, as ω= dxi∧dpi. In that case the coordinates xi, pi∈U are called can on ical. In a usual procedure the can on ical 1− form θis first defined asθ=pidxi, and then the can on ical 2– formωi sdefined asω=−dθ.A symplecticphase–space manifold is a pair(P, ω).

StepB. Finda Hamilton i an vector-fieldXHon(P, ω).

Let(P, ω)beasymplecticmanifold. A vector-fieldX: P→T Piscalled HamiltonianifthereisasmoothfunctionF: P−→RsuchthatiXω=dF (iXωdenotes the interior product or contraction of the vector-fieldXandthe 2– formω).Xislocally Hamilton i an i fiXωis closed. Let the smooth real–valued HamiltonianfunctionH: P→R, representing the total biomechanical energy H(x, p)=T (p) + V (x) (Tand Vdenote kinetic and potential energy of the system, respectively), be given in local can on ical coordinates xi, pi∈U, Uopen in P. The Hamilton i an vector-fieldXH, condition by iXHω=dH, is actually defined via symplectic matrix J, in a local chart U, as where I denotes then × nid entity matrix and∇is the gradient operator.

StepC. Finda Hamilton i an phase–flowφtofXH.

Let(P, ω)beasymplecticphase–spacemanifoldandXH=J∇HaH a milton i an vector-fieldcorrespondingtoasmoothreal–valued HamiltonianfunctionH: P→R, on it. If a unique one–parametergroupofdiffeomorphisms φt: P→Pexistsso that d dt|t=0 φtx=J∇H(x), itiscalledtheHamiltonian phase–flow.

Letf: M→Mbeadiffeomorphism.ThenT∗fpreservesthecanonical1−form θonT∗M,i.e.,(T∗f)∗θ= θ.ThusT∗fissymplecticdiffeomorphism.

on(P, ω)represents an integral curve of the Hamilton i an vector-fieldXH=J∇H, if in the local can on ical coordinates xi, pi∈U, UopeninP, Hamiltoniancanonicalequationshold:

An integral curve is said to be maximal ifitis not a restriction of an integral curvedefinedonalargerintervalofR. It follows from the standard theorem on theexistenceanduniquenessofthesolutionofasystemofODEswithsmooth r.h.s, that if the manifold (P, ω) is Hausdorff, then for any point x = (xi, pi) ∈U, UopeninP, thereexistsamaximalintegralcurveofXH=J∇H, passing for t=0, throughpointx. IncaseXHiscomplete, i.e., XHisCpand(P, ω) is compact, themaximalintegralcurveofXHistheHamiltonianphase–flow φt: U→U.

The phase–flow φtis symplecticif ωis constant along φt, i.e., φ∗ tωdenotes the pull–back ofωbyφt), if and only if(‘iff’for short)LXHω= 0 (whereLXHωdenotestheLiederivativeofωuponXH). Symplecticphase– flowφtconsistsofcanonicaltrans formations on(P, ω),i.e., diffeomorphismsin can on ical coordinates xi, pi∈U, Uopen on all (P, ω) which leave ωinvariant. In this case the Liouvilletheoremis valid: φtpreservesthe phase volume on (P, ω).

includingcontravariantequation(22.13)–the velocity vector-field, an dco variant equation(22.14)–the force 1– form(field), together with initial joint angles and momenta(22.15). HereR=R(x, p)denotestheRaileighnonlinear(biquadratic) dissipation function, and Fi= Fi(t, x, p) are covariant driving torques ofequivalentmuscularactuators, resemblingmuscularexcitationandcontractiondynamicsinrotational form. The velocity vector-field (22.13)and the force 1− form (22.14) together define the generalized Hamilton i an vector-field XH; the HamiltonianenergyfunctionH= H(x, p)is its generating function.

is Hausdorff.Therefore, forx=(xi, pi)∈Up, where Up is an open coordinate chart inT∗M, there exists a unique one–parametergroupofdiffeomorphisms φt: T∗M→T∗M, thatistheautonomousHamiltonianphase–flow:

φt:T∗M→T∗M: (p(0), x(0))→(p(t), x(t)), (22.16) (φt◦φs=φt+s, φ0=identity),

In this sub section we develop time-dependent Hamilton i an biomechanics. For this, wefirstneedtoextendourautonomousHamiltonianmachinery, using the general concepts of bundles, jetsandconnections.

While standard autonomous Lag ran gian biomechanics is developed on the configurationmanifoldX, the time–dependent biomechanics necessarily includes also therealtimeaxisR, so we have an extended configurationmanifoldR × X. Slightly more generally, the fundamental geometrical structure is the so-called configurationbundleπ: X→R. Time-dependent biomechanics is thus formally developed either on the extended configurationmanifoldR × X, or on the configurationbundleπ:X→R, usingtheconceptofjets, which are based on the idea of higher–ordertangency, or higher–order contact, at some designated point(i.e., certain anatomical joint)on a biomechanical configurationmanifold X(see[Iva 09, Iva 10]).

In general, tangent and cotangent bundles, T Mand T∗M, of a smooth manifold M, are special cases of a more general geometrical object called fibre bundle, denotedπ: Y→X, where the wordfiber Vofamapπ:Y→Xisthepreimageπ−1(x)ofanelementx ∈X. It is a space which locally looks like a product of two spaces(similarlyasamanifoldlocallylookslikeEuclideanspace), but may possess adifferentglobalstructure. To get a visual intuition behind this fundamental geometrical concept, we can say that afibrebundleYisahomeomorphicgeneralizationofaproductspaceX× V(see Figure 22.6), whereX and Varecalledthebaseand thefibre, respectively.π: Y→Xiscalled the projection, Yx= π−1(x)denotes afibreoverapointxofthebaseX, while the mapf= π−1:X→Ydefines the cross–section, producing the graph(x, f(x)) inthebundleY(e.g., incaseofatangentbundle, f=˙xrepresentsavelocity vector–field).

Fig.22.6. A sketch of a locally trivialfibrebundleY≈X× Vasageneralization ofaproductspaceX× V ; left–main components; right–a few details(see text for explanation).

More generally, a biomechanical configuration bundle, π: Y−→X, is a locally trivialfibred(or, projection)manifoldoverthebaseX. It is endowed with an atlas offibredbundlecoordinates(xλ, yi), where(xλ)arecoordinatesofX.

Now, a pair of smooth manifold maps, f 1, f 2: M→N(see Figure 22.7), are said to bek−tangent(ortangentoforderk, orhaveakthordercontact)ata pointxonadomainmanifoldM, denotedbyf 1∼f 2,iff f 1(x)=f 2(x) called 0 −tangent, ∂xf 1(x)=∂xf 2(x), called 1 −tangent,

∂xxf 1(x)=∂xxf 2(x), called 2 −tangent, ... etc. to the or derk

Inthiswaydefinedk−tangencyisanequivalencerelation.

Fig.22.7. An intuitive geometricalpicturebehindthek−jetconcept, based on the idea of a higher–ordertangency(or, higher–order contact).

A k−jet(or, a jet of or derk), denoted by jk xf, of a smooth map f: M→Nat a point x ∈M(see Figure 22.7), i sdefined as an equivalenceclassof k−tangent mapsatx,

Forexample, considerasimplefunctionf: X→Y, x→y=f(x), mapping theX−axis into theY −axisinR 2. Atachosenpointx ∈Xwehave:

a 0− jetisagraph:(x, f(x)); a 1− jet is a triple:(x, f(x), f′(x)); a 2− jetisaquadruple:(x, f(x), f′(x), f′′(x)), and soon, up to the or derk(wheref′(x) =df(x) dx, etc).

xf: X→Yiscalledthek−jetmanifoldJk(X, Y ). Formally, givenabiomechanicalbundleY−→X, itsfirst–orderjetmanifold J1 Ycomprisesthesetofequivalenceclassesj1 xs, x ∈X, ofsectionss : X−→Y so that sections sands′belong to the same class iff tsof sections si: R →Mof the configuration bundle M→R, which are i dent ified by their values si(t), as well as by the values of their partial derivatives ∂tsi= ∂tsi(t) attimepointst∈R. The 1– jetmanifoldJ 1(R, M)is coordinated by(t, xi,˙xi), that is by(time, coordinates and velocities)at every active human joint, so the 1– jetsarelocaljointcoordinatemaps ts : R →M, t→(t, xi,˙xi).

μλ.It has the local coordinates(xλ, yi, yi λ.

xsareidentifiedbytheirvaluesandthe values of theirfirst and second–orderderivativesatthepointx ∈X.

ts of section ssi: R→Moftheconfigurationbundleπ:M→R, which are i dent ifiedbytheirvaluessi(t), as well as the values of theirfirst and second partial derivatives, ∂tsi= ∂tsi(t) and ∂tts i= ∂tts i(t), respectively, at time points t∈R. The 2– jetmanifoldJ 2(R, M)is coordinated by(t, xi,˙xi, ¨xi), that is by (time, coordinates, velocities and accelerations)at every active human joint, so the 2– jetsarelocaljointcoordinatemaps:

ts : R →M, t→(t, xi,˙xi, ¨xi).

We can now formulate the time-dependentbiomechanicsinwhichthebiomechanicalphasespaceistheLegendremanifoldΠ(see[II 06c, II 07e]), endowed withtheholonomiccoordinates(t, yi, pi)with the transition functions

Note that every connectionΓ=dt ⊗(∂t+ Γi∂i)onthebundleY−→Xgives risetotheHamiltonianconnection#ΓonΠ−→X, given by

HΓ= pidyi−piΓidt.

H= pidyi−Hdt = pidyi−piΓidt −# HΓ dt. (22.17)

We call H and# H in the decomposition (22.17) the Hamilton i an and the Hamilton i an function respectively. Letγbea Hamilton i an connection on Π−→Xassociatedwith the Hamilton i an form(22.17). Its at isfiest he relations[II 06c, II 07e]

Fromequations(22.18)we see that, in the case of biomechanics, one and only one Hamilton i an connection is associated with a given Hamilton i an form.

dt(f◦s).

Thedissipative Hamilton i an system(22.20)–(22.21)is the basis for our time & fitness-dependent biomechanics. Thescalarfunctionfin(22.21)onthe biomechanical Legendrephase-spacemanifoldΠis now interpretedasanindividualneuro-muscularfitnessfunction. Thisfitnessfunctionisa‘determinant’for the performance of muscular drives for the driven, dissipativeHamiltonianbiomechanics. These muscular drives, forallactiveDOF, are given by time & fitness-dependent Pfaffian form: Fi= Fi(t, y, p, f). In this way, we obtain ourfinal model for time & fitness-dependent Hamilton i an biomechanics:

Physiologically, theactivemusculardrivesFi=Fi(t, y, p, f)consist of[II 06a, II 06b]):

which are linear in˙yiandquadraticinyi.

2. Muscular mechanics, giving the driving torques and forces Fmusc i (t, y,˙y)with(i=1, . . . , n)for human biomechanics, describes the intern alex citation and contraction dynamics of equivalent muscular actuators[Hat 78].

(a) Theexcitationdynamicscan be described by an impulse force–time relation i denote the maximal isometric muscular torques and forces, whileτ i denote the associated time characteristics of particular muscular actuators. This relation represents a solution of the Wilkie’s muscular active–state element equation[Wil 56] whereμ = μ(t)represents the active state of the muscle,Γdenotes the element gain, A corresponds to the maximum tension the element can develop, and S=S(r)is the‘desired’active state as a function of the motor unit stimulus rater. This is the basis for biomechanical force controller.

(b)The contraction dynamics has classically been described bytheHill’shyperbolicforce–velocity relation[Hil 38] whereaiandbidenotetheHill’spar a meters, corresponding to the energy dissipated during the contraction and the phosphagenicenergyconversion rate, respectively, whileδijistheKronecker’sδ−tensor.

i .

Now, for the purpose of bio medical engineering and rehabilitation, human biomechanics has developed the so–called hybrid rotational actu at or. It includes, along withmuscularandviscousforces, theD.C. motor drives, asusedinrobotics k = ik(t) −Jk¨yk(t) −Bk ˙yk(t), with lkik(t) + Rkik(t) + Ck ˙yk(t) = uk(t), wherek=1, . . . , n, ik(t)anduk(t)denotecurrentsandvoltagesintherotors of the drives, Rk, lkand Ckareresistances, in duc tance sand capacitances inthe rotors, respectively, while Jkand Bkcorrespond to inertia moments and viscous damping soft he drives, respectively.

whereBij[y(t), t]representscontinuousstochasticdiffus i onfluctuations, and Wj(t)is anN−variable Wienerprocess(i.e., generalized Brown i an motion) [II 07a], with dWj(t) = Wj(t + dt) −Wj(t), (forj= 1, . . . , n = no. ofactiveDOF).

In this section we have proposed the time-dependent Hamilton i an form of human biomechanics. Starting with the Covariant Force Law: Fi= mijajon the biomechanical configurationmanifoldM, we havefirst developed the autonomous Hamilton i an biomechanics:

onthesymplecticphasespacethatisthecotangentbundleT MofM. Then we have introduced powerful geometrical machinery consisting offibrebundles andjetmanifoldsassociatedtothebiomechanicalmanifoldM. Using the jet formalism, we derived time-dependent, dissipative, Hamilton i an equations:

for the general time-dependent human biomechanical system.

This is the time-dependent generalization of an ‘ordinary’ autonomous human biomechanics, in which total mechanical+biochemical energy is not conserved. In our view, this time-dependent energetic approach is much more realistic than the autonomous one, as in living systems energy is never conserved.

Recall(see[II 06a, II 06b, II 06c])that realistic human bio dynamics(RHB)is a science of human(andhumanoidrobot)motion in its full complexity. Itis governed bybothNewtoniandynamicsandbiologicalcontrollaws.

There are over 200 bones in the human skeleton driven by about 640 muscular actuators(see, e.g.[Mar 98,II 06b]). While the muscles generate driving torques in the moving joints,7sub cortical neural system performs both local and global (loco)motion control: first reflexly controlling contractions of individual muscles, andthenorchestratingallthemusclesintosynergeticactionsinordertoproduceefficientmovements. Whilethelocalreflexcontrolofindividualmusclesis performed on the spinal control level, the global integration of all the muscles into coordinated movements is performed within the cerebellum[II 06a, II 06b].

All hierarchical sub cortical neuro–muscular physiology, from the bottom level of asingle musclefiber, tothetoplevelofcerebellarmuscularsynergy, acts as a temporal⟨out|in⟩reaction, in such away that the higher level acts as a command/control space for the lower level, itself representing an abstract image of the lower one:

⇝excitationcontractioncoupling⇝muscle tension generating.

Its purpose is the generation of muscular forces, to be transferred into driving torques within the joint anatomical geometry. 2. At the spinal level,⟨out|in⟩is given by auto genetic–reflexstimulus–response control[Hou 79]. Here we have a neural image of all individual muscles. The main purpose of the spinal control level is to give both positive and negative feed backs to stabilize generated muscular forces within the ‘homeostatic’ (or, more appropriately,‘homeokinetic’)limits. The individual muscular actions are combined intoflexor–extensor(or agonist–antagonist)pairs, mutually controlling each other. This is the mechanism of reciprocal innervation of agonists and inhibition of antagonists. It has a purely mechanical purpose to form the so–calledequivalentmuscularactuators(EMAs), which would generatedrivingtorquesTi(t)for all movable joints. 3. Atthecerebellarlevel,⟨out|in⟩is given by sensory–motor integration [HBB 96]. Here we have an abstract edimageofallautogeneticreflexes. The mainpurposeofthecerebellarcontrollevelisintegration andfinetuningof

7 Here we need to emphasize that human joints are sign ifican tly moreflexiblethan humanoid robot joints. Namely, each humanoid joint consists of a pair of coupled segmentswithonlyEulerianrotationaldegreesoffreedom. On the other hand, in each human synovial joint, besidesgrossEulerianrotationalmovements(roll, pitch and yaw), we also have some hidden and restricted translations along(X, Y, Z)−axes. Forexample, in the knee joint, patella(kneecap)moves for about 7–10 cmfrom maximal extension to maximal flexion). It is well–known that even greater are translational amplitudes in the shoulder joint. In other words, within the realm of rigid body mechanics, a segment of a human armor leg is not properly represented as a rigid bodyfixedatacertainpoint, but rather as a rigid body hanging on rope–like ligaments. More generally, the whole skeleton mechanically represents a system of flexiblycoupledrigidbodies. This implies the more complex kinematics, dynamics and control then in the case of humanoid robots.

theactionofallactiveEMAsintoasynchronizedmovement, by supervising theindividualautogeneticreflexcircuits. At the same time, to be able to perform in new and unknown conditions, the cerebellum is continuously adapting its own neural circuitry by unsupervised (self–organizing) learning. Its action is subconscious and automatic, both in humans and in animals.

Naturally, we can ask the question: Can we assign asingle⟨out|in⟩measure to all the sen euro–muscular stimulus–response reactions? We think that we can do it; so in this Letter, we propose the concept of adaptive sensory–motor transition amplitude as a unique measure for this temporal ⟨out|in⟩relation. Conceptually, this⟨out|in⟩−amplitude can be formulated as the‘neural path integral’:

Here, the integral is taken overall activated(or,‘fired’)neuralpathwaysxi= xi(t)of the cerebellum, connecting its input sensory−state with its output motor−state, symbolically described by adaptive neural measureD[w, x], defined by the weighted product(of discrete time steps) in which thesynapticweightswi= wi(t), included in all active neural pathways xi= xi(t), are updated by the standard learning rule new value(t + 1)= old value(t)+innovation(t).

More precisely, the weight swi in(23.49)are updated according to one of the two standard neural learning schemes, in which the micro–time level is traversed in discrete steps, i.e., ift = t 0, t 1, ..., tnthent + 1 = t 1, t 2, ..., tn+1:8 wi(t + 1)=wi(t) −η∇J(t), (22.25) innovation(t) = |reward(t) −penalty(t)|.

whereηis asmall constant, called the step size, or the learning rate, and ∇J(n)denotes the gradient of the‘performancehyper–surface’atthet−th iteration.

Theoretically, equations (22.22–23.55) define an ∞−dimensional neural network [II 08c]. Practically, in a computer simulation we can use 107≤n ≤108, roughly corresponding to the number of neurons in the cerebellum[II 07b, II 07d].

The exponent termS[x]in equation(22.22)represents the auto genetic–reflex action, describingreflexly–induced motion of allactiveEMAs, from their initial stimulus−state to their final response−state, along the family of extremal (i.e., Euler–Lagrangian)pathsxi

Having, thus, defined the spinal reflex control level, we proceed to model the top subcorticalcommander/controller, the cerebellum. The cerebellum is responsible for coordinating precisely timed ⟨out|in⟩activity by integrating motor output with ongoing sensory feedback (see Figure 22.8). It receives extensive projections from sensory–motor areas of the cortex and the periphery and directs it back to premotor and motor cortex. This suggests a role in sensory–motor integration and the timing and execution of human movements. The cerebellum stores patterns of motor control for frequently performed movements, and therefore, its circuits are changed by experience and training[II 06b]. It was termed the adjustable pattern generator in the work of James Houk and collaborators [HBB 96]. Also, it has become the inspiring‘brain–model’in robotic research[II 06a].

The cerebellum is known to be involved in the production and learning of smooth coordinated movements (see [II 06b] and references therein). Two classes of inputs carry information into the cerebellum: themossyfibers(MFs)and the climbingfibers(CFs). TheMFsprovidebothplantstateandcontextual information. TheCFs, on the other hand, are thought to provide information thatreflecterrorsinrecentlygeneratedmovements. This information is used to adjust the programs encoded by the cerebellum. TheMFscarryplantstate, motorefference, and other context ual signals into the cerebellum. Thesefibers impinge on granule cells, whose axons give rise to parallel fibers (PFs). Through thecombinationofinputsfrommultipleclassesofMFsandlocalinhibitoryinterneurons, thegranule cells are thought to provide a sparse expansive encoding of the incoming state information. The large number of PFs converge on a much smallersetofPurkinjecells(PCs), whilethePCs, in turn, provide inhibitory signalstoasinglecerebellarnuclearcell. Using this principle, the Cerebellar Model Arithmetic Computer, orCMAC–neuralnetworkhasbeenbuiltandimplementedinrobotics[II 06b, II 06a], using trial-and-error learning to produce bursts of muscular activity for controlling robot arms.

Fig. 22.8. Schematic ⟨out|in⟩organization of the primary cerebellar circuit. In essence, excitatoryinputs, conveyedbycollateralaxonsofMossyandClimbingfibersactivate directlyneuronesintheDeepcerebellarnuclei. The activity of these latter is also modulatedbytheinhibitoryactionofthecerebellarcortex, mediatedbythePurkinje cells.

that artificial neural nets cannot satisfactorily model the function of this sophisticated‘super–bio–computer’, as its dimensionality is virtually infinite. Despite a lot of research dedicated to its structure and function(see[HBB 96] and references there cited), the real nature of the cerebellum still remains a‘mystery’.

The main function of the cerebellum as a motor controller is depicted in Figure 22.9. A coordinated movement is easy to recognize, but we know little about how it is achieved. In search of the neural basis of coordination, a model of spinocerebellar interactions was presented in which the structural and functional

Fig.22.9. The cerebellum as a motor controller.

organizing principle is a division of the cerebellum into discrete micro–complexes. Each micro–complex is the recipient of a spec ificmotorerrorsignal, that is, a signal that conveys information about an inappropriate movement. Thesesignalsareencodedbyspinalreflexcircuitsandconveyedtothecerebellarcortex through climbingfibreafferents. This organization reveals salient features of cerebellar information processing, butalso highlights the importance of systems level analysis for a fuller understanding of the neural mechanisms that underlie behavior.

D[w, q, p] ei Saff[q, p] (22.27) where wi= wi(t) denote the cerebellar synaptic weights positioned along its neural pathways, being continuously updated using the Hebbian–like self–organizing learning rule(22.24). Given the transition amplitude out|in(22.27), thecerebellarsensory–motor transition probability i sdefined as its absolute square, |⟨out|in⟩|2.

i (t); tin≤t≤tout, for all discrete time steps, t=1, ..., n −→∞, and we are allowing for the affineHamiltonianHaff(q, p)to depend upon all the(M≤N)EMA–angles and angular momenta collectively. Here, we actually system aticallytookadiscretizeddifferentialtimelimitoftheform tσ −tσ−1≡dτ(bothσand τdenotediscrete time steps) and wrote

(tσ−tσ−1)≡ ˙qi. For technical details regarding thepathintegralcalculationsonRiemannian andsymplecticmanifolds(includingthestandardregularization procedures).

Now, motorlearning occurring in the cerebellum can be observed usingfunctionalMRimaging, showingchangesinthecerebellaractionpotential, related to the motor tasks. To account for these electro–physiological currents, we need toaddthesourcetermJi(t)qi(t)to the affineHa milton i an action(22.26),

(thecurrentJi= Ji(t)actsasasourceJiAiofthecerebellarelectricalpotential Ai= Ai(t)), which, subsequently givesthecerebellarpathintegral with the action potential source, coming either from the motor cortex or from other sub cortical areas.

while their subsequent discretization gives the standard thermodynamic partition function, where Ejistheenergyeigenvalue corresponding to the affineHa milton i an Haff(q, p), Tis the temperature–likeenvironmentalcontrolparameter, and the sum runs over allenergyeigenstates(labelledbytheindexj). From(23.54), we can further calculate all statistical and thermodynamic system properties, as for example, transitionentropyS= kB ln Z, etc.

Our cerebellar path integral controller is closely related to entropic motor control [HN 08a, HN 08b], which deals with neuro-physiological feedback information and environmental uncertainty. The probabilistic nature of human motor action can be characterized by entropies at the level of the organism, task, and environment. Systematic changes in motor adaptation are characterized as task–organism and environment–organism tradeoffs in entropy. Such compensatory adaptations lead to a view of goal–directed motor control as the product of an underlying conservation of entropy across the task–organism–environment system. In particular, an experiment conducted in[HN 08b]examined the changes in entropy of the coordinationofisometricforceoutputunderdifferent levels of task demands and feedback from the environment. The goal of the study was to examine the hypothesis that human motor adaptation can be characterized as a process of entropy conservation that i sreflectedinthecompensationofentropybetween the task, organism motor output, and environment. Information entropy of the coordination dynamics relative phase of the motor output was made conditional on the idealized situation of human movement, for which the goal was always achieved. Conditional entropy of the motor output decreased as the error tolerance and feedback frequency were decreased. Thus, as the likelihood of meeting the task demands was decreased increased task entropy and/or the amount of information from the environment is reduced increased environmental entropy, the subjects of this experiment employed fewer coordination patterns in the force output toachieve the goal. The conservation of entropy supports the view that context dependent adaptations in human goal–directed action are guided fundamentally by natural law and provides a novel means of examining human motor behavior. This is fundamentally related to the He is enberg uncertainty principle [II 08a]and further supports the argument for the primacy of a probabilistic approach toward the study of bio dynamic cognition systems.

The action–amplitudeformalismrepresentsakindofageneralizationofthe Haken-Kelso-Bunz(HKB)model of self-organization in the individual’smotor system[HKB 85], including: multi-stability, phase transitions and hysteresis effects, presenting a contrary view to the purely feedback driven systems. HKB usestheconceptsofsynergetics(order parameters, control parameters, instability, etc)and the mathematical tools of non linearly coupled(nonlinear)dynamical systems to account for self-organized behavior both at the cooperative, coordinative level and at the level of the individual coordinating elements. The HKBmodelstandsasabuildingblockuponwhichnumerousextensionsand elaborations have been constructed. In particular, it has been possible to derive it from a realistic model of the cortical sheet in which neural areas undergo a reorganization that is mediated by intra-and inter-cortical connections. Also, theHKBmodeldescribesphasetransitions(‘switches’)in coordinated human movement as follows: (i) when the agent begins in the anti-phase mode and speed of movement is increased, a spontaneous switch to symmetrical, in-phase movement occurs;(ii)this transition happens swiftly at a certain critical frequency; (iii)after the switch has occurred and the movement rate is now decreased the subject remains in the symmetrical mode, i.e. she does not switch back; and (iv) no such transitions occur if the subject begins with symmetrical, in-phase movements. TheHKBdynamicsoftheorderparameterrelativephaseasisgivenby a nonlinearfirst-orderODE:

˙φ = (α + 2βr 2) sin φ −βr2 sin 2φ, whereφis the phase relation(that characterizes the observed patterns of behavior, changes abruptly at the transition and is only weakly dependent on parameters outside the phase transition), ris the oscillator amplitude, while α, βare coupling parameters (from which the critical frequency where the phase transition occurs can be calculated).

Today it is well known that disembodied cognition is a myth, albeit one that has had profound influencein Western science since Rene Descartes and others gave itcredenceduringtheScientificRevolution. In fact, the mind-body separation had much more to do with explanation of method than with explanation of the mind and cognition, yet it is with respect to the latter that its impact is most widely felt. Wefindittobeanunsustainable assumption in the realm of crowd behavior. Mental intention is (almost immediately) followed by a physical action, that is, a human or animal movement[Sch 07]. In animals, this physical action would be jumping, running,flying, swimming, biting or grabbing. In humans, it can be talking, walking, driving, or kicking, etc. Mathematical description of human/animalmovementintermsofthecorrespondingneuro-musculo-skeletal equations of motion, for the purpose of prediction and control, is formulated within the realm of bio dynamics(see[II 06a, II 06b, II 06c]).

In this section we will focus on intelligence and problemsolving.

Intelligence is a property of human mind that encompasses many related mental abilities, such as the capacities to reason, plan, solve problems, think abstractly, comprehend ideas and language, and learn. According to Encyclopedia Britannica, intelligenceistheabilitytoadapteffectivelytotheenvironment, either by making a change in oneself or by changing the environment or finding a new one. Differentinvestigatorshaveemphasizeddifferent aspects of intelligence in their definitions. Forexample, ina 1921 symposiumonthedefinit i on of intelligence, the American psychologist L. Term an emphasized theabilitytothinkabstractly, whileanotherAmericanpsychologist, E. Thorndike, emphasized learning and the ability to give good responses to questions.

Part 10 · pp. 275–304

In the philosophy of arguments, reason is the ability of the human mind to form and operate on concepts in abstraction, in varied accordance with rationality and logic—terms with which reason shares heritage. Reason is thus a very importantwordinWesternintellectualhistory, to describe a type or aspect of mental thought which has traditionally been claimed as distinctly human, and not to be found elsewhere in the animal world. Discussion and debate about the nature, limits and cause sofreasoncouldalmostbesaidtodefinet he main lines of historical philosophical discussion and debate. Discussion about reason especially concerns:

(b)its ability to help people decide what is true, and (c)its origin. The concept of reason is connected to the concept of language, as reflectedin the meanings of the Greek word ‘logos’, later to be translated by Latin ‘ratio’ and then French ‘raison’, from which the English word derived. As reason, rationality, and logic are all associated with the ability of the human mind to predict effects as based upon presumed causes, the word‘reason’also denotes a ground or basis for a particular argument, andhenceisusedsynonymouslywith the word ‘cause’.

It is sometimes said that the contrast between reason and logic extends back to the time of Plato and Aristotle. Indeed, although they had no separate Greek word for logic as opposed to language and reason, Aristotle’ssyllogism(Greek ‘syllogismos’) i dent ified logic clearly for the first time as a distinct field of study: the most peculiarly reasonable(‘logikˆe’)part of reasoning, so to speak.

No philosopher of any note has ever argued that logic is the same as reason. They are generally thought to be distinct, although logic is one important aspect of reason. But the tendency to the preference for‘hard logic’, or‘solid logic’, in modern times has incorrectly led to the two terms occasionally being seen as essentially synonymous or perhaps moreoftenlogicisseenasthedefining and pure form of reason.

However machines and animals can unconsciously perform logical operations, and many animals (including humans) can unconsciously, associate different perceptions as causes and effectsand then make decisions or even plans. Therefore, to have any distinct meaning at all,‘reason’ must be the type of thinking which links language, consciousness and logic, and at this time, only humans are known to combine these things.

However, note that reasoning is defined very differently depending on the context of the understanding of reason as a form of knowledge. The logical definit i on is the act of using reason to derive a conclusion from certain premises using a given methodology, and the two most commonly used explicit methods to reach aconclusionaredeductivereasoningandinductivereasoning. However, within idea list philosophical contexts, reasoning is the mental process which informs our imagination, perceptions, thoughts, and feelings with whatever intelligibility these appear to contain; and thus links our experience with universal meaning. The spec ificsofthemethodsofreasoningareof interest to such disciplines as philosophy, logic, psychology, and artificialintelligence.

Indeductivereasoning, given true premises, the conclusion must follow and it cannot be false. In this type of reasoning, the conclusion is inherent in the premises. Deductivereasoningthereforedoesnotincreaseone’sknowledgebase and is said to be non–ampliative. Classicexamplesofdeductivereasoning are foundinsuchsyllogismsasthefollowing:

1. One must exist/live to perform the act of thinking. 2. I think. 3. Therefore, Iam.

In inductive reasoning, on the other hand, when the premises are true, then the conclusion follows with some degree of probability. This method of reasoning is ampliative, as it gives more information than what was contained in the premises themselves. A classical example comes from David Hume:

1. The sun rose in the east every morning up until now. 2. Therefore the sun will also rise in the east tomorrow.

Athirdmethodofreasoningiscalledabductivereasoning, or inference to the best explanation. This method is more complex in its structure and can involve bothinductiveanddeductivearguments. The main characteristic of abduction is that it is an attempt to favor one conclusion above other sbyeitherattemptingtofalsify alternative explanations, or showing the likelihood of the favored conclusion given a set of more or less dis put able assumptions.

A fourth method of reasoning is analogy. Reasoning by analogy goes from a particular to another particular. The conclusion of an analogy is only plausible. An a logical reasoning is very frequent in commonsense, science, philosophy and the humanities, but sometimes it is accepted only as an auxiliary method.A refined approach is case–based reasoning.

A plan represents a proposed or intended method of getting from one set of circumstances to another. They are often used to move from the present situation, towards the achievement of one or more objectives or goals.

Informalorad–hoc plans are created by individual humans in all of their pursuits. Structured and formal plans, used by multiple people, are more likely to occur in projects, diplomacy, careers, economic development, militarycampaigns, combat, or in the conduct of other business.

It is common for less formal plans to be created as abstract ideas, and remain in that form as they are maintained and put to use. More formal plans as used for business and military purposes, while initially created with and as an abstract thought, are likely to be written down, drawn up or otherwise stored in a form that is accessible to multiple people across time and space. This allows more reliable collaboration in the execution of the plan.

The term planning implies the working out of sub–components in some degree of detail. Broader–brush enunciationsofobjectivesmayqualifyasmetaphorical road–maps.

Planning literally just means the creation of a plan; it can be as simple as making a list. It has acquired a technical meaning, however, to cover the area of government legislation and regulations related to the use of resources.

The problemsolving forms part of thinking. Considered the most complex of all intellectual functions, problem solving has been defined as higher–order cognitive process that requires the modulation and control of more routine or fundamental skills. It occurs if an organism or an artificial intelligence system does not know how to proceed from a given state to a desired goal state. It is part of the larger problem process that includes problemfindingandproblemshaping.

The nature of human problem solving has been studied by psychologists over the past hundred years. There are several methods of studying problemsolving, including: introspection, behaviorism, computer simulation and experimental methods.

Beginning with the early experimental work of the Gestalt i sts in Germany. Various reasons account for the choice of simple novel tasks: they had clearly definedoptimalsolutions, theyweresolvablewithinarelativelyshorttimeframe, researchers could trace participants’problem–solving steps, and soon. The researchers made the underlying assumption, of course, that simple tasks such astheTowerofHanoicapturedthemainpropertiesof‘real world’problems, and that the cognitive processes underlying participants’attempts to solve simple problems were representative of the processes engaged in when solving‘real world’problems. Thus researchers used simple problems for reasons of convenience, and thought generalizations to more complex problems would become possible. Perhaps the best–known and most impressive example of this line of research remainstheworkbyNewellandSimon[NS 72].

See more on problemsolving below.

Learning is the process of acquiring knowledge, skills, attitudes, or values, through study, experience, or teaching, that causes a change of behavior that is persistent, measurable, and spec ifiedorallows an individual to formulate anew mental construct or revise a prior mental construct(conceptual knowledge such as attitudes or values). It is a process that depends on experience and leads to long–term changes in behavior potential. Behavior potential describes the possible behavior of an individual(not actual behavior)in a given situation in order to achieve a goal. But potential is not enough; if individual learning isnot periodically reinforced, it becomes shallower and shallower, and eventually will be lost in that individual.

Shortterm changes in behavior potential, suchas fatigue, do not constitute learning. Some long–term changes in behavior potential result from aging and development, rather than learning.

Education is the conscious attempt to promote learning in others. The primary function of ‘teaching’ is to create a safe, viable, productive learning environment. Management of the total learning environment to promote, enhance and motivate learning is a paradigm shift from a focus on teaching to a focus on learning.

The stronger the stimulation for the brain, the deeper the impression that is left in the neuronal network. Therefore a repeated, very intensive experience perceived through all of the senses(audition, sight, smell)of an individual will remainlongerandprevailoverotherexperiences. The complex interactions of neurons that have formed a network in the brain determine the direction of flowof the micro–voltage electricity thatflowsthrough the brain when a person thinks. The characteristics of the neuronal network shaped by previous impressions is what we call the person’s‘character’.

The most basic learning process is imitation, one’spersonal repetition of an observed process, suchas a smile. Thus an imitation will take one’sti me(attention to the details), space(a location for learning), skills(or practice), and other resources(forexample, a protected area). Through copying, most infants learn how to hunt (i.e., direct one’s attention), feed and perform most basic tasks necessary for survival.

The so–called Bloom’sTaxonomydivides the learning process into a six–level hierarchy, where knowledge is the lowest order of cognition and evaluation the highest[Blo 80]:

1. Knowledge is the memory of previously–learnt materials such as facts, terms, basic concepts and answers. 2. Comprehension is the understanding of facts and ideas by organization, comparison, translation, interpretation, and description. 3. Application is the use of new knowledge to solve problems. 4. Analysis is the examination and division of information into parts by identifying motives or causes. A person can analyze by making inference sand findingevidencetosupportgeneralizations. 5. Synthesis is the compilation of information in anew way by combining elements into patterns or proposing alternative solutions. 6. Evaluation is the presentation and defense of opinions by making judgments about information, validity of ideas or quality of work based on: attention,1

1 The cognitive process of selectively concentrating on one thing while ignoring other things.

Recall that a language is a system of signals, suchas voice sounds, gestures or written symbols that encode or decode information. Human spoken and written languages can be described as a system of symbols(sometimes known as lexemes)andthegrammars (rules)by which the symbols are manipulated. The word‘language’is also used to refer to common properties of languages. Human languages are usually referred to as natural languages, and the science of studying them is linguistics, with Ferdin and de Saussure and Noam Chomsky as the most influentialfigures. Besides symbols, languages also often conform to a

2 An example of non–associative learning in which there is a progressive diminution of behavioral response probability with repetition of a stimulus. 3 A changeover time in the responsiveness of the sensory system to a constant stimulus. 4 An example of non–associativelearninginwhichtheprogressiveamplificat i on of a responsefollowsrepeatedadministrationsofastimulus. 5 Learning of conditioned behavior as being formed by pairing two stimuli to condition an animal into giving a certain response. 6 Operantconditioningisdistinguished from Pavlovianconditioninginthatoperant conditioning deals with the modificat i on of voluntary behavior through the use of consequences, while Pavlovianconditioningdealswiththeconditioningofinvoluntaryreflexive behavior so that it occurs under newantecedent conditions. There are fourcontextsofoperantconditioning: • (i)Positive reinforcement occurs when a behavior(response)is followed by a favorable stimulus (commonly seen as pleasant) that increases the frequency of that behavior. In the Skinner box experiment, a stimulus such as food or sugar solutioncanbedeliveredwhentheratengagesinatargetbehavior, suchas pressing a lever. (ii)Negative reinforcement occurs when a behavior(response)is followed by the removal of an aversive stimulus(commonly seen as unpleasant)thereby increasing that behavior’sfrequency. Inthe Skinnerboxexperiment, negative reinforcementcanbealoudnoisecontinuouslysoundinginsidetherat’scage until it engages in the target behavior, suchas pressing a lever, upon which the loud noise is removed. (iii)Positive punishment(also called‘Punishment by contingent stimulation’) occurs when a behavior(response)is followed by an aversive stimulus, suchas introducing a shock or loud noise, resulting in a decrease in that behavior. (iv)Negative punishment(also called‘Punishment by contingent withdrawal’) occurs when a behavior(response)is followed by the removal of a favorable stimulus, suchas taking away a child’stoyfollowinganundesiredbehavi or, resulting in a decrease in that behavior.

7 Earning that occurs as a function of observing, retaining and replicating behavior observed in others. 8 He process of symbolic activity, sometimes via a language.

rough grammar, or system of rules, used to manipulate the symbols. While a set of symbols may be used for expression or communication, it is primitive and relatively unexpressive, because there are no clear or regular relationships between the symbols. Some of the areas of the human brain involved in language processing are: Broca’s area, Wernicke’s area, Supramarginal gyr us, Angular gyr us, Primary Auditory Cortex.

Mathematics and computer science use artificialentitiescalledformallanguages(including programming languages and markup languages, but also some that are far more theoretical in nature). These often take the form of character strings, produced by some combination of formal grammar and sem antics of arbitrary complexity.

Recall that abstraction is the process of reducing the information content of a concept, typically in order to retain only information which is relevant for a particular purpose. Forexample, abstract ing a leather soccer ball to a ball retains only the information on general ball attributes and behavior. Similarly, abstract ing an emotional state to happiness reduces the amount of information conveyed about the emotional state.

Abstraction typically results in complexity reduction leading to a simpler conceptualization of a domain in order to facilitate processing or understanding of many spec ificscenariosinagenericway.

In philosophical terminology, abstraction is the thought process wherein ideas aredistancedfromobjects.

Abstraction uses a strategy of simplificat i on, wherein formerly concrete details are left ambiguous, vague, orundefined; thuseffectivecommunicationabout things in the abstract requires an intuitive or common experience between the communicator and the communication recipient.

Abstractionssometimeshaveambiguousreferents; forexample,‘happiness’ (when used as an abstraction)can refer to as many things as there are people and events or states of being which make them happy. Likewise,‘architecture’ refers not only to the design of safe, functional buildings, but also to elements of creation and innovation which aim at elegant solutions to construction problems, to the use of space, and at its best, to the attempt to evoke an emotional response in the builders, owners, viewers and users of the building.

Abstraction in philosophy is the process(or, to some, the alleged process)in concept–formation of recognizing some set of common features in individuals, and on that basis forminga concept of that feature. The notion of abstraction is important to understanding some philosophical controversies surrounding empiricismandtheproblemofuniversals. It has also recently become popular in formallogicunderpredicateabstraction.

Some research into the human brain suggests that the left and right hemisphere sdifferin their handling of abstraction. One side handles collections of examples(e.g., examples of a tree) whereas the other handles the concept itself.

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications.

Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were i dent ifiedanddefined as abstract structures. Forexample, geometry has its origins in the calculation of distances and areas in the real world; statistics has its origins in the calculation of probabilities in gambling; and algebra started with methods of solving problems in arithmetic.

Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. Take the historical development of geometry as an example; the first steps in the abstraction of geometry were made by the ancient Greeks, with Euclidbeing thefirstperson(as far as we know)to document the axioms of plane geometry. In the 17th century Descartes introduced Cartesian coordinates which allowed the development of analytic geometry. Further steps in abstraction were taken by Lobachevsky, Bolyai and Gauss who generalized the concepts of geometry to develop non–Euclideangeometries. Later in the 19 thcentury mathematicians generalised geometry even further, developing such areas as geometry inn dimensions, projective geometry, affinegeometry,finitegeometry anddifferentialgeometry. Finally Felix Klein’s‘Erlangenprogram’i dent ified the underlying theme of all of these geo me tries, defining each of them as the studyofpropertiesinvariantunderagivengroupofsymmetries. This level of abstraction revealed deep connections between geometry and abstract algebra.

The advantages of abstraction are: (i)It reveals deep connections bet we end ifferent areas of mathematics; (ii)Known results in one area can suggest conjecture sina related area; and (iii)Techniques and methods from one area can be applied to prove results in a related area.

An abstract structure is a formal object that i sdefined by a set of laws, properties, and relationships in away that is logically if not always historically independent of the structure of contingent experiences, forexample, those involving physical objects. Abstract structures are studied not only in logic and mathematics but in thefieldsthatapplythem, as computer science, and in thestudiesthatreflectonthem, as philosophy and especially the philosophy of mathematics. Indeed, modernmathematicshasbeendefined in a very general senseasthestudyofabstractstructuresbytheBourbakigroup.

The main disadvantage of abstraction is that highly abstract concepts are moredifficult to learn, and require a degree of mathematical maturity and experience before they can be assimilated.

In computer science, abstraction is a mechanism and practice to reduce and factor out details so that one can focus on a few concepts at a time.

fortunate effect of finessing some of the vexing philosophical issues of abstraction. Forexample, in both computing and in mathematics, numbers are concepts in the programming languages, as founded in mathematics. Implementation details depend on the hardware and software, but this is not a restriction because the computing concept of number is still based on the mathematical concept.

Roughly speaking, abstraction can be either that of control or data. Control abstraction is the abstraction of actions while data abstraction is that of data structures. Forexample, control abstraction in structured programming is the use of sub programs and format ted control flows. Data abstraction is to allow for handling data bits in meaningful manners. For example, it is the basic motivation behind data–type. Object–oriented programming can be seen as an attempt to abstract both data and code.

Now, recall that creativity is a mental process involving the generation of new ideas or concepts, or new associations between existing ideas or concepts. From as cient ificpointofview, the products of creative thought(sometimes referred to as divergent thought)are usually considered to have both originality and appropriateness. An alternative, more everyday conception of creativity is that it is simply the act of making something new. Althoughintuitivelyasimple phenomenon, it is in fact quite complex. It has been studied from the perspectives of behavioral psychology, social psychology, psycho metrics, cognitive science, artificialintelligence, philosophy, history, economics, design research, business, and management, among others. The studies have covered everyday creativity, exceptional creativity and even artificialcreativity. Unlike many phenomena in science, there is no single, authoritativeperspectiveordefinit i on of creativity. Unlike many phenomena in psychology, there is no standardized measurement technique.

Creativity has been attributed variously to divine intervention, cognitive processes, the social environment, personality traits, and chance(‘accident’, ‘ser end i pity’). It has been associated with genius, mental illness and humor. Some say it is a trait we are born with; others say it can be taught with the application of simple techniques. Although popularly associated with art and literature, it is also an essential part of innovation and invention and is important in professions suchas business, economics, architecture, industrial design, science and engineering.

Despite, or perhaps because of, the ambiguity and multi–dimensional nature of creativity, entire industries have been spawned from the pursuit of creative ideas and the development of creativity techniques. This mysterious phenomenon, though undeniably important and constantly visible, seems to lie tantalizingly beyondthegraspofscientificinvestigation.

J. Guilford’s group developed the so–called ‘Torrance Tests of Creative Thinking’. They involved simple tests of divergent thinking and other problem–solving skills, which were score don[Gui 67]:

1. Fluency: the total number of interpret able, meaningful, and relevant ideas generated inresponse to the stimulus; 2. Flexibility: thenumberofdifferent categories of relevant responses; 3. Originality: the statistical rarity of the responses among the test subjects; and 4. Elaboration: the amount of detail in the responses.

On the other hand, personality is a collection of emotional, thought and behavioral patterns unique to a person that is consistent overtime. Personality psychology is a branch of psychology which studies personality and individual different processes – that which makes us into a person. One emphasis is on trying to create a coherent picture of a person and all his or her major psychological processes. Another emphasis views it as the study of individual differences. These two views work together in practice. Personality psychologists are interested in broadview of the individual. This often leads to an interest in the most salient individualdifferencesamongpeople.

The word personality originates from the Latin persona, which means ‘mask’. Inthe History of theater of the ancient Latinworld, them ask was not used as a plot device to disguise the identity of a character, but rather was a convention employed to represent, or typify that character.

Gordon Allportdelineateddifferent kinds of traits, which he also called dispositions. Central traits are basic to an individual’s personality, while secondary traits are more peripheral. Common traits are those recognized within a culture and thus may vary from culture to culture. Cardinal traits are those by which an individual maybe strongly recognized.

Raymond Cattell’sre search propagated a two–tiered personality structure with sixteen‘primary factors’(16Personality Factors)andfive‘secondary factors’(see Table 23.1). Cattell referred to these 16 factors as primary factors, as opposed to the so–called‘Big Five’factors which he considered global factors. All of the primary factors correlate with global factors and could therefore be considered sub factors within them.

A different model was proposed by Hans Eysenck, who believed that just three traits: extroversion, neuroticismandpsychoticism–weresufficient to describe human personality. Eysenckwasoneof thefirst psychologists to study personality with the method of factor analysis, a statistical technique introduced by Charles Spear man and expanded by Raymond Cattell. Eysenck’sresultssuggestedtwomainpersonalityfactors[Eys 92a, Eys 92b]. Thefirst factor was the tendency to experience negative emotions, and Eysenckreferredtoitas‘neuroticism’. The second factor was the tendency enjoy positive events, especially social events, and Eysencknamedit‘extra version’. The two personality dimensions were described in his 1947 book‘Dimensions of Personality’. It is common practice in personality psychology to refer to the dimensions by thefirstletters, EandN. Eand Nprovideda 2– dimensional space to describe individual differences in behavior. An analogy can be madetohowlatitudeandlongitude

Warm, outgoing, attentive to others, kindly, easygoing, participating, likes people(Affectothymia)

Concrete thinking, lower general mental capacity, less intelligent, unable to handle abstract problems (Lower Scholastic Mental Capacity)

Abstract–thinking, more intelligent, bright, higher general mental capacity, fast learner(Higher Scholastic Mental Capacity)

Reactive emotionally, changeable, affected by feelings, emotionally less stable, easily upset (Lower Ego Strength)

Deferential, cooperative, avoids conflict, submissive, humble, obedient, easily led, docile, accommodating (Submissiveness)

Dominant, forceful, assertive, aggressive, competitive, stubborn, bossy (Dominance)

Serious, restrained, prudent, taciturn, introspective,silent(Desurgency)

Expedient, non conforming, disregards rules, self indulgent (Low Super Ego Strength)

Rule–conscious, dutiful, conscientious, conforming, moralistic, staid, rule bound (High Super Ego Strength)

Socially bold, venturesome, thick skinned,uninhibited(Parmia)

Utilitarian, objective, unsentimental,toughminded,self–reliant,no– nonsense,rough(Harria)

Trusting, unsuspecting, accepting, unconditional, easy(A lax i a)

Grounded, practical, prosaic, solution oriented,steady,conventional(Praxernia)

Forthright, genuine, art less, open, guileless, naive, unpretentious, involved(Artlessness)

Private, discreet, non disclosing, shrewd, polished, worldly, astute, diplomatic (Shrewdness)

Apprehensive, self doubting, worried, guilt prone, insecure, worrying, self blaming(Guilt Proneness)

Traditional, attached to familiar, conservative, respecting traditional ideas (Conservatism)

Self–reliant, solitary, resourceful, individualistic, self sufficient (Self- Sufficiency)

Perfection is tic, organized, compulsive, self-disciplined, socially precise, exacting willpower, control, self-sentimental (High Self–Concept Control) describe a point on the face of the earth. Also, Eysenck noted how these two dimensions were similar to the four personality types first proposed by the ancient Greekphysician Galen:

1. High Nand HighE=Cholerictype; 2. High Nand LowE=Melancholictype; 3. Low Nand HighE=Sanguinetype; and 4. Low Nand LowE=Phlegmatic type.

The third dimension,‘psychoticism’, was added to the model in the late 1970s, baseduponcollaborationsbetweenEysenckandhiswife, SybilB.G. Eysenck, the current editor of Personality and Individual Differences (see [Eys 69, Eys 76]).

The major strength of Eysenck’smo del was to provide detailed theory of the causes of personality(see his 1985 book‘Decline and Fall of the Freudian Empire’). For example, Eysenck proposed that extra version was caused by variability in cortical arousal; “introverts are characterized by higher levels of activity than extraverts and so are chronically more cortically aroused than extraverts’. While itseemscounterintuitivetosupposethatintrovertsaremorearousedthanextraverts, the put at i veeffectthishasonbehavior is such that the introvert seeks lower levels of stimulation. Conversely, theextravertseekstoheightentheir arousal to a more optimal level(aspredictedbytheYerkes–Dodson Law)by increased activity, social engagement and other stimulation–seeking behaviours.

DifferencesbetweenCattellandEysenckemergedduetopreferencesfordifferentformsoffactoranalysis, with Cattell using oblique, Eysenckorthogonal, rotation to analyze the factors that emerged when personality questionnaires were subject to statistical analysis. Today, the Big Five factors have the weight of a considerable amount of empirical research behind them. Building on the work of Cattell and others, Lewis Goldberg proposed a five–dimensional personality model, nicknamed the‘Big Five’personality traits:

Extroversion(i.e., ‘extroversionvs. introversion’above; outgoing and physical–stimulation–oriented vs. quiet and physical–stimulation–averse);

1. Neuroticism(i.e., emotional stability; calm, unperturbable, optimisticvs. emotionally reactive, prone to negative emotions); 2. Agreeableness(i.e., affable, friendly, conciliatoryvs. aggression aggressive, dominant, disagreeable); 3. Conscientiousness (i.e., dutiful, planful, and orderly vs. spontaneous, flexible, and unreliable); and 4. Openness to experience(i.e., opentonewideasandchangevs. traditional and staid).

A character structure is a system of relatively permanent motivational and other traits that are manifested in the characteristic ways that an individual relates to others and reacts to various kinds of challenges. The word‘structure’indicates that these several characteristics and/or learned patterns of behavior are linked in such away as to produce a state that can be highly resistant to change. The idea has its roots in the work of Sigmund Freud and several of his followers, the most important of whom(in this respect)is Erich Fromm.

Among the earliest factors that determine an individual’seventualcharacter structure are his or her genetic characteristics and early childhood nurture and education. A child who is well nurtured and taught in a relatively benign and consistent environment by loving adults who intend that the child should learn how to make objective appraisals regarding the environment will be likely to form a normal or productive character structure. On the other hand, a child whose nurture and/or education are not ideal, living in a treacherous environment and interacting with adults who do not take the long–term interests of the child to heart will be more likely to forma pattern of behavior that suits the child to avoidthechallengesputforthbyamalignsocialenvironment. The means that the child invents to make the best of a hostile environment. Although this may serve the child well while in that bad environment, it may also cause the child to react in inappropriate ways, ways damaging to his or her own interests, when interacting with people in a more ideal social context. Major trauma that occurs later in life, even in adulthood, cansometimeshaveaprofoundeffect. However, character may also develop in a positive way according to how the individual meets the psychosocial challenges of the lifecycle(Erik son).

Freud’sfirstpaperoncharacterdescribedtheanalcharacterconsistingof stubbornness, stinginessandextremeneatness. He saw this as a reaction formation to the child’shavingtogiveuppleasureinanaleroticism. The positive version of this character is the conscientious, inner directed obsessive. Freud also described the erotic character as both loving and dependent. And the narcissistic character as the natural leader, aggressive and independent because of not internalizingastrongsuperego.

For Erich Fromm, character develops as the way in which an individual structuresmodesofassimilationandrelatedness. The character types are almost identicaltoFreud’sbutFrommgivesthemdifferentnames, receptive, hoarding, exploitative. Fromm adds the marketing type as the person who continually adapts the self to succeed in the new service economy. For Fromm, character types can be productive or unproductive. Fromm notes that character structures develop in each individual to enable him or her to interact successfully within a given society, to adapt to its mode of production and social norms may be very counter–productive when used in adifferentsociety.

On the other hand, wisdom is the ability, developed through experience, insight andreflection, to discern truth and exercise good judgment. It is sometimes conceptualized as an especially well developed form of commonsense. Most psychologists regard wisdom as distinct from the cognitive abilities measured by standardized intelligence tests. Wisdom is often considered to be a trait that can be developed by experience, but not taught. When applied to practical matters, the term wisdom is synonymous with prudence. Some see wisdom as a quality that even a child, otherwise immature, may possess independent of experience or complete knowledge. The status of wisdom or prudence as a virtue is recognized in cultural, philosophical and religious sources. Some define wisdom inautilitariansense, as foreseeing consequences and acting to maximize the long–term common good.

Astandardphilosophicaldefinit i on says that wisdom consists of making the best use of available knowledge. As with all decisions, a wise decision maybe made within complete information. The technical philosophical term for the opposite of wisdom is folly. Forexample, inhisMetaphysics, Aristotledefines wisdom as knowledge of causes: why things exist in a particular fashion.

Beyond the simple expedient of experience(which maybe considered the most difficult way to gain wisdom as through the‘school of hard knocks’), there area variety of other avenues to gaining wisdom which vary according to different philosophies. Forexample, the so–called free thinkers believe that wisdom may come from pure reason and perhaps experience. Recall that free thought is a philosophical doctrine that holds that beliefs should be formed on the basis of science and logical principles and not be comprised by authority, tradition or any other dogmatic or otherwise fallacious belief system that restricts logical reasoning. The cognitive application of free thought is known as free thinking, and practitioners of free thought are known as free thinkers. Free thought holds that individuals should neither accept nor reject ideas proposed as truth without recourse to knowledge and reason. Thus, free thinkers strive to build their beliefs on the basis of facts, sci ent ificinquiry, and logical principles, independent of the factual/logical fallacies and intellectually-limitingeffectsofauthority, cognitive bias, conventional wisdom, popular culture, prejudice, sectarianism, tradition, urbanlegendandallotherdogmaticorotherwisefallaciousprinciples. When applied to religion, the philosophy of free thought holds that, given presently– known facts, established sci ent ific theories, and logical principles, there is insufficient evidence to support the existence of supernatural phenomena. A line from ‘Clifford’sCredo’by the 19 thCenturyBritishmathematicianandphilosopher William Clifford perhaps best describes the premise of free thought:“It is wrong always, everywhere, and for anyone, tobelieveanythinguponinsufficientevidence.”Since many popular beliefs are based on dogmas, free thinkers’opinions are often at odds with commonly–established views.

On the other hand, there is also a common belief that wisdom comes from intuition or, ‘super logic’, as it is called by Tony Buzan, inventor of mind maps. For example, holistsbelievethatwisepeoplesense, work with and align themselves and others to life. In this view, wise people help others appreciate the fundamental inter connectedness of life. Also, some religions hold that wisdom may be givenasagiftfromGod. Forexample, Buddha taught that a wise person is endowed with good bodily conduct, good verbal conduct and good mental conduct and a wise person does actions that are unpleasant to do but give good results and doesn’t do actions that are pleasant to do but give bad results; this is called karma. According to Hinduscriptures, spiritual wisdom–jnanaalonecanlead to liberation. Confucius stated that wisdom can be learned by three methods: (i)reflection(the noblest),(ii)imitation(the easiest)and(iii)experience(the bitterest).

Despite the variety of concepts of intelligence, the most influentialapproachto understanding intelligence (i.e., with the most supporters and the most published research over the longest period of time) is based on psychometric testing, which regards intelligence as cognitive ability.

Recall that psychometric sis thefieldofstudyconcernedwiththetheory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, and personality traits. Thefieldisprimarily concernedwiththestudyofdifferencesbetweenindividuals. It involves two major research tasks, namely:

(i)the construction of instruments and procedures for measurement; and (ii) the development and refinement of theoretical approaches to measurement. Much of the early theoretical and applied work in psycho metrics was undertaken in an attempt to measure intelligence.

The origin of psychometric shas connections to the relatedfieldofpsychophysics. Charles Spearman, a pioneer in psycho metrics who developed approaches to the measurement of intelligence, studied under Wilhelm Wundt and was trained in psycho physics. The psychometric i an Louis Thurstone later developed and applied a theoretical approach to the measurement referred to as the law of comparative judgment, an approach which has close connections to the psychophysicaltheorydevelopedbyErnstWeberandGustavFechner(see below). Inaddition, Spearman and Thurstone both made important contributions to the theory and application of factor analysis, a statistical method that has been used extensively in psycho metrics. More recently, psychometric theory has been applied in the measurement of personality, attitudes and beliefs, academic achievement, and in health–relatedfields. Measurementoftheseunobservable phenomena i sdifficult, and much of the research and accumulated art in this disciplinehasbeendevelopedinanattempttoproperlydefineandquantify such phenomena. Critics, including practitioners in the physical sciences and socialactivists, havearguedthatsuchdefinitionandquantificat i on is impossibly difficult, and that such measurements are often misused. Proponents of psychometric techniques can reply, though, that their critics often misuse data by not applying psychometric criteria, and also that various quantitative phenomena in the physical sciences, suchas heat and forces, cannot be observed directly but must be inferred from their manifestations. Figures whom a design ificant contributionstopsychometricsincludeKarlPearson, Louis Thurstone, Georg Rasch and Arthur Jensen.

Intelligence, narrowly defined by psycho metrics, can be measured by intelligence tests, also called intelligence quotient (IQ)9tests. Such intelligence tests take many forms, but the common tests (Stanford–Binet,10Raven’s Progressive Matrices,11

9 AnintelligencequotientorIQisascorederivedfromasetofstandardizedtestsof intelligence. Intelligence tests come in many forms, and some tests use a single type of item or question. Most tests yield both an overall score and individual sub–tests scores. Regardless of design, allIQ tests measure the same general intelligence. Component tests are generally designed and chosen because they are found to be predictable of later intellectual development, suchas educational achievement. I Qalsocorrelates with job performance, socioeconomic advancement, and‘social pathologies’. Recent work has demonstrated links between IQ and health, longevity, and functional literacy. However, I Qtestsdonotmeasureallmeaningsof‘intelligence’, suchas creativity. IQ scores are relative(like placement in a race), not absolute(like the measurement of aruler). TheaverageIQscoresformanypopulationswererisingduringthe 20th century: a phenomenon called the Flynn effect. It is not known whether these changes in scores reflect real changes in intellectual abilities. On average, IQ scores are stable over a person’slifetime, but some individuals undergo large changes. Forexample, scores can be affected by the presence of learning disabilities. 10 The modernfieldofintelligencetestingbeganwiththeStanford-BinetIQtest. The Stanford-Binetitselfstartedwith the French psychologist Alfred Binetwhowascharged bytheFrenchgovernmentwithdevelopingamethodofidentifyingintellectuallydeficient children for placement in special education programs. As Binet indicated, case studies maybe more detailed and at times more helpful, but the time required to test large numbers of people would be huge. Unfortunately, the tests he and his assistant Victor Henri developed in 1896 were largely disappointing. 11 Raven’sProgressiveMatricesarewidelyusednon–verbal intelligence tests. In each test item, one is asked tofind the missing part required to complete a pattern. Each Set of items gets progressively harder, requiring greater cognitive capacity to encode and analyze. The test is considered by many intelligence experts to be one of the most g–loaded in existence. Thematricesareofferedinthreedifferent forms ford ifferent ability levels, and for age ranges fromfivethroughadult:(i)Colored Progressive Matrices(younger children and special groups);(ii)Standard Progressive Matrices (average6to80 yearolds); and(iii)Advanced Progressive Matrices(above average adolescents and adults). According to their author, Raven’s Progressive Matrices and Vocabulary tests measure the two main components of general intelligence (originally i dent ifiedbySpearman): the ability to think clearly and make sense of complexity, which is known a seductive ability(fromthe Latinroot‘educere’, meaning‘to draw out’; and the ability to store and reproduce information, known as reproductive ability. Adequate standardization, ease of use (without written or complex instructions), and minimal cost per person tested are the main reasons for its widespread international use in most countries of the world. It appears to measure a type of reasoning ability which is fundamental to making sense out of the ‘booming buzzing confusion’ in all walks of life. Thus, it has among the highest predictive valid ities of any test in most occupational groups and, even more importantly, in predicting social mobility...the level of job a person will attain and retain. Although it is sometimes criticized for being costly, this is based on a failure to calculate cost per person tested with re–usable test booklets that can be used up to 50 times each. The authors of the Manual recommend that, when used in selection, RPM scores are set in the context of information relating to Raven’s framework for the assessment of Competence. Some of the most fundamental research

Wechsler Adult Intelligence Scale,12Wechsler–Bellevue I ,13and others)all measure the same dominant form of intelligence, gor‘general intelligence factor’. The abstraction of gstemsfromtheobservationthat scores on all forms of cognitive tests positively correlate with one another. g can be derived as the principal intelligence factor from cognitive test scores using the multivariate correlation statistical method of factor analysis(FA).

Recall that correlation, also called correlationcoefficient, indicates the strength and direction of a linear relationship between two random variables(see Figure 23.1). In other words, correlation is a measure of the relation between two or more statistical variables. In general statistical usage, correlation(or, co– rrelation)refers to the departure of two variables from independence, although correlation does not imply their functional causal relation. In this broad sense thereareseveralcoefficients, measuring the degree of correlation, adapted to the nature of data. Anumberofdifferentcoefficientsareusedfordifferent situations. Correlationcoefficientscanrangefrom−1.00to+1.00. The value in cognitive psychology has been carried out with the RPM. The tests have been shown to work–scale–measure the same thing–in a vast variety of cultural groups. There is no truth in the assertion that the low mean scores obtained in some groups arise from a general lack of familiarity with the way of thought measured by the test. Two remarkable, and relatively recent, findings are that, on the one hand, the actual scores obtained by people living in most countries with a tradition of literacy – from China, Russia, and India through Europe to Kuwait – are very similar at any point in time. On the other hand, in all countries, the scores have increased dramatically over time... such that 50% of our grandparents would be assigned to special education classes if they were judged against today’s norms. Yet none of the common explanations (e.g., access to television, changes in education, changes in family size etc.)hold up. The explanation seems to have more in common with those put forward to explain the parallel increase in life expectancy... which has doubled over the same period of time. 12 Wechsler Adult Intelligence Scale or WAIS is a general IQ test, published in February 1955 asarevisionoftheWechsler–Bellevue test(1939), standardized for use with adults over the age of 16.In thistestintelligenceisquantified as the global capacity ofthe individual to act purposefully, to think rationally, and todealeffectivelywith the environment. 13 David Wechsler (January 12, 1896, Les ped i, Romania – May 2, 1981, New York, New York)wasaleadingRomanian–American psychologist. He developed well–known intelligence scales, suchastheWechslerAdultIntelligenceScale(WAIS)and the Wechsler Intelligence Scale for Children(WISC). The Wechsler Adult Intelligence Scale(WAIS)was developedfirst in 1939 andthencalledtheWechsler–Bellevue Intelligence Test. From these he derived the Wechsler Intelligence Scale for Children (WISC)in 1949 and the Wechsler Pre school and Primary Scale of Intelligence(WPPSI) in 1967. Wechsler originally created these tests to find out more about his patients at the Bellevue clinic and he found the then–currentBinetIQtestunsatisfactory. The tests are still based on his philosophy that intelligence is“the global capacity to act purposefully, to think rationally, and to deal effectivelywith (one’s) environment.”

Fig.23.1. Example of positive linear correlations between 1000 pairs of numbers. Note that each set of points correlates maximally with itself, as shown on the diagonal. Also, note that we have not plot the upper part of the correlation matrix as it is symmetrical.

of−1.00represents a perfect negative correlation while a value of+1.00represents a perfect positive correlation. The perfect correlation indicates an existence of functional relation between two statistical variables. A value of 0.00 represents a lack of correlation. Geometrically, the correlation coefficient can also be viewed asthecosineoftheanglebetweenthetwovectorsofsamplesdrawnfromthe two random variables.

The most widely–used type of correlation simple linear coefficientisPearson r, also called linear or product–moment correlation, which assumes that the two variables are measured on at least interval scales, and it determines the extent to which values of the two variables are‘proportional’to each other. The value of correlation coefficientdoesnotdependonthespecificmeasurementunits used. Proportional means linearly related using regression line or least squares line. If the correlation coefficient is squared, then the resulting value(r 2, the coefficient of determination)will represent the proportion of common variation in the two variables(i.e., the‘strength’or‘magnitude’of the relationship). In order to evaluate the correlation between variables, it is important to know this ‘magnitude’or‘strength’as well as the sign ificanceofthecorrelation.

The sign ificancel evel calculated foreach correlation is a primary source of information about the reliability of the correlation. The sign ificance of correlation coefficient of particular magnitude will change depending on the size of the sample from which it was computed. The test of sign ificanceisbasedonthe assumption that each of the two variables is normally distributed and that their bivariate (‘combined’) distribution is normal (which can be tested by examining the 3D bivariate distribution his to gram). However, Monte–Carlo studies suggest that meeting those assumptions(especially the second one)is not absolutely crucial if our sample size is not very small and when the departure from normality is not very large. It is impossible to formulate precise recommendations based on those Monte–Carlo results, but many researchers follow a rule of thumb that if our sample size is 50 or more then serious biases are unlikely, and i four sample size is over 100 then you should not be concerned at all with the normality assumptions.

Recall that the normal distribution, alsocalledGaussiandistribution, is an extremely important probability distribution in manyfields. It is a family of distributions of the same general form, differingintheirlocationandscaleparameters: the mean(‘average’) μ and standard deviation(‘variability’) σ, respectively. The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one. It is often called the bell curve because the graph of its probability density function pdf, given bytheGaussianfunction

2πe−x 2/2 is the pdf for the standard normal distribution). The corresponding cumulativedistributionfunctioncdfisdefined astheprobabilitythatavariableXhasavaluelessthanorequaltox, and it is expressed in terms of the pdf as du.

Now, thecorrelationrX, Ybetweentwonormallydistributedrandomvariables Xand Ywith expected values μXand μYand standard deviations σXand σY i sdefined as:

where Edenotestheexpectedvalueofthevariableandcovmeanscovariance. Since μX= E(X), σ2

X= E(X2)−E2(X) and similarly for Y , we can write (see, e.g.,[CCW03])

AssumethatwehaveadatamatrixX={xiα}formed out of the sample{xi} of nnormallydistributedsimulator tests called observable–vectors or manifest variables, defined on the sample{α=1, . . . , N}of pilot(for the statistical sign ifican cet he practical user’scriterionisN≥5n). The maximum likelihood estimatorofthePearsoncorrelationcoefficientrikbetweenanytwomanifest variablesxiandxkisdefined as 14 is the arithmeticmeanofthevariablexi.15Correlation matrix Ris the matrix R ≡Rik= {rik}includingn × nPearson correlationcoefficientsrikcalculated between n manifest variables {xi}. Therefore, R is symmetrical matrix with ones on the main diagonal. The correlation matrix Rrepresentsthetotalvariability of all included manifest variables. In other words it stores all information about xβ(ti)of the same length(i = 1, ..., T ), on edefines the correlation function by where¯xdenotesatimeaverageovertheperiodstudied. For two sets of Ntime– seriesxα(ti)each (α, β= 1, ..., N)all combinations of the elementsCαβcanbe used as entries of theN× NcorrelationmatrixC. BydiagonalizingC, i.e., solving the eigenvalueproblem:

onegetstheeigenvaluesλk(k=1, ..., N)and thecorrespondingeigenvectorsvk= {vk α}. 15 The following algorithm(in pseudo code)estimatesbivariatecorrelationcoefficient with good numerical stability: Begin y); End.

all simulator tests and all pilot. Now, if the number of included simulator tests is small, this information is meaningful for the human mind. But if we perform one hundred tests(onfivehundredpilot), then the correlation matrix contains tenthousandPearson correlationcoefficients. This is the reason for seeking the ‘latent’factor structure, underlyingthewholeco–variability contained in the correlation matrix.

Therefore, the correlation i sdefined only if both of the standard deviations are finite and both of them are non zero. It is a corollary of the Cauchy–Schwarz inequality 16 that the correlation cannot exceed1 in absolute value. The correlation is1 in the case of an increasing linear relationship,−1in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables. The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables (see Figure 23.1). If the variables are independent then the correlation is 0, but the converse is not true because the correlation coefficientdetectsonlylinear depend enc ies between two variables. Forexample, suppose the random variable Xisuniformlydistributedontheinterval from−1to 1, andY=X 2. Then YiscompletelydeterminedbyX, sothatXandYaredependent, but their correlation is zero; this means that they are uncorrelated. The correlation matrix ofnrandomvariablesX 1, ..., Xnisthen × nmatrixwhoseijentryisrXiXj. If the measures of correlation used are product–momentcoefficients, the correlation matrix isthesameasthecovariancematrix of the standardized random variables Xi/σXi(for i = 1, ..., n). Consequently it is necessarily anon–negative definitematrix. The correlation matrix is symmetrical(the correlation between Xiand Xjisthesameasthecorrelationbetween XjandXi).

16 The Cauchy–Schwarzinequality, namedafterAugustinLouisCauchy(the father of complex analysis)and Hermann Amandus Schwarz, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, appliedtovariancesandcovariances. The Cauchy–Schwarzinequalitystatesthatifx andy are elements of real or complex inner product spaces then

The two sides are equal iffxandyarelinearlydependent(or in geometrical sense they are parallel). This contrasts with a property that the inner product of two vectors is zero if they are orthogonal (or perpendicular) to each other. The inequality hence confers the notion of the angle between the two vectors to an inner product, whereconceptsofEuclideangeometrymaynothavemeaningfulsense, and just ifies thatthenotionthatinnerproductspacesaregeneralizationsofEuclideanspace.

Animportant consequence of the Cauchy–Schwarz inequality is that the inner product is a continuous function.

As a higher derivation of the correlation matrix analysis and itseigenvectors, the so–called principal components, the factor analysis(FA)is a multivariate statistical technique used to explain variability among a large set of observed random variables in terms of fewer unobserved random‘latent’variables, called factors. The observed, or‘manifested’variables are modelled as linear combinations of the factors, plus‘error terms’. AccordingtoFA, classicalbivariate correlation analysis is an artificialextractionfromarialmultivariateworld, especially in human sciences. FA originated in psycho metrics, and is used in social sciences, marketing, product management, operations research, and other applied sciences that deal with large multivariate quantities of data.

Forexample,17suppose a psychologist proposes a theory that there are two kinds of intelligence,‘verbal intelligence’and‘mathematical intelligence’. Note thattheseareinherentlyunobservable. Evidence for the theory is sought in the examination scores of 1000 students in each of10differentacademicfields. Ifa student is chosen randomly from a large population, then the student’s 10 scores are random variables. The psychologist’stheorymaysaythattheaverage score in each of the 10 subjects for students with a particular level of verbal intelligence and a particular level of mathematical intelligence isa certain number times the level of verbal intelligence plus a certain number times the level of mathematical intelligence, i.e., it is a linear combination of those two ‘factors’. The numbers by which the two‘intelligences’are multiplied are posi ted by the theory to be the same for all students, and are called‘factor loadings’. Forexample, the theory may hold that the average student’saptitudeinthefieldofamphibologyis

{ 10 × the student’s verbal intelligence } + { 6 × the student’s mathematical intelligence}.

Thenumbers 10 and6 arethefactorloadingsassociatedwithamphibology. Other academic subjects may haved ifferentfactorloadings. Two students having identical degrees of verbal intelligence and identical degrees of mathematical intelligence may haved ifferentaptitudesinamphibologybecauseindividualaptitudesdifferfromaverageaptitudes. Thatdiffere nce is called the‘error’—an unfortunate misnomer in statistics that means the amount by which an individual differs from what is average. The observable data that go into factor analysis would be 10 scores of each of the1000 students, a total of 10,000 numbers. The factor loadings and levels of the two kinds of intelligence of each student must be inferred from the data. Even the number of factors(two, in this example) must be inferred from the data.

In the example above, for i = 1, ..., 1, 000 the ith student’sscoresare

...

...

...

...

17 Thisoversimplifiedexampleshouldnotbetakentoberealistic. Usually we are dealing with many factors.

miistheithstudent’s‘mathematical intelligence’,ℓk, jarethefactorloadings forthekthsubject, forj=1, 2;ϵk, iisthedifferencebetweentheithstudent’s scoreinthekthsubjectandtheaverage scoreinthekthsubjectofallstudents whose levels of verbal and mathematical intelligence are the same as those of the ith student. In matrix notation, we have where Xisa 10 × 1, 000 matrix of observable random variables,μisa 10 × 1 columnvectorofunobservableconstants(in this case constants are quantities notdifferingfromoneindividualstudent to the next; and random variables are those assigned to individual students; the randomness arises from the random way in which the students are chosen), L is a 10×2 matrix of factor loadings (unobservable constants), Fis a 2×1, 000 matrix of unobservable random variables, ϵisa 10 ×1, 000 matrixofunobservablerandomvariables.

Observe that by doubling the scale on which‘verbal intelligence’, thefirst component in each column ofF, is measured, and simultaneously halving the factor loadings for verbal intelligence makes no difference to the model. Thus, no general ity is lost by assuming that the standard deviation of verbal intelligence is 1. Likewise for‘mathematical intelligence’. Moreover, for similar reasons, no generalityislostbyassumingthetwofactorsareuncorrelatedwitheachother. The‘errors’ϵare taken to be independent of each other. The variances of the ‘errors’associated with the 10different subjects are not assumed to be equal.

MathematicalbasisofFAisprincipalcomponentsanalysis(PCA), which is a technique for simplifying a dataset, by reducing multidimensional datasets to lower dimensions for analysis. Technically speaking, PCAisalineartransformation 18 that transforms the data to anew coordinate system such that the greatest variance by any projection of the data comes to lie on thefirstcoordinate(called thefirstprincipalcomponent), the second greatest variance on

18 Recall that a linear transformation(also called linear map or linear operator)isa function between two vector spaces that preserves the operations of vector addition and scalar multiplication. In the language of abstract algebra, a linear transformation isahomomorphismofvectorspaces, or a morph is min the category of vector spaces over a givenfield.

Let Vand WbevectorspacesoverthesamefieldK. A function(operator) f: V→Wissaidtobealineartransformationifforanytwovectorsx, y∈Vand anyscalara ∈K, the following two conditions are sat isfied:

additivity:f(x + y) = f(x) + f(y), and homogeneity:f(ax) = af(x).

This is equivalent torequiringthatforanyvectorsx 1, ..., xmandscalarsa 1, ..., am, the following equality holds:

the second coordinate, and soon. PCAcanbeusedfordimensionalityreduction 19 in a dataset while retaining those characteristics of the dataset that contribute most to its variance, by keeping lower–order principal components and ignoring higher–order ones. Such low–order components often contain the‘most important’aspects of the data. PCAisalsocalledthe(discrete)Karhunen– Lo`evetransform(or KLT, named after Kari Karhunen and Michel Lo`eve) or the Hotellingtransform(in honor of Harold Hotelling 20). PCA has the distinction of being the optimal linear transformation for keeping the sub space that has largest variance. This advantage, however, comes at the price of greater computational requirement if compared, forexample, tothediscretecosinetransform. Unlike other linear transforms, thePCAdoesnothaveafixedsetofbasisvectors. Its basis vectors depend on the dataset.

With the first k−1 components, the kth component can be found by subtracting thefirstk −1principalcomponentsfromx,

19 Dimensionality reduction in statistics can be divided into two categories: feature selection and feature extraction.

Feature selection approaches try tofindasubsetoftheoriginalfeatures. Two strategies arefilter(e.g., information gain)and wrapper(e.g., genetic algorithm) approaches. It is sometimes the case that data analysis such as regression or class ificat i on can be carried out in the reduced space more accurately than in the original space. On the other hand, feature extraction is applying a mapping of the multidimensional space into a space of fewer dimensions. This means that the original features pace is transformed by applying e.g., a linear transformation via a principal components analysis.

Dimensionality reduction is also a phenomenon discussed widely in physics, whereby a physical system exists in three dimensions, but its properties behave like those of a lower–dimensional system. 20 Harold Hotelling(Fulda, Minnesota, September 29,1895- December 26,1973) was a mathematical statistician. His name is known to all statisticians because of Hotelling’sT–square distribution and its use in statistical hypothesis testing and confidenceregions. He also introduced can on ical correlation analysis, and is the eponymofHotelling’slaw, Hotelling’slemma, and Hotelling’sruleineconomics.

and then obtaining the reduced–space data matrix Y by projecting X down into thereducedspacedefined by only thefirstLsingularvectorsWL,

T .

XXT= WΣ2WT.

Theeigenvectors with the largest eigenvalues correspond to the dimensions that have the strongest correlation in the dataset.

Now, FA is performed as PCA 22 with subsequent orthogonal (non–correlated) or oblique(correlated)factor rotation for the simplest possible interpretation (see, e.g.,[KM 78a]).

21 Recall that in linear algebra, the singular value decomposition(SVD) is an important factorizationofarectangularrealorcomplexmatrix, with several applications in signal processing and statistics. TheSVDcanbeseenasageneralizationofthe spectraltheorem, whichsaysthatnormalmatricescanbeunitarilydiagonalized usingabasisofeigenvectors, to arbitrary, not necessarily square, matrices.

Suppose Misanm × nmatrixwhoseentriescomefromthefieldK, which is either thefieldofrealnumbers, or thefieldofcomplexnumbers. Then there exists afactorizationoftheform:

where Uis an m×m unitary matrix over K, the matrix Σis m×n with non negative numbers on the diagonal and zero soffthe diagonal, andV∗denotes the con jug ate transposeofV , ann × nunitarymatrixoverK. Suchafactorizationiscalleda singular–valuedecompositionofM.

The matrix Vthuscontainsasetoforthonormal‘input’or‘analyzing’basis vector directions forM. The matrix Ucontainsasetoforthonormal‘output’basis vector directions forM. The matrixΣcontains the singular values, which can be thoughtofasscalar‘gain controls’by which each corresponding input is multiplied to give a corresponding output. Acommon convention is to order the valuesΣi i in non–increasing fashion. In this case, the diagonal matrixΣis uniquely determined byM(althoughthematricesUandVarenot). 22 ThealternativeFAapproachistheso–called principal factor analysis(PFA, also called principal axis factor ing, PAF, and common factor analysis, PFA). PFAisa form of factor analysis which seeks the least number of factors which can account for the common variance(correlation)of a set of variables, whereas the more common principal components analysis(PCA)in its full form seeks the set of factors which can account for all the common and unique(spec ificpluserror)variance in a set of variables. PFAusesaPCAstrategybutappliesittoacorrelationmatrixinwhich the diagonal elements are not 1’s, asinPCA, butiteratively–derived estimates of the communal ities.

FA is used to uncover the latent structure (dimensions) of a set of variables. It reduces attribute space from a larger number of variables to a smaller number of factors and as such is a‘non–dependent’procedure(that is, it does not assume a dependent variable is spec ified). Factor analysis could be used for any of the following purposes:

1.To reduce a large number of variables to a smaller number of factors for modelling purposes, where the large number of variables precludes modelling all the measures individually. As such, factor analysis is integrated in structural equation modelling(SEM),23helping create the latent variables modeled by

1. Image factor ing: based on the correlation matrix of predicted variables rather than actual variables, where each variable is predicted from the others using multiple regression. 2. Maximum likelihood factor ing: based on a linear combination of variables to form factors, where the parameter estimates are those most likely to have resulted in the observed correlation matrix, usingMLEmethodsandassumingmultivariate normality. Correlations are weighted by each variable’suniqueness.(As discussed below, uniqueness is the variability of a variable minus its communal ity.)MLF generates a chi–square goodness–of–fittest. The researcher can increase the number of factors one at a time until a satisfactory goodness offitisobtained. Warning: for large samples, even very small improvements in explaining variance can be sign ificant by the goodness-of-fitte stand thus lead the researcher to select too many factors. 3. Alpha factor ing: based on maximizing the reliability of factors, assuming variables are randomly sampled from a universe of variables. All other methods assume cases to be sampled and variablesfixed. 4. Unweighted least squares (ULS) factor ing: based on minimizing the sum of squared differencesbetweenobservedandestimatedcorrelationmatrices, not counting the diagonal. 5. Generalized least squares(GLS)factor ing: basedonadjustingULSbyweighting the correlations inversely according to their uniqueness(more unique variables are weighted less). LikeMLF, GLSalsogeneratesachi–square goodness–of–fittest. The researcher can increase the number of factors one at a time until a satisfactory goodness offitisobtained.

23 Structural equation modelling(SEM)grows out of and serves purposes similar to multiple regression, but in a more powerful way which takes into account the modelling of interactions, nonlinearities, correlated in dependents, measurement error, correlated error terms, multiple latent in dependents each measured by multiple indicators, and one or more latent dependents also each with multiple indicators. SEM maybe used as a more powerful alternative to multiple regression, path analysis, factor analysis, time series analysis, andanalysisofcovariance. That is, theseproceduresmaybeseenasspecialcasesofSEM, or, to put it another way, SEMisan extension of the general linear model(GLM)of which multiple regression is apart.

SEM is usually viewed as a confirmatory rather than exploratory procedure, using one of three approaches: a)Strictly confirmatoryapproach: A model is tested using SEM goodness–of–fittests

SEM. However, factor analysis can be and is often used on a standalone basis for similar purposes.

with a structural(path)model spec ifiedbytheresearcher. However a so the run examined models mayfitthe data as well or better, an accepted model is only a not–disc onfirmedmodel. b)Alternative models approach: One may test two or more causal models to determine which has the best fit. There are many goodness–of–fit measures, reflecting different considerations, and usually three or four are reported by the researcher. Although desirable in principle, thisAMapproachrunsintothereal-world problem that in most spec ific research topic areas, the researcher does not find in the literature two well-developed alternative models to test. c)Model development approach: In practice, much SEM research combines confirmatoryandexploratorypurposes: amodelistestedusingSEMprocedures, found to be deficient, and an alternative model is then tested based on changes suggested by SEMmodificationindexes. This is the most common approach found in the literature. The problem with the model development approach is that models confirmed in this manner are post–hoc ones which may not best able(may notfitnewdata, having been created based on the uniqueness of an initial dataset). Researchers may attempt to overcome this problem by using across–validation strategy under which the model is developed using a calibration data sample and then confirmed using an independent validation sample.

Regardless of approach, SEM cannot itself draw causal arrows in models or resolve causalambiguities. Theoretical insight and judgment by the researcher is still of utmost importance.

TheSEMprocesscentersaroundtwosteps: valid ating the measurement model andfittingthestructuralmodel. The former is accomplished primarily through confirmatory factor analysis, while the latter is accomplished primarily through path analysis with latent variables. One starts by specifying a model on the basis of theory. Each variable in the model is conceptualized as a latent one, measured by multiple indicators. Several indicators are developed foreach model, with a view to winding up with at least three per latent variable after confirmatoryfactoranalysis. Based on a large(n > 100) representative sample, factor analysis(common factor analysis or principal axis factor ing, not principle components analysis)is used to establish that indicators seem to measure the corresponding latent variables, represented by the factors. The researcher proceeds only when the measurement model has been validated. Two or more alternative models(one of which maybe the null model) are then compared in terms of modelfit, which measures the extent to which the covariancespredictedbythemodelcorrespondtotheobservedcovariancesinthe data. The so–calledmodificat i on indices and other coefficientsmaybeusedbythe researcher to alter one or more models to improvefit.

AdvantagesofSEMcomparedtomultipleregressionincludemoreflexibleassumptions (particularly allowing interpretation even in the face of multicollinearity), use of confirmatoryfactoranalysistoreduce measurement error by having multiple indicators per latent variable, theattractionofSEM’sgraphicalmodellinginterface, thedesirabilityoftestingmodelsoverallratherthancoefficientsindividually, the ability to test models with multiple dependents, the ability to model mediating variables, the ability to model error terms, the ability to testcoefficientsacrossmultiplebetween–subjects groups, and ability to handled ifficultdata(time series with auto correlated error, non–normal data, incomplete data).

2. To select a subset of variables from a larger set, based on which original variables have the highest correlations with the principal component factors. 3. To create a set of factors to be treated a sun correlated variables as one approach to handling multi–col linear it yin such procedures as multiple regression 4.To validate a scale or index by demonstrating that its constituent items load on the same factor, and to drop proposed scale items which cross–load on more than one factor. 5. To establish that multiple tests measure the same factor, thereby giving just ificat i on for administering fewer tests. 6. To identify clusters of cases and/or outliers. 7. To determine network groups by determining which sets of people cluster together.

The so–called exploratory factor analysis(EFA) seeks to uncover the underlying structure of a relatively large set of variables. The researcher’s`a priori assumption is that any indicator maybe associated with any factor. This is the most common form of factor analysis. There is no prior theory and one uses factor loading stointuitthefactorstructureof the data.

On the other hand, the so–called confirmatoryfactoranalysis(CFA)seeks to determine if the number of factors and the loadings of measured(indicator) variables on them conform to what is expected on the basis of pre–established theory. Indicator variables are selected on the basis of prior theory and factor analysis is used to see if they load as predicted on the expected number of factors. The researcher’s `a priori assumption is that each factor (the number and labels of which may be spec ified `a priori) is associated with a spec ified subset of indicator variables. A minimum requirement of confirmatoryfactoranalysisisthatone hypothesize beforehand the number of factors in the model, but usually also the researcher will pos it expectations about which variables will load on which factors(see, e.g.,[KM 78b]). The researcher seeks to determine, for instance, if measures created to represent a latent variable really belong together.

The factor loadings, alsocalledcomponentloadingsinPCA, are the correlation coefficients between the variables (rows) and factors (columns) in the factor matrix. Analogous to Pearson’sr, the squared factor loading is the percent of variance in that variable explained by the factor. To get the percent of variance in all the variables accounted for by each factor, add the sum of the squared factor loadings for that factor(column)and divide by the number of variables (note that the number of variables equals the sum of their variances as the variance of a standardized variable is 1). This is the same as dividing the factor’s eigenvaluebythenumberofvariables.

The factor scores, alsocalledcomponentscoresinPCA, factor scores are the scores of each case(row)on each factor(column). To compute the factor score for a given case for a given factor, one takes the case’sstandardizedscore on each variable, multiplies by the corresponding factor loading of the variable for the given factor, and sums these products. Computing factor scores allows one to look for factor outliers. Also, factor scores maybe used as variables in subsequent modelling.

Rotation serves to make the output more understandable and is usually necessary to facilitate the interpretation of factors. Thesumofeigenvaluesis not affectedbyrotation, but rotation willaltertheeigenvalues(and percent of variance explained)of particular factors and will change the factor loadings. Since alternative rotations may explain the same variance(have the same to tale i gen value)but haved ifferentfactorloadings, and since factor loadings are used to intuit the meaning of factors, this means that different meanings may be ascribed to the factors depending on the rotation – a problem some cite as a drawback to factor analysis. If factor analysis is used, the researcher may wish to experiment with alternative rotation methods to see which leads to the most interpret able factor structure.

Var imax rotation is an orthogonal rotation of the factor axes to maximize the variance of the squared loadings of a factor(column)on all the variables(rows) in a factor matrix, whichhastheeffectofdifferentiatingtheoriginalvariables by extracted factor. Each factor will tend to have either large or small loadings of any particular variable. A var imax solution yields results which make it as easy as possible to identify each variable with asingle factor. This is the most common rotation option.

The oblique rotations allow the factors to be correlated, and so a factor correlation matrix is generated when oblique is requested. Two most common oblique rotation methods are:

Pro max rotation rotation–an alternative non–orthogonal rotation method which is computation ally faster than the direct obl i min method and therefore is sometimes used for very large datasets.

1.Offers a much more objective method of testing intelligence in humans; 2. Allows for a satisfactory comparison between the results of intelligence tests; and 3. Provides support for theories that would bedifficult to prove otherwise.

Charles Spearmanpioneered the use of factor analysis inthefieldofpsychology and is sometimes credited with the invention of factor analysis. He discovered that schoolchildren’sscoresonawidevarietyofseeminglyunrelatedsubjects were positively correlated, which led him to postulate that a general mental ability, org, underlies and shapes human cognitive performance. Hispostulate now enjoys broad support in thefieldofintelligenceresearch, where it is known asthegtheory.

Raymond Cattell expanded on Spearman’s idea of a two–factor theory of intelligence after performing his own tests and factor analysis. He used a multi–factor theory to explain intelligence. Cattell’stheoryaddressed alternate factors in intellectual development, including motivation and psychology. Cattell also developed several mathematical methods for adjusting psychometric graphs, suchas his ‘scree’ test and similarity coefficients. His research lead to the development of his theory offluid and crystallized intelligence. Cattell was a strong advocate of factor analysis and psycho metrics. He believed that all theory should be derived from research, which supports the continued use of empirical observation and objective testing to study human intelligence.

Starting with the correlation matrix R including the number of sign ificant correlations, the goal of exploratory factor analysis (FA) is to detect latent underlying dimensions(i.e., the factor structure)among these to fall manifest variables. Instead of the correlation matrix, the factor analysis canstartfromthecovariance matrix (see Figure 4), which is the symmetrical matrix with variances of all manifest variables on the main diagonal and theircovariancesinothermatrix cells. For the purpose of the present project the correlation matrix is far more meaningful starting point. Three main applications of factor analytic techniques are(see, e.g.[CL 71]):

1. to reduce the number of manifest variables, 2. to classify manifest variables, and 3. to score each individual soldier on the latent factor structure.

Factor analysis model expands each of the manifest variables xiwiththemeans µifromthedatamatrixX = {xiα}as a linear vector–function xi= µi + Lij fj+ ei, (i = 1, . . . , n; j= 1, . . . , m) (23.1) where n and m denote the numbers of manifest and latent variables, respectively, fjdenotesthejthcommon–factor vector(with zero mean and unity–matrix covariance), L=Lijisthematrixoffactorloadingslij, an dei corresponds to theithspecific–factor vector(spec ificvariancenotexplainedbythecomm on factors, with zero mean and diagonal–matrixcovariance).

That portion of the variance oftheithmanifestvariablexi contributed by them common factors fj, the sum of squares of the loadings lij, is called the ith communality.

Now, inthecorrelationmatrixRthevariancesofallvariablesareequal to 1.0. Therefore, the total variance in that matrix is equal to the number of variables. Extractionoffactorsisbasedonthesolutionofeigenvalueproblem, i.e., characteristicequationforthecorrelationmatrixR, whereλiareeigenvaluesofR, representing the variances extracted by the factors, and xinow represent the corresponding eigenvectors, representing principal components or factors. The question then is, how many factors do we want to extract? Note that as we extract consecutive factors, they account for less and less variability. The decision of when to stop extracting factors basically depends on when there is only very little‘random’variability left. According to the widely used Kaiser criterion we can retain only factors wit he i gen values greater than 1. In essence this is like saying that, unless a factor extracts at least as much as the equivalent of one original variable, we drop it. The proportion of variance of a particular item that is due to common factors(shared with other items)is calledcommunality. Therefore, an additional task facing us when applying this model is to estimate the communal ities for each variable, that is, the proportion of variance that each item has in common with other items. The proportion of variance that is unique to each item is then the respective item’stotalvariance minus the communal ity. A common starting point is to use the squared multiple correlation of an item with all other items as an estimate of the communal ity. The correlations between the manifest variables and the principal components are called factor loadings. Thefirst factor is generally more highly correlated with the variables than the second, third and other factors, as these factors are extracted successively and will account for less and less variance overall.

Therefore, the principal component factor analysis of the sample correlation matrix Risspecifiedintermsofitsm < neigenvalue–eigenvector pairs(λj, xj) whereλj≥λj+1. Thematrixofestimatedfactorloadingslijisgivenby

Factor extraction can be performed also by other methods, collectively called principal factors, including:(i)Maximum likelihood factors,(ii)Principal axis method, (iii) Centroid method, (iv) Multiple R 2- communalities, and (v) Iterated Minrescommunalities. However, we shall stick on the principal components becauseoftheirobviouseigen–structure.

In any case, matrix of factor loadings L is determined only up to an orthogonal matrixO. The communal ities, givenbythediagonalelementsofLLTarealso unaffectedbythechoiceofO. This ambiguity provides the rationale for‘factor rotation’, since orthogonal mat ric esc or respond to‘coordinate’rotations.

We could plot, theoretically, the factor loadings i nam−dimensional scatter– plot. In that plot, each variable is represented as a point. In this plot we could rotate the axes in any direction without changing the relative locations of the points to each other; however, the actual coordinates of the points, that is, the factor loadings would of course change. There are various rotational strategies that have been proposed. The goal of all of these strategies is to get a clear pattern of loadings, that is, factors that are somehow clearly marked by high loadings for some variables and low loadings for others. This general pattern is also sometimes referred to as simple structure(a more formalized definit i on can be found in most standard textbooks). Typical rotational strategies are Varimax, Quartimax, and Equimax(see Anderson,1984). Basically, the extraction of principal components amounts toa variance maximizing Varimax–rotation of the original space of manifest–variables. We want to get a pattern of loadings on

Part 11 · pp. 305–334

each factor that is as diverse as possible, lending itself to easier interpretation. After we have found the line on which the variance is maximal, there remains some variability around this line. In principal components analysis, after the first factor has been extracted, that is, after the first line has been drawn through the data, we continue and define another line that maximizes the remaining variability, and soon. In this manner, consecutive factors are extracted. Because each consecutive factor i sdefined to maximize the variability that is not captured by the preceding factor, consecutive factors are independent of each other. Put another way, consecutivefactorsareuncorrelatedororthogonaltoeachother.

Basically, therotationofthematrixofthefactorloadingsLrepresentsits post–multiplication, i.e.L∗= LObytherotation matrixO, whichitselfresemblesoneofthematricesincludedintheclassicalrotationalLiegroupsSO(m) (containing the spec ificm−fold combination of sin uses and cos in uses. The linear factor equation (23.1) represents the orthogonal factor model, provided that vectorsfjandeiareindependent(orthogonal to each other, i.e., having zero covariance).

The most frequently used Kaiser’s Normal Varimax rotation procedure selects theorthogonaltransformationTthat‘spreads out’the squares of the loadings on each factor as much as possible, i.e., maximizes the total’squared’variance ijdenotetherotatedfactorloadingsfromtherotatedfactormatrixL∗. Besides orthogonal rotation, there is another concept of oblique(non orthogonal, or correlated) factors, which could help to achieve more interpret able simple structure. Specifically, computational strategies have been developed to rotate factors so as to best represent clusters of manifest variables, without the constraint of orthogonal ity of factors. Oblique rotation produces the factor structure made from the smaller set of mutually correlated factors. An oblique rotation to the simple structure corresponds to nonrigid rotation of the factor axes(i.e., principal components)in the factors pace such that the rotated axes l∗ obl(no longer perpendicular)pass(nearly)through the clusters of manifest variables. Although the purest mathematical background does not exist for then on–orthogonal factor rotation, theparsimonyprinciple:“explain the maximum of the common variabilityofthedatamatrixX={xiα}withthe minimum number of factors”, is fully developed only in this form of factor analysis, and the factor–correlationmatrixL∗ obl resembles the correlation matrix between manifest variables in the latent, factor space with double–reduced numberofobservables.

oblfj+ ei, (i = 1, . . . , n; j= 1, . . . , m), wherethevectorsfjandeiareinterdependent(correlated to each other). With oblique rotation, using common procedures, like Kaiser–Harris Orthoblique, Obl i min, Oblimax, Quart i min, Pro max(see[And 84]), we could

1. perform a hierarchical (iterated) factor analysis, obtaining second–order factors, third–order factors, etc.,finishingwithasinglegeneralfact or(forexample using principal component analysis of the factor–correlation matrix L∗ obl are highly correlated we can assume a linear functional link between them; connecting all correlated factors on the certain hierarchical level, we can make a block–diagram out of them depicting a linear system; this is the real point of the exploratory factor analysis.

The factor scores Sjα(where jlabels factors and α labels individual pilot) are incidental parameters that characterize general performance of the individuals (see [CL 71, And 84]). Factor scores with zero mean and unity-matrixcovarianceare usually automatically evaluated in principal–component, orthogonal and oblique factor analysis, according to the formula:

Calculate correlation matrix: R = Corr Mat[data]; Print[”R=”, R//Matrix Form]

Calculateeigenvaluesofthecorrelationmatrix: λ = Eigenvalues[R]//Matrix Form

Correspondingeigenvectors: vec = Eigenvectors[R]; Print[vec//Transpose//Matrix Form]

Determine sign ificantprincipalcomponents according to the criterionλ ≥2: Print[“PRINCIPALCOMPONENTS” →{vec[[1]], vec[[2]]}//Transpose//Matrix Form]

Defineo per at or matrix: No Fact = 2;P= Array[p, NoV ars, No Fact]; Table[p[i, j]= 1, {i, NoV ars}, {j, No Fact}]; Table[p[i, j]= 0, {i, 2, NoV ars, 2}, {j, 2, No Fact, 2}];

The factor scores can be used further for multivariate regression in the latent space(instead in the original manifests pace)for reducing the number of predictors in the general regression analysis(see[CL 71]).

To develop correlation and factor dynamics model, we are using geometrical analogy with nonrelativistic quantum mechanics(see [Dir 82]). A time dependent state of a quantum system is determined by a normalized(complex), time– dependent, wavepsi–functionψ=ψ(t), i.e. aunitDirac’s‘ket’vector|ψ(t)⟩, anelementoftheHilbertspaceL 2(ψ) with a coordinate basis(qi), under the action ofthe Herm it i an operators, obtainedbytheprocedureofquantization of classical mechanical quantities, forwhichrealeigenvaluesaremeasured. The state–vector |ψ(t)⟩, describing the motion of de Broglie’s waves, has a statistical interpretation as the probability amplitude of the quantum system, for the square of its magnitude determines the density of the probability of the system detected at various points of space. The summation over the entire space must yield unity and this is the normalization condition for the psi–function, determining the unit length of the state vector|ψ(t)⟩.

In the coordinate q–representation and the Schr¨odinger S–picture we consider an action of an evolution operator(innormalunitsPlanckconstantℏ= 1) i.e., aone–parameterLie–groupofunitarytransformationsevolvingaquantum system. The action represents an exponential map of the system’stotalenergy operator–Hamilton i anˆH=ˆH(t). It moves the quantum system from one instant of time, t 0, tosomefuturetimet, on the state–vector |ψ(t)⟩, rotating it: |ψ(t)⟩=ˆS(t, t 0)|ψ(t 0)⟩. In this case the Hilbertcoordinatebasis(qi)isfixed,

Perform oblique rotation: Q = Transpose[P]; S= R.P; G = Q.S; Do[k= 1

Factor structure matrix: Print[“F= ”, F//Matrix Form]

Inter-factor correlation matrix: Print[“C= ”, C//Matrix Form]

Factor projection matrix: Print[“Φ = ”, Φ//Matrix Form] with initial condition given at one instant oft i met0as|ψ(t 0)⟩= |ψ⟩.

whichrepresentsthecharacteristicequationfortheHamiltonianoperatorˆH andgivesitsrealeigenvalues(stationary energy states)Enandcorresponding orthonormaleigenfunctions(i.e., probability amplitudes)|ψn⟩.

To model the correlation and factor dynamics we start with the characteristic equation for the correlation matrix makingheuristicanalogywiththestationarySchr¨odingerequation(23.3). This analogy allows a‘physical’interpretationofthecorrelationmatrixRasanoperatorofthe‘totalcorrelationorcovariationenergy’of the statistical system (the simulator–testdatamatrixX={xiα}),eigenvaluesλncorrespondingto the‘stationary energy states’, andeigenvectorsxnresembling‘probability amplitudes’of the system.

So far we have considered one instant oft i met 0. Including the time–flow intothestationarySchr¨odingerequation(23.3)we get the time–dependent Schr¨odingerequation(23.2)andreturningbackwithourheuristicanalogy, we get the basic equation of then–dimensional correlation dynamics with initial condition attimet0 given as a stationary manifest–vectors xk(t 0) = xk(k= 1, . . . , n).

In more realistic case of‘many’observables(i.e., verybign), instead of the correlation dynamics (23.4), we can use the reduced–dimension factor dynamics, represented by analogous equation in the factors pace spanned by the extracted (oblique)factorsF = f i, with inter–factor–correlationmatrixC=cij(i, j= 1, . . . , no. of factors) ∂tfi(t) = C(t) fi(t), (23.5) subject to initial condition atti met0 givenasstationaryvectorsfi(t 0) = fi.

Now, according to the fundamental existence and uniqueness theorem for linearautonomousordinarydifferentialequations, if A=A(t)is ann × nreal matrix, then the initial value problem x(t) = x0 etA, forallt ∈R.

Therefore, analytical solutions of our correlation and factor–correlation dynamics equations(23.4)and(23.5)are given respectively by exponential maps

Thus,foreacht∈R,thematrixxexp[t R],respectivelythematrixfexp[t C], maps xk →xkexp[t R], respectively fi →fiexp[t C].

fact= {exp[t C]}t∈Rare 1– parameterfamilies(groups)oflinearmapsofRnintoRn, representing the correlationflow, respectively the factor–correlationflowofsimulator–tests. The linearflowsgt

1. identity map: g 0= I, and 2. composition: gt 1+t 2= gt 1◦gt 2.

1. Ifgtxk=xkforallt∈R, thenγ(xk)={xk}and it is called a point orbit. Point or bits correspond to equilibrium points in the manifest and the factor space, respectively. 2. If there exists aT> 0 suchthatgTxk= xk, thenγ(xk)is called a periodic orbit. Periodic or bits describe a system that evolves periodically in time in the manifest and the factors pace, respectively. 3. Ifgtxk̸= xkforallt̸ = 0, thenγ(xk)is called an on–periodicorbit.

1.If gtfi= fifor all t ∈R, then γ(fi) = {fi} and it is called a point orbit. Point or bits correspond to equilibrium points in the manifest and the factor space, respectively. 2. If there exists aT>0suchthatgTfi=fi, thenγ(fi)is called a periodic orbit. Periodic or bits describe a system that evolves periodically in time in the manifest and the factors pace, respectively. 3. Ifgtfi̸= fiforallt̸ = 0, thenγ(fi)is called an on–periodicorbit.

Now, to interpret properly the meaning of(really discrete)timeinthecorrelationmatrixR= R(t)and factor–correlation matrixC = C(t), we can perform a successive time–series{t, t + Δt, t + 2Δt, t + kΔt, · · · }of simulator tests(and subsequent correlation and factor analysis), anddiscretizeourcorrelation(respectively, factor–correlation) dynamics, to get xk(t + Δt)=xk(0) + R(t) xk(t)Δt, and fi(t + Δt)=fi(0) + C(t) fi(t)Δt, respectively. Finally we can represent the discrete correlation and factor– correlation dynamics in the form of the (computation ally applicable) three–point iterativedynamicsequation, respectively in the manifests pace in which the time–iteration variable s labels the time occurrence of the simulator tests(and subsequent correlation and factor analysis), starting with the initial state, labelleds = 0.

In the psychometric view, the concept of intelligence is most closely i dent ified with Spearman’sg, orGf(‘fluidg’). However, psychometric i an scan measure a wide range of abilities, which are distinct yet correlated. One common view isthattheseabilitiesarehierarchicallyarrangedwithgatthevertex(ortop, over laying all other cognitive abilities).25

On the other hand, critics of the psychometric approach, such as Robert Sternberg from Yale, point out that people in the general population have a somewhat different conception of intelligence than most experts. In turn, they argue that the psychometric approach measures only apart of what is commonly understood as intelligence. Other critics, suchas Arthur Eddington,26argue that the equipment used in an experiment often determines the results and that proving

25 Intelligence, IQ, and gare distinct terms. As already said above, intelligence is the term used in ordinary discourse to refer to cognitive ability. However, it is generally regarded as too imprecise to be useful for as cient ifictreatmentofthesubject. The intelligence quotient (IQ) is an index calculated from the scores on test items judged by experts to encompass the abilities covered by the term intelligence. I Qmeasures a multidimensional quantity: itisanamalgamofdifferent kinds of abilities, the proportionsofwhichmaydifferbetweenIQtests. ThedimensionalityofIQscores can be studied by factor analysis, whichrevealsasingledominantfactorunderlying thescoresonallIQtests. This factor, which is a hypothetical construct, iscalledg. Variation ing corresponds closely to the intuitive notion of intelligence, andthusg is sometimes called general cognitive ability or general intelligence. 26 Sir Arthur Stanley Eddington, OM(December 28,1882— November 22,1944) wasanastrophysicistoftheearly 20 thcentury. The Eddingtonlimit, the natural thate.g., intelligence exists does not prove that current equipment measure it correctly. Sceptics often argue that so much sci ent ific knowledge about the brain is still to be discovered that claiming the conventional IQ test methodology to be infallibleisjustasmallstepforwardfromclaimingthatcraniometry 27 was the infallible method for measuring intelligence(which had sci ent ificmeritsbased on knowledge available in the nineteenth century).

A more fundamental criticism is that both the psychometric model used in these studies and the conceptualization of cognitive abilityitselfarefundamentallyoffbeam. These views were expressed by none other than Charles Spearman, the ‘discoverer’ ofg – himself. Thus he wrote: “Every normal man, woman, and child isa genius at something. It remains to discover at what. This must be amostdifficultmatter, owing to the very fact that it occurs in only a minute proportion of all possible abilities. It certainly cannot be detected by any of the testing procedures at present in current usage. But these procedures are capable, I believe, of vast improvement.”In this context he noted that it is more important to ask‘What does this person think about?’than‘How well can he or she think?’Spearman went on to observe that the tests from which hisghad emerged had no place in schools since they did notreflectthediversetalents of the children and thu sdeflectedteachersfromtheirfundamentaleducational role, which is to nurture and recognize these diverse talents.

He also noted, as paraphrased here, that the so–called ‘cognitive ability’ is not primarily cognitive but affectiveandconative. In constructing meaning out of confusion(Spearman’seductive ability)onefirstfollowsfeelingsthatbeckonor attract. One then has to engage in‘experimental interactions with the environment’ to check out those, largely non–verbal, ‘hunches’. This requires determination and persistence—con a tion. Now, all of the seared ifficult and demanding activities which will only be undertaken whilst one is undertaking activities one cares about. So thefirst question is:‘What kinds of activity is this person strongly motivated to undertake’(and the kinds of activity which people may be strongly motivated to undertake are legion and mostly unrelated to those assessed in conventional‘intelligence’tests). And the second question is:‘How manyofthecumulativeandsubstitutablecomponentsofcompetencerequired tocarryouttheseactivitieseffectivelydoesthispersondisplaywhilstcarrying out that activity?’So one cannot, in reality, assess a person’sintelligence, or even their eductive ability, except in relation to activities they care about. What one see sine.g., the Raven Progressive Matricesisthecumulativeeffectofhow well they do all these things in relation to a certain sort of task. The problem is that this is not — and cannot be — ‘cognitive ability’ in any general sense of the limittotheluminositythatcanberadiatedbyaccretionontoacompactobject, is named in his honor. He is famous for his work regarding the Theory of Relativity. Eddingtonwroteanarticlein 1919, Reportontherelativitytheoryofgravitation, whichannouncedEinstein’stheoryofgeneralrelativitytotheEnglish–speaking world. 27 Craniometryisthetechniqueofmeasuringthebonesoftheskull. Craniometry was once intensively practiced in anthropology/ethnology.

word but only in relation to this kind of task. As Roger Sperry 28 has observed, what is neurologic ally localized is not ‘cognitive ability’ in any general sense but the emotional predisposition to ‘think’ about a particular kind of thing (for more details, seee.g., papersofJohnRaven 29[ Rav 02]).

Most experts accept the concept of asingle dominant factor of intelligence, general mental ability org, while others argue that intelligence consists of a set of relatively independent abilities[APS 98]. Theevidenceforgcomes from factor analysis of tests of cognitive abilities. The methods of factor analysis do not guarantee asingle dominant factor will be discovered. Other psychological tests, which do not measure cognitive ability, such as personality tests, generate multiple factors.

Proponents of multiple–intelligencetheoriesoftenclaimthatgis, at best, a measure of academic ability. Other types of intelligence, they claim, might be just as important outside of a school setting. Robert Sternberghasproposeda‘Triarchic Theory of Intelligence’. Howard Gardner’s theory of multiple intelligences breaks intelligence down into at least eight different components: logical, linguistic, spatial, musical, kinesthetic, naturalist, intra–personal and inter–personal intelligences. Daniel Golemanandseveralother researchers have developed the concept of emotional intelligence and claim it is at least as important as more traditional sorts of intelligence. These theories grew from observations of human developmentand of brain injury victims who demonstrate an acute loss of a particular cognitive function (e.g., the ability to think numerically, or the ability to understand written language), without showing any loss in other cognitive areas.

28 Roger Wolcott Sperry(August 20,1913– April 17,1994) was an euro psychologist who, together with David Hunter Hubel and Torsten Nils Wiesel, won the 1981 Nobel Prize in Medicine for his work with split–brain research. Before Sperry’s experiments, some research evidence seemed to indicate that areas of the brain were largely undifferentiated and interchangeable. In his early experiments Sperry challenged this view by showing that after early development circuits of the brain are largely hard wired. In his Nobel–winning work, Sperryseparatedthecorpuscallosum, the area of the brain used to transfer signals between the right and left hemispheres, to treat epileptics. Sperry and his colleagues then tested these patients with tasks that were known to be dependent on spec ific hemispheres ofthe brain and demonstrated that the two halves of the brain may each contain consciousness. In his words, each hemisphere is “...indeed a conscious system in its own right, perceiving, thinking, remembering, reasoning, willing, and emoting, all at a characteristically human level, and . . . both the left and the right hemisphere maybe conscious simultaneously in different, even in mutually confli cting, mental experiences that run along in parallel.”This research contributedgreatlytounderstandingthelateralizationofbrainfunctions. In 1989, Sperryalsoreceived National Medal of Science. 29 John Carlyle RavenfirstpublishedhisProgressiveMatricesintheUnitedKingdom in 1938. His three sons established Scotl and–basedtestpublisherJCRavenLtd. in 1972.In 2004, Harcourt Assessment, Inc. adivisionofHarcourtEducationacquired JCRavenLtd.

In response, g theorists have pointed out that g’s predictive validity 30 has been repeatedly demonstrated, forexample in predicting important non–academic outcomes suchas job performance, while no multiple–intelligencestheoryhas shown comparable validity. Meanwhile, they argue, the relevance, and even the existence, ofmultipleintelligenceshavenotbeenborneoutwhenactually tested[Hun 01]. Furthermore, gtheoristscontendthat proponents of multiple– intelligences(see, e.g.,[Ste 95]) have not disproved the existence of a general factor of intelligence[Kli 00]. The fundamental argument for a general factor is that test scores on a wide range of seemingly unrelated cognitive ability tests (suchas sentence completion, arithmetic, and memorization) are positively correlated: people who score highly on one test tend to score highly on all of them, and gthusemergesinafactoranalysis. This suggests that the tests are not unrelated, but that they all tap acommon factor.

Clearly, biologically realized ‘cognitive intelligence’ is the most complex property of human mind and can be perceived only by itself. Our problem is what we call or may call cognitive intelligence. From the formal, computational perspective, cognitiveintelligenceisoneofilldefinedconcepts. Itsdefinitionsareimmersed in numerous sci ent ific contexts and mirrors their historical evolutions, as well as, different‘interests’of researchers. Its weakness is usually based on its abstract multi faces image and, on the other hand, a universal utility character.

The classical behavioral/biologistsdefinit i on of intelligence reads:“Intelligence is the ability to adapt to new conditions and to successfully cope with lifesituations.”Thisdefinit i on seems to be the best, but‘intelligence’here depends on available physical tools and spec ificlifeexperience(individual hidden knowledge, preferences and access to information), therefore it is not enough selective to be measured, compared or designed. In general, cognitive intelligence is a human–like intelligence. Unfortunately there are many opinions what human–like intelligence means. Forexample,(i)cognitive intelligence uses a human mental introspective experience for the modelling of intelligent system thinking; and(ii)cognitive intelligence may use brain models to extract brain’s intelligence property.

Therefore, cognitive intelligence can be seen as a product of human self– conscious recognition of efficient mental processes, defined a’priori as intelligent. In order to get a consensus on the notion of cognitive intelligence is useful to have an agreement on which intelligence is not cognitive. A not–cognitive intelligence could be considered as an intelligence being developed using not human

30In psycho metrics, predictive validity is the extent to which a scale predicts scores on some criterion measure. Forexample, the validity of a cognitive test for job performance is the correlation between test scores and, say, supervisor performance ratings. Such a cognitive test would have predictive validity if the observed correlation were statistically sign ificant. Predictive validity shares similarities with concurrent validity in that both are generally measured as correlations between a test and some criterion measure.

analogies; e.g., itispossibletoconstructverydifferent models offlyingobjects starting from the observation ofstorks, balloons, beet les or clouds–maybe this observation can be useful.

Thedifferencebetweenhumanandartificial intelligence theories is similar to the difference between a birds theory of fly and the airplanes fly theory, the both can lead to a more general theory offlybutthislastneedsagoal–oriented and a higher abstraction level of the conceptualization/ontology.

AccordingtotheTOGAmeta–theoryparadigms,31for sci ent ificandpracticalmodellingpurposes, it is reasonable to separate conceptually the following fiveconcepts: information, knowledge, preferences, intelligence and emotions. If properlydefined, all of them can be independently i dent ifiedanddesigned.

tificialagent’sactivity(suchas: addresses, tel. numbers, encyclopedicdata, various lists of names and results of measurements). Every information has always a source domain. It is a relative concept. Information is a concept from the onto logy of mode ler/problem–solver/decision–maker. Knowledge: every abstract property of human/artificial agent which has ability to process/transform a quantitative/qualitative information into other information, or into another knowledge. It includes: instructions, emergency procedures, exploitation/user manuals, sci ent ificmaterials, models and theories. Every knowledge has its reference domain where it is applicable. It has to include the source domain of the processed information. It is a relative concept. Preference: an ordered relation among two properties of the domain of activity of a cognitive agent, it indicates a property with higher utility. Preference relations serve to establish an intervention goal of an agent. Cognitive preferences are relative.A preference agent which manages preferences of an intelligent agent can be external or its internal part. Goal: a hypothetical state of the domain of activity which has maximal utility in a carrierofdifferent structures of knowledge expressed in computer languages and usuallyfocusedontherealizationofpredefinedobjectives(a design-goal). It may include build-in preferences and information and/or request spec ificIPKasdata. (ii)from the software engineers perspective: a data-processing tool(more precise technicaldef. you mayfindontheWeb).

Suchconceptualmodularityshould enable to construct: emotional intelligence, social intelligence, skill intelligence, organizational intelligence, and many other X–intelligences, where X denotes a type of knowledge, preferences or a carrier system involved.

Forexample, business intelligence and emotional intelligence, rather are applications of intelligence either for business activities or for the second, under emotional/(not conscious)constrains and‘biological requests’.

In the above context, an abstract intelligent agent can be considered as a functionalkernelofanynaturalorartificialintelligentsystem.

Although there is no general theory of cognitive development, the most historically influentialtheorywasdevelopedbyJeanPiaget. Piagettheoryprovided many central concepts in thefieldofdevelopmentalpsychology. His theory concerned the growth of intelligence, whichforPiagetmeanttheabilitytomore accurately represent the world, and perform logical operations on representations of concepts grounded in the world. His theory concerns the emergence and acquisition of schemata, schemes of how one perceives the world, in ‘developmental stages’, times when children are acquiring new ways of mentally representing information. Pia get theory is considered‘constructivist, meaning that, unlike nativisttheories(which describe cognitive development as the unfolding of innate knowledge and abilities)orempiricisttheories(which describe cognitive development as the gradual acquisition of knowledge through experience), asserts that we construct our cognitive abilities through self–motivated action in the world.

Piaget divided schemes that children use to understand the world through four main stages, roughly correlated with and becoming increasingly sophisticated with age:

1. Sensorimotor stage(years 0–2), 2. Pre operational stage(years 2–7), 3. Concrete operational stage(years 7–11), and 4. Formal operational stage(years 11– adulthood).

Infants are born with a set of congenitalreflexes, according to Piaget, as well as a drive to explore their world. Their initial schemas are formed through differentiationofthecongenitalreflexes(see assimilation and accommodation, below).

The sensorimotor stage is the first of the four stages. According to Piaget, this stage marks the development of essential spatial abilities and understanding of the world in six sub–stages:

1. The first sub–stage occurs from birth to six weeks and is associated primarily with the develop men to freflexes. Threeprimaryreflexesaredescribedby Piaget: sucking of objects in the mouth, following moving or interesting objects with the eyes, and closing of the hand when an object makes contact with the palm (palmar grasp). Over these first six weeks of life, these reflexes begin to become voluntary actions; forexample, thepalmarreflexbecomes intentional grasping. 2. The second sub–stage occurs from six weeks to four months and is associated primarily with the development of habits. Primary circular reactions or repeating of an action involving only ones own body begin. An example of this type of reaction would involve something like an infant repeating the motion of passing their hand before their face. Also at this phase, passive reactions, causedbyclassicaloroperantconditioning, can begin. 3. The third sub–stage occurs from four to nine months and is associated primarily with the development of coordination between vision and prehension. Three new abilities occur at this stage: intentional grasping for a desired object, secondary circular reactions, anddifferentiationsbetweenendsand means. At this stage, infants will intentionally grasp the air in the direction of a desired object, often to the amusement of friends and family. Secondary circular reactions, or the repetition of an action involving an external object begin; forexample, moving a switch to turn on alight repeatedly. The differentiationbetweenmeansalsooccurs. This is perhaps one of the most important stages of a child’sgrowthasitsignifiest he dawn of logic. Towards the late part of this sub–stage infants begin to have asense of object permanence, passingtheA–not–Berrortest. 4. The fourth sub-stage occurs from nine to twelve months and is associated primarily with the development of logic and the coordination between means and ends. This is an extremely important stage of development, holding what Pia get calls the‘firstproperintelligence’. Also, this stage marks the beginning of goal orientation, the deliberate planning of steps to meet an objective. 5. Thefifthsub–stage occurs from twelveto eighteen months and is associated primarily with the discovery of new means to meet goals. Piagetdescribes the child at this juncture as the ‘young scientist’, conducting pseudo– experiments to discover new methods of meeting challenges. 6. The sixth sub–stage is associated primarily with the beginnings of insight, or true creativity. This marks the passage into thepre operational stage.

The Pre operational stage is the second of four stages of cognitive development. By observing sequences of play, Piaget was able to demonstrate that towards the end of the second year a qualitatively quite new kind of psychological functioning occurs. Operation in Pia get i an theory is any procedure for mentally acting on objects. Thehallmarkofthepreoperationalstageissparseandlogicallyinadequatementaloperations.

1. Symbolic functioning, which is characterised by the use of mental symbols words or pictures which the child uses to represent something which is not physically present. 2. Centration, which is characterized by a child focusing or attending to only one aspect of a stimulus or situation. Forexample, in pouring a quantity ofliquidfromanarrowbeakerintoashallowdish, a pre schoolchild might judge the quantity of liquid to have decreased, because it is‘lower’, that is, the child attends to the height of the water, but not to the compensating increase in the diameter of the container. 3. Intuitive thought, which occurs when the child is able to believe in something without knowing why she or he believes it. 4. Egocentrism, which is aversion of centration, this denotes a tendency of child to only think from own point of view. 5. Inability to Conserve; Through Piaget’sconservationexperiments(conservation of mass, volume and number)Pia get concluded that children in the pre operational stage lack perception of conservation of mass, volume, and number after the original form has changed. Forexample, a child in this phase will believe that a string of beads setup in a‘O–O–O–O–O’pattern will have the same number of beads as a string which has a‘O–O–O–O–O’ pattern, because they are the same length, or that at all, thin 8- ounce cup has more liquid in it than a wide, fat 8– ounce cup.

The concrete operational stage is the third of four stages of cognitive development in Piaget’stheory. This stage, whichfollowsthePreoperationalstageand occurs from the ages of7to 11, is characterized by the appropriate use of logic. The six important processes during this stage are:

1. Decentering, where the child takes into account multiple aspects of a problem to solve it. Forexample, the child will no longer perceive an exceptionally wide but short cup to contain lessthan a normally-wide, taller cup. 2. Reversibility, where the child understands that numbers or objects can be changed, then returned to their original state. For this reason, a child will be able to rapidly determine that4 + 4 which they can answer to be 8, minus 4 will equal four, the original quantity. 3. Conservation: understanding that quantity, length or number of items is unrelated to the arrangement or appearance of the object or items. For instance, when a child is presented with two equally–sized, full cups they will be able to discern that if water is transferred to a pitcher it will conserve the quantity and be equal to the otherfilledcup. 4. Serial is a tion: the ability to arrange objects in an order according to size, shape, or any other characteristic. Forexample, if giv end ifferent–shaded objects they may make a color gradient. 5. Class ificat i on: the ability to name and identify sets of objects according to appearance, size or other characteristic, including the idea that one set of objects can include another. A child is no longer subject to the illogical limitations of animism(the belief that all objects are animals and therefore have feelings). 6. Elimination of Egocentrism: the ability to view things from another’sperspective(eveniftheythinkincorrectly). For instance, show a child a comic inwhichJaneputsadollunderabox, leaves the room, andthenJillmoves the doll to a drawer, and Janecomesback; a child in this stage will not say that Janewillthinkthedollisinthedrawer.

The formal operational stage is the fourth and final of the stages of cognitive development of Piaget’s theory. This stage, which follows the Concrete Operational stage, commences at around 11 years of age (puberty)and continues into adulthood. It is characterized by acquisition of the ability to think abstractly and draw conclusions from the information available. During this stage the young adult functions in a cognitively normal manner and therefore is able to understand such things as love,‘shades of gray’, and values. Lucidly, biological factors may be traced to this stage as it occurs during puberty and marks the entering into adulthood in physiologically, cognitive, moral (Kohlberg), psycho sexual (Freud), and social development (Erik son). Many people do not successfully complete this stage, but mostly remain in concrete operations.

Recall that psycho physics is a sub discipline of psychology, founded in 1860by Gustav Fechnerwiththepublicationof‘Elementeder Psychophysik’, dealing with the relationship between physical stimuli and their subjective correlates, or percepts. Fechner described research relating physical stimuli with how they are perceived and set out the philosophical foundations of thefield. Fechnerwanted to develop a theory that could relate matter to the mind, by describing the relationship between the world and the way it is perceived(Snodgrass,1975). Fechner’s work formed the basis of psychology as a science. Wilhelm Wundt, the founder of thefirst laboratory for psychological research, builtuponFechner’s work.

The Weber–Fechnerlawattemptstodescribe the relationship between the physical magnitudes of stimuli and the perceived intensity of the stimuli. Ernst Weber was one of thefirst people to approach the study of the human response to a physical stimulus in a quantitative fashion. Gustav Fechnerlateroffered an elaboratetheoreticalinterpretationofWeber’sfindings, which he called simply Weber’slaw, though his admirers made the law’snameahyphenate. Fechner believedthatWeberhaddiscoveredthefundamentalprincipleofmind/body interaction, a mathematical analog of the function Rene Descartes once assigned tothepinealgland.

increase. Weber found that the response was proportional to a relative increase in the weight. That is to say, if the weight is 1 kg, an increase of a few grams will not be noticed. Rather, when the mass is increased by a certain factor, an increase in weight is perceived. If the mass is doubled, the threshold is also doubled. This kind of relationship can be described by a linear ordinary differential equation as, wheredpisthedifferentialchangeinperception, dSisthedifferentialincrease in the stimulus and Sis the stimulus at the instant.A constant factor kisto be determined experimentally. Integrating the above equation gives: p = k ln S +C, where Cis the constant of integration. To determine C, we can put p = 0, which means no perception; then we get, C=−k ln S 0, whereS0 is that threshold of stimulus below which it is not perceived at all. In this way, we get the solution

Therefore, the relationship between stimulus and perception is logarithmic. This logarithmicrelationshipmeans that if a stimulus varies as a geometric progression(i.e. multiplied by afixedfactor), the corresponding perception is altered in an arithmetic progression (i.e. in add it ive constant amounts). Forexample, if a stimulus is trip led in strength(i.e,3 × 1), the corresponding perception may be two times as strong as its original value(i.e.,1 + 1). If the stimulus is again trip led in strength(i.e.,3 × 3 × 1), the corresponding perception will be three times as strong as its original value(i.e.,1 + 1 + 1). Hence, formultiplications in stimulus strength, the strength of perception only adds. Thislogarithmicrelationship is valid, not just for the sensation of weight, but for other stimuli and our sensory perceptions as well.

In case of vision, we have that the eye senses brightness logarithmically. Hence stellarmagnitudeismeasuredonalogarithmicscale. This magnitude scale was inventedbytheancientGreekastronomerHipparchusinabout 150B.C. He ranked the stars he could see in terms of their brightness, with1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits. An increase in5 magnitudes corresponds to a decrease in brightness by a factor 100.

In case of sound, we have still another logarithmic scale is thedecibelscale of sound intensity. And yet another is pitch, which, however, differsfrom the other cases in that the physical quantity involved is not a‘strength’. In the case of perception of pitch, humans hear pitch in a logarithmic or geometric ratio–based fashion: For notes spaced equally apart to the human ear, the frequencies are relatedbyamultiplicativefactor. For instance, the frequency of corresponding notes of adjacent octaves differ by a factor of 2. Similarly, the perceived difference in pitch between 100 Hzand150 Hzisthesameasbetween1000 Hzand1500 Hz. Musical scales are always based on geometric relationships for this reason. Notationandtheoryaboutmusicoftenreferstopitchintervalsinanadditive way, which makes sense ifoneconsidersthelogarithmsof the frequencies, as log(a × b) = log a + log b.

Psycho physicists usually employ experimental stimuli that can be objectively measured, such as pure tones varying in intensity, or lights varying in luminance. All the senses have been studied: vision, hearing, touch (including skin and enteric perception), taste, smell, and the sense of time. Regardless of the sensory domain, there are three main topics in the psychophysical class ificat i on scheme: absolute thresholds, discrimination thresholds, and scaling.

The most common use of psycho physics is in producing scales of human experience of various aspects of physical stimuli. Take for an example the physical stimulus of frequency of sound. Frequency of a sound is measured in Hertz (Hz), cycles per second. But human experience of the frequencies of sound is not the same as the frequencies. For one thing, there is a frequency below which no sounds can be heard, no matter how intense they are(around 20 Hzdepending on the individual) and there is a frequency above which no sounds can be heard, no matter how intense they are(around 20,000Hz, again depending on the individual). For another, doubling the frequency of a sound(e.g., from 100 Hzto 200Hz)does not lead to a doubling of experience. The perceptual experience of the frequency of sound is called pitch, and it is measured by psycho physicists inmels.

More analytical approaches allow the use of psychophysical methods to study neurophysiological properties and sensory processing mechanisms. This is of particularimportanceinhuman research, where other(more invasive) methods are not used due to ethical reasons. Areas of investigation include sensory thresholds, methods of measurement of sensitivity, and signal detection theory.

Perception is the process of acquiring, interpreting, selecting, and organizing sensory information. Methods of studying perception range from essentially biological or physiological approaches, through psychological approaches to the often abstract‘thought–experiments’of mental philosophy.

Experiments in psycho physics seek to determine whether the subject can detect a stimulus, identify it, differentiatebetweenitandanotherstimulus, and describe the magnitude or nature of thi sdifference[Sno 75]. Often, the classic methods of experimentation are argued to be inefficient. This is because a lot of sampling and data has to be collected at points of the psychometric function that is known(the tails). Stair case procedures can be used to quickly estimate threshold. However, thecostofthisefficiency, is that we do not get the same amount of information regarding the psychometric function as we can through classical methods; e.g., we cannot extract an estimate of the slope(derivative) of the function.

A psychometric function describes the relationship between a parameter of a physical stimulus and the responses of a person who has to decide about a certain aspect of that stimulus. The psychometric function usually resembles a sigmoid function with the percentage of correct responses(or a similar value) displayedontheordinateandthephysicalparameterontheabscissa. Ifthe stimulus parameter is very far towards one end of its possible range, the person will always be able to respond correctly. Towards the other end of the range, the person never perceives the stimulus properly and therefore the probability of correct responses is at chance level. Inbetween, there is a transition range where the subject has an above–chance rate of correct responses, but does not always respond correctly. The inflectionpointofthesigmoidfunctio nor the point at which the function reaches the middle between the chance level and 100% is usually taken as sensory threshold. Acommon example is visual acuity testing with an eye chart. Thepersonseessymbolsofdifferentsizes(the size is the relevant physical stimulus parameter)and has to decide which symbol it is. Usually, there is one line on the chart where a subject can identify some, but not all, symbols. This is equal to the transition range of the psychometric function and the sensory threshold corresponds to visual acuity.

On the other hand, a sensory threshold is a theoretical concept which states: “A stimulus that is less intense than the sensory threshold will not elicit any sensation.” Whilst the concept can be applied to all senses, it is most commonly applied to the detection and perception of flavours and aromas. Several different sensorythresholdshavebeendefined:

1. Absolute threshold: the lowest level at which a stimulus can be detected. 2. Recognition threshold: the level at which a stimulus can not only be detected butalsorecognised. 3.Differentialthreshold: the level at which an increase in a detected stimulus can be perceived. 4. Terminal threshold: the level beyond which a stimulus is no longer detected.

In other words, a threshold is the point of intensity at which the participant can just detect the presence of, ord iffere nce in, a stimulus. Stimuli with intensities below the threshold are considered not detect able, however stimuli at values close to threshold will often be detect able some proportion of the time. Due to this, a threshold is considered to be the point at which a stimulus, or change in a stimulus, is detected some proportion pof the time. An absolute threshold is the level of intensity of a stimulus at which the subject is able to detect the presence of the stimulus some proportion of the time(aplevelof 50% is often used). An example of an absolute threshold is the number of hairs on the back of one’shandthatmustbetouchedbeforeit can be felt, a participant may be unable to feel asingle hair being touched, but maybe able to feel two or three as this exceeds the threshold. Adiffere nce threshold is the magnitude of thedifferencebetweentwostimuliofdifferingintensitiesthattheparticipantis able to detect some proportion of the time(again,50%is often used). To test this threshold, severaldifferencemethodsareused. The subject maybe asked to adjust one stimulus until it is perceived as the same as the other, maybe askedtodescribethemagnitudeofthediffere nce between two stimuli, or may be asked to detect a stimulus against a background. Absoluteanddifference thresholds are sometimes considered similar because there is always background noise interfering with our ability to detect stimuli, howeverstudyofdifference thresholds still occurs, forexample in pitch discrimination tasks(see[Sno 75]).

The sensory analysis applies principles of experimental design and statistical analysis to the use of human senses(sight, smell, taste, touch and hearing)for the purposes of evaluating consumer products. The discipline requires panels of human assessors, on whom the products are tested, and recording the responses made by them. By applying statistical techniques to the results it is possible to make inference sand insights about the products under test. Most large consumer goods companies have departments dedicated to sensory analysis. Sensory Analysis can generally be broken down into three sub–sections:

1.EffectiveTesting(dealing with objective facts about products); 2.AffectiveTesting(dealing with subjective facts suchas preferences); and 3. Perception(the biochemical and psychological aspects of sensation).

The signal detection theory(SDT)is a means to quantify the ability to discern between signal and noise. It has applications in manyfieldssuch as quality control, telecommunications, and psychology (see[Abd 06]). The concept is similar to the signal to noise ratio used in the sciences, and it is also usable in alarm management, where it is important to separate important events from background noise. According to the theory, there area number of psychological determinersofhowwewilldetectasignal, and where our threshold levels will be. Experience, expectations, physiological state(e.g, fatigue)and other factors affect thresholds. For instance, a sen try in wartime will likely detect fainter stimuli than the same sen try in peace time. SDTisusedwhenpsychologistswantto measure the way we make decisions under conditions of uncertainty, such as how we would perceive distances in foggy conditions. SDT assumes that ‘the decision maker is not a passive receiver of information, but an active decision–maker who makes difficult perceptual judgements under conditions of uncertainty’. In foggy circumstances, we are forced to decide how far an object is away from us based solely upon visual stimulus which is impaired by the fog. Since the brightness of the object, suchas a trafficlight, is used by the brain to discriminate the distance of an object, and the fog reduces the brightness of objects, we perceive the object to be much further away than it actually is. To apply signal detection theory to a dataset where stimuli were either present or absent, and the observer categorized each trial as having the stimulus present or absent, the trials are sorted into one of four categories, depending upon the stimulus and response:

appeared that the processes underlying creative problem solving ind ifferent domainsdifferedfromeachother[Ste 95]. These realization shave led to rather differentresponsesinNorthAmericaandinEurope.

In North America, initiated by the work of Herbert Simon on learning by doing in semantically rich domains, researchers began to investigate problemsolving separately ind ifferentnaturalknowledgedomains–suchas physics, writing, or chess playing–thus relinquishing their attempts to extract a global theory of problemsolving(see, e.g.,[SF 91]). Instead, these researchers have frequently focused on the development of problemsolving within a certain domain, that is on the development of expertise.

Areas that have attracted rather intensive attention in North America include such diverse fieldsas: reading, writing, calculation, political decision making, managerialproblemsolving, lawyers’reasoning, personal problemsolving, mathematical problemsolving, mechanical problemsolving, problemsolving in electronics, computer skills, game playing, and social problem solving. In particular, GeorgeP´olya’s 1945 book‘How to Solve It’, is asmall volume describing methods of problem–solving. It suggests the following steps when solving a mathematical problem:

1. First, you have to understand the problem. 2. After understanding, then make a plan. 3. Carry out the plan. 4. Look back on your work. How could it be better?

If this technique fails, Polya advises: “If you cannot solve a problem, then there is an easier problem you can solve:findit.”Or,“If you cannot solve the proposed problem try to solvefirstsomerelatedproblem. Could you imagine a more accessible related problem?”

His small book contains a dictionary–stylesetofheuristics, many of which have to do with generating a more accessible problem, like the ones given in the table below:

The technique‘have I used everything’is perhaps most applicable to formal educational examinations(e.g., nmendiggingmditches, see footnote below) problems. The book has achieved‘classic’status because of its considerable influence. Marvin Min sky said in his influentialpaper‘Steps Toward Artificial Intelligence’:“And everyone should know the work of George Poly a on how to solve problems.”Polya’sbookhashadalargeinfluenceonmathematicstextbooks. Most formulations of a problemsolving framework inU.S. textbooks attributesomerelationshiptoPolya’sproblemsolvingstages. Other books on problemsolving are often related to less concrete and more creative techniques, likee.g., lateral thinking, mind mapping and brain storming(see below).

On the other hand, in Europe, two main approaches have surfaced, oneinitiatedbyDonaldBroadbentintheUK[Bro 77,BB 95] and the other one by Dietrich D¨or ner in Germany [Dor 75, DV 85, DW 95]. The two approaches have in common an emphasis on relatively complex, semantically rich, computerized laboratory tasks, constructed to resemble ‘real–life’ problems. The approaches differ can you find a problem analogous to your problem and solve that?

can you find a problem more general than your problem...?

can you solve your problem by deriving ageneralizationfromsomeexamples?

can you vary or changeyour problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem?

can youfindasubproblemorsideproblem whose solution will help you solve your problem?

can youfindaproblemrelatedtoyours that has already been solved and use that to solve your problem?

can you find a problem more specialized?

can you decompose the problem and ”recombineitselementsinsomenewmanner”?

can you start with the goal and work backwards to something you already know?

can you draw a picture of the problem?

can you add some new element to your problem to get closer to a solution?

somewhat in their theoretical goals and methodology, however. The tradition initiatedbyBroadbentemphasizesthedistinctionbetweencognitiveproblem– solving processes that operate under awareness versus outside of awareness, and typically employs mathematically well–definedcomputerizedsystems. ThetraditioninitiatedbyD¨or ner, on the other hand, has an interest in the interplay of the cognitive, motivational, and social components of problemsolving, and utilizes very complex computerized scenarios that contain up to 2,000 highly interconnected variables.

Tosumup, researchers’realization that problem–solvingprocessesdiffer across knowledge domains and across levels of expertise(see, e.g.[Ste 95]) and that, consequently,findingsobtainedinthelaboratorycannotnecessarilygeneralize to problem–solving situations outside the laboratory, has during the past two decades led to an emphasis on real–world problemsolving. ThisemphasishasbeenexpressedquitedifferentlyinNorthAmericaandEurope, however. Whereas North American research has typically concentrate don studying problem solving in separate, natural knowledge domains, much of the European research has focused on novel, complex problems, and has been performed with computerized scenarios(see[Fun 95], for an overview).

Polytely (multiple goals), including in expressive ness, opposition and transience;

Dynamism(time considerations), including temporal constraints, temporal sensitivity, pha seeffects, and dynamic unpredictability.

Theresolutionofdifficult problems requires a direct attack on each of these characteristics that are encountered.

32 Trial and error(also known in computer science literature as generate and test and as‘guess and check’when solving equations in elementary algebra)is a method of problemsolving for obtaining knowledge, both pro positional knowledge and knowhow.

This approach can be seen as one of the two basic approaches to problemsolving and is contrasted with an approach using insight and theory.

In trial and error, one selects(or, generates)a possible answer, applies it to the problem and, ifitis not successful, selects(or generates)another possibility that is subsequently tried. The process ends when a possibility yields a solution.

In some versions of trial and error, the option that is a priori viewed as the most likely one should be triedfirst, followed by the next most likely, and soon until a solution is found, or all the options are exhausted. In other versions, options are simply tried at random.

This approach is most successful with simple problems and in games, and is often resorted to when no apparent rule applies. This does not mean that the approach need be careless, for an individual can be methodical in manipulating the variables inaneffort to sort through possibilities that may result in success. Nevertheless, this method is often used by people who have little knowledge in the problem area.

Trial and error has a number of features: solution-oriented: trial and error makes no attempt to discover why a solution works, merely that it is a solution.

problem-spec ific: trial and error makes no attempt to generalize a solution to other problems.

non-optimal: trial and error is an attempt tofindasolution, not all solutions, and not the best solution.

needs little knowledge: trial and error can proceed where there is little or no knowledge of the subject.

Forexample, trial and error has traditionally been the main method offinding new drugs, suchas antibiotics. Chemists simply try chemicals at random until they findonewiththedesiredeffect.

The sci ent ificmethodcanberegardedascontain ing an element of trial and error in its formulation and testing of hypot he ses. Also compare genetic algorithms, simulated anne a ling and reinforcement learning–all varieties of search which apply the basic idea of trial and error.

Biological Evolution is also a form of trial and error. Randommutations and sexual genetic variations can be viewed as trials and poor reproductivefitnessas the error. Thus after along time‘knowledge’of well–adapt edge no mes accumulates simply by virtue of them being able to reproduce.

Bogosortcanbeviewedasatrialand error approach to sorting a list. In mathematics the method of trial and error can be used to solve formulae–it is as lower, less precise method than algebra, but is easier to understand. 33 Brain storming is a creativity technique of generating ideas to solve a problem. The main result of a brain storm session maybe a complete solution to the problem, a list of ideas for an approach to a subsequent solution, or a list of ideas resulting in a plan tofindasolution. Brain storming was originated in 1953 in the book‘Applied Imagination’ by Alex Osborn, an advertising executive. Other methods of generating ideas are individual ideation and the morphological analysis approach.

Project Management – identifying client objectives, risks, deliver a bles, work packages, resources, roles and responsibilities, tasks, issues

Business planning–develop and improve the product idea. Trial preparation by attorneys. Brain storming can be done either individually or in a group. In group brain storming, the participants are encouraged, and often expected, to share their ideas with one another as soonas they are generated. Complexproblemsorbrainstorm sessions with a diversity of people maybe prepared by a chairman. The chairman is the leader and facilitator of the brain storm session.

The key to brain storming is to not interrupt the thought process. As ideas come to mind, they are captured and stimulate the development of better ideas. Thus a group brain storm session is best conducted in a moderate–sized room, and participants sit so that they can all look at each–other.Aflipchart, blackboard, or overhead projector is placed in a prominent location. The room is free of telephones, clocks, or any other distractions. 34 Morphological analysis was designed for multi-dimensional, non-quantifiable problems where causal modelling and simulation do not function well or at all. Fritz Zwickydevelopedthisapproachtoseeminglynon-reducible complexity[Zwi 69]. Using the technique of cross consistency assessment(CCA), the system however does allow for reduction, not by reducing the number of variables involved, but by reducing the number of possible solutions through the elimination of the illogical solution combinations in a grid box. 35 The technique of focal objects for problem solving involves synthesizing the seemingly non–matchingcharacteristicsofdifferent objects into something new.

Forexample, to generate new solutions to gardening take some ideas at random, such swimming and a couch, and invent ways for them to merge. Swimming might be used withthe idea of gardening to create a plant oxygen tank for underwater divers. Acouchmightbeusedwiththeideaofgardeningtoinventnewgenesthatwould grow plants into the shape of a couch. The larger the number of diverse objects included, the greater the opportunity for inventive solutions.

Another way to think of focal objects is as a memory cue: ifyou’retry ing tofind allthedifferent ways to use a brick, give yourself some random‘objects’(situations, concepts, etc.)and see if you canfindause. Given‘blender’, forexample, I would try to think of all the ways a brick could be used witha blender(as a lid?). Another concept for the brick game:findpatternsinyoursolutions, and then break those patterns. If you keepfindingwaystobuildthingswithbricks, think of ways to use bricks that don’t involve construction. Pattern–breaking, combined with focal object cues, can lead to very divergent solutions. 36 Lateral thinking is a term coined by Edwardde Bono[Bon 73], a Maltesepsychologist, physician, and writer, although it may have been an idea whose time was ready; thenotionoflateraltruthisdiscussedbyRobertM. Pir sig in Zen and the Art of Motorcycle Maintenance. deBonodefines Lateral Thinking as methods of thinking concerned with changing concepts and perception. Forexample:

It took two hours fort women to dig a holefivefeetdeep. How deep would it have been if ten men had dug the hole for two hours?

The answer appears to be 25 feet deep. This answer assumes that the thinker has followed asimple mathematical relationship suggested by the description given, but we can generate some lateral thinking ideas about what affectsthesizeofthehole which may lead tod ifferentanswers:

A hole may need to be of a certain size or shape so digging might stop early at a required depth.

The deeper a hole is, themoreeffortisrequiredtodigit, sincewastesoilneeds to be lifted higher to the ground level. There is a limit to how deep a hole can be dugbymanpowerwithoutuseofladdersorhoistsforsoilremoval, and 25 feet is beyond this limit.

Deeper soil layers maybe harder to dig out, or we may hit bedrock or the water table.

Each man digging needs space to use a shovel.

It is possible that with more people working on a project, each person may become less efficient due to increased opportunity for distraction, the assumption he can slack off,more people to talk to, etc.

More men could work in shifts to dig faster for longer. Therearemoremenbutaretheremoreshovels? Thetwohoursdugbytenmenmaybeunderdifferentweatherconditionsthan the two hours dug by two men.

Rain couldfloodtheholetopreventdigging. Temperature conditions may freeze the men before theyfinish. Would we rather have5 holes each5 feet deep? The two men maybe an engineering crew with digging machinery. What if one man in each group is a manager whowill not actually dig? The extra eight men might not be strong enough to dig, or much stronger than thefirst two.

The most useful ideas listed above are outside the simple mathematics implied by the question. Lateral thinking is about reasoning that is not immediately obvious and about ideas that may not be obtainable by using only traditional step–by–step logic.

Techniques that apply lateral thinking to problems are characterized by the shifting of thinking patterns away from entrenched or predictable thinking to new or unexpected ideas. Anew idea that is the result of lateral thinking is not always a helpful one, but when a good idea is discovered in this way it is usually obvious in hindsight, which is a feature lateral thinking shares with a joke.

Lateral thinking can be contrasted with critical thinking, which is primarily concerned with judging the truth value of statements and seeking error. Lateral Thinking is more concerned with the movement value of statements and ideas, how to move from them to other statements and ideas.

For example the statement ‘cars should have square wheels’ when considered with critical thinking would beevaluated as a poor suggestion, as there are many engineering problems with square wheels. The Lateral Thinking treatment of the same statement would be to see where it leads. Square wheels would produce predictable bumps. If bumps can be predicted then suspension can be designed to compensate. Anotherwaytopredictbumpswouldbealaserorsonaronthefrontofthecar examining the road surface ahead. This leads to the idea of active suspension with a sensor on the car that has normal wheels. The initial statement has been left behind. 37 Recall that a mind map is a diagram used to represent words, ideas, tasks or other items linked to and arranged radiallyaround a central key word or idea. It is used to generate, visualize, structure and classify ideas, and as an aid in study, organization, problemsolving, and decisionmaking.

It is an image–centered diagram that represents semantic or other connections between portions of information. By presenting these connections in a radial, nonlinear graphicalmanner, it encourages a brain storming approach to any given organizational task, eliminating the hurdle of initially establishing an intrinsically appropriate or relevant conceptual framework to work within.

A mind map is similar to a semantic network or cognitive map but there are no formal restrictions on the kinds of links used.

Most often the map involves images, words, and lines. The elements are arranged intuitively according to the importance of the concepts and they are organized into groupings, branches, or areas. The uniform graphic formulation of the semantic structure of information on the method of gathering knowledge, may aid recall of existing memories.

People have been using image centered radial graphic organization techniques referredtovariablyasmentalorgenericmindmapsforcenturiesinareassuchas engineering, psychology, and education, although the claim to the origin of the mind maphasbeenmadebyaBritishpopularpsychologyauthor, Tony Buzan.

The mind map continues to be used in various forms, and for various applications including learning and education(where it is often taught as’Webs’or’Webbing’), planning and in engineering diagram ming.

When compared with the earlier original concept map(which was developed by learning experts in the 1960s) the structure of a mind map is a similar, but simplified, radial by having one central keyword.

Mind maps have many applications in personal, family, educational, and business situations, including note-taking, brain storming(where in ideas are inserted into the map radially around the center node, withouttheimplicitprioritizationthatcomes from hierarchy or sequential arrangements, and where in grouping and organizing is reserved for later stages), summarizing, rev is ing and general clarifying of thoughts. Forexample, one could listen to a lecture and takedown notes using mind maps for the most important points or key words. One can also use mind maps asa mnemonic technique or to sort out acomplicated idea. Mind maps are also promoted as away to collaborate in color pen creativity sessions. 38 Recall that analogy is either the cognitive process of transferring information from a particular subject (the analogue or source) to another particular subject (the target), or a linguistic expression corresponding to such a process. In a narrower sense, analogy is an inference or an argument from a particular to another particular, as opposed to deduct i on, in duc tion, and abduction, where at least one of the premises or the conclusion is general. The word analogy can also refer to the relation between the source and the target themselves, which is often, though not necessarily, a similarity, as in the biological notion of analogy.

Niels Bohr’smodeloftheatommadeananalogybetweentheatomandthe solar system. Analogy plays a sign ificant role in problemsolving, decisionmaking, perception, memory, creativity, emotion, explanation and communication. It lies behind basic tasks suchas the i dent ificat i on of places, objects and people, forexample, in face perception and facial recognition systems. It has been argued that analogy is ‘the core of cognition’. Specificallyanalogicallanguagecomprisesexemplificat i on, comparisons, metaphors, similes, all ego ries, andparables, but not me tony my. Phrases like and soon, and the like, asif, and the very word like also rely on an an a logical understanding by the receiver of a message including them. Analogy is important not only in ordinary language and commonsense, whereproverbsandidiomsgive many examples of its application, but also in science, philosophy and the humanities. The concepts of association, comparison, correspondence, homomorphism, iconicity, isomorphism, mathematicalhomology, metaphor, morphologicalhomology, resemblance, and similarity are closely related to analogy. Incognitivelinguistics, the notion of conceptual metaphor maybe equivalent to that of analogy.

Analogy has been studied and discussed since classical antiquity by philosophers, scientists and lawyers. The last few decades have shown a renewed interest in analogy, most notable in cognitive science.

Decisionmaking is possibly one of the most important aspects of human activities. And an adequate situation awareness(SA)is one of the necessary elements for optimal decision making [SI 09]. Recall from The Law of Mental Training, that the most general SA–definit i on (due toM. Endsley) reads: “Situation awareness is the perception of the elements in the environment within a volume of time and space, the comprehension of their meaning, and projection of their status

The logical and mathematical tradition speaks of an arrow, homomorphism, mapping, or morphism from what is typically the more complex domain or source to what is typically the less complex codomainor target, using all of these words in the sense of mathematical category theory.

The tradition that appears to be more common in cognitive psychology, literary theory, and specialization swi thin philosophy outside of logic, speaks of a mapping from what is typically the more familiar area of experience, the source, to what is typically the more problematic area of experience, the target. 39 Research is often described as an active, diligent, and systematic process of inquiry aimed at discovering, interpreting, and rev is ing facts. This intellectual investigation produces a greater understanding of events, behaviors, or theories, and makes practical applications through laws and theories. The term research is also used to describe a collection of information about a particular subject, and is usually associated with science and the sci ent ificmethod.

The word research derives from Middle French; its literal meaning is ’to investigate thoroughly’.

Thomas Kuhn, in his book‘The Structure of Sci ent ificRevolutions’, traces an interesting history and analysis of the enter prize of research.

Basic research (also called fundamental or pure research) has as its primary objective the advancement of knowledge and the theoretical understanding of the relations among variables. It is exploratory and often driven by the researcher’scuriosity, interest, or hunch. It is conducted without any practical end in mind, although it may have unexpected results pointing to practical applications. The terms“basic” or“fundamental”indicate that, through theory generation, basic research provides the foundation for further, sometimes applied research. As there is no guarantee of short-term practical gain, researchers oftenfinditdifficult to get funding for basic research. Research is a subset of invention.

Applied research is done to solve spec ific, practical questions; its primary aim is not to gain knowledge for its own sake. It can be exploratory, but is usually descriptive. It is almost always done on the basis of basic research. Applied research can be carried out by academic or industrial institutions. Often, an academic institution suchas a university will have a spec ificappliedresearchprogramfunded by an industrial partner interested in that program. Common areas of applied research include electronics, informatics, computer science, material science, process engineering, drugdesign...

in the near future.” It is obvious that the environment and the characteristics of the space within it are dynamic and nonlinear. Therefore, SAisbothdynamic and nonlinear. Further, thereisalimitonmaximumconceivableSA. Similar to any natural system, increase in SA cannot continue forever despite a continuous supply of new useful information. The factors that limit this continuous increase inSAinclude:(i)limitations inhuman perception and memory capacity, memory loss, mental fatigue and vigilance; and(ii)cognitive distractions. Based on these two assumptions, Sharma and Ivancevic (2009) have defined both a simple, intuitiveSA-model and i tsn−dimensionalgeneralization.

whereSn+1isthenextSA, BistheSAgrowth-rateparameter, representing the amount of useful informationflowperunittime, SnisthepresentSA, and (1 −Sn)keepsthemaximumconceivableSAwithinlimitsbyaccommodating the influenceoffactorssuchasperceptuallimitations, memory capacity, memory loss, mental fatigue, vigilance and cognitive distractions. The model(23.6) describes the next SA, Sn+1, as a function of the present SA, Sn, and the growthrateparameterB. In this model, the term(1 −Sn)keepstheSAwithinlimits becauseasSnincreases,(1−Sn)decreases. In this simple model, SAisexpressed as a fraction between zero and 1. ZerorepresentsanabsenceofanySA, whereas 1 represents the maximum conceivable SA. It is worthwhile noting that the growth-rateparameterBisanimportantfeatureofthemodel.

The nonlinear dynamical characteristics of SA, embodied in the simple model (23.6) show that the supply rate of useful information is an important parameter that influencesthequalityofSA. A higher rate of information supply does not necessarilyimproveSA. On the contrary, if the supply rate is increased beyond a certain threshold, SAstartsbifurcating. A further increase in this rate results inSAbecomingchaotic. SuchbifurcationandchaoticbehaviorresultinSA degradation. Therefore, itisarguedhereinthatthehighestlevelofSAcanbe achieved by maintaining the supply rate of useful information just below the thresholdwhereSAfirststartsbifurcating.

ThesimplisticSA–model(23.6)is further expanded to accommodate various SAdimensions, suchas perceptual limitations, memory capacity, memory loss, mental fatigue, vigilance and cognitive distractions. The basic model (1) is generalized into an n−dimensional SA dynamics model, defined by the vector–matrix evolution equation in both discrete time(denotedbyindext)and continuous time(denoted by a continuous–time growth–ratefunctionBik(t)):

whereupperindextdenotesadiscretetime–evolution step, while lower indices i, k = 1, ..., n represent various SA dimensions. In (23.7), St i represents the vector withnormalizedSAdimensionsatadiscretetime–stept:

Bik(t)is then × ngrowth-parameter matrix with time-varying elements:

ThecomprehensivestudyoftheSA–models(23.6)and(23.7)by Sharma and Ivancevic [SI 09] has suggested that the quality of SA depends on the growth-rate parameter, which represents the amount of useful information supplied per unit time. Further, the study showed theeffectofvariationofthisparameteronSA usinglogisticmapsand a bifurcation diagram. At the beginning, the increase in the parameter value resulted in a steady growth inSA. However, once the value attained a certain threshold, SAstartedbifurcating. An additional increase in this value resulted in more frequent bifurcations. When the value was raised even more, SAstartedbehavinginachaoticmanner. A further increase in the value resulted in bifurcation and chaos again. The presence of such complex chaotic behaviorwasre-affirmedbycomputingtheLyapunovexponentforthelogistic modelofSAdynamics. One possible explanation for such bifurcation and chaos may be that the supply rate of useful information may have exceeded the human cognitive limitations. During bifurcation and chaos, SAis in an unstable state and keeps oscillating. ThestudysuggestedthatsuchunstableSArepresents uncertainty and, therefore, is undesirable. The results suggest that the supply of useful information at a higher rate(more information)does not always enhance SA. Instead, such supply may often disrupt or reduce SA. Therefore, it is argued that the growth-rate parameter just below thefirst bifurcation threshold should beselectedtoachievethehighestSAlevel. This is animportantfindinginSA, which may assist in improving theeffectivenessofhumandecisionmaking. The findinghasawideapplicationrang ing from commercial to military decision making. This new knowledge is invaluable to the individuals responsible for managing the community, enter prize or the nation to optimize their decisions. The knowledge is also useful in the design of command and control systems.

This knowledge is also crucial in modern tennis: it allows you to focus on the ball and yet to know what your opponent is doing or planning to do.

In this section we present several aspects of the current research in quantum brain.

Recent developments/effortstounderstandaspectsofthebrainfunctionatthe sub–neural level are discussed in[II 08c]. Microtubules(MTs), proteinpolymers constructingthecytoskeletonofaneuron, participate in a wide variety of dynamical processes in the cell. OfspecialinterestforthissubsectionistheMTs participation in bio–information processes such as learning and memory, by possessing a well–known binaryerror–correcting code[K 1(13, 26, 5)] with 64 words. In fact, MTsandDNA/RNAareuniquecellstructuresthatpossessacodesystem. ItseemsthattheMTs’code system is strongly related to akind of mental code in the following sense. TheMTs’periodicparacrystallinestructuremake them able to support a super position of coherent quantum states, as it has been recentlyconjecturedbyHameroffand Penrose[HP 96], representing an external or mental order, forsufficienttimeneededforefficientquantumcomputing [II 07b, II 08c].

Livingorganismsarecollectiveassembliesofcellswhichcontaincollective assembliesoforganizedmaterial, includingmembranes, organelles, nuclei, and the cytoplasm, the bulk interior medium of living cells. Dynamic rearrangements of the cytoplasmwithineucaryoticcells, the cells of all animals and almost all plants on Earth, account for their changing shape, movement, etc. This extremely important cytoplasmic structural and dynamical organization is due to the presence of networks of interconnected protein polymers, which are referred to as the cytosceletondue to their bone–like structure [HP 96]. The cytoskeleton consists of MT’s, act in micro-filaments, intermediatefilaments and an organizing complex, the centro some with its chief component the centriole, built from two bundles of microtubulesinaseparatedTshape. Parallel–arrayed MTsareinterconnected by cross–bridging proteins(MT–Associated Proteins: MAPs)tootherMTs, organelle filaments and membranes to form dynamic networks[HP 96]. MAPs may be contractile, structural, or enzymatic. Avery important role is played by contractile MAPs, like dynein and kinesin, through their participation in cell movements as well as in intra-neural, or axoplasmictransportwhichmoves material and thus is of fundamental importance for the maintenance and regulation of synapses(see, e.g.,[Ecc 64]). ThestructuralbridgesformedbyMAPsstabilize MTsandpreventtheirdisassembly. TheMT–MAP‘complexes’orcyto-sceletal networks determine the cell architecture and dynamic functions, such amitosis, or cell division, growth, differentiation, movement, and for us here the very crucial, synapse formation and function, all essential to the living state. It is usually saidthatmicrotubulesareubiquitousthroughtheentirebiology[HP 96].

MTsarehollowcylinderscomprisedofanexteriorsurfaceofcross–section diameter 25 nm (1 nm = 10−9 meters) with 13 arrays (pro tofilaments) of protein dimerscalledtubulines. The interior of the cylinder, of cross–section diameter 14nm, contains ordered water molecules, which implies the existence of an electric dip ole moment and an electricfield. Thearrangementofthedimers is such that, if one ignores their size, they resemble triangular lattice son the MTsurface. Eachdimerconsistsoftwohydrophobicproteinpockets, and has anunpairedelectron. There are two possible positions of the electron, calledα and βcon formations. When the electron is in the β−con formation there is a 29o distortion of the electric dip ole moment as compared to theαcon formation.

In standard models for the simulation oftheMTdynamics(see[II 07b, II 08c] and references there in), the‘physical’DOF–relevant for the description of the energy transfer–is the projection of the electric dip ole moment on the longitudinal symmetry axis(x−axis)oftheMTcylinder. The 29 odistortion oftheβ−conformationleadstoadisplacementunalongthex−axis, which isthustherelevantphysicalDOF. There has been speculation for quite some timethatMTsareinvolvedininformationprocessing: it has been shown that the particular geometrical arrangement (packing)of thetubulinproto-filaments obeys an error–correctingmathematicalcodeknownastheK 2(13, 26, 5)− code. Error correcting codes are also used in classical computers to protect against errors while in quantum computers special error correcting algorithms are used to protect against errors by preserving quantumcoherenceamongqubits.

Information processing occurs viainteractionsamongtheMTproto–filament chains. The system maybe considered assimilartoamodelofinteractingIsing chains on a triangular lattice, thelatterbeingdefined on the plane stemming fromfilletingopenandflatteningthecylindricalsurfaceofMT. Classically, the various dimers can occur in either α or βcon formations. Each dimer is influenced bytheneighboringdimersresultinginthepossibilityofatransition. This is the basis for classical information processing, which constitutes the picture of a (classical)cellular auto mat on.

ThequantumnatureofanMTnetworkresultsfromtheassumptionthateach dimerfindsitselfinasuperpositionofαandβcon formations. Viewed as a two–state quantum mechanical system, theMTtubulindimerscoupletoconformational changes with 10−9−10−11 sec transitions, corresponding to an angular frequencyω∼O(1010) −O(1012)Hz(see[II 07b, II 08c]).

Part 12 · pp. 335–364

The quantum computer character of the MT network [Pen 89] results from the assumptionthateachdimerfindsitselfinasuperpositionofαandβcon formations[Ham 87]. Thereisamacroscopic coherent state among the various chains, which lasts for O(1 sec) and constitutes the‘pre conscious’ state. The interaction of the chains with(non–criticalstringy)quantum gravity, then, induces self– collapse of the wave function of the coherent MTnetwork, resulting in quantum computation[II 07b, II 08c].

withxaspatialcoordinatealongthelongitudinalsymmetryaxisoftheMT. Thereisatimevariabletduetofluctuationsofthedisplacementsu(x)asa resultofthedipoleoscillationsinthedimers.

Theeffectsoftheneighboringdimers(including neighboring chains)canbe phenomenologicallyaccountedforbyaneffectivepotentialV (u). In the kink– sol it on model 40 a double–well potential was used, leading to a classical kink solutionfortheu(x, t)field. More complicated interactions are allowed in this picture, wheremoregenericpolynomialpotentialshavebeenconsidered.

The effects of the surrounding water molecules can be summarized by a viscous forcetermthatdampsoutthedimeroscillations, withγdetermined phenom eno logically at this stage. This friction should be viewed as an environment aleffect, which however does not lead to energy dissipation, as a result of then on–trivialsolitonicstructureoftheground–state and then on–zero constant force due to the electric field. This is a wellknown result, directly relevant to energy transfer in biological systems.

whereξ=x −vt, u′(ξ)=du/dξ,visthevelocityofthesoliton,ρ∝γ[II 07b, II 08c], andP(u)isapolynomialinu, of a certain degree, stemming from the variationsofthepotentialV (u)describinginteractionsamongtheMTchains. In the mathematical literature there has been a class ificat i on of solutions of equations of this form. For certain forms of the potential the solutions include kinksolitonsthatmayberesponsiblefordissipation–free energy transfer in biological cells:

u(x, t) ∼c1 (tanh[c2(x −vt)] + c3) , (23.11)

40 Recallthatkinksaresolitary(non–dispersive)waves arising in various1D (bio)physical systems.

wherec 1, c 2, c3 areconstantsdependingontheparametersofthedimerlattice model. There are sol i tons of the form u(x, t) = c′ i, i = 1, . . . 3 are appropriate constants.

A semi classic alquantizationofsuchsolitonic states has been considered by Ellis etal. (see [II 07b, II 08c]). The result of such a quantization yields a modified sol it on equation for the(quantum corrected)fielduq(x, t):

xuq(x, t) + M(1)[uq(x, t)] = 0, (23.12)

∂z2U(n)(z)|z=uq(x,t), U(n)≡dnU/dzn.

The quantity Udenotesthepotentialoftheoriginalsoliton Hamilton i an, and G(x, y, t) is a bilocal field that describes quantum corrections due to the modified bos onfieldaroundthesoliton. ThequantitiesM(n)carry information about the quantum corrections. Forthekinksoliton(23.11)the quantum corrections (23.12)have been calculated, thereby providing us with a concrete example of a large–scale quantum coherent state.

Atypicalpropagationvelocityofthekinksolitonsisv∼2m/sec, although, modelswithv∼20 m/sec have also been considered. This implies that, for moderately long microtubules of length L ∼10−6m, such kinks transport energy without dissipation in

Such time scales are comparable to, or smaller in magnitude than, thedecoherencetimescaleoftheabove–described coherent(solitonic)statesuq(x, t). This implies the possibility that fundamental quantum mechanical phenomena may then be responsible for friction less energy(and signal)transfer across micro tubular arrangements in the cell(see[II 07b, II 08c]).

Neuro–and psycho–dynamics have its physical behaviorbothonthemacroscopic, classical, inter–neuronal level[II 07d], and on the microscopic, quantum, intra–neuronal level[II 07b, II 08a]. Onthemacroscopiclevel, various models of neural networks(NNs, for short)have been proposed as goal–oriented models of the spec ificneuralfunctions, like for instance, function–approximation, pattern–recognition, class ificat i on, or control. In the physically–based, Hopfield– typemodelsofNNs[Hop 82, Hop 84] the information is stored as a content– addressable memory in which synapticstrengths are modified after the Hebbian rule(see[Heb 49]). Its retrieval is made whenthenetworkwiththesymmetric couplings works as the point–attractor with the fixed points. Analysis of both activation and learning dynamics of Hopfield–Hebbian NNs using the techniques of statistical mechanics(see, e.g.[II 08b]), provides us with the most important information of storage capacity, role of noise and recall performance [II 06b, II 07d].

Conversely, an indispensable role of quantum theory in the brain dynamics was emphasized recently. On the general microscopic intra–cellular level, energy transfer across the cells, without dissipation, had been first conjectured to occur in biological matter by [FK 83]. The phenomenon conjectured by them was based on their1 Dsuperconductivitymodel: in one dimensional electron systems with holes, the formation of solitonic structures due to electron–hole pairing results in the transfer of electric current without dissipation. In a similar manner, Fr¨olich and Kremerconjecturedthatenergyin biological matter could be transferred without dissipation, if appropriate solitonic structures are formed inside the cells. This idea has lead theorists to construct various models for the energy transfer across the cell, based on the formation of kink classical solutions.

The interior of living cells is structurally and dynamically organized by cytoskeletons, i.e., networks of protein polymers. Of these structures, microtubules(MTs, for short)appear to be the most fundamental. Their dynamics has been studied by a number of authors in connection with the mechanism responsible for dissipation-free energy transfer. Hameroffand his colleagues [HP 96, HHT 02] have conjectured another fundamental role for the MTs, namely being responsible for quantum computations in the human neurons. Penrose [Pen 89, Pen 94, Pen 97, Pen 98] further argued that the latter is associated with certain aspects of quantum theory that are believed to occur in the cytoskeleton MTs, in particular quantum super position and subsequent collapse of the wave functionofcoherentMTnetworks. IthasbeensuggestedthattheneuralMTs are the micro sites for the emergence of stable, macroscopicquantumcoherent states, i dent ifiablewiththepreconscious states; stringy–quantum space-time effects trigger an organized collapse of the coherent states down to a spec ificor conscious state.

In particular, MTsinthecytoskeletonsofeukaryoticcellsprovideawide range of micro–skeletal and micro–muscular functionalities. Some evidence has indicated that they can serve as a medium for intracellular signaling processing. Fortheinherentsymmetrystructuresandtheelectricpropertiesoftubulin dimers, themicrotubule(MT)was treated as a1 Dferroelectricsystem(see [II 07b, II 08c]). Thenonlineardynamicsofthedimerelectricdipoleswasdescribedbyvirtueofthedouble–well potential and the physical problem was further mapped onto the pseudo–spin system, takingintoaccounttheeffectof the external electricfieldontheMT.

More precisely, MTsare polymers of tubulinsubunits(dimers)arranged on a hexagon all at tice. Eachtubulindimercomprisestwomonomers, theα−tubulin andβ−tubulin, and can be found in two states. Inthefirst state a mobile negative charge is located into theα−tubulinmonomerandinthesecondinto theβ−tubulinmonomer. Each tubulindimerismodelledasanelectricaldipole coupled to its neighbors by electrostatic forces. The location of the mobile charge in each dimer depends on the location of the charges in the dimer’s neighborhood. Mechanicalforcesthatactonthemicrotubuleaffectthedistancesbetweenthe dimersandaltertheelectrostaticpotential. Changes in this potential affect themobilenegativechargelocationineachdimerandthechargedistribution inthemicrotubule. Theneteffectisthatmechanicalforcesaffectthecharge distributioninmicrotubules. Various models of the mind have been based on theideathatneuronMTscanperformcomputation. From this point of view, information processing is the fundamental issue for understanding the brain mechanisms that produce consciousness. Thecytoskeletonpolymerscouldstore and process information through their dynamic coupling mediated by mechanical energy. Theproblemofinformationtransferandstorageinbrainmicrotubules was analyzed, considering them as a communication channel. Therefore, we have two space-timebiophysicalscalesofneuro-and psycho-dynamics: classical and quantum. Naturally the question arises: are these two scales somehow interrelated, is there a space-time self-similarity between them?

The purpose of the present section is to prove the formal positive answer to the self-similarity question. We try to describe neurodynamics on both physical levels by the unique form of asingle equation, namelyopenLiouvilleequation: NN–dynamics using its classical form, and MT–dynamics using its quantum form in the He is enberg picture. If this formulation is consistent, that would prove the existenceoftheformalneuro-biological space-time self-similarity. Even more, this would prove the existence of a Neurodynamical Law, which acts on different scales of brain’sfunctioning.

SupposethatonthemacroscopicNN–levelwehaveaconservativeHamiltonian system acting in a 2ND symplectic phase space T∗Q = {qi(t), pi(t)}, i = 1 . . . N (which is the cotangent bundle of the NN–configuration manifold Q = {qi}), with a Hamilton i an function H= H(qi, pi, t) : T∗Q × R →R(see[II 06a, II 06c]). The conservative dynamics i sdefinedbyclassicalHamilton’scanonicalequations:

˙qi= ∂piH –contravariant velocity equation , ˙pi= −∂qiH –covariant force equation , (23.14)

(here and henceforth over dot denotes the total time derivative). Within the frameworkoftheconservativeHamiltoniansystem(23.58)we can apply the formalismofclassicalPoissonbrackets: foranytwofunctionsA=A(qi, pi, t) andB=B(qi, pi, t)their Pois son bracket is(using the summation convention) defined as[II 06c, II 07e]

Forexample, if A=ρ(qi, pi, t)is a density function of ensemble phase–points (or, aprobabilitydensitytoseeastatex(t)=(qi(t), pi(t))of ensemble at a momentt), then equation

Conserved quantity here is the Hamilton i an function H= H(qi, pi, t), which the sum of kinetic and potential energy. Forexample, incaseofanNDharmonic oscillator, we have the phase space M= T∗RN≃R2N, with thesymplectic form ω= dpi ∧dqiandHamiltonian(total energy)function:

˙qi= pi, ˙pi= −δijqj, (whereδijistheKroneckersymbol). (23.18) whichimplieasthattheparticularfunctionsf= pipj + qiqjandg= piqj+ pjqi

(for i, j=1, ..., N)–are constants of motion. This system is integr able in an open set ofT∗Rnwith Nintegrabilityfunctions:

N+ (qN)2.

Weperformtheformalquantization of the conservative equation(23.16)inthe He is enberg picture: allvariablesbecomeHermitianoperators(denoted by‘∧’), thesymplecticphasespaceT∗Q={qi, pi}becomestheHilbertstatespace H = Hˆqi⊗Hˆpi(whereHˆqi= Hˆq 1⊗... ⊗HˆqNandHˆpi= Hˆp 1⊗... ⊗HˆpN ), the classical Poisson bracket[ ,]becomesthequantumcommutator{ ,}multiplied by−i(in normal units)[II 08a] denotesthevonNeumanndensitymatrixoperator, where each quantum state |Ψa>occurswithprobabilityP(a);ˆρ=ˆρ(ˆqi, ˆpi, t)is closely related to another von Neumannconcept: entropy

We now move to the open(non conservative)system: onthemacroscopicNN– level the opening operation equals to the adding of a covariant vector of external (dissipativeand/or motor)forcesFi=Fi(qi, pi, t)to(the right-hand-side of) thecovariantHamilton’sforceequation, sothatHamilton’sequationsobtain the open(dissipativeand/or forced)form[II 06a, II 06c]:

In the framework of the open Hamilton i an system(23.21)dynamics of any function A(qi, pi, t)i sdefined by the open(dissipativeand/or forced)evolution equation:

In particular, if A = ρ(qi, pi, t) represents the density function of ensemble phase– points then its dynamics is given by the open (dissipative and/or forced) Liouville equation[II06c,II07e]:

Equation(23.23)representstheopenclassicalmodelofourmicroscopicNNdynamics.

For example, in case of our ND oscillator, Hamilton i an function (23.17) is not conservedanymore, thecanonicalequations(23.18)become and the system is not integr able anymore.

where σi= σi(t) are the continual neuronal variables of ith neurons (representing either membrane action potentials in case of GRN, or oscillator phases in case of CNO); Jijareindividualsynapticweights; fi= fi(σi, Jij)are the deterministic forces(given, inGRN-case, by whereζi(t)denoteuncorrelatedGaussiandistributedrandomforcesandthe parameterTcontrolstheamountofnoiseinthesystem, rangingfromT=0 (deterministic dynamics)toT= ∞(completely random dynamics).

More convenient description of the neural random process (23.24) is provided bytheFokker-Planckequationdescribingthe time evolution of the probability densityP(σi)[II 07a, II 08b]

Now, in the case of deterministic dynamics T= 0, equation (23.25) can be easily putintotheformoftheconservativeLiouvilleequation(23.16), by making the substitutions:

whereH=H(σi, Jij). Further, wecanformallyidentifythestochasticforces, i.e., the second-order noise-term T iρwithF i[ρ, σi] , to get the open Liouvilleequation(23.23).

Therefore, on the NN–level deterministic dynamics corresponds to the conservative system(23.16). Inclusionofstochasticforcescorrespondstothesystem opening(23.23), implyingthemacroscopicarrowoftime.

whereˆFi=ˆFi(ˆqi, ˆpi, t)representsthecovariantquantumoperatorofexternal frictionforcesintheHilbertstatespaceH = Hˆqi⊗Hˆpi.

Equation (23.26) represents the open quantum–decoherence model of ourmicroscopicMT–dynamics.

where the target–space density matrixˆρ(ˆqi, ˆpi)is viewed as a function of coordinatesˆqithatparameterizethecouplings of the generalizedσ−models on the bosonicstringworld-sheet, andtheirconjugatemomentaˆpi, whileˆgij=ˆgij(ˆqi) is the quantum operator of the positive definitemetric in the space of couplings. Therefore, the covariant quantum operator of external friction forces is in EMN– formulation given asˆFi(ˆqi,ˆ˙qi) =ˆgijˆ˙qj.

Equation (23.27) establishes the conditions under which a large–scale coherent state appearing in the MT-network, which can be considered responsible for loss– freeenergytransferalongthetubulins.

duetothepositivedefinitenessofthemetricˆgij, and thus automatically and naturally implying a microscopic arrow of time[II 08a, II 08c].

BoththemacroscopicNN–equation(23.23)andthemicroscopicMT–equation (23.26)havethesameopenLiouvilleform, which implies the arrow of time [II 07e, II 08a]. Thesedemonstratestheexistenceofaformalneuro-biological space-time self–similarity.

Therefore, wehavedescribedneuro–and psycho–dynamicsofbothNNand MTensembles, belongingtocompletelydifferentbiophysicalspace-time scales, brain’s neural networks and brain’s microtubules, by the unique form of the open Liouvilleequation, which implies the arrow of time. In this way the existence of theformalneuro-biological space-time self-similarity has been proved.

This proof impliestheexistenceofauniqueNeurodynamicalLaw, which acts on different scales of brain’s functioning. In both cases of macroscopic continuous neural networks and microscopic discretemicrotubuleswehaveaprocesswhich isexpressibleintheopenLiouvilleequationform.

The conservative brain model was formulated within the framework of the quantumfieldtheory(QFT)by[RU 67]. The motivations at the basis of the formulation of the quantum brain model by Umezawa and Ricciarditracebackto the laboratory observations leading Lashley to remark (in 1940) that “masses of excitations...within general fieldsofactivity, without regard to particular nerve cells are involved in the determination of behavior”. In 1960’s, K. Pribram, also motivated by experimental observations, startedtoformulatehisholographic hypothesis(see[Pri 71]). AccordingtoW. Freeman[Fre 90, Fre 91, Fre 92, Fre 96], “information appears indeed in such observations to be spatially uniform in much the way that the information densityisuniforminahologram”. While the activity of the single neuron is experimentally observed inform of discrete and stochasticpulsetrainsandpointprocesses, the‘macroscopic’activity of large assemblyofneuronsappearstobespatiallycoherentandhighlystructuredin phase and amplitude.

Motivated by such an experimental situation, Umezawa and Ricciardiformulatedin[RU 67] the quantum brain model as a many–body physics problem, usingtheformalismofQFTwithspontaneousbreakdownofsymmetry(which had been successfully tested in condensed matter experiments). Such a formal ism provides the only available theoretical tool capable to describe long–range correlations suchas the ones observed in the brain–presenting almost simultaneous responses in several regions to some external stimuli. The understanding of these long–range correlations in terms of modern biochemical and electro chemical processes is still lacking, which suggests that these responses could not be explained in terms of single neuron activity[Pri 71].

Lagrangian dynamicsinQFTis, in general, invariant under some group Gof continuoustransformations, asproposedbythefamousNoethertheorem. Now, spontaneous symmetry breakdown, one of the corner–stonesofHaken’ssynergetics(see[Hak 93]), occurs when the minimum energy state(the ground, or vacuum, state)of the system is not invariant under the full group G, but under one of its sub groups. Then it can be shown that collective modes, the so–called Nambu–Gold stone(NG)bosonmodes, are dynamically generated. Propagating over the whole system, these modes are the carriers of the long–range correlation, in which the order manifests itself as a global property dynamically generated. The long–range correlation modes are responsible for keeping the ordered pattern: they are coherently condensed in the ground state(similar toe.g., inthe crystal case, where they keep the atoms trapped in their lattice sites). The long– range correlation thus forms a sort of net, extending over all the system volume, which traps the system components in the ordered pattern. This explains the “holistic”macroscopiccollectivebehavior of the system components.

More precisely, according to the Gold stone theorem in QFT (see [II 08a, II 08c]), the spontaneous breakdown of the symmetry implies the existence of long–range correlation NG–modes in the ground state of the system. These modes are mass less modes in the infinitevolumelimit, but they may acquire afinite, non-zero massduetoboundaryorimpurityeffects. In the quantum brain model these modes are called dipole–wave–quanta (DWQ). The density of their condensation in the ground states acts as a code classifying the state and the memory there recorded. Stateswithdifferentcodevaluesareunitarilyinequivalentstates, i.e., thereisnounitarytransformationrelatingstatesofdifferentcodes.41

Thecanonicalquantizationprocedureofadissipativesystemrequirestoinclude in the formalism also the system representing the environment (usually the heat bath) in which the system is embedded. One possible way to do that is to depict the environment as the time–reversal image of the system [II 08a, II 08c]: the environment is thus described as the double of the system in the time–reversed dynamics(the system image in the mirror of time).

WithintheframeworkofdissipativeQFT, the brain system is described in terms of an infinitecollectionofdampedharmonicoscillatorsAκ(the simplest prototype of a dissipative system) representing theDWQ. Now, the collection of dampedharmonicoscillatorsisruledbytheHamiltonian[II 08a]:

whereΩκis the frequency andΓκis the damping constant. The˜Aκmodes are the‘time–reversed mirror image’(i.e., the‘mirror modes’)ofthe Aκmodes. They are the doubled modes, representing the environment modes, in such a way that κ generic ally labels their degrees–of–freedom. In particular, we consider the dampedharmonicoscillator(DHO) asasimpleprototypefordissipativesystems(with intention that thus get results also apply to more general systems). The damped oscillator(23.31)isa non–Hamilton i an system and therefore the customarycanonicalquantization procedure cannot be followed. However, one can face the problem by resorting to wellknown tools suchas the density matrixρandtheWignerfunction W= W(x, p, t).

Recall (from the previous sub section) that the density matrix equation of motion, i.e., quantum Liouvilleequation, is given by

41 WeremarkthatthespontaneousbreakdownofsymmetryispossiblesinceinQFT there exist infinitely many ground states or vacua which are physically distinct (technically speaking, they are“unitarilyinequivalent”). In quantum mechanics(QM), on the contrary, all the vacuaarephysicallyequivalent and thus there cannot be symmetry breakdown.

W(x, y, t) e(−ipy)dy.

i ∂tW(x, y, t)=HoW(x, y, t), with

In this way, Vitielloconcludedthatthe density matrix and the Wignerfunction formalism required, even in the conservative case (with zero mechanical resistance γ),the introduction of a‘doubled’set of coordinates, x±, or, alternatively, x and y. One may understand this as related to the introduction of the‘couple’of indices necessary to label the density matrix elements(23.34).

Let us now consider the case of the particleinteractingwith a thermal bath at temperatureT . Letfdenotetherandomforceontheparticleatthepositionx due to the bath. The interaction Hamilton i an between the bath and the particle is written as

Now, in the Feynman–Vernonformalism(see [II 08a, II 08c]), theeffectiveaction A[x, y]for the particle is given by where the symbol⟨.⟩denotes the average with respect to the thermal bath; ‘(.)+’and‘(.)−’denote time ordering and anti–time ordering, respectively; the coordinates x±are defined as in (23.33). If the interaction between the bath and thecoordinatex(23.37)wereturnedoff,then the operator fofthebathwould develop in time according to where Hγis the Hamilton i an of the isolated bath (deco up led from the coordinate x).f(t)is then the force operator of the bath to be used in(23.38).

x , are givenintermsoftheretardedandadvancedGreenfunctionsGret(t −s)and Gadv(t −s)by respectively. In (23.39), N(t−s) is the quantum noise in the fluctuating random force given by

2⟨f(t)f(s) + f(s)f(t)⟩.

Equations (23.39–23.41), are exact results for linear passive damping due to the bath. At quantum level no nzeroyaccountsforquantumnoiseeffectsinthe fluctuatingrandomforceinthesystem–environment coupling arising from the imaginary part of the action(see[SVW 95]). When in(23.40)we use are derived. They−oscillator is the time–reversedimageofthex−oscillator (23.31). Fromthemanifoldsofsolutionstoequations(23.31)and(23.43), we could choose those for which they coordinate is constrained to be zero, they simplify to

Thus we get the classical damped oscillator equation from a Lagrangiantheory at the expense of introducing an ‘extra’ coordinate y, later constrained to vanish. Notethattheconstrainty(t) = 0 is not in violation of the equations of motion since it is a true solution to(23.31)and(23.43).

Therefore, thegeneralschemeofthedissipativequantumbrainmodelcan be summarized as follows. The starting point is that the brain is permanently coupled to the environment. Clearly, the spec ificdetailsofsuchacouplingmay be very intricate and changeable so that the yared ifficult to be measured and known. One possible strategy is to average theeffectsof the coupling and represent them, at some degree of accuracy, by means of some‘effective’interaction. Another possibility is to take into account the environmental influenceonthe brain by a suitable choice of the brain vacuum state. Such a choice is triggered by the external input (breakdown of the symmetry), and it actually is the end point of the internal (spontaneous) dynamical process of the brain (self–organization). The chosen vacuum thus carries the signature (memory) of the reciprocal brain– environment influence at a given time under given boundary conditions. A change in the brain–environment reciprocal influence then would correspond to a change in the choice of the brain vacuum: the brain state evolution or ‘story’ is thus the story of the trade of the brain with the surrounding world. The theory should then provide the equations describing the brain evolution‘throughthevacua’, each vacuum foreach instant of time of its history.

The brain evolution is thus similar to a time–ordered sequence of photo grams: each photo gram represents the‘picture’of the brain at a given instant of time. Putting together these photo grams in‘temporal order’one gets a movie, i.e. the story(the evolution)of open brain, which includes the brain–environment interact i oneffects.

The evolution of a memory spec ifiedbyagivencodevalue, sayN, canbe then represented as a trajectory of given initial condition running overtime– dependent vacuum states, denoted by|0(t)⟩N , each one minimizing the free energy functional. These trajectories are known to be classical trajectories in the infinitevolumelimit: transition from one representation to another in equivalent one would be strictly forbidden in a quantum dynamics.

Since we have now two–modes(i.e., non–tildeandtildemodes), the memory state |0(t)⟩Nturns out to bea two–mode coherent state. This is known to bean entangled state, i.e., it cannot be factor i zed into two single–mode states, the non– tildeandthetildeone. Thephysicalmeaningofsuchanentanglementbetween non-tilde and tilde modes is in the fact that the brain dynamics is permanently a dissipativedynamics. Theentanglement, which is an unavoidable mathematical result of dissipation, represents the impossibility of cutting the links between the brain and the external world.42

In the dissipative brain model, noise and chaos turn out to be natural ingredients of the model. In particular, in the infinite volume limit the chaotic behavior of the trajectories in memory space may account for the high perceptive resolution in the recognition of the perceptual inputs. Indeed, smalldifferences in thecodesassociatedtoexternalinputsmayleadtodivergingdifferences in the corresponding memory paths. On the other side, it also happens that codes differing only in afinite number of their components(in the momentum space) may easily be recognized as being the‘same’code, which makes possible that ‘almost similar’ inputs are recognized by the brain as ‘equal’ inputs (as in pattern recognition).

Therefore, the brain maybe viewed as a complex system with(infinitely) many macroscopic configurations (the memory states). Dissipation is recognized to be the root of such a complexity.

Classical physics has provided a strong foundation for understanding brain function through measuring brain activity, modelling the functional connectivity of networksofneuronswithalgebraicmatrices, and modelling the dynamics of neuronsandneuralpopulationswithsetsofcoupleddifferentialequations(see [II 08a, II 08c] and references therein). Various tools from classical physics enabled recognition and documentation of aspects of the physical states of the brain; the

42 Weremarkthattheentanglementispermanentinthelargevolumelimit. Due to bound a ryeffects, however, aunitarytransformationcoulddisentanglethetilde and non–tildesectors: this may result in a pathological state for the brain. Itis known that forced isolation of a subject produces pathological states of various kinds. Wealsoobservethatthetildemodeisnotjustamathematicalfiction. It corresponds to a real excitation mode (quasi–particle) of the brain arising as an effect of its interaction with the environment: the couples of non–tilde/tildedwqquanta represent the correlation modes dynamically created in the brain as a response to the brain–environment reciprocal influence. Itistheinteractionbetweentilde and non–tildemodesthatcontrolstheirreversibletimeevolutionofthebrain: these collectivemodesareconfined to live in the brain. They vanish as soonas the links between the brain and the environment are cut.

structures and dynamics of neurons, the operations of membranes and organelles that generate and channel electric currents; andthemolecularandioniccarriersthatimplementtheneuralmachineriesofelectrogenesisandlearning. They support description of brain functions at several levels of complexity through measuring neural activity in the brains of animal and human subjects engaged in behavioral exchanges with their environments. One of the key properties of brain dynamics are the coordinated oscillations of populations of neurons that change rapidly in concert with changes in the environment[II 06b, II 07b]. Also, most experiment alneurobiologistsandneural theorists have focused on sensorimotor functions and their adaptations through various forms of learning and memory. Reliance has been placed on measurements of the rates and intervals of trains of action potentials of small numbers of neurons that are tuned to perceptual in variances and modelling neural interactions with discrete networks of simulated neurons. These and related studies have given a vivid picture of the cortex as a mosaic of modules, each of which performs a sensory or motor function; they have not given a picture of comparable clarity of the integration of modules.

According to [FV 06], many–body quantum field theory appears to be the only existing theoretical tool capable to explain the dynamic origin of long–range correlations, their rapid and efficient formation and dissolution, their interim stability in ground states, themultiplicityofcoexistingandpossiblynon–interfering ground states, their degree of ordering, and their rich textures relating to sensory and motor facets of behaviors. It is historical fact that many–body quantum field theory has been devised and constructed in past decades exactly to understand features like ordered pattern formation and phase transitions in condensed matter physics that could not be understood in classical physics, similar to those in the brain.

The domain of validity of the‘quantum’is not restricted to the microscopic world[Ume 93]. Therearemacroscopicfeaturesofclassicallybehavingsystems, which cannot be explained without recourse to the quantum dynamics. This fieldtheoreticmodelleadstothe view of the phase transition as a condensation thatiscomparabletotheformationoffogandraindropsfromwatervapor, andthatmightservetomodelboththegammaandbetaphasetransitions. According to such a model, the production of activity with long–range correlation in the brain takes place through the mechanism of spontaneous breakdown of symmetry(SBS), which has for decades been shown to describe long-range correlation in condensed matter physics. The adoption of such afieldtheoretic approach enables modelling of the whole cerebral hemisphere and its hierarchy of components down to the atomic level as a fully integrated macroscopic quantum system, name lyasamacroscopicsystemwhichisa quantum system not in the trivial sense that it is made, like all existing matter, by quantum components such as atoms and molecules, but in the sense that some of its macroscopic properties can best be described with recourse to quantum dynamics (see [FV 06] and references there in).

It is well–known that non-equilibrium phase transitions[Hak 93] arephenomenawhichbringaboutqualitativephysicalchangesatthemacroscopiclevelin presence of the same microscopic forces acting among the constituents of a system. Phase transitions can also be associated with autonomous robot competence levels, as informal spec ifications of desired classes of behaviors for robots over all environments they will encounter, as described by R. Brooks’ subsumption architecture approach. The distributed network of augmentedfinite–state machines can exist ind ifferent phases or modalities of their state–space variables, which determine the systems intrinsic behavior. The phase transition represented by this approach is triggered by either internal(a set–point)or external(a command)control stimuli, suchas a command to transition from asleep mode to awakemode, or walking to running.

On the other hand, it is well–known that humans possess more degrees of freedomthanareneededtoperformanydefinedmotortask, but are required toco-ordinate them in order to reliably accomplish high-level goals, while faced with intense motor variability. In an attempt to explain how this takes place, Todorov and Jordan [TJ 02] have formulated an alternative theory of human motorco-ordinationbasedontheconceptofstochasticoptimalfeedbackcontrol. They were abletoconciliate the requirement of goal achievement (e.g., grasping an object)with that of motor variability(biomechanical degrees of freedom). Moreover, their theory accommodates the idea that the human motor control mechanism uses internal‘functionalsynergies’to regulate task–irrelevant(redundant)movement.

Until recently, research concerning sensory processing and research concerning motor control have followed parallel but independent paths. The part itioning of the two lines of research in practice partly derived from and partly fostered a bipartite view of sensorimotor processing in the brain –that a sensory/perceptual system creates a general purpose representation of the world which serves as the input to the motor systems(and other cognitive systems)that generate action/behavior as an output. Recent results from research on vision in natural tasks have seriously challenged this view, suggesting that the visual system does not generate a general–purpose representation of the world, but rather extracts information relevant to the task at hand. At the same time, researchers in motor control have developed an increasing understanding of how sensory limitations and sensory uncertainty can shape the motor strategies that humans employ to perform tasks. Moreover, many aspects of the problem of sensorimotor control are spec ificto the mapping from sensory signals to motor out puts and do not exist in either domain in isolation. Sensory feedback control of hand movements, coordinate transformations of spatial representations and the influence of processing speed and attention on sensory contributions to motor control are just a few of these. In short, to understand how human(and animal)actors use sensory information to guide motor behavior, we must study sensory and motorsystemsasanintegratedwholeratherthanasdecomposablemodulesin a sequence of discrete processing steps.

Cognitive neuroscience investigations, includingfMRIstudiesofhumanco– action, suggest that cognitive and neural processes supporting co–action include joint attention, action observation, task sharing, and action coordination(see [II 07d]). For example, when two actors are given a joint control task (e.g., tracking a moving target on screen) and potentially confli cting controls (e.g., one person in charge of acceleration, the other–deceleration), their joint performance depends on how well they can anticipate each other’s actions. In particular, better coordination is achieved when individuals receive real–time feedback about the timing of each other’sactions.

A developingfieldincoordinationdynamics involves the theory of social coordination, whichattemptstorelatetheDCtonormalhumandevelopmentof complex social cues following certain patterns of interaction. This work is aimed at understanding how human social interaction is mediated by meta-stability of neural networks. fMRIand EEG are particularly useful in mapping thalamocorticalresponsetosocialcuesinexperimentalstudies. In particular, anewtheory calledthePhicomplexhasbeendevelopedbyS. Kel so and collaborators, to provide experimental results for the theory of social coordination dynamics(see therecentnonlineardynamicspaperdiscussingsocialcoordinationandEEG dynamics. According to this theory, a pair of phi rhythms, likely generated in the mirror neuron system, is the hallmark of human social coordination. Using a dual-EEGrecordingsystem, the authors monitored the interactions of eight pairs of subjects as they moved theirfingerswithandwithoutaviewofthe other individual in the pair.

As quantum probability inhuman cognition and decisionmaking has recently become popular, let us brieflydescribethisfundamentalconcept(see[II 07e, II 07d, II 08a]for more details).

Recall that a random variable Xisdefinedbyitsdistributionfunctionf(x). Its probabilistic description is based on the following rules:(i)P(X=xi)is the probabilitythatX= xi; and(ii)P(a ≤X≤b)istheprobabilitythatXliesin a closed interval[a, b]. Its statistical description is based on:(i)μXorE(X)is the mean or expectation of X; and (ii) σXis the standard deviation of X. There are two cases of random variables: discrete and continuous, each having its own probability(and statistics)theory.

A discrete random variable Xhas only a count able number of values {xi}. Its distributionfunctionf(xi)has the following properties:

f(xi) dx = 1.

Statistical description of Xisbased on its discrete mean value μXand standard deviationσX, given respectively by

X.

f(x) dx = 1.

X.

Now, let us observe the similarity between the two descriptions. The same kind of similarity between discrete and continuous quantum spectrum stroke P. Dirac

Σ– meaning ‘both integral and sum at once’: summing over a discrete spectrum and integration over a continuous spectrum.

To emphasize this similarity even further, as well as to set–up the stage for the path integral, recall the notion of a cumulative distribution function of a randomvariableX, thatisafunctionF: R −→R, defined by

F(a) = P(X) ≤a.

according to as xisadiscreteorcontinuousrandom variable. In either case, F(a) ≤F(b)whenever a ≤b. Also, that is, F(x)ismonotonicand its limit to the left is0 and the limit to the right is 1. Furthermore, its cumulative probability is given by f(x) = ∂xF(x).

Now, recallthatMarkovstochasticprocessisarandomprocesscharacterized by a lack of memory, i.e., the statistical properties of the immediate future are uniquely determined by the present, regardless of the past[II 06c, II 08b].

Forexample, arandomwalkisanexampleoftheMarkovchain, i.e., a discrete–time Markovprocess, such that the motion of the system in consideration is viewed as a sequence of states, in which the transition from one state to another depends only on the preceding one, ortheprobabilityofthesystembeinginstatekdependsonlyonthepreviousstatek−1. The property ofa Markovchainofprimeimportancein biomechanics is the existence of an invariant distribution of states: westartwithaninitialstatex0 whose absolute probability is 1. Ultimately the states should be distributed according to a spec ifiedd is tri but i on.

(whereqi, piarecoordinatesandmomenta, whileH=H(q, p)is the total system energy) – and pure stochastic dynamics (Markov process), there is so–called hybrid dynamics, particularly Brown i an dynamics, inwhichsomeofDOFare represented only through theirs toc has tic influenceonothers. As an example, suppose a system of particlesinteractswithaviscous medium. Insteadofspecifyingadetailedinteractionofeachparticlewiththeparticlesoftheviscous medium, werepresentthemediumasastochasticforceactingontheparticle. Thestochasticforcereducesthedimensionallyofthedynamics.

Recall that the Brown i an dynamics represents the phase–space trajectories of a collection ofparticlesthatindividuallyobeyLangevinrateequationsinthe fieldofforce(i.e., the particles interact with each other via some deterministic force). For a free particle, the Langevinequationreads[II 06c, II 08b]:

where m denotes the mass of the particle and vits velocity. The right–hand side represent the coupling to a heat bath; the effect of the random force R(t) is to heat the particle. To balance over heating(on the average), the particle is subjected to frictionβ.In humanoid dynamics this is performed with the Rayleigh–Van der Pol’sdissipation. Formally, thesolutiontotheLangevinequationcanbe written as where the integral on the right–hand side is a stochasticintegraland the solution v(t)is a random variable. Thestochasticpropertiesofthesolutiondependsignifican tly on the stochastic properties of the random force R(t). In the Brown i an dynamics the random force R(t) is Gaussian distributed. Then the problem boils down to finding the solution to the Langevin stochastic differential equation with the supplementary condition(zero and mean variance) where<.>denotes the mean value, Tistemperature, kB−equipartition(i.e., uniform distribution of energy)coefficient, Diracδ(t)−function.

1. Assign an initial position and velocity. 2. Draw a random number from a Gaussiandistributionwithmean zero and variance. 3. Integratethevelocitytogetvn+1. 4. Add the random component to the velocity.

Another approach to taking account the coupling of the system to a heat bath istosubjecttheparticlestocollisionswithvirtualparticles. Such collisions are imagined to affectonlymomentaoftheparticles, hence they affectthekinetic energy and introducefluctuationsinthetotalenergy. Eachstochasticcollision is assumed to be an instantaneous event affectingonlyoneparticle.

On the other hand, the so–called Itostochasticintegralrepresentsakindof classical Riemann–Stieltjesintegralfromlinearfunctionalanalysis, which is(in 1 Dcase)for an arbitrary time–functionG(t)defined as the mean square limit t

Now, the general ND Markov process can be defined by Ito stochastic differential equation(SDE), dxi(t)=Ai[xi(t), t]dt + Bij[xi(t), t] dWj(t), xi(0)=xi 0, (i, j= 1, . . . , N) in which xi(t) is the variable of interest, the vector Ai[x(t), t] denotes deterministic drift, the matrix Bij[x(t), t]represents continuous stochastic diffus i onfluctuations, andWj(t)is an N−variable Wienerprocess(i.e., generalized Brown i an motion), anddWj(t) = Wj(t + dt) −Wj(t).

3. When both A i[x(t), t]andBij[x(t), t]are zero, i.e., the state–space consists ofintegersonly, itreducestotheMasterequationofdiscontinuousjumps

The Mark ovassumptioncannowbeformulated in terms of the conditional probabilitiesP(xi, ti): if the time sti increase from right to left, the conditional probability is determined entirely by the knowledge of the most recent condition. Markovprocessisgeneratedbyaset of conditional probabilities whose probability–densityP=P(x′, t′|x′′, t′′)evolutionobeysthegeneralChapman– Kolmogorovintegro–differentialequation including deterministic drift, diffus i onfluctuationsanddiscontinuousjumps (given respectively in thefirst, second and third terms on ther.h.s.).

It is this general Chapman–Kolmogorov integro–differential equation, with its conditional probability density evolution, P= P(x′, t′|x′′, t′′), that we are going

Σ,providing us with the physical insight behind the abstract (conditional) probability densities.

General nonlinear attractor dynamics, both deterministic and stochastic, as well as possibly chaotic, developed in the framework of Feynman path integrals, have recently[IA 07] beenappliedtoLewinianfield–the or etic psychodynamics(see [Lew 97]), resulting in the development of a new concept of life–space foam (LSF) as a natural medium for motivational and cognitive psychodynamics. According totheLSF–formalism, theclassicLewinianlifespacecanbemacroscopically represented as a smooth manifold with steady force–fields and behavioral paths, while at the microscopic level it is more realistically represented as a collection

Σ(denotingintegrationovercontinuouspaths and summation over discrete Markov jumps) acting inside it and generating microscopic fluctuationdynamics.

of wildlyfluctuatingforce–fields,(loco)motion paths and local geo me tries(and topologieswithholes).

WehaveusedthenewLSFconcepttodevelopmodellingframeworkformotivationaldynamics(MD)and induced cognitive dynamics(CD)(see[IA 07]). Motivation processes both precede and coincide with every goal–directed action. Usually these motivation processes include the sequence of the following four feedforward phases[IA 07]:(*)

1. Intention FormationF, including: decisionmaking, commitment building, etc. 2. Action Initiation I, including: handling conflictofmotives, resistance to alternatives, etc. 3. Maintaining the Act i onM, including: resistance to fatigue, distractions, etc. 4. Term in at i onT , including parking and avoidingaddiction, i.e., staying in control.

With each of the phases{F, I, M, T }in(*), we can associate a transition pro pag at or–an ensemble of(possibly crossing)feedforward paths propagating through the‘wood of obstacles’(includingtopologicalholesintheLSF, see Figure 23.3), sothatthecompletetransitionisaproductofpropagators(as well as sum over paths). All the phases–propagators are controlled by a unique Monitor feedback process.

A set of least–action principles is used to model the smoothness of global, macro–level LSF paths, fields and geometry, according to the following prescription. The action S[Φ], psycho–physical dimensions of Energy × T ime = Effort and depending on macroscopicpaths,fieldsandgeometries(commonlydenoted by an abstract fieldsymbolΦi)i sdefined as a temporal integral from the initial time instant tin i to thefinaltimeinstanttfin,

Fig.23.3. Transition–pro pag at or corresponding to each of the motivational phases {F, I, M, T }, consisting of an ensemble of feedforward paths propagating through the ‘wood of obstacles’. The paths affected by driving and restraining force–fields, as well as by the local LSF–geometry. Transition goes from Intention, occurring at a sample time instant t 0,to Action, occurring at some late rti met 1. Each pro pag at or is controlled by itsownMonitorfeedback.

where the integral istakenoverallncoordinatesxj=xj(t)oftheLSF, and ∂xjΦi are time and space partial derivatives of the Φi−variables over coordinates. The standard least action principle performed at a macroscopic (global) time–level from some initial time tin i to the finaltimetfin. In this way, we have obtained macro–objectsintheglobalLSF: asinglepathdescribedbyNewtonian–like equation of motion, asingleforce– fielddescribedbyMaxwellian–likefieldequations, and asingle obstacle–free Riemann i an geometry(withglobaltopologywithoutholes).

To model the corresponding local, micro–level LSFstructuresofrapidlyfluctuatingcognitivedynamics, an adaptive path integral is formulated, defining a multi–phase and multi–path(multi–fieldandmulti–geometry) transition amplitudefromthestateofIntentiontothestateofAction, where the Lebesgueintegration is performed over all continuous Φi con= paths + fields + geo me tries, while summation is performed overall discrete processes and regional topologies Φj dis. The symbolic differential D[wΦ] in the general path integral(23.48), represents an adaptive path measure, defined as a weighted product(with i = 1, ..., n = con + dis)

The adaptive path integral (23.48)–(23.49) represents an ∞−dimensional neural network, with weights wup dating by the general rule[IA 07] new value(t + 1)= old value(t)+innovation(t).

Were mark here that the traditional neural networks approaches are known for their classes of functions they can represent. This limitation has been attributed to their low-dimensionality(the largest neural networks are limited to the order of 105 dimensions. The proposed path integral approach represents anew family of function-representation methods, whichpotentiallyoffers a basis for a fundamentally more expansive solution.

where˙xirepresents motivational (loco)motion velocity vector with cognitive processing speed. Thefirst bracket term in(23.50)representsthekineticenergyT , generated bytheRiemannianmetrictensorgij, while the second bracket term, ϕi(xi), denotes the family of potential force–fields, driving the(loco)motions xi=xi(t)(the strengths of thefieldsϕi(xi)dependontheirpositionsxiin LSF. The corresponding Euler–Lag ran gian equation gives the Newtonian–like equation of motion dtL ˙xi= Lxi, with L = T−ϕi(xi).

Now, accordingtoLewin, thelifespacealsohasasophisticatedtopological structure. As a Riemann i an smooth n−manifold, theLSF–manifold Σ gives rise toitsfundamentaln−groupoid, orn−categoryΠn(Σ)(see[II 06c, II 07e]). In Πn(Σ), 0– cells are points in Σ; 1– cells are paths in Σ (i.e., parameterized smooth mapsf:[0, 1]→Σ);2– cellsaresmoothhomotopies(denoted by≃)of paths relative to end points(i.e., parameterizedsmoothmapsh:[0, 1] × [0, 1]→Σ); 3– cellsaresmoothhomotopiesofhomotopiesofpathsinΣ(i.e., parameterized smoothmapsj:[0, 1] × [0, 1] × [0, 1]→Σ). Categoricalcompositionisdefined bypastingpathsandhomotopies. In this way, thefollowingrecursivehomotopy dynamicsemergesontheLSF–manifoldΣ(**):

0 −cell : x0 • x 0∈M; in the higher cells below: t, s ∈[0, 1];

f: [0, 1] →M, f: x 0→x 1,x 1= f(x 0),f(0) = x 0,f(1) = x 1; e.g., linear path: f(t) = (1 −t) x0 + t x 1; or

Euler–Lagrangian f−dynamics with endpoint conditions(x 0, x 1) :

dtf ˙xi= fxi, with x(0) = x0, x(1) = x1, (i = 1, ..., n);

h : [0, 1] × [0, 1] →M, h : f→g, g= h(f(x 0)), h(x 0, 0) = f(x 0),h(x 0, 1) = g(x 0),h(0, t) = x 0,h(1, t) = x1 e.g., linearhomotopy: h(x 0, t) = (1 −t) f(x 0) + t g(x 0); or homotopybetweentwoEuler–Lagrangian (f, g) −dynamics with the same endpoint conditions(x 0, x 1) :

dtg ˙xi= gxi with x(0) = x0, x(1) = x1;

j(x0, t, 0) = h(f(x0)),j(x0, t, 1) = i(f(x0)), j(x0, 0, s) = f(x0),j(x0, 1, s) = g(x0), j(0, t, s) = x0,j(1, t, s) = x1 e.g.,linearcompositehomotopy:j(x0, t, s) = (1 −t) h(f(x0)) + t i(f(x0)); or,homotopybetweentwohomotopiesbetweenabovetwoEuler-

Lagrangian (f, g) −dynamics with the same endpoint conditions(x 0, x 1).

On the micro–LSFlevel, insteadofasinglepathdefinedbytheNewtonian–like equation of motion (23.51), we have an ensemble of fluctuating and crossing paths with weighted probabilities (of the unit total sum). This ensemble of micro–paths i sdefined by the simplest instance of our adaptive path integral(23.48), similar tothe Feynman’soriginalsumoverhistories, whereD[wx]is a functional measure on the space of all weighted paths, and the exponential depends on the actionS[x]given by(23.50). This procedure can be redefined in a mathematically cleaner way ifweWick–rotate thetimevariablet to imaginary values, t→τ= it, therebymakingallintegralsreal:  where Ejisthemotionenergyeigenvalue(reflectingeachpossiblemotivational energetic state), Tis the temperature–likeenvironmentalcontrolparameter, andthesumrunsoverallmotionenergyeigenstates(labelledbytheindexj). From(23.54), we can further calculate all thermodynamic–like and statistical propertiesofMDandCD, as forexample, transition entropy, S= kB ln Z, etc.

From CD–perspective, our adaptive path integral (23.52) calculates all (alternative)pathways of informationflowduringthetransitionIntention−→Action. In the connection ist language,(23.52)represents activation dynamics, to which our Monitor process gives akind of back propagation feedback, acommon type of supervised learning ws(t + 1)=ws(t) −η∇J(t), (23.55) where ηis a small constant, called the step size, or the learning rate, and ∇J(n) denotes the gradient of the‘performancehyper–surface’atthet−thiteration.

to choose? In our path–integral language this reads: Which path(alternative) should be give nthehighestprobabilityweightw? This problem can be either iteratively solved by the learning process (23.55), controlled by the MONIT OR feedback, which we term algorithmic approach, or by the local decisionmaking process under uncertainty, which wetermheuristicapproach[IA 07]. This qualitative analysis is based on the micro–levelinterpretationoftheNewtonian–like action S[x], given by (23.50) and figuring both processing speed˙x and LTM (i.e., the force–field ϕ(x), see next sub section). Here we consider three different cases:

1. If the potentialϕ(x)is notverydependentuponpositionx(t), then the more direct paths contribute the most, as longer paths, with higher mean square velocities[ ˙x(t)]2make the exponent more negative(after Wickrotation(23.53)). 2. On the other hand, suppose thatϕ(x)doesindeeddependonpositionx. For simplicity, let the potential increaseforthelargervaluesofx. Thena direct path does not necessarily give the largest contribution to the overall transition probability, because the integrated value of the potential is higher than over another paths. 3. Finally, consider a path that deviates widely from the direct path. Then ϕ(x) decreases over that path, but at the same time the velocity˙x increases. In this case, we expect that the increased velocity˙xwouldmorethancompensate for the decreased potential over the path.

Therefore, the most important path(i.e., thepathwiththehighestweightw) would be the one for which any smaller integrated value of the surrounding field potential ϕ(x) is more than compensated for by an increase in kinetic–like energy

2˙x 2.In principle, this is neither the most direct path, nor the longest path, but rather a middle way between the two. Formally, it is the path along which the average Lagrangian is minimal, i.e., the path that requires minimal memory(both LTM and WM) and processing speed. This mechanical result is consistent with the‘cognitivefiltertheory’of selective attention[Bro 77], whichpostulatesalowlevelfilterthatallowsonlya limitednumberofperceptstoreachthebrainatanytime. In this theory, the importance of conscious, directed attention is minimized. The type of attention involving low levelfilte ring corresponds to the concept of early selection.

Although we termed this‘heuristicapproach’in the sense that we can instantly feel both the processing speed˙xandtheLTMfieldϕ(x)involved, there is clearly a psycho–physical rule in the background, namely the averaging minimum relation(23.56).

From the decisionmaking point of view, all possible paths(alternatives) represent the consequences of decisionmaking. They are, by default, short– term consequences, as they are modelled in the micro–time–level. However, the path integral formal ism allows calculation of the long–term consequences, just by extending the integration time, tfin−→∞. Besides, this averaging decision mechanics–choosing the optimal path–actually performs the‘averaging lift’ intheLSF: from the micro–level to the macro–level.

For example, one of the simplest types of performance–degrading disturbances intheLSFiswhatwetermmotivationalfatigue–a motivational drag factor that slows the actors’progress towards their goal. There are two fundamentally different sources of this motivational drag, both leading to apparently the same reduction in performance: (a) tiredness / exhaustion and (b) sat i a tion (e.g., boredom). Both involve the same underlying mechanism(theraisingvalenceofthe alternatives to continuing the action) but the alternatives will differ considerably, depending on the properties of the task, from self–preservation/recuperation in the exhaustion case through to competing goals in the sat i a tion case.

The spatial representation of this motivational drag is relatively simple: uni– dimensionalLSF–coordinatesmaybesufficient for most purposes, which makes it attractive for the initial validation of our predictive model. Similarly uncomplicated spatial representations can be achieved for what we term motivational boost derived from the proximity to the goal(including the well–known phenomenon of‘the home stretch’): the closer the goal(e.g., afinishingline)is perceived to be, the stronger its‘pulling power’[Lew 97]. Combinationsofmotivationaldragandmotivationalboosteffectsmaybeofparticularinterest in a range of applications. These combinations can be modelled within relatively simpleuni–dimensionalLSF–coordinate systems.

InthissectionweextendtheLSF–formal ism to incorporate geometrical chaos [IJP 08,II 07a, II 08b]andassociatedtopologicalphasetransitions.

It is well–known that onthebasisoftheergodichypothes is, statistical mechanics describes the physics of many-degrees of freedom systems by replacing timeaveragesoftherelevantobservableswithensembleaverages. Therefore, insteadofusingstatisticalensembles, wecaninvestigatetheHamiltonian(microscopic)dynamics of a system undergoing a phase transition. The reason for tackling dynamics is twofold. First, thereareobservables, like Lyapunovexponents, thatareintrinsicallydynamical. Second, thegeometrizationofHamiltoniandynamicsintermsofRiemanniangeometryprovidesnewobservablesand, in general, an interesting framework to investigate the phenomenon of phase transitions. The geometrical formulation of the dynamics of conservative systems[II 06c, II 08b]wasfirstusedbyKrylovinhisstudiesonthedynamical foundations of statistical mechanics and subsequently became a standard tool tostudyabstractsystemsinergodictheory.

wherepiarethecanonical(loco)motionmomenta.

Dynamics of NDOF mechanical–like systems with action (23.57) and Hamilton i an (23.58) are commonly given by the set of geodesic equations [II 06c, II 07e] jkare the Chris toffel symbols of the affine Levi–Civita connection of the RiemannianLSF–manifoldΣ.

Alternatively, adescriptionoftheextremaoftheHamilton’saction(23.57) canbeobtainedusingtheEisenhartmetriconanenlargedLSFspace-time manifold(given by{q 0≡t, q 1, . . . , qN}plusonerealcoordinateqN+1), whose arc–length is ds 2= −2V (q)(dq 0)2+ aijdqidqj+ 2dq0 dqN+1. (23.60)

The manifold has a Lorentz i an structure and the dynamical trajectories are those geode-sics satisfying the condition ds 2= Cdt 2, where Cis a positive constant. In this geometrical framework, the instability of the trajectories is the instability ofthegeodesics, and it is completely determined by the curvature properties oftheLSF–manifoldΣaccordingtotheJacobiequationofgeodesicdeviation [II 06c, II 07e] jkmarethecomponentsoftheRiemanncurvaturetensorof theLSF–manifoldΣ.

0k0Jk= 0, (i = 1, . . . , N), (23.62)

Lyapunovexponentsmeasurethestrengthofdynamicalchaos.

Now, torelatetheseresultstotopologicalphasetransitionswithintheLSF– manifoldΣ,recall that any two high–dimensionalmanifoldsΣvandΣv′have thesametopologyiftheycanbecontinuouslyanddifferentiablydeformed into one another, that is if the yared iffeomorphic. Thusbytopologychange the‘lossofdiffeomorphicityismeant. In this respect, the so–calledtopologicaltheorem[FP 04] says that non–analytic ity is the‘shadow’of a more fundamental phenomenon occurring in the system’sconfigurationmanifold(in our casetheLSF–manifold): atopologychangewithinthefamilyofequipotential hypersurfaces

Σv= {(q 1, . . . , qN) ∈RN|V (q 1, . . . , qN) = v}, where Vandqiarethemicroscopicinter action potential and coordinates respectively. ThistopologicalapproachtoPTsstemsfromthenumericalstudy of the dynamical counterpart of phase transitions, andpreciselyfromtheobservationofdiscontinuousorcuspypatternsdisplayedbythelargestLyapunov exponentλ1at the transition energy. Lyapunovexponentscannotbemeasured in laboratory experiments, atvariancewiththermodynamicobservables, thus, beinggenuinedynamicalobservablestheyareonlybeestimatedinnumerical simulations of the microscopic dynamics. If there are critical points of Vin configurationspace, thatispointsqc=[ qN]such that∇V (q)|q=qc=0, according to the Morse lemma, in the neighborhood of any critical point qcthere always exists a coordinate system˜q(t) = [˜q 1(t), .., ˜qN(t)]for which where kis the index of the critical point, i.e., the number of negative eigenvalues of the Hess i an of the potential energy V . In the neighborhood of a critical point oftheLSF–manifoldΣ,(23.64)yields which giveskunstabledirectionswhich contribute to the exponential growth of thenormofthetangentvectorJ.

This means that the strength of dynamical chaos within the individual’s LSF– manifoldΣ,measuredbythelargestLyapunovexponentλ1given by(23.63), isaffectedbytheexistenceofcriticalpointsqcofthepotentialenergyV (q). However, as V (q) is bounded below, it is a good Morse function, with no vanishingeigenvaluesofitsHessianmatrix. According to Morsetheory, the existence ofcriticalpointsofVisassociatedwithtopologychangesofthehypersurfaces {Σv}v∈R.

More precisely, letVN(q 1, . . . , qN): RN→R, be a smooth, bounded from below,finite–range and confiningpotential 43. DenotebyΣv=V−1(v), v∈R,

43 These requirements for Varefulfilledbystandardinteratomicandintermolecular interaction potentials, as well as by classical spin potentials.

Part 13 · pp. 365–390

its level sets, orequipotentialhypersurfaces, intheLSF–manifoldΣ.Then let ¯v=v/Nbethepotentialenergyperdegree of freedom. IfthereexistsN 0, and if for any pair of values¯vand¯v′belongingtoagivenintervalI¯v=[¯v 0, ¯v 1] and foranyN> N0 thenthesequenceoftheHelmoltzfreeenergies{FN(β)}N∈N– where β= 1/T(Tis the temperature) and β∈Iβ= (β(¯v 0), β(¯v 1)) – is uniformly convergentatleastinC 2(Iβ)[thespaceoftwicedifferent i able functions in the interval Iβ],sothatlimN→∞FN∈C 2(Iβ)and neitherfirstnorsecondorder phase transitions can occur in the(inverse)temperature interval(β(¯v 0), β(¯v 1)), where the inverse temperature i sdefined as[FP 04] isoneofthepossibledefinitionsofthemicro-can on i calc onfigurationalentropy. The intensive variable¯vhasbeenintroducedtoeasethecomparisonbetween quantitiescomputedatdifferentN–values.

Thistheoremmeansthatatopologychangeof the{Σv}v∈Ratsomevc is a necessary condition for a phase transition to take place at the corresponding energy value. Thetopologychangesimpliedherearethosedescribed withintheframeworkofMorsetheorythrough‘attachment of handles’tothe LSF–manifoldΣ.

IntheLSFpath–integral language[IA 07], wecansaythatsuitabletopology changesofequipotentialsubmanifoldsoftheindividual’sLSF–manifoldΣcan entail thermodynamic–like phase transitions[Hak 93], according to the general formula:

ThestatisticalbehavioroftheLSF–(loco)motion system(23.57)with the standard Hamilton i an (23.58) is encompassed, in the can on ical ensemble, by itspartitionfunction, given by the phase–space path integral[II 07e]

2π .

Σv= {(q1, . . . , qN) ∈RN|V (q1, . . . , qN) = v}.

dqiV (q)e−βV (q).

defined on the {Σv}v∈R. Once the microscopic interaction potential V (q) is given, the configurationspaceofthesystemisautomaticallyfoliatedintothefamily {Σv}v∈Roftheseequipotentialhypersurfaces. Now, from standard statistical mechanical arguments we know that, at any given value of the inverse temperatureβ,thelargerthenumberN, the closer toΣv≡Σuβare the micro states that sign ifican tly contribute to the averages, computed through ZN(β), ofthermodynamicobservables. The hyper surfaceΣuβis the one associated with the average potential energy computed at a given β. Thus, at any β, if Nis very large the effective support of the can on ical measure shrinks very close to a single Σv= Σuβ.Hence, thebasicoriginofaphasetransitionliesinasuitabletopology change of the{Σv}, occurring at some vc. This topology change induces the singular behavior of the thermodynamic observable sata phase transition. Itis conjecturedthatthecounterpartofaphasetransitionisabreakingofdiffeomorphicityamongthesurfaces Σv, it is appropriate to choose adiffeomorphism invariant to probe if and howthetopologyoftheΣvchanges as a function of v. Fortunately, suchatopologicalinvariantexists, the Euler characteristic of the LSF–manifoldΣ,defined by[II 06c, II 07e]

44The Betti numbers bkare the dimensions of the de Rham’s cohomology vector spaces Hk(Σ; R)(thereforethebkareintegers).

is the invariant volume measure of the LSF–manifold Σ and a is the determinant oftheLSFmetrictensoraij.Fortechnicaldetailsofthistopologicalapproach, see[II07e,II08b].

The domain of validity of the‘quantum’is not restricted to the microscopic world[Ume 93]. Therearemacroscopicfeaturesofclassicallybehavingsystems, which cannot be explained without recourse to the quantum dynamics. This field the or etic model leads to the view of the phase transition as a condensation that is comparable to the formation of fog and rain drops from water vapor, and that mightservetomodelboththegammaandbetaphasetransitions. According to such a model, the production of activity with long-range correlation in the brain takes place through the mechanism of spontaneous breakdown of symmetry(SBS), which has for decades been shown to describe long-range correlation in condensed matter physics. The adoption of such afieldtheoreticapproach enables modelling of the whole cerebral hemisphere and its hierarchy of components down to the atomic level asafullyintegratedmacroscopic quantum system, name lyasamacroscopicsystemwhichisa quantum system not in the trivial sense that it is made, like all existing matter, by quantum components such as atoms and molecules, but in the sense that some of its macroscopic properties can best be described with recourse to quantum dynamics (see [FV 06] and references there in).

To model the dynamics of the two–actorco–action, we propose to associate each of the actors with a set of their own time dependent trajectories, which constitutes ann−dimensionalRiemannianLSF–manifold,Σα= {xi(ti)} andΣβ={yj(tj)}, respectively. Theirassociatedtangentbundlescontain theirindividualnD(loco)motion velocities, T Σα={ ˙xi(ti)=dxi/dti}and T Σβ={ ˙yj(tj)=dyj/dtj}. Further, following the general formal ism of[IA 07], outlined in the introduction, we use the modelling machinery consisting of: (i)Adaptive joint action at the top–master level, describing the externally– appearing deterministic, continuous and smooth dynamics, and (ii) Corresponding adaptive path integral(23.73)at the bottom–slave level, describing a wildly fluctuatingdynamics including both continuous trajectories and Markov chains. This lower–leveljointdynamicscanbefurtherdiscretizedintoapartitionfunctionofthecorrespondingstatisticaldynamics.

, where IN≤ti, tj, t ≤OUT. (23.69)

Thefirst term in(23.69)represents potential energy of the cognitive/ motivational interaction between the two agentsαi andβj.45It is a double integraloveradeltafunctionofthesquareofintervalI2 between two points onthepathsintheirLife–Spaces; thus, interaction occurs only when this interval, representing the motivational cognitive distance between the two agents, vanishes. Note that the cognitive(loco)motions of the two agentsαi[xi(ti)] andβj[yj(tj)], generallyoccuratdifferenttimestiandtjunlessti=tj, when cognitive synchronization occurs.

The second term in(17.11)represents kinetic energy of the physical interaction. Namely, when the cognitive synchronization in thefirsttermtakesplace, the second term of physical kinetic energy is activated in the common manifold, which is one of the agents’Life Spaces, sayΣα= {xi(ti)}.

A[x, y; ti, tj]: MENTALINTENTION from mutual cognitive intention to joint physical action, in which the joint action starts after the mutual cognitive intention is synchronized. In simple words, “we canefficientlyacttogetheronlyafterwehavetuned–up our intentions.”

Conversely, if we have a need to represent coaction of three actors, say αi,βj and γk(e.g., αiin charge of acceleration, βj– deceleration and γk−steering), we canassociateeachofthemwithannDRiemannianLife–Spacemanifold,Σα= {xi(ti)}, Σβ= {yj(tj)}, and Σγ= {zk(tk)}, respectively, with the corresponding tangentbundlescontainingtheirindividual(loco)motion velocities, T Σα= { ˙xi(ti)=dxi/dti}, T Σβ={ ˙yj(tj)=dyj/dtj}andT Σγ={ ˙zk(tk)=dzk/dtk}. Then, instead of(17.11)we have rs(t, q,˙q)˙qr ˙qs dt, (23.70) where IN≤ti, tj, tk, t ≤OUT, with I2

45 Although, formally, this term contains cognitive velocities, it still represents‘potential energy’from the physical point of view.

the bilateral co–action (17.11).46It actually happens in the common 3nD Finsler manifoldΣJ=Σα∪Σβ∪Σγ, parameterizedbythelocaljointcoordinates dependent on the common time t. That is, ΣJ= {qr(t), r= 1, ..., 3n}. Geometry of the joint manifold ΣJis defined by the Finsler metric function ds = F(qr, dqr), defined by

F2(q,˙q) = grs(q,˙q) ˙qr˙qs, (23.71)

Σ D[w, x, y] eiA[x, y; ti, tj], (23.73) wheretheLebesgueintegrationisperformedoverallcontinuouspathsxi= xi(ti) and yj= yj(tj), while summation is performed over all associated discrete Markovfluctuationsandjumps. The symbol i cdifferentialinthepathintegral (23.73)represents an adaptive path measure, defined as a weighted product ijdxidyj, (i, j= 1, ..., n). (23.74)

Σ D[w; x, y, z; q] eiA[ti,tj,tk;t], (23.75) ijkrdxidyjdzkdqr,(i, j, k= 1, ..., n;r= 1, ..., 3n). (23.76)

The proposed path integral approach represents anew family of more expansive function-representation methods, which is now capable of representing input/output behavior of more than one actor. However, as we add the second and

46Recallthatthenecessaryconditionforchaosincontinuoustemporalorspatiotemporalsystemsistohavethreevariableswithnonlinearcouplingsbetweenthem.

subsequent actors to the model, the requirements for the rigorous geometrical representations of theirrespectiveLSFsbecomenontrivial. For asingle actor or a two–actorco–actiontheRiemanniangeometrywassufficient, but it becomes insufficient for modelling the n–actor (withn ≥3) joint action, due to an intrinsic chaotic coupling between the individual actors’LSFs. To model ann–actor jointLSF, wehavetousetheFinslergeometry, whichisageneralizationofthe Riemann i an one. This progression may seem trivial, both from standard psychological point of view, and from computational point of view, but it is not trivial from the geometrical perspective.

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1− jetspace,244,258 2− jetspace,245,259 12– week programme,119 3– body problem,194 abilitytothinkabstractly,273 abstract intelligent agent,317 abstraction,279 achieved,265 act in,126 Action Initiation,361 action potential,130,152 action–at–a–distanceelectrodynamics,

113 activation dynamics,365 activation heat,128 active state,131 active–state element,261 adaptive Kalmanfiltering,153 adaptiveNLS,101 adaptive quantum systems,99 adenosinediphosphate,135 adenosine triphosphate,126 afferentnerves,164 affineHa milton i an control action,268 affinetransformation,205 algorithmic approach,365 alpha,134 Amari’sneuralfieldequation,178 amplitude,235 anaerobic glycolysis(or, glucosebreakanalogousART2 system,98 Andronov,199 Andronov–Hopfbifurcation,199 Anosovdiffeomorphism,200 Anosovflow,200 Anosovmap,200 antagonistic muscle-pairs,141,142 anticipation,70 area–preserving map,218 Arnold cat map,200 associative cortex,47 associative memory,155 athleticism,55 atmospheric convection,209 ATPase,136 attack,85 attention,278 attractor,208 attractor&repellerdynamics,179 auditory cortex,150 auto genetic,134 autogeneticmotorservo,133 autonomic nervous system,160 averaging decision mechanics,366 Axiom–A systems,201 axon,157 back propagation,365 Baker map,220 base,251 bas in of attraction,208 basinsofattraction,218 basket cell,182 behaviorism,276 bell curve,293 Bernoulli map,200 Bernoulli systems,198 bi–stable perception,110 bifurcation diagram,216 billiard,198 binding by assembly,108 binding by convergence,108 binding by synchrony,109 binding problem,108 binding unit,108 BinetIQtest,291 binocularrivalrymodel,111 bio–diversity,215 biophotons,112 bipolarneurons,158 Birkhoff,196,197 Birkhoffcurve shorteningflow,197 blitzchess,91 blitz-chess player,8 Bloom’sTaxonomy,277 Bose–Einsteincondensate,109 Bourbakigroup,281 brain plasticity,155 brain stem,47 Brown i an dynamics,357 Buddha,287 bullet chess,9,91 butterflyeffect,211 can on ical micro-circuits,104 canonicalquantization,347 Cant or set,203 carrying capacity,214 Cartesiantranslationcoordinates,75 Cartwright,199 case–based reasoning,275 Cauchy–Schwarzinequality,295 cavityresonators,111 central bio dynamic adjunct i on,156 central nervous system,159 cerebellarcortex,150 cerebellum,39,176 cerebral cortex,103,150 cerebral hemispheres,176 champion model,51,52 chaos,194 chaos theory,196 Chapman–Kolmogorov equation,359 differentialequation,359 character structure,285 circle map,219 classical brain,101 classicalperceptron,101 classify,304 climbingfibre,182 co–area formula,371 co–existence of alternatives,235 coarsesystem,199 code,347 coefficient of determination,291 cognition,179 cognitive agent,316 cognitive architecture,316 cognitive binding problem,108 cognitive test scores,291 Cohen–Grossbergactivationequations,

98 Cohen–Grossbergtheorem,98 collapse of the tunnel ing state,107 common factor analysis,300 communalities,300 communication,278 completely unpredictable,192 complex nonlinearity,220 complex reaction,8,79 complex–valuedorderparameter,181 complexity measure,186 computational neuroscience,220 con a tion,313 condensed,347 confirmatoryfactoranalysis,302 Confucius,287 connectionism,102,103 connectionistbraintheory,102 connectivity matrix,99 consciousness,111 consequences,366 conservation law,213 continuous Hopfieldcircuit,98 continuous model for neural activity in cortical structures,178 contractile elements,125 contraction,141 Contraction dynamics,142 contraction dynamics,130,262 contravariantaccelerationfunctor,252 contravariantequation,255 convectiveB´enardfluidflow,210 corpuscallosum,314 correlation,291 correlationcoefficient,291 correlationflow,311 correlation function,293 cotangentbundle,252 counter–attack,85 counter–propagation network,98 covariantderivative,248 covariantequation,255 covariantforcefunctor,250 covariantforcelaw,250 craniometry,313 creatinekinase,136 creatine phosphate,136 creative imagination,47 creative visualization,49 Crick–Kochbindinghypothesis,156 critical dynamics,99 Critique of Pure Reason,108 cross bridges,126 crossstriations,125 crossed neural control,38 cumulative distribution function,293, data–processingneurons,226 deepnuclei,183 degrees of freedom,59 dendrites,157 dendron,105 depolarization,166 desired,265 deterministic chaos,210 deterministic chaotic system,192,193 diencephalon,176 diffeomorphism,203 differenceequation,215 diffus i onfluctuations,262 diffus i on processes,132 dimensionality reduction,298 Direct obl i min rotation,303 direct–training,51 discrete Amar i equation,178 discrete–time models,215 discrete–time steps,196 disguises,8 dissipation,352 icsequations,141 dissipativestructures,141 distribution function,355 dual,252 Duffing,199 Duffingmap,219 dynamic force–velocity relation,128 dynamicalhomeostasis,99 dynamicalinvariant,198 dynamicalsystems,192 dynamics and topologyof brain networks, early selection,366 Ecclesmodelofsynapticactivation,133 edge of chaos,99 Eisenhartmetric,368 Embden–Meyerhoffpathway,136 emotional intelligence,314 empirical mean,298 encoding,46 endomysium,125 entangled state,352 entericnervoussystem,162 entropicmotorcontrol,269 entropy,198 equivalent muscular actu at or,134 equivalentmuscularactuators,141,142,

255,261 ergodichypothesis,197 Ermentroutcanonicalmodel,222 Euclideangeometry,295 Euclideanpathintegral,236 Euclideanspace,295 Eulercharacteristic,194 Euler-Lagrangianequation,236 evolutionequations,260 excitation,141 Excitationdynamics,142 excitationdynamics,130,261 excitatory,169 exocytosis,105 experience,287 exploratory,307 exploratory factor analysis,302 extroversion,283 factor analysis,283,291 factor loadings,303 factor matrix,303 factor rotation,300 factor scores,303 factor–correlationflow,311 factors,296 fasciculus,125 feature extraction,298 feature selection,298 Feigenbaumcascade,212 Feigenbaumconstant,212 Feigenbaumnumber,216 Feynmanpathintegral,113 Feynman–Vernonformalism,349 fifth–order transmission cascade,132 figure–groundvase,110 filamentary–overlap function,130 filter,298 Fitz Hugh–Nagumomodel,226 fixedpointattractors,177 fixed–point,203,207,209 flow,204 flux,132 Flynneffect,289 focalobjects,329 Fokker–Planckequation,359 folding,202 force 1– form,255 force generator,132 force–velocity curve,12,53 force-time,142 forceddissipativeHamiltonianmusculoskeletaldynamics,142 formal language,279 Fr¨olichwaves,109 fractalattractor,211 fractaldimension,209 fractalpattern,216 fractalset,203 freeCa–i on concentration,130 free thinkers,287 free thinking,287 free thought,287 frontallobe,150 functional causal relation,291 fusiform,125

Gaussianfunction,293 general cognitive ability,312 general intelligence,312 general sports performance,51 generalizedH´enonmap,218 genetic algorithms,328 glia,159 global factors.,283 globularcluster,217 glucose–1–phosphate,136 glycolysis,136 Goldstonetheorem,347 Golgicells,182 Golgitendonorgans,134,149 graph,241,256 greymatter,150 growth rate,214

H–cobordismtheorem,201 H´enonmap,196,217 H´enonstrangeattractor,217 Hadamard,198 Haken’ssynergetics,347 haloorbit,198 Hamilton i an,260 Hamilton i an function,260 Hamilton i an system,218 Hammingdistance,99 Hayashi,199 heat equation,129 Hebbrules,153 Hebbiantheory,170 He is enberg uncertainty principle,270 heuristicapproach,365 hightennisefficiency,116 higher–ordertangency,241,256 Hindmarsh–Rosethalamicneuron,222 Hinduscriptures,287 hiper–cube≡neuro–muscular control space,133 Hodgkin–Huxley,167 Hodgkin–Huxleyequation,98 Hodgkin–HuxleyHH–neuronmodel,131 Hodgkin–Huxleymodel,224 Hodgkin–Huxleyneuralmodel,105,223 holists,287 holographichypothesis,346 homeostasis,174 homoclinicpoint,195,204 homoclinictangle,194,207 homomorphismofvectorspaces,297 Hopf,199 Hopfieldmodel,154 Hotellingtransform,298 Hotelling’slaw,298 Hotelling’slemma,298 Hotelling’srule,298 Hotelling’sT–square distribution,298 human and animal behavior,48 human brain,150 human mind,273 hybrid rotational actu at or,262 hyperbolicforce–velocity,262 hyperbolicforce–velocity curve,128 hyperbolicforce-velocity,142 hyperpolarization,166 hypothalamus,47 imitation,277,287 inferiorcerebellarpeduncle,183 information theory,200 inhibitory,169 inorganicphosphate,135 integrate–and–firen eur on,221 integrate–and–fire–or–burstneuron,221 intelligencequotient,288 Intention Formation,361 interceptivetennistactics,70 introspection,276 intuition,193,287 invariant set,203,205 inverse–selection,51 inverted–Uhypothesis,93 irregular and unpredictable,191 irreversible processes,141 isometric steady–state contraction,127 iteratedmap,196 Itostochasticintegral,358

Jacob i equation,248 Jacob i variationfield,248 jet,243,257 jetmanifold,258 jets,241,256 jnana,287

Kaplan–Yorkemap,220 Karhunen–Lo`evetransform,298 karma,287 Kepler,194 kinetic chain,69 knowledge of the manifold,108 Kolmogorov,198,200 Krylov,199 Kuramotomodel,180,188

Lagrange’spoints,197 language,273 Laplace–Belt ram i operator,249 Laplacianoperator,100 latency relaxation,128 lateralcisternae,126 learning,273 learning on the spot,116 learning rate,,265,365 leg power,53 lever–arm,35 life space foam,360 lightning chess,9,91 limbicsystem,47 limit cycle,209 linear,291 linear map,297 linear operator,297 linear super position,231 linear transformation,297 linguistics,279 Liouvilleequation,359 Liouvilletheorem,342 Little wood,199 localmetricform,240,253 log is tic equation,214 log is tic growth,214 log is tic map,215,217 long–range correlation,347 long–term consequences,366 long–term memory,46 loop-diagram,173 Lorentz i an structure,368 Lorenzmask,211 Lorenzsystem,211,217 Lyapunov,199 Lyapunovexponent,191 Lyapunovexponents,209,369 Lyapunovtime,191

Mach–Zehnderinterferometer,235 macroscopicmuscle–load dynamics,128 main laws of tennis science,10 Maintaining the Action,361 maintenance heat,128 Malthusmodel,214 Malthusianparameter,214,216 manifest variables,293 map,215 mapsink,209 Markovassumption,359 Markovchain,357 Markovpartitions,200 Markovstochasticprocess,357 Master equation,359 maximal possible racket-head speed,70 maximum likelihood esti mat or,293 mean,292 mean–fieldtheory,99 mechanical+biochemical energy is not conserved,263 medullaoblongata,176 mental abilities,273 mental focus,114 mental interpretation,113 mental potential energy,114 mental speed,9,91 mental synchronization,114 Meta physics,286 metrictensor,114 microscopic theory of muscularcontraction,126 microtubules,109,336 mid brain,176 middlecerebellarpeduncle,183 mindmaps,287 mind reading,8 Mittag–Leffler,195 modelfit,302 moment–arm,35 Monte Carlo method,217 Monte–Carlo,292 morphism,297 Morris–Lecarneuron,223 Morse,198 Morselemma,369 Morsetheory,198,202,370 mossyfibers,183 mostefficienttechnique,57 motion action principle,363 motor cortex,151 motor nervefibers,160 motorservo,150 motor units,11

142 muscular power,14 myocybernetics,130 myofibrils,126 myoglobin,136 myosin,126

Navier–Stokesequations,210 neocortex,103 neocorticalbiophysics,106 neocorticaldynamics,105 neuralattractordynamics,178 neural communication,152 neural network,153 neural path integral,265 neural–networks complexity,186 neurons,156 Neuro Quantology,97 neuroticism,283 Newton–Eulerdynamics,57 Newton-Eulerdynamics,75 Newtonian–like action,363 noise,265 non–periodicorbit,311 non–wandering set,204 nonlinear Schr¨odingerequation,101 nonrelativisticquantummechanics,309 nonrigid,307 normal distribution,292 normalized force across the series elastic oblique factor model,307 occipitallobe,150 olfactorycortex,150 operantconditioning,278 orbit,204 oscillator neural networks,180 otolithorgans,36 Ott–Grebogi–Yorkemap,220 parabolic length–tension curve,128 paradigm shift,277 parameters pace,99 parasympathetic nervous system,162 par i etal cortex,50 parietallobe,150 parsimonyprinciple,307 partitionfunction,200,371 passivesarcomeretension,131 path that requires minimal memory,366 pattern vector,101 Pavlov,170 Pavlov’sconditional–reflexlearning,170 Pavlov i an conditioning,278 peduncles,183 Penfield’smotorhomunculus,38 Penfield’ssensoryhomunculus.,38 pennate,125 perception energy,111 perceptronlearningalgorithm,101 perimysium,125 perioddoublingbifurcations,216 period–doubling bifurcation,212 periodicorbit,195,207,311 peripheral nervefibers,160 peripheral nervous system,159 personality,282 personality tests,314 Pfaffian forms,253 phase space,60 phase–spaceflow,196 phases,361 photons,230 physical kinetic energy,114 physical power,34 Piagettheory,317 pinballgame,191 plan,273 Planck’sconstant,233 plans,8 plyometrics,63 Poincar´e,192 Poincar´econjecture,201 Poincar´esection,194,196 Poincar´emap,218 Poincar´esection,217 Poincar´e–Bendixsontheorem,193 Poincar´e–Hopfindextheorem,193 point at infinity,232 pointorbit,311 pons,176 Pontryagin,199 population models,213 positionalstiffness,134 power sports,34 powerful mind,10 Prandtlnumber,211 predictive validity,315 primary factors,283 principal axis factor ing,300 principal components analysis,297,298 principal factor analysis,300 principal factors,306 principal intelligence factor,291 principle of cognitive optical illusions,

110 probability,275 probability density function,293 problemsolving,276 processing speed,363,366 product–moment,291 projection,251 Pro max rotation rotation,304 psychological tests,314 psychometric function,323 psychometric testing,287 psycho metrics,287 psycho physics,320 psychoticism,283 quadraticI&Fneuron,222 quantum brain,101 quantum jump,229 quantum neural network,100 quantumstateket–vector,231 quantum super position equation,100 quantum trigger,107 quantumtunnelingeffect,107 quasi-particletunneling,107 random variable,355 random walk,357 rate of relaxation,178 Raven’sProgressiveMatrices,290 Rayleigh–Vander Pol’sdissipation reason,273 reasoning ability,290 reciprocal activation,150 reciprocal inhibition,150 recurrent neural networks,99 recurrent, distributedparameter,132 recursivehomotopydynamics,364 recursivelyhierarchicalstructure,150 red,126 reduce,304 reflection,287 reflex,172 reflexarc,40,62 reflexmovements,40 reinforcement learning,328 reliable predictor,214 resonate–and–firen eur on,222 Ricciflow,248 Riemann curvature tensor,248 Riemann sphere,231 Riemann i an metric tensor,363 rotation(orientation)coordinates,76 rotational Hill’spar a meters,142 route to chaos,200 Rubinface,110 Rudolphine Tables,194 Ruelle,200 saltatoryconduction,167 sample,293 sarcolemma,125 sarcomere,126 sarcoplasm,125 sarcoplasmicreticulum,125 Schr¨odingerequation,99,111,231 Schr¨odingerequation,,233 Schr¨odingerpicture,231 Schwanncells,159 sci ent ificmethod,328 score,305 second–orderjetmanifold,258 selection methods,52 selective attention,366 self–limiting process,214 self–organized,265 sensitization,278 sensory adaptation,278 sensory analysis,324 sensory memory,45 sensory threshold,323 servoregulatoryloops,134 Shannon,200 Shannonentropy,187 short–term consequences,366 short–term memory,45 shortening heat,128 signal,265 signal detection theory,324 simulated anne a ling,328 Sinai,200 singularvaluedecomposition,299 situation awareness,92,333 situational awareness,9 slidingfilament mechanism,126 slidingfilament model,127 slowfibers,136 Smale,197 Smalehorseshoemap,202 small–world geometry,187 small–world networks,187 social hierarchy,187 social learning,278 social network,187 somatic nervous system,160 soma to sensory cortex,151 space of all weighted paths,365 spectraltheorem,299 spinal cord,40 spinal nerves,165 spindle receptors,134,149 spinesynapse,105 split–brain,314 squeezing,202 stable and unstable manifold,204 stablemanifold,218 standard deviation,292 standard map,218 standard normal distribution,292 standard problem–solving techniques,

328 Stanford–Binet,289 stationary states,228 step size,265,365 stochasticintegral,358 stochastictorquesandforces,262 Storage,46 strangeattractor,193,200,209–211 streamofphotons,233 strengths,363 stretch potent i a tion,130 stretch–reflex,57,62 stretch–shortening cycle,54,64 stretch-reflexinaction,8 stretching,202 stroboscopic section,208 structural equation modelling,301 structurally stable,200 sum over histories,365 superiorcerebellarpeduncle,183 supervised gradient descent learning,265 syllogism,274 symbolic dynamics,200,205,206 sympathetic nervous system,162 syncytium,109

Ttubules,126 Takens,200 talent model,51,52 talents,51 tangentdynamicsequation,368 temporallobe,150 tennisblitz-chess,8 tennis performance criteria,51 Termination,361 the angle between the two vectors,295 the same factor structure,52 theoretical ecology,215 theory of cognitive development,317 theory of turbulence,200 thermo-dynamic–likepartitionfunction,

365 thermodynamic relation,128 ther moe last i cheat,129 theta–neuron,222 thought–particles,113 three–pointiterativedynamicsequation, approach,316 topologicaltheorem,369 topology,192 training methods,52 transcendentalunityofapperception,108 transducerneurons,132,226 transition energy,369 transition entropy,365 transition pro pag at or,361 Transition–pro pag at or,362 triad,126 unipolarneurons,158 unitaryevolution,231 unitaryquantumevolution,233 unpredictability,194 unstablemanifold,218 unsupervised,265 Upanishads,97 vacuum state,351 Vander Pol,199 Vander Poloscillator,202 Varimaxrotation,303 velocity phase–space,252 velocity phase-spacemanifold,251 velocity vector-field,252,255 vertebratebrain,175 vestibularorgan,35 visual cortex,150 wave–particledualityofmatter,229 wave-function collapse,111 weakly–connected neural network,156 Weber–Fechnerlaw,321 Wechsler Adult Intelligence Scale,290 Wechsler–Bellevue I,291 whip-like movement,8 white matter,150 white musclefibers,126,136 Wickrotation,236 Wiener,199 Wienerprocess,262,359 Wignerfunction,348 Wilson-Cowanmodel,223 wisdom,286 working memory,45 wrapper,298