Natural biodynamics

Glossary of Frequently Used Symbols xv 1 . Introduction 1 1.1 The Problem of Natural Biodynamics . . . . . . . . . . . . 1 1.2 A Brief History of Biodynamics . . . . . . . . . . . . . . . . 3 1.3 Mechanical Basis of Biodynamics . . . . . . . . . . . . . . . 8 1.3.1 Natural Galilei Group . . . . . . . . . . . . . . . . . 9 1.3.2 Newtonian Equations of Motion . . . . . . . . . . . . 10 1.3.3 Calculus of Variations . . . . . . . . . . . . . . . . . 11 1.3.4 Lagrangian Equations of Motion . . . . . . . . . . . 12 1.3.5 Hamiltonian Equations of Motion . . . . . . . . . . . 13 1.3.6 Lagrangian Flows on Biodynamic Manifolds . . . . . 14 1.4 Conservative versus Dissipative Hamiltonian Dynamics . . . 15 1.4.1 Dissipative Systems . . . . . . . . . . . . . . . . . . . 16 1.4.2 Thermodynamic Equilibrium . . . . . . . . . . . . . 18 1.4.3 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 19 1.4.4 The Second Law of Thermodynamics . . . . . . . . . 19 1.4.5 Geometry of Phase Space . . . . . . . . . . . . . . . 21 1.5 Neural Basis of Biodynamics . . . . . . . . . . . . . . . . . 23 2 . Natural Language of Biodynamics 25 2.1.1 Preliminaries from Calculus, Algebra and Topology . 25 2.1.1.1 Notes From Calculus . . . . . . . . . . . . . 26 2.1 Categorical Metalanguage . . . . . . . . . . . . . . . . . . . 25 XiX xx Natural Biodynamics 2.1.1.2 Notes from Set Theory . . . . . . . . . . . . 2.1.1.3 Notes from General Topology . . . . . . . . 2.1.1.4 Commutative Diagrams . . . . . . . . . . . 2.1.1.5 Groups and Related Algebraic Structures . . 2.1.2 Categories . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Functors . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Natural Transformations . . . . . . . . . . . . . . . . 2.1.5 Limits and Colimits . . . . . . . . . . . . . . . . . . . 2.1.6 The Adjunction . . . . . . . . . . . . . . . . . . . . . 2.1.7 n-Categories . . . . . . . . . . . . . . . . . . . . . . 2.1.7.1 Generalization to ‘Big’ n-Categories . . . . 2.1.7.2 Topological Structure of n-Categories . . . 2.1.8 Algebra in Abelian Categories . . . . . . . . . . . . . 2.1.9 Fundamental Biodynamic Adjunction . . . . . . . . . 2.2 The Basics of Dynamics . . . . . . . . . . . . . . . . . . . . 2.2.1 Ordinary Differential Equations . . . . . . . . . . . . 2.2.2 Linear Autonomous Dynamics . . . . . . . . . . . . . 2.2.2.1 The Flow of a Linear ODE . . . . . . . . . . 2.2.2.2 Canonical Linear Flows in R2 . . . . . . . . 2.2.2.3 Topological Equivalence . . . . . . . . . . . 2.3 Chaos and Synergetics in Biodynamics . . . . . . . . . . . . 2.3.1 Prototype of Chaotic and Synergetic Systems . . . . 2.3.2 Chaotic Systems and Biomorphs . . . . . . . . . . . 2.3.2.1 Simulation Examples: Chaotic Systems . . . 2.3.2.2 Simulation Examples: Biomorphic Systems . 2.3.3 Controlling Chaos within the Chaos Theory . . . . . 2.3.3.1 Exploiting Critical Sensitivity . . . . . . . . 2.3.3.2 Lyapunov exponents and KY-dimension . . 2.3.3.3 Kolmogorov-Sinai entropy . . . . . . . . . . 2.3.3.4 Chaos Control by Ott, Grebogi and Yorke . 2.3.3.5 Floquet Stability Analysis and OGY Control 2.3.3.6 Jerk Functions of Simple Chaotic Flows . . 2.3.4 The Basic Hamiltonian Model of Biodynamics . . . . 2.3.5 The Basics of Haken’s Synergetics . . . . . . . . . . . 2.3.5.1 Phase Transitions . . . . . . . . . . . . . . . 2.3.5.2 Mezoscopic Derivation of Order Parameters 2.3.6 Macro-Synergetic Control of Biodynamics . . . . . . 28 28 32 35 39 43 45 47 47 49 49 54 57 59 60 60 63 63 65 67 68 73 75 75 81 82 82 85 87 89 92 97 101 102 104 106 109 3 . Natural Geometry of Biodynamics 111 Contents xxi 3.1 Motivation for Geometry in Biodynamics . . . . . . . . . . 3.2 Biodynamic Manifold M . . . . . . . . . . . . . . . . . . . . 3.2.1 Definition of the Manifold M . . . . . . . . . . . . . 3.2.2 Smooth Maps Between Manifolds . . . . . . . . . . . 3.3 Biodynamic Bundles . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Tangent Bundle of the Manifold M . . . . . . . 3.3.2 The Cotangent Bundle of the Manifold M . . . . . . 3.4 Sections of Biodynamic Bundles . . . . . . . . . . . . . . . . 3.4.1 Biodynamic Evolution and Flow . . . . . . . . . . . . 3.4.2 Vector-Fields and Their Flows . . . . . . . . . . . . 3.4.2.1 Vector-Fields on M . . . . . . . . . . . . . . 3.4.2.2 Integral Curves as Biodynamic Trajectories 3.4.2.3 Biodynamic Flows on M . . . . . . . . . . . 3.4.2.4 Categories of ODES . . . . . . . . . . . . . . 3.4.3 Differential Forms on M . . . . . . . . . . . . . . . . 3.4.3.1 1-Forms on M . . . . . . . . . . . . . . . . 3.4.3.2 &Forms on M . . . . . . . . . . . . . . . . 3.4.3.3 Exterior Differential Systems . . . . . . . . . 3.4.3.4 Exterior Derivative on M . . . . . . . . . . 3.4.3.5 De Rham Complex and Homotopy Operators 3.4.3.6 Stokes Theorem and De Rham Cohomology 3.4.3.7 Euler-Poincar6 Characteristics of M . . . . 3.4.3.8 Duality of Chains and Forms on M . . . . . 3.4.3.9 Other Exterior Operators on M . . . . . . . 3.4.4 Geometry of Nonlinear Dynamics . . . . . . . . . . . 3.5 Lie Categories in Biodynamics . . . . . . . . . . . . . . . . 3.5.1 Lie Derivative in Biodynamics . . . . . . . . . . . . . 3.5.1.1 Lie Derivative on Functions . . . . . . . . . 3.5.1.2 Lie Derivative of Vector Fields . . . . . . . . 3.5.1.3 Derivative of the Evolution Operator . . . . 3.5.1.4 Lie Derivative of Differential Forms . . . . . 3.5.1.5 Lie Derivative of Various Tensor Fields . . . 3.5.1.6 Lie Algebras . . . . . . . . . . . . . . . . . . 3.5.2 Lie Groups in Biodynamics . . . . . . . . . . . . . . 3.5.2.1 Lie Groups and Their Lie Algebras . . . . . 3.5.2.2 Actions of Lie Groups on M . . . . . . . . . 3.5.2.3 Basic Biodynamic Groups . . . . . . . . . . 3.5.2.4 Groups of Joint Rotations . . . . . . . . . . 3.5.2.5 Special Euclidean Groups of Joint Motions . 114 117 118 120 121 121 123 124 125 126 126 127 131 132 133 134 135 137 138 140 142 143 144 146 148 152 152 152 155 157 158 159 161 162 162 167 169 171 175 xxii Natural Biodynamics 3.5.3 Group Structure of the Biodynamic Manifold M . . 3.5.3.1 Purely Rotational Biodynamic Manifold . . 3.5.3.2 Reduction of the Rotational Manifold . . . . 3.5.3.3 The Complete Biodynamic Manifold . . . . 3.5.3.4 Realistic Human Spine Manifold . . . . . . . 3.5.4 Lie Symmetries in Biodynamics . . . . . . . . . . . . 3.5.4.1 Lie Symmetry Groups . . . . . . . . . . . . 3.5.4.2 Prolongations . . . . . . . . . . . . . . . . . 3.5.4.3 Special Biodynamic Equations . . . . . . . . 3.6 Riemannian Geometry in Biodynamics . . . . . . . . . . . . 3.6.1 Local Riemannian Geometry on M . . . . . . . . . . 3.6.1.1 Riemannian Metric on M . . . . . . . . . . 3.6.1.2 Geodesics on M . . . . . . . . . . . . . . . . 3.6.1.3 Riemannian Curvature on M . . . . . . . . 3.6.2 Global Riemannian Geometry on M . . . . . . . . . 3.6.2.1 The Second Variation Formula . . . . . . . . 3.6.2.2 Gauss-Bonnet Formula . . . . . . . . . . . . 3.6.2.3 Ricci Flow on M . . . . . . . . . . . . . . . 3.6.2.4 Structure Equations on M . . . . . . . . . . 3.6.2.5 Basics of Morse Theory . . . . . . . . . . . . 3.6.2.6 Basics of (Co)Bordism Theory . . . . . . . . 3.7 Symplectic Geometry in Biodynamics . . . . . . . . . . . . 3.7.1 Symplectic Algebra . . . . . . . . . . . . . . . . . . . 3.7.2 Symplectic Geometry on M . . . . . . . . . . . . . . 3.8 Impulse Biodynamics and Synthetic Geometry . . . . . . . 3.8.1 Delta Spikes . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Kick Dynamics . . . . . . . . . . . . . . . . . . . . . 3.8.2.1 Deterministic Delayed Kicks . . . . . . . . . 3.8.2.2 Random Kicks and Langevin Equation . . . 3.8.3 Distributions and Synthetic Differential Geometry . . 3.8.3.1 Distributions . . . . . . . . . . . . . . . . . . 3.8.3.2 Synthetic Calculus in Euclidean Spaces . . . 3.8.3.3 Spheres and Balls as Distributions . . . . . . 3.8.3.4 Stokes Theorem for Unit Sphere . . . . . . . 3.8.3.5 Time Derivatives of Expanding Spheres . . . 3.8.3.6 The Wave Equation . . . . . . . . . . . . . . 3.9 A Quick Look at Modern Geometrodynamics . . . . . . . . 3.9.1 Einstein Equations . . . . . . . . . . . . . . . . . . . 3.9.2 n-Categories in Physics . . . . . . . . . . . . . . . . 180 180 182 184 184 185 185 188 194 195 196 197 201 202 205 205 208 209 212 213 215 217 217 218 220 220 222 222 223 226 227 229 231 233 234 235 237 237 237 Contents xxiii 3.9.3 Quantum Geometry Framework . . . . . . . . . . . . 240 3.10 3D Modelling and Animation in Biodynamics . . . . . . . . 242 3.10.1 Essentials of Human Animation . . . . . . . . . . . . 242 3.10.1.1 Motion CaptureBased Human Animation . 243 3.10.1.2 Virtual Muscular Dynamics in 3D-Graphics 244 245 3.10.2.1 Power Basis Form of a Curve . . . . . . . . 246 3.10.2.2 Bezier Curves . . . . . . . . . . . . . . . . . 247 3.10.2.3 Rational Bezier Curves . . . . . . . . . . . . 249 3.10.3 B-Spline Basis Functions . . . . . . . . . . . . . . . . 249 3.10.3.1 DeBoor-Cox Recursive Definition . . . . . . 250 3.10.3.2 Derivatives of B-Spline Basis Functions . . . 250 3.10.4 B-Spline Curves and Surfaces in Geometric Modelling 251 3.10.4.1 Definition of B-Spline Curves . . . . . . . . 251 3.10.4.2 Properties of B-Spline Curves . . . . . . . . 251 3.10.4.3 Derivatives of a B-Spline Curve . . . . . . . 252 3.10.4.4 Definition of B-Spline Surfaces . . . . . . . 253 3.10.4.5 Properties of B-Spline Surfaces . . . . . . . 253 3.10.4.6 Derivatives of a B-Spline Surface . . . . . . 254 3.10.5 NURBS Curves and Surfaces . . . . . . . . . . . . . 255 3.10.5.1 Definition of NURBS Curves . . . . . . . . . 255 3.10.5.2 Properties of NURBS Curves . . . . . . . . 255 3.10.5.3 Definition of NURBS Surfaces . . . . . . . . 256 3.10.5.4 Properties of NURBS Surfaces . . . . . . . . 257 3.11 Kinematics of Biomechanical Chains . . . . . . . . . . . . . 257 3.11.1 3D Transformation Matrix . . . . . . . . . . . . . . . 257 3.11.2 A Multilink Kinematic Chain . . . . . . . . . . . . . 258 259 3.11.4 Transformation Matrix Used in Computer Graphics . 260 3.10.2 Curves and Surfaces in Geometric Modelling . . . . . 3.11.3 CNS Representation of the Body Posture . . . . . . . 4 . Natural Mechanics of Biodynamics 4.1 Lagrangian Formalism in Biodynamics . . . . . . . . . . . . 4.2 Hamiltonian Formalism in Biodynamics . . . . . . . . . . . 4.2.1 Nonlinear Dynamics in Hamiltonian Form . . . . . . 4.2.1.1 Real 1-DOF Hamiltonian Dynamics . . . . 4.2.1.3 Library of Basic Hamiltonian Systems . . . 4.2.1.4 n-DOF Hamiltonian Dynamics . . . . . . . 4.2.1.2 Complex One-DOF Hamiltonian Dynamics 4.2.2 Hamiltonian Geometry in Biodynamics . . . . . . . . 261 261 265 267 267 276 279 286 289 xxiv Natural Biodynamics 4.2.3 Hamilton-Poisson Geometry in Biodynamics . . . . . 4.2.3.1 Hamilton-Poisson Biodynamic Systems . . . 4.2.4 Completely Integrable Hamiltonian Systems . . . . . 4.2.4.1 Liouville Theorem . . . . . . . . . . . . . . . 4.2.4.2 Action-Angle Variables . . . . . . . . . . . . 4.2.5 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5.1 Ergodicity in Hamiltonian Systems . . . . . 4.2.5.2 Dynamical Systems and Hyperbolicity . . . 4.2.5.3 Ergodic Theory and Nontrivial Recurrence . 4.2.5.4 Nonuniformly Hyperbolic Trajectories . . . 4.2.5.5 Systems with Nonzero Lyapunov Exponents 4.3 Quantum Formalism in Nano-Biodynamics . . . . . . . . . 4.3.1 Quantum Mechanics in Biological Matter . . . . . . 4.3.2 Dirac’s Canonical Quantization . . . . . . . . . . . . 4.3.2.1 Quantum States and Operators . . . . . . . 4.3.2.2 Quantum Pictures . . . . . . . . . . . . . . . 4.3.2.3 Spectrum of a Quantum Operator . . . . . . 4.3.2.4 General Representation Model . . . . . . . . 4.3.2.5 Direct Product Space . . . . . . . . . . . . . 4.3.2.6 State-Space for n Quantum Particles . . . . 4.3.2.7 Quantum Measurement and Penrose Paradox 4.3.3 The Problem of Quantum Measurement and Entropy 4.3.3.2 Quantum Object . . . . . . . . . . . . . . . 4.3.3.1 The Classical Apparatus . . . . . . . . . . . 4.3.3.3 Adiabatic Measurement Lagrangians . . . . 4.3.3.4 The Stern-Gerlach Experiment . . . . . . . 4.3.3.5 Work and Heat . . . . . . . . . . . . . . . . 4.3.3.6 Statistical Thermodynamics . . . . . . . . . 4.3.3.7 Friction in a Macroscopic Apparatus . . . . 4.3.3.8 Low Velocity Projective Measurements . . . 4.3.3.9 Information and Entropy . . . . . . . . . . . 4.3.4 Von Neumann’s Density Matrix Quantization . . . . 4.3.4.1 Dissipative Quantum Formalism . . . . . . . 4.3.5 Geometric Quantization . . . . . . . . . . . . . . . . 4.3.5.2 Geometric Prequantization . . . . . . . . . . Variational Formalism in Biodynamics . . . . . . . . . . . . 4.4.1 Biodynamic Action Functional . . . . . . . . . . . . . 4.3.5.1 Motivation . . . . . . . . . . . . . . . . . . . 4.4 4.4.2 Lagrangian Action . . . . . . . . . . . . . . . . . . . 292 294 298 298 299 301 301 302 305 306 307 310 310 311 312 318 320 324 325 326 328 330 331 331 333 334 335 336 338 340 341 342 344 345 345 347 351 351 352 Contents XXV 4.4.3 Hamiltonian Action . . . . . . . . . . . . . . . . . . . 353 4.4.4 Noether Theorem . . . . . . . . . . . . . . . . . . . . 354 4.4.5 Hamiltonian-Action Formulation of Biodynamics . . 356 4.4.6 Feynman Quantum Action . . . . . . . . . . . . . . . 359 4.5 Nonholonomic Biodynamics . . . . . . . . . . . . . . . . . . 364 4.5.1 Lagrangian Approach . . . . . . . . . . . . . . . . . . 364 4.5.2 Hamiltonian Approach . . . . . . . . . . . . . . . . . 366 4.5.3 Biodynamic Example: Bicycle Dynamics . . . . . . . 368 4.6 Stochastic Formalism in Biodynamics . . . . . . . . . . . . 370 4.6.1 Markov Stochastic Processes . . . . . . . . . . . . . . 372 4.6.2 Statistical Mechanics of Oscillator Chains . . . . . . 374 4.7 Muscular Excitation-Contraction Dynamics . . . . . . . . . 376 4.7.1 Human Musculo-Skeletal System . . . . . . . . . . . 376 4.7.1.1 Human Skeleton . . . . . . . . . . . . . . . . 376 4.7.1.2 Human Joints . . . . . . . . . . . . . . . . . 378 4.7.1.3 Human Muscular System . . . . . . . . . . . 379 4.7.1.4 Human Energy Flow . . . . . . . . . . . . . 381 4.7.1.5 Equivalent Muscular Actuator . . . . . . . . 385 4.7.2 Molecular Muscular Dynamics . . . . . . . . . . . . . 385 4.7.3 Mezoscopic Muscular Dynamics . . . . . . . . . . . . 386 4.7.3.1 Myocybernetics . . . . . . . . . . . . . . . . 392 4.7.4 Macroscopic Muscular Dynamics . . . . . . . . . . . 392 4.7.4.1 Soft Tissue Dynamics of Relaxed Muscles . 392 4.7.4.2 Classical Hill’s Model . . . . . . . . . . . . . 395 4.7.4.3 Biodynamics of Load-Lifting . . . . . . . . . 396 4.8 Lie Functors in Biodynamics . . . . . . . . . . . . . . . . . 403 4.8.1 Lie-Lagrangian Biodynamic Functor . . . . . . . . . 403 4.8.1.1 Joint Kinematics . . . . . . . . . . . . . . . 403 4.8.1.2 Exterior Lagrangian Dynamics . . . . . . . . 405 4.8.2 Lie-Hamiltonian Biodynamic Functor . . . . . . . . 410 4.8.2.1 The Abstract Functor Machine . . . . . . . 412 4.8.2.2 Muscle-Driven Hamiltonian Biodynamics . . 413 4.8.3 Stochastic-Lie-Hamiltonian Biodynamic Functor . . 414 4.9 Mechanics of Spinal Injuries . . . . . . . . . . . . . . . . . . 419 4.9.1 Spinal Dislocations, Disclinations and Fractures . . . 420 4.9.2 Measuring the Risk of Local Intervertebral Injuries . 420 4.9.2.1 Biodynamic Jerk Functions . . . . . . . . . 424 4.9.3 Measuring the Risk of Vertebral Fractures . . . . . . 426 4.8.4 Fuzzy-Stochastic-Lie-Hamiltonian Functor . . . . . 416 xxvi Natuml Biodynamics 4.9.3.1 Research on Bone Injuries . . . . . . . . . . 426 5 . Natural Topology of Biodynamics 427 5.1 Category of (Co)Chain Complexes in Biodynamics . . . . . 427 5.1.1 (Co)Homologies in Abelian Categories Related to M 428 5.1.2 Reduction and Euler-Poincar6 Characteristic . . . . 430 5.2 Natural Duality in Biodynamics . . . . . . . . . . . . . . . . 431 5.2.1 Geometric Duality Theorem for M . . . . . . . . . . 431 5.2.1.1 Lie-Functorial Proof . . . . . . . . . . . . . 432 5.2.1.2 Geometric Proof . . . . . . . . . . . . . . . . 433 5.2.2 Topological Duality Theorem for M . . . . . . . . . 437 5.2.2.1 Cohomological Proof . . . . . . . . . . . . . 437 5.2.2.2 Homological Proof . . . . . . . . . . . . . . 439 5.2.3 Lagrangian Versus Hamiltonian Duality . . . . . . . 439 5.2.4 Globally Dual Structure of Rotational Biodynamics . 440 5.3 Topological Phase Transitions and Hamiltonian Chaos . . . 441 5.3.1 Phase Transitions in Hamiltonian Systems . . . . . . 441 5.3.2 Geometry of the Largest Lyapunov Exponent . . . . 444 5.3.3 Euler Characteristics of Hamiltonian Systems . . . . 447 5.4 The Covariant Force Functor . . . . . . . . . . . . . . . . . 452 6 . Natural Control and Self-organization in Biodynamics 453 6.1 The Basics of Classical Control and Stability . . . . . . . . 455 6.1.1 Introduction to Feedback Control . . . . . . . . . . . 455 6.1.2 Linear Stationary Systems and Operators . . . . . . 460 6.1.2.1 Basics of Kalman State-Space Theory . . . 460 6.1.2.2 Regulator Problem . . . . . . . . . . . . . . 461 6.1.2.3 End Point Control Problem . . . . . . . . . 462 6.1.2.4 Servomechanism Problem . . . . . . . . . . 463 6.1.2.5 Repetitive Mode Problem . . . . . . . . . . 463 6.1.2.6 Feedback Changes the Operator . . . . . . . 464 6.1.3 Stability and Boundedness . . . . . . . . . . . . . . . 465 6.1.4 Lyapunov’s Stability Method . . . . . . . . . . . . . 468 6.1.5 Graphical Techniques for Nonlinear Systems . . . . . 469 6.1.5.1 Describing Function Analysis . . . . . . . . 470 6.2 The Basis of Modern Geometric Control . . . . . . . . . . . 472 6.2.1 Feedback Linearization . . . . . . . . . . . . . . . . . 472 6.2.1.1 Exact Feedback Linearization . . . . . . . . 472 Contents xxvii 6.2.1.2 Relative Degree . . . . . . . . . . . . . . . . 6.2.1.3 Approximative Feedback Linearization . . . 6.2.2 Controllability . . . . . . . . . . . . . . . . . . . . . . 6.2.2.1 Linear Controllability . . . . . . . . . . . . . 6.2.2.2 Nonlinear Controllability . . . . . . . . . . . 6.2.2.3 Controllability Condition . . . . . . . . . . . 6.2.2.4 Distributions . . . . . . . . . . . . . . . . . . 6.2.2.5 Foliations . . . . . . . . . . . . . . . . . . . 6.3 Modern Control Techniques for Mechanical Systems . . . . 6.3.1 Abstract Control System . . . . . . . . . . . . . . . . 6.3.2 Controllability of a Linear Control System . . . . . . 6.3.3 Affine Control System and Local Controllability . . . 6.3.4 Hamiltonian Control and Maximum Principle . . . . 6.3.4.1 Hamiltonian Control Systems . . . . . . . . 6.3.4.2 Pontryagin’s Maximum Principle . . . . . . 6.3.4.3 Affine Control Systems . . . . . . . . . . . . 6.4 Locomotion Systems and Human Gait . . . . . . . . . . . . 6.4.1 Control of Locomotion Systems . . . . . . . . . . . . 6.4.1.1 Stratified Kinematic Controllability . . . . . 6.4.1.2 The Distributions Approach . . . . . . . . . 6.4.1.3 The Exterior Differential Systems Approach 6.4.1.4 On the Existence and Uniqueness of Solutions 6.4.1.5 Trajectory Generation Problem . . . . . . . 6.5 Biodynamic Control Policy, Learning and Self-organization 6.5.1.1 Direct Learning of the Control Policy . . . . 6.5.1.2 6.5.1.3 Learning of Motor Control Components . . 6.5.3 Neuro-Muscular Excitation-Contraction Dynamics . 6.5.3.1 Motor Units . . . . . . . . . . . . . . . . . . 6.5.3.2 Darwinian Oscillatory Neural Net . . . . . . 6.5.3.3 Recurrent Neuro-Muscular Model . . . . . . 6.5.3.4 Autogenetic Reflex Motor-Servo . . . . . . . 6.5.3.5 Biodynamics Control . . . . . . . . . . . . . 6.4.2 Gait Biodynamics . . . . . . . . . . . . . . . . . . . . 6.5.1 Control Policy Learning by Robots . . . . . . . . . . Indirect Learning of the Control Policy . . . 6.5.2 Pathways to Self-organization in Biodynamics . . . 6.5.4 Lie-Adaptive Biodynamic Control . . . . . . . . . . Kalman Filtering and Inertial Navigation 6.6 Essentials of Biodynamic Measurement: . . . . . . . . . . 476 478 481 481 482 484 485 486 487 487 488 489 490 490 493 494 496 496 497 499 499 i 501 502 504 508 508 509 510 512 512 514 514 514 517 519 520 524 526 xxviii Natural Biodynamics 6.6.0.1 Kalman Filter Basics . . . . . . . . . . . . . 526 6.6.0.2 Inertial Navigation . . . . . . . . . . . . . . 532 6.6.0.3 Adaptive Estimation in Biomechanics . . . . 536 6.7 Humanoid Robotics . . . . . . . . . . . . . . . . . . . . . . 537 6.7.1 Honda Humanoid Series . . . . . . . . . . . . . . . . 537 6.7.2 Cerebellar Robotics . . . . . . . . . . . . . . . . . . . 538 7 . Natural Brain Dynamics and Sensory-Motor Integration 541 7.1 Introduction to Brain . . . . . . . . . . . . . . . . . . . . . 542 7.2 Human Nervous System . . . . . . . . . . . . . . . . . . . . 549 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.2.6 7.2.7 7.2.8 7.2.9 Building Blocks of the Nervous System . . . . . . . . 550 7.2.1.1 Neuronal Circuits . . . . . . . . . . . . . . . 552 7.2.1.2 Basic Brain Partitions and Their Functions 556 7.2.1.3 Nerves . . . . . . . . . . . . . . . . . . . . . 558 7.2.1.4 Action potential . . . . . . . . . . . . . . . . 559 7.2.1.5 Synapses . . . . . . . . . . . . . . . . . . . . 561 Reflex Action: the Basis of CNS Activity . . . . . . . 566 The Spinal Cord Pathways . . . . . . . . . . . . . . . 568 7.2.3.1 Spinal Lower Motor Neurons . . . . . . . . . 569 7.2.3.2 Central Pattern Generators in the Spinal Cord575 7.2.3.3 Influence of Higher Centers . . . . . . . . . 576 First Look at the Brain . . . . . . . . . . . . . . . . 578 Motor Pathways . . . . . . . . . . . . . . . . . . . . . 581 7.2.5.1 Primary Motor Cortex . . . . . . . . . . . . 581 7.2.5.2 Motor Association/Premotor Cortical Areas 584 Subcortical Motor ‘Side Loops’ . . . . . . . . . . . . 587 7.2.6.1 The Cerebellum . . . . . . . . . . . . . . . . 588 7.2.6.2 The Basal Ganglia . . . . . . . . . . . . . . 594 7.2.6.3 Cerebellar Movement Control . . . . . . . . 596 Human Senses and their Pathways . . . . . . . . . . 600 The Human-Like Vision . . . . . . . . . . . . . . . . 603 7.2.8.2 Retina . . . . . . . . . . . . . . . . . . . . . 606 7.2.8.3 Cornea . . . . . . . . . . . . . . . . . . . . . 607 7.2.8.4 Iris . . . . . . . . . . . . . . . . . . . . . . . 608 7.2.8.5 Pursuit Eye Control and Motion Perception 610 7.2.8.6 Optical Flow . . . . . . . . . . . . . . . . . . 613 The Visual Pathway . . . . . . . . . . . . . . . . . . 615 7.2.8.1 Extramular SO( 3) -Muscles . . . . . . . . . 604 7.2.9.1 Light Reflex and 3D Vision . . . . . . . . . 619 Contents xxix 7.2.10 Differential Geometry of the Striate Cortex . . . . . 7.2.1 1 Auditory and Vestibular Pathways . . . . . . . . . . 7.2.11.1 The Inner Ear . . . . . . . . . . . . . . . . . 7.2.11.2 Auditory Transduction . . . . . . . . . . . . 7.2.11.3 Central Auditory Pathways . . . . . . . . . 7.2.11.4 The Vestibular System . . . . . . . . . . . . 7.2.11.5 The Semicircular Canals . . . . . . . . . . . 7.2.11.6 The Vestibulo-Ocular Reflex . . . . . . . . . 7.2.11.7 The Utricle and Saccule . . . . . . . . . . . 7.2.11.8 Mechanics of the Semicircular Canals . . . . 7.2.11.9 Endolymph Flow in the Semicircular Canals 7.2.12 Somatosensory Pathways . . . . . . . . . . . . . . . . 7.2.12.1 The Discriminative Touch System . . . . . . 7.2.12.2 The Pain and Temperature System . . . . . 7.2.12.3 The Proprioceptive System . . . . . . . . . . 7.3 The Sensory-Motor Adjunction . . . . . . . . . . . . . . . . 7.3.1 Summary on Sensory-Motor Pathways . . . . . . . . 7.3.2 Sensory-Motor Control . . . . . . . . . . . . . . . . . 7.3.2.1 Multisensory Integration for Motor Planning 7.3.2.2 The Sensory-Motor Adjunction . . . . . . . 7.3.3 The Central Biomechanical Adjunction . . . . . . . . 7.3.3.1 Postural Control Experiments . . . . . . . . 7.3.3.2 Learning Arm Movement Control . . . . . . 7.3.4 Mechanism of Brain Injuries . . . . . . . . . . . . . . 7.3.4.1 Basic Dynamics of Brain Injuries . . . . . . 7.3.4.2 Research on Head and Brain Injuries . . . . Microscopic Neurodynamics of Microtubules . . . . . 7.4.1.1 Biochemistry of Microtubules . . . . . . . . 7.4.1.2 Kink Soliton Model of MT-Dynamics . . . . Mezoscopic Neurodynamics of Action Potentials . . . 7.4.2.1 Hodgkin-Huxley Model . . . . . . . . . . . . 7.4.2.2 FitzHugh-Nagumo Model . . . . . . . . . . 7.5 Biological Neural Nets . . . . . . . . . . . . . . . . . . . . . 7.5.1 Phase Dynamics of Oscillatory Neural Nets . . . . . 7.5.1.1 Kuramoto Synchronization Model . . . . . . 7.5.1.2 Lyapunov Chaotic Synchronization . . . . . 7.5.2 Complex Networks Dynamics . . . . . . . . . . . . . 7.5.2.1 Continuum Limit of the Kuramoto Network 7.4 Brain Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 7.4.2 620 622 622 623 624 626 626 627 628 629 630 632 633 634 635 638 638 642 645 649 649 652 655 658 658 661 661 662 662 664 667 667 671 672 672 675 676 678 679 Natural Baodynamics 7.5.2.2 Path-Integral Approach to Complex Nets . 679 7.5.3 Complex Adaptive Systems . . . . . . . . . . . . . . 681 7.5.4 Noise Delayed Bifurcation in Coupled Neurons . . . 684 7.5.4.1 The Theta-Neuron . . . . . . . . . . . . . . 684 7.5.4.2 Coupled Theta-Neurons . . . . . . . . . . . 685 7.5.5 Classification of ‘Spiking’ Neuron Models . . . . . . . . 688 7.5.6 Weakly Connected and Canonical Neural Nets . . . . 692 7.5.7 Quantum Brain Model . . . . . . . . . . . . . . . . . 694 7.5.8 Open Liouville Neurodynamics and Self-similarity . 698 7.5.8.1 Hamiltonian Framework . . . . . . . . . . . 700 7.5.8.2 Conservative Classical System . . . . . . . . 700 7.5.8.3 Conservative Quantum System . . . . . . . 700 7.5.8.4 Open Classical System . . . . . . . . . . . . 701 7.5.8.5 Continuous Neural Network Dynamics . . . 702 7.5.8.6 Open Quantum System . . . . . . . . . . . . 703 7.5.8.7 Non-Critical Stringy MT-Dynamics . . . . 704 7.5.8.8 Equivalence of Neurodynamic Forms . . . . 704 7.6 Artificial Neural Nets in Biodynamics . . . . . . . . . . . . 705 7.6.1 Biological Versus Artificial Neural Nets . . . . . . . . 705 7.6.2 Common Discrete ANNs . . . . . . . . . . . . . . . . 707 7.6.2.1 Multilayer Perceptrons . . . . . . . . . . . . 707 7.6.2.2 Summary of Supervised Learning Methods . 719 7.6.2.3 Other Standard ANNs . . . . . . . . . . . . 720 7.6.2.4 Fully Recurrent ANNs . . . . . . . . . . . . 727 7.6.2.5 Dynamical Games and Recurrent ANNs . . 728 7.6.2.6 Complex-Valued ANNs . . . . . . . . . . . . 731 7.6.3 Common Continuous ANNs . . . . . . . . . . . . . . 732 7.6.3.1 Neurons as Functions . . . . . . . . . . . . . 733 7.6.3.2 Basic Activation and Learning Dynamics . . 735 7.6.3.3 Standard Models of Continuous Nets . . . . 736 7.7 Distinguished ANN Models . . . . . . . . . . . . . . . . . . 741 7.7.1 Generalized Kohonen’s SOM . . . . . . . . . . . . . . 741 7.7.1.1 The Winner Relaxing Kohonen Algorithm . 742 7.7.1.2 The Magnification Factor . . . . . . . . . . . 743 7.7.2 Dynamics of Hopfield’s Associative Recurrent Nets . 745 7.7.2.1 Ising-Spin Neurons . . . . . . . . . . . . . . 745 7.7.2.2 Graded-Response Neurons . . . . . . . . . . 746 7.7.2.3 Hopfield’s Overlaps . . . . . . . . . . . . . . 747 7.7.1.3 Magnification Exponent . . . . . . . . . . . 744 Contents m i 7.7.2.4 Overlap Dynamics . . . . . . . . . . . . . . 749 7.7.2.5 Hebbian Learning Dynamics . . . . . . . . . 750 7.7.3 A Self-organizing Bidirectional Competitive Net . . 753 7.8 Fuzzy Logic in Biodynamics . . . . . . . . . . . . . . . . . . 755 7.8.1 The Concept of Fuzziness . . . . . . . . . . . . . . . 755 7.8.1.1 ‘Fuzzy Thinking’ . . . . . . . . . . . . . . . 755 7.8.1.2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . 756 7.8.1.3 fizziness of the Real World . . . . . . . . . 757 7.8.1.4 Fuzzy Entropy . . . . . . . . . . . . . . . . . 758 7.8.2 Fuzzy Inference Engine . . . . . . . . . . . . . . . . . 761 7.8.3 Fuzzy Logic Control . . . . . . . . . . . . . . . . . . 764 7.8.3.1 Fuzzy Control of Biodynamic Jerks . . . . . 768 7.8.3.2 Characteristics of Fuzzy Control . . . . . . . 769 7.8.3.3 Evolving Connectionist Systems . . . . . . . 770 7.8.4 High-Resolution FAM Agents . . . . . . . . . . . . . 771 7.8.4.1 Generic Nonlinear MIMO Systems . . . . . 772 7.8.4.2 Alternative MIMO Systems . . . . . . . . . 775 7.8.4.3 Biodynamics Example: Tennis Game . . . . 777 7.9 Natural System in a Category . . . . . . . . . . . . . . . . . 783 7.9.1 Categorical Patterns and Hierarchical Links . . . . . 783 7.9.2 A General Natural System . . . . . . . . . . . . . . . 786 7.9.3 The Category of Neurons . . . . . . . . . . . . . . . 787 7.9.4 Memory Evolutive System . . . . . . . . . . . . . . . 787 7.9.5 Neural System in a Category . . . . . . . . . . . . . 789 7.10 Brain-Mind Functorial Machines . . . . . . . . . . . . . . . 792 7.10.1 Neurodynamic 2-Functor . . . . . . . . . . . . . . . 792 7.10.2 Solitary ‘Thought Nets’ and the Emerging Mind . . . 795 7.10.2.1 Synergetic ‘Thought Solitons’ . . . . . . . . 795 7.11 Body-Mind Adjunction and Natural Psychodynamics . . . 800 801 7.11.1.1 Six Faces of the Life Space Foam . . . . . . 806 7.11.1.2 General Formalism . . . . . . . . . . . . . . 806 7.11.1.3 Motion and Decision Making in LSFPaths . 810 7.11.1.4 ForceFields and Memory in LSFfields . . . 814 7.11.1.5 Geometries, Topologies and Noise in LSF,,,, 816 7.12 Brain-Like Control in a Nutshell . . . . . . . . . . . . . . . 818 7.12.1 Functor Control Machine . . . . . . . . . . . . . . . . 820 7.12.2 Spinal Control Level . . . . . . . . . . . . . . . . . . 822 7.12.3 Cerebellar Control Level . . . . . . . . . . . . . . . . 827 7.11.1 Natural Psychodynamics in the Life Space Foam . . Natural Biodynamics 7.12.4 Cortical Control Level . . . . . . . . . . . . . . . . . 7.12.5 A Note on Muscular Training 7.12.6 Errors in Motion Control: Locomotor Injuries . . . . . . . . . . . . . . . . . Appendix A A.l Basic Formulas from Tensor Analysis . . . . . . . . . . . . . A.l.l General Functional Transformation . . . . . . . . . . A.l.l.l Transformation of Coordinates . . . . . . . . A.1.1.2 Scalar Invariants . . . . . . . . . . . . . . . A.1.1.3 Vectors and Covectors . . . . . . . . . . . . A.1.1.4 Second-Order Tensors . . . . . . . . . . . . A . 1.1.5 Higher-Order Tensors . . . . . . . . . . . . A.1.1.6 Tensor Symmetry . . . . . . . . . . . . . . . A . 1.2 Euclidean Tensors . . . . . . . . . . . . . . . . . . . . A.1.2.1 Basis Vectors and the Metric Tensor in R" . A.1.2.2 Tensor Products in R" . . . . . . . . . . . . A.1.3 Tensor Derivatives on Riemannian Manifolds . . . . A.1.3.1 Christoffel's Symbols . . . . . . . . . . . . . A.1.3.2 Geodesics . . . . . . . . . . . . . . . . . . . A.1.3.3 The Covariant Derivative . . . . . . . . . . . A.1.3.4 Vector Differential Operators . . . . . . . . A.1.3.5 The Absolute Derivative . . . . . . . . . . . A.1.4 The Covariant Force Law in Biodynamics . . . . . . A.1.5 The Essence of Natural Hamiltonian Biodynamics . . A.1.6 Neuro-Hamiltonian Control in Biodynamics . . . . . A.2 Frequently Used Neurophysiological Terms . . . . . . . . . . A.3.1 Nuclear Magnetic Resonance in 2D Medical Imaging A.3.2 3D Magnetic Resonance Imaging of Human Brain . . A.3.3 Diffusion MRI in 3D Volume . . . . . . . . . . . . . . A.3.4 Imaging Diffusion with MRI . . . . . . . . . . . . . . A.3.5 3D Diffusion Tensor . . . . . . . . . . . . . . . . . . A.3.6 Brain Connectivity Studies . . . . . . . . . . . . . . . A.3.7 Brain Waves and Independent Component Analysis . . . . . . A.4.1 Complex Numbers and Vectors . . . . . . . . . . . . A.4.1.1 Quaternions and Rotations . . . . . . . . . . A.3 Modern 3D Neuroimaging . . . . . . . . . . . . . . . . . . . A.4 Complex Functions, Manifolds and Hilbert Spaces A.4.2 Complex Functions . . . . . . . . . . . . . . . . . . . A.4.3 Complex Manifolds . . . . . . . . . . . . . . . . . . . 830 833 836 837 837 837 838 839 839 840 842 842 844 844 845 846 846 847 847 848 849 853 855 856 857 880 880 881 882 883 884 887 888 889 889 890 894 898 Contents xxxiii A.4.4 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . 902 A.5 Classical Lie Theory . . . . . . . . . . . . . . . . . . . . . . 904 A.5.1 Basic Tables of Lie Groups and Their Lie Algebras . 904 A.5.2 Representations of Lie group