Natural Biodynamcis

by Ivancevic, Vladimir G.; Ivancevic, Tijana T.

• ISBN13: 9789812565341
• ISBN10: 9812565345
• Format: Hardcover
• Copyright: 2006-04-01
• Publisher: World Scientific Pub Co Inc

Summary

This comprehensive volume is a graduate-level text in human biodynamics, written in the unified categorical language of modern differential geometry and topology. Combining mathematics, physics and robotics with human physiology, this is the first book that describes all levels of human biodynamics, from musculo-skeletal mechanics to the higher brain functions. The book develops and uses a variety of research methods, ranging from chaos theory and Haken's synergetics, through quantum mechanics, to nonlinear control and artificial intelligence, to provide the means to understand, predict and control the behavior of human-like systems in their full neuro-musculo-skeletal complexity. The applications of this unique scientific methodology range from prediction of human neuro-musculo-skeletal injuries to brain-like control of humanoid robots.

Table of Contents

Preface	vii
Glossary of Frequently Used Symbols	xv
1. Introduction	1	(24)
1.1 The Problem of Natural Biodynamics	1	(2)
1.2 A Brief History of Biodynamics	3	(5)
1.3 Mechanical Basis of Biodynamics	8	(7)
1.3.1 Natural Galllei Group	9	(1)
1.3.2 Newtonian Equations of Motion	10	(1)
1.3.3 Calculus of Variations	11	(1)
1.3.4 Lagrangian Equations of Motion	12	(1)
1.3.5 Hamiltonian Equations of Motion	13	(1)
1.3.6 Lagrangian Flows on Biodynamic Manifolds	14	(1)
1.4 Conservative versus Dissipative Hamiltonian Dynamics	15	(8)
1.4.1 Dissipative Systems	16	(2)
1.4.2 Thermodynamic Equilibrium	18	(1)
1.4.3 Nonlinearity	19	(1)
1.4.4 The Second Law of Thermodynamics	19	(2)
1.4.5 Geometry of Phase Space	21	(2)
1.5 Neural Basis of Biodynamics	23	(2)
2. Natural Language of Biodynamics	25	(86)
2.1 Categorical Metalanguage	25	(35)
2.1.1 Preliminaries from Calculus, Algebra and Topology	25	(14)
2.1.1.1 Notes From Calculus	26	(2)
2.1.1.2 Notes from Set Theory	28	(1)
2.1.1.3 Notes from General Topology	28	(4)
2.1.1.4 Commutative Diagrams	32	(3)
2.1.1.5 Groups and Related Algebraic Structures	35	(4)
2.1.2 Categories	39	(4)
2.1.3 Functors	43	(2)
2.1.4 Natural Transformations	45	(2)
2.1.5 Limits and Colithits	47	(1)
2.1.6 The Adjunction	47	(2)
2.1.7 n—Categories	49	(8)
2.1.7.1 Generalization to 'Big' n—Categories	49	(5)
2.1.7.2 Topological Structure of n—Categories	54	(3)
2.1.8 Algebra in Abelian Categories	57	(2)
2.1.9 Fundamental Biodynamic Adjunction	59	(1)
2.2 The Basics of Dynamics	60	(8)
2.2.1 Ordinary Differential Equations	60	(3)
2.2.2 Linear Autonomous Dynamics	63	(5)
2.2.2.1 The Flow of a Linear ODE	63	(2)
2.2.2.2 Canonical Linear Flows in R²	65	(2)
2.2.2.3 Topological Equivalence	67	(1)
2.3 Chaos and Synergetics in Biodynamics	68	(43)
2.3.1 Prototype of Chaotic and Synergetic Systems.	73	(2)
2.3.2 Chaotic Systems and Biomorphs	75	(7)
2.3.2.1 Simulation Examples: Chaotic Systems	75	(6)
2.3.2.2 Simulation Examples: Biomorphic Systems	81	(1)
2.3.3 Controlling Chaos within the Chaos Theory	82	(19)
2.3.3.1 Exploiting Critical Sensitivity	82	(3)
2.3.3.2 Lyapunov exponents and KY—dimension	85	(2)
2.3.3.3 Kolmogorov—Sinai entropy	87	(2)
2.3.3.4 Chaos Control by Ott, Grebogi and Yorke	89	(3)
2.3.3.5 Floquet Stability Analysis and OGY Control	92	(5)
2.3.3.6 Jerk Functions of Simple Chaotic Flows	97	(4)
2.3.4 The Basic Hamiltonian Model of Biodynamics	101	(1)
2.3.5 The Basics of Haken's Synergetics	102	(7)
2.3.5.1 Phase Transitions	104	(2)
2.3.5.2 Mezoscopic Derivation of Order Parameters	106	(3)
2.3.6 Macro—Synergetic Control of Biodynamics	109	(2)
3. Natural Geometry of Biodynamics	111	(150)
3.1 Motivation for Geometry in Biodynamics	114	(3)
3.2 Biodynamic Manifold M	117	(4)
3.2.1 Definition of the Manifold M	118	(2)
3.2.2 Smooth Maps Between Manifolds	120	(1)
3.3 Biodynamic Bundles	121	(3)
3.3.1 The Tangent Bundle of the Manifold M	121	(2)
3.3.2 The Cotangent Bundle of the Manifold M	123	(1)
3.4 Sections of Biodynamic Bundles	124	(28)
3.4.1 Biodynamic Evolution and Flow	125	(1)
3.4.2 Vector-Fields and Their Flows	126	(7)
3.4.2.1 Vector-Fields on M	126	(1)
3.4.2.2 Integral Curves as Biodynamic Trajectories	127	(4)
3.4.2.3 Biodynamic Flows on M	131	(1)
3.4.2.4 Categories of ODEs	132	(1)
3.4.3 Differential Forms on M	133	(15)
3.4.3.1 1—Forms on M	134	(1)
3.4.3.2 k—Forms on M	135	(2)
3.4.3.3 Exterior Differential Systems	137	(1)
3.4.3.4 Exterior Derivative on M	138	(2)
3.4.3.5 De Rham Complex and Homotopy Operators	140	(2)
3.4.3.6 Stokes Theorem and De Rham Cohomology	142	(1)
3.4.3.7 Euler-Poincare Characteristics of M	143	(1)
3.4.3.8 Duality of Chains and Forms on M	144	(2)
3.4.3.9 Other Exterior Operators on M	146	(2)
3.4.4 Geometry of Nonlinear Dynamics	148	(4)
3.5 Lie Categories in Biodynamics	152	(43)
3.5.1 Lie Derivative in Biodynamics	152	(10)
3.5.1.1 Lie Derivative on Functions	152	(3)
3.5.1.2 Lie Derivative of Vector Fields	155	(2)
3.5.1.3 Derivative of the Evolution Operator	157	(1)
3.5.1.4 Lie Derivative of Differential Forms	158	(1)
3.5.1.5 Lie Derivative of Various Tensor Fields	159	(2)
3.5.1.6 Lie Algebras	161	(1)
3.5.2 Lie Groups in Biodynamics	162	(18)
3.5.2.1 Lie Groups and Their Lie Algebras	162	(5)
3.5.2.2 Actions of Lie Groups on M	167	(2)
3.5.2.3 Basic Biodynamic Groups	169	(2)
3.5.2.4 Groups of Joint Rotations	171	(4)
3.5.2.5 Special Euclidean Groups of Joint Motions	175	(5)
3.5.3 Group Structure of the Biodynamic Manifold M	180	(5)
3.5.3.1 Purely Rotational Biodynamic Manifold	180	(2)
3.5.3.2 Reduction of the Rotational Manifold	182	(2)
3.5.3.3 The Complete Biodynamic Manifold	184	(1)
3.5.3.4 Realistic Human Spine Manifold	184	(1)
3.5.4 Lie Symmetries in Biodynamics	185	(10)
3.5.4.1 Lie Symmetry Groups	185	(3)
3.5.4.2 Prolongations	188	(6)
3.5.4.3 Special Biodynamic Equations	194	(1)
3.6 Riemannian Geometry in Biodynamics	195	(22)
3.6.1 Local Riemannian Geometry on M	196	(9)
3.6.1.1 Riemannian Metric on M	197	(4)
3.6.1.2 Geodesics on M	201	(1)
3.6.1.3 Riemannian Curvature on M	202	(3)
3.6.2 Global Riemannian Geometry on M	205	(12)
3.6.2.1 The Second Variation Formula	205	(3)
3.6.2.2 Gauss—Bonnet Formula	208	(1)
3.6.2.3 Ricci Flow on M	209	(3)
3.6.2.4 Structure Equations on M	212	(1)
3.6.2.5 Basics of Morse Theory	213	(2)
3.6.2.6 Basics of (Co)Bordism Theory	215	(2)
3.7 Symplectic Geometry in Biodynamics	217	(3)
3.7.1 Symplectic Algebra	217	(1)
3.7.2 Symplectic Geometry on M	218	(2)
3.8 Impulse Biodynamics and Synthetic Geometry	220	(17)
3.8.1 Delta Spikes	220	(2)
3.8.2 Kick Dynamics	222	(4)
3.8.2.1 Deterministic Delayed Kicks	222	(1)
3.8.2.2 Random Kicks and Langevin Equation	223	(3)
3.8.3 Distributions and Synthetic Differential Geometry	226	(11)
3.8.3.1 Distributions	227	(2)
3.8.3.2 Synthetic Calculus in Euclidean Spaces	229	(2)
3.8.3.3 Spheres and Balls as Distributions	231	(2)
3.8.3.4 Stokes Theorem for Unit Sphere	233	(1)
3.8.3.5 Time Derivatives of Expanding Spheres	234	(1)
3.8.3.6 The Wave Equation	235	(2)
3.9 A Quick Look at Modern Geometrodynamics	237	(5)
3.9.1 Einstein Equations	237	(1)
3.9.2 n—Categories in Physics	237	(3)
3.9.3 Quantum Geometry Framework	240	(2)
3.10 3D Modelling and Animation in Biodynamics	242	(15)
3.10.1 Essentials of Human Animation	242	(3)
3.10.1.1 Motion Capture-Based Human Animation	243	(1)
3.10.1.2 Virtual Muscular Dynamics in 3D-Graphics	244	(1)
3.10.2 Curves and Surfaces in Geometric Modelling	245	(4)
3.10.2.1 Power Basis Form of a Curve	246	(1)
3.10.2.2 Bezier Curves	247	(2)
3.10.2.3 Rational Bezier Curves	249	(1)
3.10.3 B-Spline Basis Functions	249	(2)
3.10.3.1 DeBoor-Cox Recursive Definition	250	(1)
3.10.3.2 Derivatives of B-Spline Basis Functions	250	(1)
3.10.4 B-Spline Curves and Surfaces in Geometric Modelling	251	(4)
3.10.4.1 Definition of B-Spline Curves	251	(1)
3.10.4.2 Properties of B-Spline Curves	251	(1)
3.10.4.3 Derivatives of a B-Spline Curve	252	(1)
3.10.4.4 Definition of B-Spline Surfaces	253	(1)
3.10.4.5 Properties of B-Spline Surfaces	253	(1)
3.10.4.6 Derivatives of a B-Spline Surface	254	(1)
3.10.5 NURBS Curves and Surfaces	255	(2)
3.10.5.1 Definition of NURBS Curves	255	(1)
3.10.5.2 Properties of NURBS Curves	255	(1)
3.10.5.3 Definition of NURBS Surfaces	256	(1)
3.10.5.4 Properties of NURBS Surfaces	257	(1)
3.11 Kinematics of Biomechanical Chains	257	(4)
3.11.1 3D Transformation Matrix	257	(1)
3.11.2 A Multilink Kinematic Chain	258	(1)
3.11.3 CNS Representation of the Body Posture	259	(1)
3.11.4 Transformation Matrix Used in Computer Graphics	260	(1)
4. Natural Mechanics of Biodynamics	261	(166)
4.1 Lagrangian Formalism in Biodynamics	261	(4)
4.2 Hamiltonian Formalism in Biodynamics	265	(45)
4.2.1 Nonlinear Dynamics in Hamiltonian Form	267	(22)
4.2.1.1 Real 1-DOF Hamiltonian Dynamics	267	(9)
4.2.1.2 Complex One-DOF Hamiltonian Dynamics	276	(3)
4.2.1.3 Library of Basic Hamiltonian Systems	279	(7)
4.2.1.4 n-DOF Hamiltonian Dynamics	286	(3)
4.2.2 Hamiltonian Geometry in Biodynamics	289	(3)
4.2.3 Hamilton-Poisson Geometry in Biodynamics	292	(6)
4.2.3.1 Hamilton-Poisson Biodynamic Systems	294	(4)
4.2.4 Completely Integrable Hamiltonian Systems	298	(3)
4.2.4.1 Liouville Theorem	298	(1)
4.2.4.2 Action-Angle Variables	299	(2)
4.2.5 Ergodicity	301	(9)
4.2.5.1 Ergodicity in Hamiltonian Systems	301	(1)
4.2.5.2 Dynamical Systems and Hyperbolicity	302	(3)
4.2.5.3 Ergodic Theory and Nontrivial Recurrence	305	(1)
4.2.5.4 Nonuniformly Hyperbolic Trajectories	306	(1)
4.2.5.5 Systems with Nonzero Lyapunov Exponents	307	(3)
4.3 Quantum Formalism in Nano-Biodynamics	310	(41)
4.3.1 Quantum Mechanics in Biological Matter	310	(1)
4.3.2 Dirac's Canonical Quantization	311	(19)
4.3.2.1 Quantum States and Operators	312	(6)
4.3.2.2 Quantum Pictures	318	(2)
4.3.2.3 Spectrum of a Quantum Operator	320	(4)
4.3.2.4 General Representation Model	324	(1)
4.3.2.5 Direct Product Space	325	(1)
4.3.2.6 State-Space for n Quantum Particles	326	(2)
4.3.2.7 Quantum Measurement and Penrose Paradox	328	(2)
4.3.3 The Problem of Quantum Measurement and Entropy	330	(12)
4.3.3.1 The Classical Apparatus	331	(1)
4.3.3.2 Quantum Object	331	(2)
4.3.3.3 Adiabatic Measurement Lagrangians	333	(1)
4.3.3.4 The Stern-Gerlach Experiment	334	(1)
4.3.3.5 Work and Heat	335	(1)
4.3.3.6 Statistical Thermodynamics	336	(2)
4.3.3.7 Friction in a Macroscopic Apparatus	338	(2)
4.3.3.8 Low Velocity Projective Measurements	340	(1)
4.3.3.9 Information and Entropy	341	(1)
4.3.4 Von Neumann's Density Matrix Quantization	342	(3)
4.3.4.1 Dissipative Quantum Formalism	344	(1)
4.3.5 Geometric Quantization	345	(6)
4.3.5.1 Motivation	345	(2)
4.3.5.2 Geometric Prequantization	347	(4)
4.4 Variational Formalism in Biodynamics	351	(13)
4.4.1 Biodynamic Action Functional	351	(1)
4.4.2 Lagrangian Action	352	(1)
4.4.3 Hamiltonian Action	353	(1)
4.4.4 Noether Theorem	354	(2)
4.4.5 Hamiltonian-Action Formulation of Biodynamics	356	(3)
4.4.6 Feynman Quantum Action	359	(5)
4.5 Nonholonomic Biodynamics	364	(6)
4.5.1 Lagrangian Approach	364	(2)
4.5.2 Hamiltonian Approach	366	(2)
4.5.3 Biodynamic Example: Bicycle Dynamics	368	(2)
4.6 Stochastic Formalism in Biodynamics	370	(6)
4.6.1 Markov Stochastic Processes	372	(2)
4.6.2 Statistical Mechanics of Oscillator Chains	374	(2)
4.7 Muscular Excitation-Contraction Dynamics	376	(27)
4.7.1 Human Musculo-Skeletal System	376	(9)
4.7.1.1 Human Skeleton	376	(2)
4.7.1.2 Human Joints	378	(1)
4.7.1.3 Human Muscular System	379	(2)
4.7.1.4 Human Energy Flow	381	(4)
4.7.1.5 Equivalent Muscular Actuator	385	(1)
4.7.2 Molecular Muscular Dynamics	385	(1)
4.7.3 Mezoscopic Muscular Dynamics	386	(6)
4.7.3.1 Myocybernetics	392	(1)
4.7.4 Macroscopic Muscular Dynamics	392	(11)
4.7.4.1 Soft Tissue Dynamics of Relaxed Muscles	392	(3)
4.7.4.2 Classical Hill's Model	395	(1)
4.7.4.3 Biodynamics of Load-Lifting	396	(7)
4.8 Lie Functors in Biodynamics	403	(16)
4.8.1 Lie-Lagrangian Biodynamic Functor	403	(7)
4.8.1.1 Joint Kinematics	403	(2)
4.8.1.2 Exterior Lagrangian Dynamics	405	(5)
4.8.2 Lie-Hamiltonian Biodynamic Functor	410	(4)
4.8.2.1 The Abstract Functor Machine	412	(1)
4.8.2.2 Muscle-Driven Hamiltonian Biodynamics	413	(1)
4.8.3 Stochastic-Lie-Hamiltonian Biodynamic Functor	414	(2)
4.8.4 Fuzzy-Stochastic-Lie-Hamiltonian Functor	416	(3)
4.9 Mechanics of Spinal Injuries	419	(8)
4.9.1 Spinal Dislocations, Disclinations and Fractures	420	(1)
4.9.2 Measuring the Risk of Local Intervertebral Injuries	420	(6)
4.9.2.1 Biodynamic Jerk Functions	424	(2)
4.9.3 Measuring the Risk of Vertebral Fractures	426	(1)
4.9.3.1 Research on Bone Injuries	426	(1)
5. Natural Topology of Biodynamics	427	(26)
5.1 Category of (Co)Chain Complexes in Biodynamics	427	(4)
5.1.1 (Co)Homologies in Abelian Categories Related to M	428	(2)
5.1.2 Reduction and Euler-Poincare Characteristic	430	(1)
5.2 Natural Duality in Biodynamics	431	(10)
5.2.1 Geometric Duality Theorem for M	431	(6)
5.2.1.1 Lie-Functorial Proof	432	(1)
5.2.1.2 Geometric Proof	433	(4)
5.2.2 Topological Duality Theorem for M	437	(2)
5.2.2.1 Cohomological Proof	437	(2)
5.2.2.2 Homological Proof	439	(1)
5.2.3 Lagrangian Versus Hamiltonian Duality	439	(1)
5.2.4 Globally Dual Structure of Rotational Biodynamics	440	(1)
5.3 Topological Phase Transitions and Hamiltonian Chaos	441	(11)
5.3.1 Phase Transitions in Hamiltonian Systems	441	(3)
5.3.2 Geometry of the Largest Lyapunov Exponent	444	(3)
5.3.3 Euler Characteristics of Hamiltonian Systems	447	(5)
5.4 The Covariant Force Functor	452	(1)
6. Natural Control and Self-Organization in Biodynamics	453	(88)
6.1 The Basics of Classical Control and Stability	455	(17)
6.1.1 Introduction to Feedback Control	455	(5)
6.1.2 Linear Stationary Systems and Operators	460	(5)
6.1.2.1 Basics of Kalman State-Space Theory	460	(1)
6.1.2.2 Regulator Problem	461	(1)
6.1.2.3 End Point Control Problem	462	(1)
6.1.2.4 Servomechanism Problem	463	(1)
6.1.2.5 Repetitive Mode Problem	463	(1)
6.1.2.6 Feedback Changes the Operator	464	(1)
6.1.3 Stability and Boundedness	465	(3)
6.1.4 Lyapunov's Stability Method	468	(1)
6.1.5 Graphical Techniques for Nonlinear Systems	469	(3)
6.1.5.1 Describing Function Analysis	470	(2)
6.2 The Basis of Modern Geometric Control	472	(15)
6.2.1 Feedback Linearization	472	(9)
6.2.1.1 Exact Feedback Linearization	472	(4)
6.2.1.2 Relative Degree	476	(2)
6.2.1.3 Approximative Feedback Linearization	478	(3)
6.2.2 Controllability	481	(6)
6.2.2.1 Linear Controllability	481	(1)
6.2.2.2 Nonlinear Controllability	482	(2)
6.2.2.3 Controllability Condition	484	(1)
6.2.2.4 Distributions	485	(1)
6.2.2.5 Foliations	486	(1)
6.3 Modern Control Techniques for Mechanical Systems	487	(9)
6.3.1 Abstract Control System	487	(1)
6.3.2 Controllability of a Linear Control System	488	(1)
6.3.3 Affine Control System and Local Controllability	489	(1)
6.3.4 Hamiltonian Control and Maximum Principle	490	(6)
6.3.4.1 Hamiltonian Control Systems	490	(3)
6.3.4.2 Pontryagin's Maximum Principle	493	(1)
6.3.4.3 Affine Control Systems	494	(2)
6.4 Locomotion Systems and Human Gait	496	(12)
6.4.1 Control of Locomotion Systems	496	(8)
6.4.1.1 Stratified Kinematic Controllability	497	(2)
6.4.1.2 The Distributions Approach	499	(1)
6.4.1.3 The Exterior Differential Systems Approach	499	(2)
6.4.1.4 On the Existence and Uniqueness of Solutions	501	(1)
6.4.1.5 Trajectory Generation Problem	502	(2)
6.4.2 Gait Biodynamics	504	(4)
6.5 Biodynamic Control Policy, Learning and Self-Organization	508	(18)
6.5.1 Control Policy Learning by Robots	508	(4)
6.5.1.1 Direct Learning of the Control Policy	509	(1)
6.5.1.2 Indirect Learning of the Control Policy	510	(2)
6.5.1.3 Learning of Motor Control Components	512	(1)
6.5.2 Pathways to Self-Organization in Biodynamics	512	(2)
6.5.3 Neuro-Muscular Excitation-Contraction Dynamics	514	(10)
6.5.3.1 Motor Units	514	(1)
6.5.3.2 Darwinian Oscillatory Neural Net	514	(3)
6.5.3.3 Recurrent Neuro-Muscular Model	517	(2)
6.5.3.4 Autogenetic Reflex Motor-Servo	519	(1)
6.5.3.5 Biodynamics Control	520	(4)
6.5.4 Lie-Adaptive Biodynamic Control	524	(2)
6.6 Essentials of Biodynamic Measurement: Kalman Filtering and Inertial Navigation	526	(11)
6.6.1 Kalman Filter Basics	526	(6)
6.6.2 Inertial Navigation	532	(4)
6.6.3 Adaptive Estimation in Biomechanics	536	(1)
6.7 Humanoid Robotics	537	(4)
6.7.1 Honda Humanoid Series	537	(1)
6.7.2 Cerebellar Robotics	538	(3)
7. Natural Brain Dynamics and Sensory—Motor Integration	541	(296)
7.1 Introduction to Brain	542	(7)
7.2 Human Nervous System	549	(89)
7.2.1 Building Blocks of the Nervous System	550	(16)
7.2.1.1 Neuronal Circuits	552	(4)
7.2.1.2 Basic Brain Partitions and Their Functions	556	(2)
7.2.1.3 Nerves	558	(1)
7.2.1.4 Action potential	559	(2)
7.2.1.5 Synapses	561	(5)
7.2.2 Reflex Action: the Basis of CNS Activity	566	(2)
7.2.3 The Spinal Cord Pathways	568	(10)
7.2.3.1 Spinal Lower Motor Neurons	569	(7)
7.2.3.2 Central Pattern Generators in the Spinal Cord	d575
7.2.3.3 Influence of Higher Centers	576	(2)
7.2.4 First Look at the Brain	578	(3)
7.2.5 Motor Pathways	581	(6)
7.2.5.1 Primary Motor Cortex	581	(3)
7.2.5.2 Motor Association/Premotor Cortical Areas	584	(3)
7.2.6 Subcortical Motor 'Side Loops'	587	(13)
7.2.6.1 The Cerebellum	588	(6)
7.2.6.2 The Basal Ganglia	594	(2)
7.2.6.3 Cerebellar Movement Control	596	(4)
7.2.7 Human Senses and their Pathways	600	(3)
7.2.8 The Human—Like Vision	603	(12)
7.2.8.1 Extraocular SO(3)—Muscles	604	(2)
7.2.8.2 Retina	606	(1)
7.2.8.3 Cornea	607	(1)
7.2.8.4 Iris	608	(2)
7.2.8.5 Pursuit Eye Control and Motion Perception	610	(3)
7.2.8.6 Optical Flow	613	(2)
7.2.9 The Visual. Pathway	615	(5)
7.2.9.1 Light Reflex and 3D Vision	619	(1)
7.2.10 Differential Geometry of the Striate Cortex	620	(2)
7.2.11 Auditory and Vestibular Pathways	622	(10)
7.2.11.1 The Inner Ear	622	(1)
7.2.11.2 Auditory Transduction	623	(1)
7.2.11.3 Central Auditory Pathways	624	(2)
7.2.11.4 The Vestibular System	626	(1)
7.2.11.5 The Semicircular Canals	626	(1)
7.2.11.6 The Vestibulo-Ocular Reflex	627	(1)
7.2.11.7 The Utricle and Saccule	628	(1)
7.2.11.8 Mechanics of the Semicircular Canals	629	(1)
7.2.11.9 Endolymph Flow in the Semicircular Canals	630	(2)
7.2.12 Somatosensory Pathways	632	(6)
7.2.12.1 The Discriminative Touch System	633	(1)
7.2.12.2 The Pain and Temperature System	634	(1)
7.2.12.3 The Proprioceptive System	635	(3)
7.3 The Sensory-Motor Adjunction	638	(23)
7.3.1 Summary on Sensory-Motor Pathways	638	(4)
7.3.2 Sensory-Motor Control	642	(7)
7.3.2.1 Multisensory Integration for Motor Planning	645	(4)
7.3.2.2 The Sensory-Motor Adjunction	649	(1)
7.3.3 The Central Biomechanical Adjunction	649	(9)
7.3.3.1 Postural Control Experiments	652	(3)
7.3.3.2 Learning Arm Movement Control	655	(3)
7.3.4 Mechanism of Brain Injuries	658	(3)
7.3.4.1 Basic Dynamics of Brain Injuries	658	(3)
7.3.4.2 Research on Head and Brain Injuries	661	(1)
7.4 Brain Dynamics	661	(11)
7.4.1 Microscopic Neurodynamics of Microtubules	662	(5)
7.4.1.1 Biochemistry of Microtubules	662	(2)
7.4.1.2 Kink Soliton Model of MT-Dynamics	664	(3)
7.4.2 Mezoscopic Neurodynamics of Action Potentials	667	(5)
7.4.2.1 Hodgkin-Huxley Model	667	(4)
7.4.2.2 FitzHugh-Nagumo Model	671	(1)
7.5 Biological Neural Nets	672	(33)
7.5.1 Phase Dynamics of Oscillatory Neural Nets	672	(6)
7.5.1.1 Kuramoto Synchronization Model	675	(1)
7.5.1.2 Lyapunov Chaotic Synchronization	676	(2)
7.5.2 Complex Networks Dynamics	678	(3)
7.5.2.1 Continuum Limit of the Kuramoto Network	679	(1)
7.5.2.2 Path-Integral Approach to Complex Nets	679	(2)
7.5.3 Complex Adaptive Systems	681	(3)
7.5.4 Noise Delayed Bifurcation in Coupled Neurons	684	(4)
7.5.4.1 The Theta-Neuron	684	(1)
7.5.4.2 Coupled Theta-Neurons	685	(3)
7.5.5 Classification of 'Spiking' Neuron Models	688	(4)
7.5.6 Weakly Connected and Canonical Neural Nets	692	(2)
7.5.7 Quantum Brain Model	694	(4)
7.5.8 Open Liouville Neurodynamics and Self-Similarity	698	(7)
7.5.8.1 Hamiltonian Framework	700	(1)
7.5.8.2 Conservative Classical System	700	(1)
7.5.8.3 Conservative Quantum System	700	(1)
7.5.8.4 Open Classical System	701	(1)
7.5.8.5 Continuous Neural Network Dynamics	702	(1)
7.5.8.6 Open Quantum System	703	(1)
7.5.8.7 Non-Critical Stringy MT-Dynamics	704	(1)
7.5.8.8 Equivalence of Neurodynamic Forms	704	(1)
7.6 Artificial Neural Nets in Biodynamics	705	(36)
7.6.1 Biological Versus Artificial Neural Nets	705	(2)
7.6.2 Common Discrete ANNs	707	(25)
7.6.2.1 Multilayer Perceptrons	707	(12)
7.6.2.2 Summary of Supervised Learning Methods	719	(1)
7.6.2.3 Other Standard ANNs	720	(7)
7.6.2.4 Fully Recurrent ANNs	727	(1)
7.6.2.5 Dynamical Games and Recurrent ANNs	728	(3)
7.6.2.6 Complex-Valued ANNs	731	(1)
7.6.3 Common Continuous ANNs	732	(9)
7.6.3.1 Neurons as Functions	733	(2)
7.6.3.2 Basic Activation and Learning Dynamics	735	(1)
7.6.3.3 Standard Models of Continuous Nets	736	(5)
7.7 Distinguished ANN Models	741	(14)
7.7.1 Generalized Kohonen's SOM	741	(4)
7.7.1.1 The Winner Relaxing Kohonen Algorithm	742	(1)
7.7.1.2 The Magnification Factor	743	(1)
7.7.1.3 Magnification Exponent	744	(1)
7.7.2 Dynamics of Hopfield's Associative Recurrent Nets	745	(8)
7.7.2.1 Ising-Spin Neurons	745	(1)
7.7.2.2 Graded-Response Neurons	746	(1)
7.7.2.3 Hopfield's Overlaps	747	(2)
7.7.2.4 Overlap Dynamics	749	(1)
7.7.2.5 Hebbian Learning Dynamics	750	(3)
7.7.3 A Self—Organizing Bidirectional Competitive Net	753	(2)
7.8 Fuzzy Logic in Biodynamics	755	(28)
7.8.1 The Concept of Fuzziness	755	(6)
7.8.1.1 'Fuzzy Thinking'	755	(1)
7.8.1.2 Fuzzy Sets	756	(1)
7.8.1.3 Fuzziness of the Real World	757	(1)
7.8.1.4 Fuzzy Entropy	758	(3)
7.8.2 Fuzzy Inference Engine	761	(3)
7.8.3 Fuzzy Logic Control	764	(7)
7.8.3.1 Fuzzy Control of Biodynamic Jerks	768	(1)
7.8.3.2 Characteristics of Fuzzy Control	769	(1)
7.8.3.3 Evolving Connectionist Systems	770	(1)
7.8.4 High—Resolution FAM Agents	771	(12)
7.8.4.1 Generic Nonlinear MIMO Systems	772	(3)
7.8.4.2 Alternative MIMO Systems	775	(2)
7.8.4.3 Biodynamics Example: Tennis Game	777	(6)
7.9 Natural System in a Category	783	(9)
7.9.1 Categorical Patterns and Hierarchical Links	783	(3)
7.9.2 A General Natural System	786	(1)
7.9.3 The Category of Neurons	787	(1)
7.9.4 Memory Evolutive System	787	(2)
7.9.5 Neural System in a Category	789	(3)
7.10 Brain—Mind Functorial Machines	792	(8)
7.10.1 Neurodynamic 2—Functor	792	(3)
7.10.2 Solitary 'Thought Nets' and the Emerging Mind	795	(5)
7.10.2.1 Synergetic 'Thought Solitons'	795	(5)
7.11 Body—Mind Adjunction and Natural Psychodynamics	800	(18)
7.11.1 Natural Psychodynamics in the Life Space Foam	801	(17)
7.11.1.1 Six Faces of the Life Space Foam	806	(1)
7.11.1.2 General Formalism	806	(4)
7.11.1.3 Motion and Decision Making in LSFpaths	810	(4)
7.11.1.4 Force—Fields and Memory in LSFfields	814	(2)
7.11.1.5 Geometries, Topologies and Noise in LSFgeom	816	(2)
7.12 Brain—Like Control in a Nutshell	818	(19)
7.12.1 Functor Control Machine	820	(2)
7.12.2 Spinal Control Level	822	(5)
7.12.3 Cerebellar Control Level	827	(3)
7.12.4 Cortical Control Level	830	(3)
7.12.5 A Note on Muscular Training	833	(3)
7.12.6 Errors in Motion Control: Locomotor Injuries	836	(1)
Appendix A	837	(110)
A.1 Basic Formulas from Tensor Analysis	837	(20)
A.1.1 General Functional Transformation	837	(7)
A.1.1.1 Transformation of Coordinates	838	(1)
A.1.1.2 Scalar Invariants	839	(1)
A.1.1.3 Vectors and Covectors	839	(1)
A.1.1.4 Second—Order Tensors	840	(2)
A.1.1.5 Higher—Order Tensors	842	(1)
A.1.1.6 Tensor Symmetry	842	(2)
A.1.2 Euclidean Tensors	844	(2)
A.1.2.1 Basis Vectors and the Metric Tensor in Rn	844	(1)
A.1.2.2 Tensor Products in Rn	845	(1)
A.1.3 Tensor Derivatives on Riemannian Manifolds	846	(7)
A.1.3.1 Christoffel's Symbols	846	(1)
A.1.3.2 Geodesics	847	(1)
A.1.3.3 The Covariant Derivative	847	(1)
A.1.3.4 Vector Differential Operators	848	(1)
A.1.3.5 The Absolute Derivative	849	(4)
A.1.4 The Covariant Force Law in Biodynamics	853	(2)
A.1.5 The Essence of Natural Hamiltonian Biodynamics	855	(1)
A.1.6 Neuro—Hamiltonian Control in Biodynamics	856	(1)
A.2 Frequently Used Neurophysiological Terms	857	(23)
A.3 Modern 3D Neuroimaging	880	(9)
A.3.1 Nuclear Magnetic Resonance in 2D Medical Imaging	880	(1)
A.3.2 3D Magnetic Resonance Imaging of Human Brain	881	(1)
A.3.3 Diffusion MRI in 3D Volume	882	(1)
A.3.4 Imaging Diffusion with MRI	883	(1)
A.3.5 3D Diffusion Tensor	884	(3)
A.3.6 Brain Connectivity Studies	887	(1)
A.3.7 Brain Waves and Independent Component Analysis	888	(1)
A.4 Complex Functions, Manifolds and Hilbert Spaces	889	(15)
A.4.1 Complex Numbers and Vectors	889	(5)
A.4.1.1 Quaternions and Rotations	890	(4)
A.4.2 Complex Functions	894	(4)
A.4.3 Complex Manifolds	898	(4)
A.4.4 Hilbert Space	902	(2)
A.5 Classical Lie Theory	904	(11)
A.5.1 Basic Tables of Lie Groups and Their Lie Algebras	904	(3)
A.5.2 Representations of Lie groups	907	(1)
A.5.3 Root Systems and Dynkin Diagrams	908	(5)
A.5.3.1 Definitions	908	(1)
A.5.3.2 Classification	909	(1)
A.5.3.3 Dynkin Diagrams	910	(3)
A.5.3.4 Root Systems and Lie Theory	913	(1)
A.5.4 Simple and Semisimple Lie Groups and Algebras	913	(2)
A.6 Phase Transitions, Partition Function and Noise	915	(23)
A.6.1 Equilibrium Phase Transitions	915	(6)
A.6.1.1 Classification of Phase Transitions	916	(2)
A.6.1.2 Basic Properties of Phase Transitions	918	(3)
A.6.2 Landau's Theory of Phase Transitions	921	(1)
A.6.3 Partition Function	922	(8)
A.6.3.1 Classical Partition Function	923	(1)
A.6.3.2 Quantum Partition Function	924	(1)
A.6.3.3 Vibrations of Coupled Oscillators	925	(5)
A.6.4 Noise-Induced Nonequilibrium Phase Transitions	930	(8)
A.6.4.1 General Zero-Dimensional System	931	(3)
A.6.4.2 General d—Dimensional System	934	(4)
A.7 MathematicaTm Derivation of Main Biodynamic Functions	938	(9)
A.7.1 Load-Lifting Biodynamics	938	(1)
A.7.2 Brain-Like Control Functions	939	(5)
A.7.2.1 Spinal FC-level	939	(2)
A.7.2.2 Cerebellar FC-level	941	(1)
A.7.2.3 Cortical FC-level	942	(2)
A.7.3 Anatomical Description of Human Movements	944	(3)
A.7.3.1 Movements in Synovial Joints	944	(1)
A.7.3.2 Examples of Sport Movements	945	(2)
Bibliography	947	(32)
Index	979