Natural Biodynamics

Contents......Page 20
Preface......Page 8
Glossary of Frequently Used Symbols......Page 16
1.1 The Problem of Natural Biodynamics......Page 36
1.2 A Brief History of Biodynamics......Page 38
1.3 Mechanical Basis of Biodynamics......Page 43
1.3.1 Natural Galilei Group......Page 44
1.3.2 Newtonian Equations of Motion......Page 45
1.3.3 Calculus of Variations......Page 46
1.3.4 Lagrangian Equations of Motion......Page 47
1.3.5 Hamiltonian Equations of Motion......Page 48
1.3.6 Lagrangian Flows on Biodynamic Manifolds......Page 49
1.4 Conservative versus Dissipative Hamiltonian Dynamics......Page 50
1.4.1 Dissipative Systems......Page 51
1.4.2 Thermodynamic Equilibrium......Page 53
1.4.4 The Second Law of Thermodynamics......Page 54
1.4.5 Geometry of Phase Space......Page 56
1.5 Neural Basis of Biodynamics......Page 58
2.1.1 Preliminaries from Calculus, Algebra and Topology......Page 60
2.1.1.1 Notes From Calculus......Page 61
2.1.1.3 Notes from General Topology......Page 63
2.1.1.4 Commutative Diagrams......Page 67
2.1.1.5 Groups and Related Algebraic Structures......Page 70
2.1.2 Categories......Page 74
2.1.3 Functors......Page 78
2.1.4 Natural Transformations......Page 80
2.1.6 The Adjunction......Page 82
2.1.7.1 Generalization to ‘Big’ n-Categories......Page 84
2.1.7.2 Topological Structure of n-Categories......Page 89
2.1.8 Algebra in Abelian Categories......Page 92
2.1.9 Fundamental Biodynamic Adjunction......Page 94
2.2.1 Ordinary Differential Equations......Page 95
2.2.2.1 The Flow of a Linear ODE......Page 98
2.2.2.2 Canonical Linear Flows in R2......Page 100
2.2.2.3 Topological Equivalence......Page 102
2.3 Chaos and Synergetics in Biodynamics......Page 103
2.3.1 Prototype of Chaotic and Synergetic Systems......Page 108
2.3.2.1 Simulation Examples: Chaotic Systems......Page 110
2.3.2.2 Simulation Examples: Biomorphic Systems......Page 116
2.3.3.1 Exploiting Critical Sensitivity......Page 117
2.3.3.2 Lyapunov exponents and KY-dimension......Page 120
2.3.3.3 Kolmogorov-Sinai entropy......Page 122
2.3.3.4 Chaos Control by Ott, Grebogi and Yorke......Page 124
2.3.3.5 Floquet Stability Analysis and OGY Control......Page 127
2.3.3.6 Jerk Functions of Simple Chaotic Flows......Page 132
2.3.4 The Basic Hamiltonian Model of Biodynamics......Page 136
2.3.5 The Basics of Haken’s Synergetics......Page 137
2.3.5.1 Phase Transitions......Page 139
2.3.5.2 Mezoscopic Derivation of Order Parameters......Page 141
2.3.6 Macro-Synergetic Control of Biodynamics......Page 144
3. Natural Geometry of Biodynamics......Page 146
3.1 Motivation for Geometry in Biodynamics......Page 149
3.2 Biodynamic Manifold M......Page 152
3.2.1 Definition of the Manifold M......Page 153
3.2.2 Smooth Maps Between Manifolds......Page 155
3.3.1 The Tangent Bundle of the Manifold M......Page 156
3.3.2 The Cotangent Bundle of the Manifold M......Page 158
3.4 Sections of Biodynamic Bundles......Page 159
3.4.1 Biodynamic Evolution and Flow......Page 160
3.4.2.1 Vector-Fields on M......Page 161
3.4.2.2 Integral Curves as Biodynamic Trajectories......Page 162
3.4.2.3 Biodynamic Flows on M......Page 166
3.4.2.4 Categories of ODES......Page 167
3.4.3 Differential Forms on M......Page 168
3.4.3.1 1-Forms on M......Page 169
3.4.3.2 &Forms on M......Page 170
3.4.3.3 Exterior Differential Systems......Page 172
3.4.3.4 Exterior Derivative on M......Page 173
3.4.3.5 De Rham Complex and Homotopy Operators......Page 175
3.4.3.6 Stokes Theorem and De Rham Cohomology......Page 177
3.4.3.7 Euler-Poincare Characteristics of M......Page 178
3.4.3.8 Duality of Chains and Forms on M......Page 179
3.4.3.9 Other Exterior Operators on M......Page 181
3.4.4 Geometry of Nonlinear Dynamics......Page 183
3.5.1.1 Lie Derivative on Functions......Page 187
3.5.1.2 Lie Derivative of Vector Fields......Page 190
3.5.1.3 Derivative of the Evolution Operator......Page 192
3.5.1.4 Lie Derivative of Differential Forms......Page 193
3.5.1.5 Lie Derivative of Various Tensor Fields......Page 194
3.5.1.6 Lie Algebras......Page 196
3.5.2.1 Lie Groups and Their Lie Algebras......Page 197
3.5.2.2 Actions of Lie Groups on M......Page 202
3.5.2.3 Basic Biodynamic Groups......Page 204
3.5.2.4 Groups of Joint Rotations......Page 206
3.5.2.5 Special Euclidean Groups of Joint Motions......Page 210
3.5.3.1 Purely Rotational Biodynamic Manifold......Page 215
3.5.3.2 Reduction of the Rotational Manifold......Page 217
3.5.3.4 Realistic Human Spine Manifold......Page 219
3.5.4.1 Lie Symmetry Groups......Page 220
3.5.4.2 Prolongations......Page 223
3.5.4.3 Special Biodynamic Equations......Page 229
3.6 Riemannian Geometry in Biodynamics......Page 230
3.6.1 Local Riemannian Geometry on M......Page 231
3.6.1.1 Riemannian Metric on M......Page 232
3.6.1.2 Geodesics on M......Page 236
3.6.1.3 Riemannian Curvature on M......Page 237
3.6.2.1 The Second Variation Formula......Page 240
3.6.2.2 Gauss-Bonnet Formula......Page 243
3.6.2.3 Ricci Flow on M......Page 244
3.6.2.4 Structure Equations on M......Page 247
3.6.2.5 Basics of Morse Theory......Page 248
3.6.2.6 Basics of (Co)Bordism Theory......Page 250
3.7.1 Symplectic Algebra......Page 252
3.7.2 Symplectic Geometry on M......Page 253
3.8.1 Delta Spikes......Page 255
3.8.2.1 Deterministic Delayed Kicks......Page 257
3.8.2.2 Random Kicks and Langevin Equation......Page 258
3.8.3 Distributions and Synthetic Differential Geometry......Page 261
3.8.3.1 Distributions......Page 262
3.8.3.2 Synthetic Calculus in Euclidean Spaces......Page 264
3.8.3.3 Spheres and Balls as Distributions......Page 266
3.8.3.4 Stokes Theorem for Unit Sphere......Page 268
3.8.3.5 Time Derivatives of Expanding Spheres......Page 269
3.8.3.6 The Wave Equation......Page 270
3.9.2 n-Categories in Physics......Page 272
3.9.3 Quantum Geometry Framework......Page 275
3.10.1 Essentials of Human Animation......Page 277
3.10.1.1 Motion CaptureBased Human Animation......Page 278
3.10.1.2 Virtual Muscular Dynamics in 3D-Graphics......Page 279
3.10.2 Curves and Surfaces in Geometric Modelling......Page 280
3.10.2.1 Power Basis Form of a Curve......Page 281
3.10.2.2 Bezier Curves......Page 282
3.10.3 B-Spline Basis Functions......Page 284
3.10.3.2 Derivatives of B-Spline Basis Functions......Page 285
3.10.4.2 Properties of B-Spline Curves......Page 286
3.10.4.3 Derivatives of a B-Spline Curve......Page 287
3.10.4.5 Properties of B-Spline Surfaces......Page 288
3.10.4.6 Derivatives of a B-Spline Surface......Page 289
3.10.5.2 Properties of NURBS Curves......Page 290
3.10.5.3 Definition of NURBS Surfaces......Page 291
3.11.1 3D Transformation Matrix......Page 292
3.11.2 A Multilink Kinematic Chain......Page 293
3.11.3 CNS Representation of the Body Posture......Page 294
3.11.4 Transformation Matrix Used in Computer Graphics......Page 295
4.1 Lagrangian Formalism in Biodynamics......Page 296
4.2 Hamiltonian Formalism in Biodynamics......Page 300
4.2.1.1 Real 1-DOF Hamiltonian Dynamics......Page 302
4.2.1.2 Complex One-DOF Hamiltonian Dynamics......Page 311
4.2.1.3 Library of Basic Hamiltonian Systems......Page 314
4.2.1.4 n-DOF Hamiltonian Dynamics......Page 321
4.2.2 Hamiltonian Geometry in Biodynamics......Page 324
4.2.3 Hamilton-Poisson Geometry in Biodynamics......Page 327
4.2.3.1 Hamilton-Poisson Biodynamic Systems......Page 329
4.2.4.1 Liouville Theorem......Page 333
4.2.4.2 Action-Angle Variables......Page 334
4.2.5.1 Ergodicity in Hamiltonian Systems......Page 336
4.2.5.2 Dynamical Systems and Hyperbolicity......Page 337
4.2.5.3 Ergodic Theory and Nontrivial Recurrence......Page 340
4.2.5.4 Nonuniformly Hyperbolic Trajectories......Page 341
4.2.5.5 Systems with Nonzero Lyapunov Exponents......Page 342
4.3.1 Quantum Mechanics in Biological Matter......Page 345
4.3.2 Dirac’s Canonical Quantization......Page 346
4.3.2.1 Quantum States and Operators......Page 347
4.3.2.2 Quantum Pictures......Page 353
4.3.2.3 Spectrum of a Quantum Operator......Page 355
4.3.2.4 General Representation Model......Page 359
4.3.2.5 Direct Product Space......Page 360
4.3.2.6 State-Space for n Quantum Particles......Page 361
4.3.2.7 Quantum Measurement and Penrose Paradox......Page 363
4.3.3 The Problem of Quantum Measurement and Entropy......Page 365
4.3.3.2 Quantum Object......Page 366
4.3.3.3 Adiabatic Measurement Lagrangians......Page 368
4.3.3.4 The Stern-Gerlach Experiment......Page 369
4.3.3.5 Work and Heat......Page 370
4.3.3.6 Statistical Thermodynamics......Page 371
4.3.3.7 Friction in a Macroscopic Apparatus......Page 373
4.3.3.8 Low Velocity Projective Measurements......Page 375
4.3.3.9 Information and Entropy......Page 376
4.3.4 Von Neumann’s Density Matrix Quantization......Page 377
4.3.4.1 Dissipative Quantum Formalism......Page 379
4.3.5.1 Motivation......Page 380
4.3.5.2 Geometric Prequantization......Page 382
4.4.1 Biodynamic Action Functional......Page 386
4.4.2 Lagrangian Action......Page 387
4.4.3 Hamiltonian Action......Page 388
4.4.4 Noether Theorem......Page 389
4.4.5 Hamiltonian-Action Formulation of Biodynamics......Page 391
4.4.6 Feynman Quantum Action......Page 394
4.5.1 Lagrangian Approach......Page 399
4.5.2 Hamiltonian Approach......Page 401
4.5.3 Biodynamic Example: Bicycle Dynamics......Page 403
4.6 Stochastic Formalism in Biodynamics......Page 405
4.6.1 Markov Stochastic Processes......Page 407
4.6.2 Statistical Mechanics of Oscillator Chains......Page 409
4.7.1.1 Human Skeleton......Page 411
4.7.1.2 Human Joints......Page 413
4.7.1.3 Human Muscular System......Page 414
4.7.1.4 Human Energy Flow......Page 416
4.7.2 Molecular Muscular Dynamics......Page 420
4.7.3 Mezoscopic Muscular Dynamics......Page 421
4.7.4.1 Soft Tissue Dynamics of Relaxed Muscles......Page 427
4.7.4.2 Classical Hill’s Model......Page 430
4.7.4.3 Biodynamics of Load-Lifting......Page 431
4.8.1.1 Joint Kinematics......Page 438
4.8.1.2 Exterior Lagrangian Dynamics......Page 440
4.8.2 Lie-Hamiltonian Biodynamic Functor......Page 445
4.8.2.1 The Abstract Functor Machine......Page 447
4.8.2.2 Muscle-Driven Hamiltonian Biodynamics......Page 448
4.8.3 Stochastic-Lie-Hamiltonian Biodynamic Functor......Page 449
4.8.4 Fuzzy-Stochastic-Lie-Hamiltonian Functor......Page 451
4.9 Mechanics of Spinal Injuries......Page 454
4.9.2 Measuring the Risk of Local Intervertebral Injuries......Page 455
4.9.2.1 Biodynamic Jerk Functions......Page 459
4.9.3.1 Research on Bone Injuries......Page 461
5.1 Category of (Co)Chain Complexes in Biodynamics......Page 462
5.1.1 (Co)Homologies in Abelian Categories Related to M......Page 463
5.1.2 Reduction and Euler-Poincare Characteristic......Page 465
5.2.1 Geometric Duality Theorem for M......Page 466
5.2.1.1 Lie-Functorial Proof......Page 467
5.2.1.2 Geometric Proof......Page 468
5.2.2.1 Cohomological Proof......Page 472
5.2.3 Lagrangian Versus Hamiltonian Duality......Page 474
5.2.4 Globally Dual Structure of Rotational Biodynamics......Page 475
5.3.1 Phase Transitions in Hamiltonian Systems......Page 476
5.3.2 Geometry of the Largest Lyapunov Exponent......Page 479
5.3.3 Euler Characteristics of Hamiltonian Systems......Page 482
5.4 The Covariant Force Functor......Page 487
6. Natural Control and Self-organization in Biodynamics......Page 488
6.1.1 Introduction to Feedback Control......Page 490
6.1.2.1 Basics of Kalman State-Space Theory......Page 495
6.1.2.2 Regulator Problem......Page 496
6.1.2.3 End Point Control Problem......Page 497
6.1.2.5 Repetitive Mode Problem......Page 498
6.1.2.6 Feedback Changes the Operator......Page 499
6.1.3 Stability and Boundedness......Page 500
6.1.4 Lyapunov’s Stability Method......Page 503
6.1.5 Graphical Techniques for Nonlinear Systems......Page 504
6.1.5.1 Describing Function Analysis......Page 505
6.2.1.1 Exact Feedback Linearization......Page 507
6.2.1.2 Relative Degree......Page 511
6.2.1.3 Approximative Feedback Linearization......Page 513
6.2.2.1 Linear Controllability......Page 516
6.2.2.2 Nonlinear Controllability......Page 517
6.2.2.3 Controllability Condition......Page 519
6.2.2.4 Distributions......Page 520
6.2.2.5 Foliations......Page 521
6.3.1 Abstract Control System......Page 522
6.3.2 Controllability of a Linear Control System......Page 523
6.3.3 Affine Control System and Local Controllability......Page 524
6.3.4.1 Hamiltonian Control Systems......Page 525
6.3.4.2 Pontryagin’s Maximum Principle......Page 528
6.3.4.3 Affine Control Systems......Page 529
6.4.1 Control of Locomotion Systems......Page 531
6.4.1.1 Stratified Kinematic Controllability......Page 532
6.4.1.3 The Exterior Differential Systems Approach......Page 534
6.4.1.4 On the Existence and Uniqueness of Solutions......Page 536
6.4.1.5 Trajectory Generation Problem......Page 537
6.4.2 Gait Biodynamics......Page 539
6.5.1 Control Policy Learning by Robots......Page 543
6.5.1.1 Direct Learning of the Control Policy......Page 544
6.5.1.2 Indirect Learning of the Control Policy......Page 545
6.5.2 Pathways to Self-organization in Biodynamics......Page 547
6.5.3.2 Darwinian Oscillatory Neural Net......Page 549
6.5.3.3 Recurrent Neuro-Muscular Model......Page 552
6.5.3.4 Autogenetic Reflex Motor-Servo......Page 554
6.5.3.5 Biodynamics Control......Page 555
6.5.4 Lie-Adaptive Biodynamic Control......Page 559
6.6.0.1 Kalman Filter Basics......Page 561
6.6.0.2 Inertial Navigation......Page 567
6.6.0.3 Adaptive Estimation in Biomechanics......Page 571
6.7.1 Honda Humanoid Series......Page 572
6.7.2 Cerebellar Robotics......Page 573
7. Natural Brain Dynamics and Sensory-Motor Integration......Page 576
7.1 Introduction to Brain......Page 577
7.2 Human Nervous System......Page 584
7.2.1 Building Blocks of the Nervous System......Page 585
7.2.1.1 Neuronal Circuits......Page 587
7.2.1.2 Basic Brain Partitions and Their Functions......Page 591
7.2.1.3 Nerves......Page 593
7.2.1.4 Action potential......Page 594
7.2.1.5 Synapses......Page 596
7.2.2 Reflex Action: the Basis of CNS Activity......Page 601
7.2.3 The Spinal Cord Pathways......Page 603
7.2.3.1 Spinal Lower Motor Neurons......Page 604
7.2.3.2 Central Pattern Generators in the Spinal Cord575......Page 610
7.2.3.3 Influence of Higher Centers......Page 611
7.2.4 First Look at the Brain......Page 613
7.2.5.1 Primary Motor Cortex......Page 616
7.2.5.2 Motor Association/Premotor Cortical Areas......Page 619
7.2.6 Subcortical Motor ‘Side Loops’......Page 622
7.2.6.1 The Cerebellum......Page 623
7.2.6.2 The Basal Ganglia......Page 629
7.2.6.3 Cerebellar Movement Control......Page 631
7.2.7 Human Senses and their Pathways......Page 635
7.2.8 The Human-Like Vision......Page 638
7.2.8.1 Extramular SO( 3) -Muscles......Page 639
7.2.8.2 Retina......Page 641
7.2.8.3 Cornea......Page 642
7.2.8.4 Iris......Page 643
7.2.8.5 Pursuit Eye Control and Motion Perception......Page 645
7.2.8.6 Optical Flow......Page 648
7.2.9 The Visual Pathway......Page 650
7.2.9.1 Light Reflex and 3D Vision......Page 654
7.2.10 Differential Geometry of the Striate Cortex......Page 655
7.2.11.1 The Inner Ear......Page 657
7.2.11.2 Auditory Transduction......Page 658
7.2.11.3 Central Auditory Pathways......Page 659
7.2.11.5 The Semicircular Canals......Page 661
7.2.11.6 The Vestibulo-Ocular Reflex......Page 662
7.2.11.7 The Utricle and Saccule......Page 663
7.2.11.8 Mechanics of the Semicircular Canals......Page 664
7.2.11.9 Endolymph Flow in the Semicircular Canals......Page 665
7.2.12 Somatosensory Pathways......Page 667
7.2.12.1 The Discriminative Touch System......Page 668
7.2.12.2 The Pain and Temperature System......Page 669
7.2.12.3 The Proprioceptive System......Page 670
7.3.1 Summary on Sensory-Motor Pathways......Page 673
7.3.2 Sensory-Motor Control......Page 677
7.3.2.1 Multisensory Integration for Motor Planning......Page 680
7.3.3 The Central Biomechanical Adjunction......Page 684
7.3.3.1 Postural Control Experiments......Page 687
7.3.3.2 Learning Arm Movement Control......Page 690
7.3.4.1 Basic Dynamics of Brain Injuries......Page 693
7.4 Brain Dynamics......Page 696
7.4.1.1 Biochemistry of Microtubules......Page 697
7.4.1.2 Kink Soliton Model of MT-Dynamics......Page 699
7.4.2.1 Hodgkin-Huxley Model......Page 702
7.4.2.2 FitzHugh-Nagumo Model......Page 706
7.5.1 Phase Dynamics of Oscillatory Neural Nets......Page 707
7.5.1.1 Kuramoto Synchronization Model......Page 710
7.5.1.2 Lyapunov Chaotic Synchronization......Page 711
7.5.2 Complex Networks Dynamics......Page 713
7.5.2.2 Path-Integral Approach to Complex Nets......Page 714
7.5.3 Complex Adaptive Systems......Page 716
7.5.4.1 The Theta-Neuron......Page 719
7.5.4.2 Coupled Theta-Neurons......Page 720
7.5.5 Classification of ‘Spiking’ Neuron Models......Page 723
7.5.6 Weakly Connected and Canonical Neural Nets......Page 727
7.5.7 Quantum Brain Model......Page 729
7.5.8 Open Liouville Neurodynamics and Self-similarity......Page 733
7.5.8.3 Conservative Quantum System......Page 735
7.5.8.4 Open Classical System......Page 736
7.5.8.5 Continuous Neural Network Dynamics......Page 737
7.5.8.6 Open Quantum System......Page 738
7.5.8.8 Equivalence of Neurodynamic Forms......Page 739
7.6.1 Biological Versus Artificial Neural Nets......Page 740
7.6.2.1 Multilayer Perceptrons......Page 742
7.6.2.2 Summary of Supervised Learning Methods......Page 754
7.6.2.3 Other Standard ANNs......Page 755
7.6.2.4 Fully Recurrent ANNs......Page 762
7.6.2.5 Dynamical Games and Recurrent ANNs......Page 763
7.6.2.6 Complex-Valued ANNs......Page 766
7.6.3 Common Continuous......Page 767
7.6.3.1 Neurons as Functions......Page 768
7.6.3.2 Basic Activation and Learning......Page 770
7.6.3.3 Standard Models of Continuous Nets......Page 771
7.7.1 Generalized Kohonen’s SOM......Page 776
7.7.1.1 The Winner Relaxing Kohonen Algorithm......Page 777
7.7.1.2 The Magnification Factor......Page 778
7.7.1.3 Magnification Exponent......Page 779
7.7.2.1 Ising-Spin Neurons......Page 780
7.7.2.2 Graded-Response Neurons......Page 781
7.7.2.3 Hopfield’s Overlaps......Page 782
7.7.2.4 Overlap Dynamics......Page 784
7.7.2.5 Hebbian Learning Dynamics......Page 785
7.7.3 A Self-organizing Bidirectional Competitive Net......Page 788
7.8.1.1 ‘Fuzzy Thinking’......Page 790
7.8.1.2 Fuzzy Sets......Page 791
7.8.1.3 Fuzziness of the Real World......Page 792
7.8.1.4 Fuzzy Entropy......Page 793
7.8.2 Fuzzy Inference Engine......Page 796
7.8.3 Fuzzy Logic Control......Page 799
7.8.3.1 Fuzzy Control of Biodynamic Jerks......Page 803
7.8.3.2 Characteristics of Fuzzy Control......Page 804
7.8.3.3 Evolving Connectionist Systems......Page 805
7.8.4 High-Resolution FAM Agents......Page 806
7.8.4.1 Generic Nonlinear MIMO Systems......Page 807
7.8.4.2 Alternative MIMO Systems......Page 810
7.8.4.3 Biodynamics Example: Tennis Game......Page 812
7.9.1 Categorical Patterns and Hierarchical Links......Page 818
7.9.2 A General Natural System......Page 821
7.9.4 Memory Evolutive System......Page 822
7.9.5 Neural System in a Category......Page 824
7.10.1 Neurodynamic 2-Functor......Page 827
7.10.2.1 Synergetic ‘Thought Solitons’......Page 830
7.11 Body-Mind Adjunction and Natural Psychodynamics......Page 835
7.11.1 Natural Psychodynamics in the Life Space Foam......Page 836
7.11.1.2 General Formalism......Page 841
7.11.1.3 Motion and Decision Making in LSFPaths......Page 845
7.11.1.4 ForceFields and Memory in LSFfields......Page 849
7.11.1.5 Geometries, Topologies and Noise in LSF,,,,......Page 851
7.12 Brain-Like Control in a Nutshell......Page 853
7.12.1 Functor Control Machine......Page 855
7.12.2 Spinal Control Level......Page 857
7.12.3 Cerebellar Control Level......Page 862
7.12.4 Cortical Control Level......Page 865
7.12.5 A Note on Muscular Training......Page 868
7.12.6 Errors in Motion Control: Locomotor Injuries......Page 871
A.1.1 General Functional Transformation......Page 872
A.1.1.1 Transformation of Coordinates......Page 873
A.1.1.3 Vectors and Covectors......Page 874
A.1.1.4 Second-Order Tensors......Page 875
A.1.1.6 Tensor Symmetry......Page 877
A.1.2.1 Basis Vectors and the Metric Tensor in R\"......Page 879
A.1.2.2 Tensor Products in R\"......Page 880
A.1.3.1 Christoffel\'s Symbols......Page 881
A.1.3.3 The Covariant Derivative......Page 882
A.1.3.4 Vector Differential Operators......Page 883
A.1.3.5 The Absolute Derivative......Page 884
A.1.4 The Covariant Force Law in Biodynamics......Page 888
A.1.5 The Essence of Natural Hamiltonian Biodynamics......Page 890
A.1.6 Neuro-Hamiltonian Control in Biodynamics......Page 891
A.2 Frequently Used Neurophysiological Terms......Page 892
A.3.1 Nuclear Magnetic Resonance in 2D Medical Imaging......Page 915
A.3.2 3D Magnetic Resonance Imaging of Human Brain......Page 916
A.3.3 Diffusion MRI in 3D Volume......Page 917
A.3.4 Imaging Diffusion with MRI......Page 918
A.3.5 3D Diffusion Tensor......Page 919
A.3.6 Brain Connectivity Studies......Page 922
A.3.7 Brain Waves and Independent Component Analysis......Page 923
A.4.1 Complex Numbers and Vectors......Page 924
A.4.1.1 Quaternions and Rotations......Page 925
A.4.2 Complex Functions......Page 929
A.4.3 Complex Manifolds......Page 933
A.4.4 Hilbert Space......Page 937
A.5.1 Basic Tables of Lie Groups and Their Lie Algebras......Page 939
A.5.2 Representations of Lie groups......Page 942
A.5.3.1 Definitions......Page 943
A.5.3.2 Classification......Page 944
A.5.3.3 Dynkin Diagrams......Page 945
A.5.4 Simple and Semisimple Lie Groups and Algebras......Page 948
A.6.1 Equilibrium Phase Transitions......Page 950
A.6.1.1 Classification of Phase Transitions......Page 951
A.6.1.2 Basic Properties of Phase Transitions......Page 953
A.6.2 Landau’s Theory of Phase Transitions......Page 956
A.6.3 Partition Function......Page 957
A.6.3.1 Classical Partition Function......Page 958
A.6.3.2 Quantum Partition Function......Page 959
A.6.3.3 Vibrations of Coupled Oscillators......Page 960
A.6.4 Noisdnduced Nonequilibrium Phase Transitions......Page 965
A.6.4.1 General Zero-Dimensional System......Page 966
A.6.4.2 General d-Dimensional System......Page 969
A.7.1 Load-Lifting Biodynamics......Page 973
A.7.2.1 Spinal FC-level......Page 974
A.7.2.2 Cerebellar FC-level......Page 976
A.7.2.3 Cortical FC-level......Page 977
A.7.3.1 Movements in Synovial Joints......Page 979
A.7.3.2 Examples of Sport Movements......Page 980
Bibliography......Page 982
Index......Page 1014