Brain Dynamics (eBook)

Foreword to the Second Edition 6
Preface 7
Contents 9
Part I Basic Experimental Facts and Theoretical Tools 14
1. Introduction 15
1.1 Goal 15
1.2 Brain: Structure and Functioning. A Brief Reminder 16
1.3 Network Models 17
1.4 How We Will Proceed 19
2. The Neuron – Building Block of the Brain 20
2.1 Structure and Basic Functions 20
2.2 Information Transmission in an Axon 21
2.3 Neural Code 23
2.4 Synapses – The Local Contacts 24
2.5 Naka–Rushton Relation 25
2.6 Learning and Memory 27
2.7 The Role of Dendrites 27
3. Neuronal Cooperativity 28
3.1 Structural Organization 28
3.2 Global Functional Studies. Location of Activity Centers 34
3.3 Interlude: A Minicourse on Correlations 36
3.4 Mesoscopic Neuronal Cooperativity 42
4. Spikes, Phases, Noise: How to Describe Them Mathematically? We Learn a Few Tricks and Some Important Concepts 48
4.1 The d-Function and Its Properties 48
4.2 Perturbed Step Functions 54
4.3 Some More Technical Considerations* 57
4.4 Kicks 59
4.5 Many Kicks 62
4.6 Random Kicks or a Look at Soccer Games 63
4.7 Noise Is Inevitable. Brownian Motion and the Langevin Equation 65
4.8 Noise in Active Systems 67
4.9 The Concept of Phase 71
4.10 Phase Noise 79
4.11 Origin of Phase Noise* 82
Part II Spiking in Neural Nets 85
5. The Lighthouse Model. Two Coupled Neurons 86
5.1 Formulation of the Model 86
5.2 Basic Equations for the Phases of Two Coupled Neurons 89
5.3 Two Neurons: Solution of the Phase-Locked State 91
5.4 Frequency Pulling and Mutual Activation of Two Neurons 95
5.5 Stability Equations 98
5.6 Phase Relaxation and the Impact of Noise 103
5.7 Delay Between Two Neurons 107
5.8 An Alternative Interpretation of the Lighthouse Model 109
6. The Lighthouse Model. Many Coupled Neurons 111
6.1 The Basic Equations 111
6.2 A Special Case. Equal Sensory Inputs. No Delay 113
6.3 A Further Special Case. Different Sensory Inputs, but No Delay and No Fluctuations 115
6.4 Associative Memory and Pattern Filter 117
6.5 Weak Associative Memory. General Case* 121
6.6 The Phase-Locked State of N Neurons. Two Delay Times 124
6.7 Stability of the Phase-Locked State. Two Delay Times* 126
6.8 Many Different Delay Times* 131
6.9 Phase Waves in a Two-Dimensional Neural Sheet 132
6.10 Stability Limits of Phase-Locked State 133
6.11 Phase Noise* 134
6.12 Strong Coupling Limit. The Nonsteady Phase- Locked State of Many Neurons 138
6.13 Fully Nonlinear Treatment of the Phase- Locked State* 142
7. Integrate and Fire Models (IFM) 148
7.1 The General Equations of IFM 148
7.2 Peskin’s Model 150
7.3 A Model with Long Relaxation Times of Synaptic and Dendritic Responses 152
8. Many Neurons, General Case, Connection with Integrate and Fire Model 158
8.1 Introductory Remarks 158
8.2 Basic Equations Including Delay and Noise 158
8.3 Response of Dendritic Currents 160
8.4 The Phase-Locked State 162
8.5 Stability of the Phase-Locked State: Eigenvalue Equations 163
8.6 Example of the Solution of an Eigenvalue Equation of the Form of ( 8.59) 166
8.7 Stability of Phase-Locked State I: The Eigenvalues 168
8.8 Stability of Phase-Locked State II: The Eigenvalues of the Integrate and Fire Model 169
8.9 Generalization to Several Delay Times 172
8.10 Time-Dependent Sensory Inputs 173
8.11 Impact of Noise and Delay 174
8.12 Partial Phase Locking 174
8.13 Derivation of Pulse-Averaged Equations 175
Appendix 1 to Chap. 8: Evaluation of (8.35) 179
Appendix 2 to Chap. 8: Fractal Derivatives 183
9. Pattern Recognition Versus Synchronization: Pattern Recognition 186
9.1 Introduction 186
9.2 Basic Equations 187
9.3 A Reminder of Pattern Recognition by the Synergetic Computer and an Alternative Approach 190
9.4 Properties of the Synergetic Computer of Type II 193
9.5 Limit of Dense Pulses 198
9.6 Pulse Rates Are Positive 203
9.7 Chopped Signals. Quasi-Attractors 205
9.8 Appendix to Sect. 9.5 208
10. Pattern Recognition Versus Synchronization: Synchronization and Phase Locking 211
10.1 The Synchronized State 211
10.2 Stability of the Synchronized State 216
10.3 Stability Analysis Continued: Solution of the Stability Equations 219
10.4 Generalization to More Complicated Dendritic Responses* 223
10.5 Stability Analysis for the General Case of Dendritic Responses* 227
10.6 From Synchronization to Phase Locking 231
10.7 Conclusion to Chaps. 9 and 10: Two Pathways to Pattern Recognition 238
Part III Phase Locking, Coordination and Spatio- Temporal Patterns 240
11. Phase Locking via Sinusoidal Couplings 241
11.1 Coupling Between Two Neurons 241
11.2 A Chain of Coupled-Phase Oscillators 244
11.3 Coupled Finger Movements 246
11.4 Quadruped Motion 249
11.5 Populations of Neural Phase Oscillators 251
12. Pulse-Averaged Equations 253
12.1 Survey 253
12.2 The Wilson–Cowan Equations 254
12.3 A Simple Example 255
12.4 Cortical Dynamics Described by Wilson–Cowan Equations 260
12.5 Visual Hallucinations 262
12.6 Jirsa–Haken–Nunez Equations 263
12.7 An Application to Movement Control 267
Part IV Conclusion 272
13. The Single Neuron 273
13.1 Hodgkin–Huxley Equations 273
13.2 FitzHugh–Nagumo Equations 276
13.3 Some Generalizations of the Hodgkin– Huxley Equations 280
13.4 Dynamical Classes of Neurons 281
13.5 Some Conclusions on Network Models 282
14. Conclusion and Outlook 283
15. Solutions to Exercises 286
Section 4.4, Exercise 286
Section 4.7, Exercise 286
Section 4.8.3, Exercise 286
Section 4.9.3, Exercise 1 288
Section 4.9.3, Exercise 2 288
Section 4.9.3, Exercise 3 289
Section 4.9.3, Exercise 4 289
Section 4.11, Exercise 289
Section 5.3, Exercise 290
Section 6.4, Exercise 1 290
Section 6.4, Exercise 2 291
Section 6.11, Exercise 1 291
Section 6.11, Exercise 2 292
Section 6.11, Exercise 3 292
Section 6.12, Exercise 292
Section 7.3, Exercise 1 293
Section 7.3, Exercise 2 294
Section 7.3, Exercise 3 294
Section 7.3, Exercise 4 295
Section 8.5, Exercise 1 295
Section 8.8, Exercise 295
Section 8, Appendix 2, Exercise 296
Section 9.4, Exercise, Example 1 296
Section 9.4, Exercise, Example 2 297
Section 9.5, Exercise 1 298
Section 9.5, Exercise 2 299
Section 9.5, Exercise 3 300
Section 9.5, Exercise 4 300
Section 9.5, Exercise 5 302
Section 9.5, Exercise 6 303
Section 9.6, Exercise 304
Section 10.1, Exercise 304
Section 10.2, Exercise 1 305
Section 10.2, Exercise 2 306
Section 10.2, Exercise 3 308
Section 10.5, Exercise 312
Section 12.6, Exercise 1 314
References 315
Preface 315
Introduction 315
The Neuron – Building Block of the Brain 317
Neuronal Cooperativity 318
Spikes, Phases, Noise: How to Describe Them Mathematically? We Learn a Few Tricks and Some Important Concepts 319
The Lighthouse Model. Two Coupled Neurons 320
The Lighthouse Model. Many Coupled Neurons 320
Integrate and Fire Models (IFM) 321
Many Neurons, General Case, Connection with Integrate and Fire Model 321
Pattern Recognition Versus Synchronization: Pattern Recognition 322
Phase Locking via Sinusoidal Couplings 322
Pulse-Averaged Equations 323
The Single Neuron 325
Conclusion and Outlook 326
Index 327