Brain dynamics : synchronization and activity patterns in pulse-coupled neural nets with delays and noise

This book addresses a large variety of models in mathematical and computational neuroscience. It is written for the experts as well as for graduate students wishing to enter this fascinating field of research. The author studies the behaviour of large neural networks composed of many neurons coupled by spike trains. An analysis of phase locking via sinusoidal couplings leading to various kinds of movement coordination is included.

Part I Basic Experimental Facts and Theoretical Tools
1 Introduction	p. 3
1.1 Goal	p. 3
1.2 Brain: Structure and Functioning. A Brief Reminder	p. 4
1.3 Network Models	p. 5
1.4 How We Will Proceed	p. 6
2 The Neuron - Building Block of the Brain	p. 9
2.1 Structure and Basic Functions	p. 9
2.2 Information Transmission in an Axon	p. 10
2.3 Neural Code	p. 12
2.4 Synapses - The Local Contacts	p. 13
2.5 Naka-Rushton Relation	p. 14
2.6 Learning and Memory	p. 16
2.7 The Role of Dendrites	p. 16
3 Neuronal Cooperativity	p. 17
3.1 Structural Organization	p. 17
3.2 Global Functional Studies. Location of Activity Centers	p. 23
3.3 Interlude: A Minicourse on Correlations	p. 25
3.4 Mesoscopic Neuronal Cooperativity	p. 31
4 Spikes, Phases, Noise: How to Describe Them Mathematically? We Learn a Few Tricks and Some Important Concepts	p. 37
4.1 The ¿-Function and Its Properties	p. 37
4.2 Perturbed Step Functions	p. 43
4.3 Some More Technical Considerations*	p. 46
4.4 Kicks	p. 48
4.5 Many Kicks	p. 51
4.6 Random Kicks or a Look at Soccer Games	p. 52
4.7 Noise Is Inevitable. Brownian Motion and the Langevin Equation	p. 54
4.8 Noise in Active Systems	p. 56
4.8.1 Introductory Remarks	p. 56
4.8.2 Two-State Systems	p. 57
4.8.3 Many Two-State Systems: Many Ion Channels	p. 58
4.9 The Concept of Phase	p. 60
4.9.1 Some Elementary Considerations	p. 60
4.9.2 Regular Spike Trains	p. 63
4.9.3 How to Determine Phases From Experimental Data? Hilbert Transform	p. 64
4.10 Phase Noise	p. 68
4.11 Origin ofPhase Noise*	p. 71
Part II Spiking in Neural Nets
5 The Lighthouse Model. Two Coupled Neurons	p. 77
5.1 Formulation of the Model	p. 77
5.2 Basic Equations for the Phases of Two Coupled Neurons	p. 80
5.3 Two Neurons: Solution of the Phase-Locked State	p. 82
5.4 Frequency Pulling and Mutual Activation of Two Neurons	p. 86
5.5 Stability Equations	p. 89
5.6 Phase Relaxation and the Impact ofNoise	p. 94
5.7 Delay Between Two Neurons	p. 98
5.8 An Alternative Interpretation of the Lighthouse Model	p. 100
6 The Lighthouse Model. Many Coupled Neurons	p. 103
6.1 The Basic Equations	p. 103
6.2 A Special Case. Equal Sensory Inputs. No Delay	p. 105
6.3 A Further Special Case. Different Sensory Inputs, but No Delay and No Fluctuations	p. 107
6.4 Associative Memory and Pattern Filter	p. 109
6.5 Weak Associative Memory. General Case*	p. 113
6.6 The Phase-Locked State of N Neurons. Two Delay Times	p. 116
6.7 Stability of the Phase-Locked State. Two Delay Times*	p. 118
6.8 Many Different Delay Times*	p. 123
6.9 Phase Waves in a Two-Dimensional Neural Sheet	p. 124
6.10 Stability Limits of Phase-Locked State	p. 125
6.11 Phase Noise*	p. 126
6.12 Strong Coupling Limit. The Nonsteady Phase-Locked State of Many Neurons	p. 130
6.13 Fully Nonlinear Treatment of the Phase-Locked State*	p. 134
7 Integrate and Fire Models (IFM)	p. 141
7.1 The General Equations of IFM	p. 141
7.2 Peskin's Model	p. 143
7.3 A Model with Long Relaxation Times of Synaptic and Dendritic Responses	p. 145
8 Many Neurons, General Case, Connection with Integrate and Fire Model	p. 151
8.1 Introductory Remarks	p. 151
8.2 Basic Equations Including Delay and Noise	p. 151
8.3 Response of Dendritic Currents	p. 153
8.4 The Phase-Locked State	p. 155
8.5 Stability of the Phase-Locked State: Eigenvalue Equations	p. 156
8.6 Example of the Solution of an Eigenvalue Equation of the Form of (8.59)	p. 159
8.7 Stability of Phase-Locked State I: The Eigenvalues of the Lighthouse Model with ¿′ &neq; 0	p. 161
8.8 Stability of Phase-Locked State II: The Eigenvalues of the Integrate and Fire Model	p. 162
8.9 Generalization to Several Delay Times	p. 165
8.10 Time-Dependent Sensory Inputs	p. 166
8.11 Impact ofNoise and Delay	p. 167
8.12 Partial Phase Locking	p. 167
8.13 Derivation ofPulse-Averaged Equations	p. 168
Appendix 1 to Chap. 8: Evaluation of (
8.35 )	p. 173
Appendix 2 to Chap. 8: Fractal Derivatives	p. 177
Part III Phase Locking, Coordination and Spatio-Temporal Patterns
9 Phase Locking via Sinusoidal Couplings	p. 183
9.1 Coupling Between Two Neurons	p. 183
9.2 A Chain of Coupled-Phase Oscillators	p. 186
9.3 Coupled Finger Movements	p. 188
9.4 Quadruped Motion	p. 191
9.5 Populations of Neural Phase Oscillators	p. 193
9.5.1 Synchronization Patterns	p. 193
9.5.2 Pulse Stimulation	p. 193
9.5.3 Periodic Stimulation	p. 194
10 Pulse-Averaged Equations	p. 195
10.1 Survey	p. 195
10.2 The Wilson-Cowan Equations	p. 196
10.3 A Simple Example	p. 197
10.4 Cortical Dynamics Described by Wilson-Cowan Equations	p. 202
10.5 Visual Hallucinations	p. 204
10.6 Jirsa-Haken-Nunez Equations	p. 205
10.7 An Application to Movement Control	p. 209
10.7.1 The Kelso Experiment	p. 209
10.7.2 The Sensory-Motor Feedback Loop	p. 211
10.7.3 The Field Equation and Projection onto Modes	p. 212
10.7.4 Some Conclusions	p. 213
Part IV Conclusion
11 The Single Neuron	p. 217
11.1 Hodgkin-Huxley Equations	p. 217
11.2 FitzHugh-Nagumo Equations	p. 218
11.3 Some Generalizations of the Hodgkin-Huxley Equations	p. 222
11.4 Dynamical Classes of Neurons	p. 223
11.5 Some Conclusions on Network Models	p. 224
12 Conclusion and Outlook	p. 225
References	p. 229
Index	p. 241

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