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Brain dynamics : synchronization and activity patterns in pulse-coupled neural nets with delays and noise / Hermann Haken. — Berlin ;New York : Springer, c2002.—(59.597/H155)

Contents

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