Haken Brain Dynamics

An Introduction to Models and Simulations
2. Auflage 2007
ISBN: 978-3-540-75238-7
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Wasserzeichen (»Systemvoraussetzungen)

E-Book, Englisch, 333 Seiten

This is an excellent introduction for graduate students and nonspecialists to the field of mathematical and computational neurosciences. The book approaches the subject via pulsed-coupled neural networks, which have at their core the lighthouse and integrate-and-fire models. These allow for highly flexible modeling of realistic synaptic activity, synchronization and spatio-temporal pattern formation. The more advanced pulse-averaged equations are discussed.

Autoren/Hrsg.

Haken, Hermann

Inhaltsverzeichnis

1;Foreword to the Second Edition;6
2;Preface;7
3;Contents;9
4;Part I Basic Experimental Facts and Theoretical Tools;14
4.1;1. Introduction;15
4.1.1;1.1 Goal;15
4.1.2;1.2 Brain: Structure and Functioning. A Brief Reminder;16
4.1.3;1.3 Network Models;17
4.1.4;1.4 How We Will Proceed;19
4.2;2. The Neuron – Building Block of the Brain;20
4.2.1;2.1 Structure and Basic Functions;20
4.2.2;2.2 Information Transmission in an Axon;21
4.2.3;2.3 Neural Code;23
4.2.4;2.4 Synapses – The Local Contacts;24
4.2.5;2.5 Naka–Rushton Relation;25
4.2.6;2.6 Learning and Memory;27
4.2.7;2.7 The Role of Dendrites;27
4.3;3. Neuronal Cooperativity;28
4.3.1;3.1 Structural Organization;28
4.3.2;3.2 Global Functional Studies. Location of Activity Centers;34
4.3.3;3.3 Interlude: A Minicourse on Correlations;36
4.3.4;3.4 Mesoscopic Neuronal Cooperativity;42
4.4;4. Spikes, Phases, Noise: How to Describe Them Mathematically? We Learn a Few Tricks and Some Important Concepts;48
4.4.1;4.1 The d-Function and Its Properties;48
4.4.2;4.2 Perturbed Step Functions;54
4.4.3;4.3 Some More Technical Considerations*;57
4.4.4;4.4 Kicks;59
4.4.5;4.5 Many Kicks;62
4.4.6;4.6 Random Kicks or a Look at Soccer Games;63
4.4.7;4.7 Noise Is Inevitable. Brownian Motion and the Langevin Equation;65
4.4.8;4.8 Noise in Active Systems;67
4.4.9;4.9 The Concept of Phase;71
4.4.10;4.10 Phase Noise;79
4.4.11;4.11 Origin of Phase Noise*;82
5;Part II Spiking in Neural Nets;85
5.1;5. The Lighthouse Model. Two Coupled Neurons;86
5.1.1;5.1 Formulation of the Model;86
5.1.2;5.2 Basic Equations for the Phases of Two Coupled Neurons;89
5.1.3;5.3 Two Neurons: Solution of the Phase-Locked State;91
5.1.4;5.4 Frequency Pulling and Mutual Activation of Two Neurons;95
5.1.5;5.5 Stability Equations;98
5.1.6;5.6 Phase Relaxation and the Impact of Noise;103
5.1.7;5.7 Delay Between Two Neurons;107
5.1.8;5.8 An Alternative Interpretation of the Lighthouse Model;109
5.2;6. The Lighthouse Model. Many Coupled Neurons;111
5.2.1;6.1 The Basic Equations;111
5.2.2;6.2 A Special Case. Equal Sensory Inputs. No Delay;113
5.2.3;6.3 A Further Special Case. Different Sensory Inputs, but No Delay and No Fluctuations;115
5.2.4;6.4 Associative Memory and Pattern Filter;117
5.2.5;6.5 Weak Associative Memory. General Case*;121
5.2.6;6.6 The Phase-Locked State of N Neurons. Two Delay Times;124
5.2.7;6.7 Stability of the Phase-Locked State. Two Delay Times*;126
5.2.8;6.8 Many Different Delay Times*;131
5.2.9;6.9 Phase Waves in a Two-Dimensional Neural Sheet;132
5.2.10;6.10 Stability Limits of Phase-Locked State;133
5.2.11;6.11 Phase Noise*;134
5.2.12;6.12 Strong Coupling Limit. The Nonsteady Phase- Locked State of Many Neurons;138
5.2.13;6.13 Fully Nonlinear Treatment of the Phase- Locked State*;142
5.3;7. Integrate and Fire Models (IFM);148
5.3.1;7.1 The General Equations of IFM;148
5.3.2;7.2 Peskin’s Model;150
5.3.3;7.3 A Model with Long Relaxation Times of Synaptic and Dendritic Responses;152
5.4;8. Many Neurons, General Case, Connection with Integrate and Fire Model;158
5.4.1;8.1 Introductory Remarks;158
5.4.2;8.2 Basic Equations Including Delay and Noise;158
5.4.3;8.3 Response of Dendritic Currents;160
5.4.4;8.4 The Phase-Locked State;162
5.4.5;8.5 Stability of the Phase-Locked State: Eigenvalue Equations;163
5.4.6;8.6 Example of the Solution of an Eigenvalue Equation of the Form of ( 8.59);166
5.4.7;8.7 Stability of Phase-Locked State I: The Eigenvalues;168
5.4.8;8.8 Stability of Phase-Locked State II: The Eigenvalues of the Integrate and Fire Model;169
5.4.9;8.9 Generalization to Several Delay Times;172
5.4.10;8.10 Time-Dependent Sensory Inputs;173
5.4.11;8.11 Impact of Noise and Delay;174
5.4.12;8.12 Partial Phase Locking;174
5.4.13;8.13 Derivation of Pulse-Averaged Equations;175
5.4.14;Appendix 1 to Chap. 8: Evaluation of (8.35);179
5.4.15;Appendix 2 to Chap. 8: Fractal Derivatives;183
5.5;9. Pattern Recognition Versus Synchronization: Pattern Recognition;186
5.5.1;9.1 Introduction;186
5.5.2;9.2 Basic Equations;187
5.5.3;9.3 A Reminder of Pattern Recognition by the Synergetic Computer and an Alternative Approach;190
5.5.4;9.4 Properties of the Synergetic Computer of Type II;193
5.5.5;9.5 Limit of Dense Pulses;198
5.5.6;9.6 Pulse Rates Are Positive;203
5.5.7;9.7 Chopped Signals. Quasi-Attractors;205
5.5.8;9.8 Appendix to Sect. 9.5;208
5.6;10. Pattern Recognition Versus Synchronization: Synchronization and Phase Locking;211
5.6.1;10.1 The Synchronized State;211
5.6.2;10.2 Stability of the Synchronized State;216
5.6.3;10.3 Stability Analysis Continued: Solution of the Stability Equations;219
5.6.4;10.4 Generalization to More Complicated Dendritic Responses*;223
5.6.5;10.5 Stability Analysis for the General Case of Dendritic Responses*;227
5.6.6;10.6 From Synchronization to Phase Locking;231
5.6.7;10.7 Conclusion to Chaps. 9 and 10: Two Pathways to Pattern Recognition;238
6;Part III Phase Locking, Coordination and Spatio- Temporal Patterns;240
6.1;11. Phase Locking via Sinusoidal Couplings;241
6.1.1;11.1 Coupling Between Two Neurons;241
6.1.2;11.2 A Chain of Coupled-Phase Oscillators;244
6.1.3;11.3 Coupled Finger Movements;246
6.1.4;11.4 Quadruped Motion;249
6.1.5;11.5 Populations of Neural Phase Oscillators;251
6.2;12. Pulse-Averaged Equations;253
6.2.1;12.1 Survey;253
6.2.2;12.2 The Wilson–Cowan Equations;254
6.2.3;12.3 A Simple Example;255
6.2.4;12.4 Cortical Dynamics Described by Wilson–Cowan Equations;260
6.2.5;12.5 Visual Hallucinations;262
6.2.6;12.6 Jirsa–Haken–Nunez Equations;263
6.2.7;12.7 An Application to Movement Control;267
6.3;Part IV Conclusion;272
6.4;13. The Single Neuron;273
6.4.1;13.1 Hodgkin–Huxley Equations;273
6.4.2;13.2 FitzHugh–Nagumo Equations;276
6.4.3;13.3 Some Generalizations of the Hodgkin– Huxley Equations;280
6.4.4;13.4 Dynamical Classes of Neurons;281
6.4.5;13.5 Some Conclusions on Network Models;282
6.5;14. Conclusion and Outlook;283
6.6;15. Solutions to Exercises;286
6.6.1;Section 4.4, Exercise;286
6.6.2;Section 4.7, Exercise;286
6.6.3;Section 4.8.3, Exercise;286
6.6.4;Section 4.9.3, Exercise 1;288
6.6.5;Section 4.9.3, Exercise 2;288
6.6.6;Section 4.9.3, Exercise 3;289
6.6.7;Section 4.9.3, Exercise 4;289
6.6.8;Section 4.11, Exercise;289
6.6.9;Section 5.3, Exercise;290
6.6.10;Section 6.4, Exercise 1;290
6.6.11;Section 6.4, Exercise 2;291
6.6.12;Section 6.11, Exercise 1;291
6.6.13;Section 6.11, Exercise 2;292
6.6.14;Section 6.11, Exercise 3;292
6.6.15;Section 6.12, Exercise;292
6.6.16;Section 7.3, Exercise 1;293
6.6.17;Section 7.3, Exercise 2;294
6.6.18;Section 7.3, Exercise 3;294
6.6.19;Section 7.3, Exercise 4;295
6.6.20;Section 8.5, Exercise 1;295
6.6.21;Section 8.8, Exercise;295
6.6.22;Section 8, Appendix 2, Exercise;296
6.6.23;Section 9.4, Exercise, Example 1;296
6.6.24;Section 9.4, Exercise, Example 2;297
6.6.25;Section 9.5, Exercise 1;298
6.6.26;Section 9.5, Exercise 2;299
6.6.27;Section 9.5, Exercise 3;300
6.6.28;Section 9.5, Exercise 4;300
6.6.29;Section 9.5, Exercise 5;302
6.6.30;Section 9.5, Exercise 6;303
6.6.31;Section 9.6, Exercise;304
6.6.32;Section 10.1, Exercise;304
6.6.33;Section 10.2, Exercise 1;305
6.6.34;Section 10.2, Exercise 2;306
6.6.35;Section 10.2, Exercise 3;308
6.6.36;Section 10.5, Exercise;312
6.6.37;Section 12.6, Exercise 1;314
7;References;315
7.1;Preface;315
7.2;Introduction;315
7.3;The Neuron – Building Block of the Brain;317
7.4;Neuronal Cooperativity;318
7.5;Spikes, Phases, Noise: How to Describe Them Mathematically? We Learn a Few Tricks and Some Important Concepts;319
7.6;The Lighthouse Model. Two Coupled Neurons;320
7.7;The Lighthouse Model. Many Coupled Neurons;320
7.8;Integrate and Fire Models (IFM);321
7.9;Many Neurons, General Case, Connection with Integrate and Fire Model;321
7.10;Pattern Recognition Versus Synchronization: Pattern Recognition;322
7.11;Phase Locking via Sinusoidal Couplings;322
7.12;Pulse-Averaged Equations;323
7.13;The Single Neuron;325
7.14;Conclusion and Outlook;326
8;Index;327