| Basic Experimental Facts and Theoretical Tools | |
| Introduction | p. 3 |
| Goal | p. 3 |
| Brain: Structure and Functioning. A Brief Reminder | p. 4 |
| Network Models | p. 5 |
| How We Will Proceed | p. 7 |
| The Neuron - Building Block of the Brain | p. 9 |
| Structure and Basic Functions | p. 9 |
| Information Transmission in an Axon | p. 10 |
| Neural Code | p. 12 |
| Synapses - The Local Contacts | p. 13 |
| Naka-Rushton Relation | p. 14 |
| Learning and Memory | p. 16 |
| The Role of Dendrites | p. 16 |
| Neuronal Cooperativity | p. 17 |
| Structural Organization | p. 17 |
| Global Functional Studies. Location of Activity Centers | p. 23 |
| Interlude: A Minicourse on Correlations | p. 25 |
| Mesoscopic Neuronal Cooperativity | p. 31 |
| Spikes, Phases, Noise: How to Describe Them Mathematically? We Learn a Few Tricks and Some Important Concepts | p. 37 |
| The -Function and Its Properties | p. 37 |
| Perturbed Step Functions | p. 43 |
| Some More Technical Considerations* | p. 46 |
| Kicks | p. 48 |
| Many Kicks | p. 51 |
| Random Kicks or a Look at Soccer Games | p. 52 |
| Noise Is Inevitable. Brownian Motion and the Langevin Equation | p. 54 |
| Noise in Active Systems | p. 56 |
| Introductory Remarks | p. 56 |
| Two-State Systems | p. 57 |
| Many Two-State Systems: Many Ion Channels | p. 58 |
| The Concept of Phase | p. 60 |
| Some Elementary Considerations | p. 60 |
| Regular Spike Trains | p. 63 |
| How to Determine Phases From Experimental Data? Hilbert Transform | p. 64 |
| Phase Noise | p. 68 |
| Origin of Phase Noise* | p. 71 |
| Spiking in Neural Nets | |
| The Lighthouse Model. Two Coupled Neurons | p. 77 |
| Formulation of the Model | p. 77 |
| Basic Equations for the Phases of Two Coupled Neurons | p. 80 |
| Two Neurons: Solution of the Phase-Locked State | p. 82 |
| Frequency Pulling and Mutual Activation of Two Neurons | p. 86 |
| Stability Equations | p. 89 |
| Phase Relaxation and the Impact of Noise | p. 94 |
| Delay Between Two Neurons | p. 98 |
| An Alternative Interpretation of the Lighthouse Model | p. 100 |
| The Lighthouse Model. Many Coupled Neurons | p. 103 |
| The Basic Equations | p. 103 |
| A Special Case. Equal Sensory Inputs. No Delay | p. 105 |
| A Further Special Case. Different Sensory Inputs, but No Delay and No Fluctuations | p. 107 |
| Associative Memory and Pattern Filter | p. 109 |
| Weak Associative Memory. General Case* | p. 113 |
| The Phase-Locked State of N Neurons. Two Delay Times | p. 116 |
| Stability of the Phase-Locked State. Two Delay Times* | p. 118 |
| Many Different Delay Times* | p. 123 |
| Phase Waves in a Two-Dimensional Neural Sheet | p. 124 |
| Stability Limits of Phase-Locked State | p. 125 |
| Phase Noise* | p. 126 |
| Strong Coupling Limit. The Nonsteady Phase-Locked State of Many Neurons | p. 130 |
| Fully Nonlinear Treatment of the Phase-Locked State* | p. 134 |
| Integrate and Fire Models (IFM) | p. 141 |
| The General Equations of IFM | p. 141 |
| Peskin's Model | p. 143 |
| A Model with Long Relaxation Times of Synaptic and Dendritic Responses | p. 145 |
| Many Neurons, General Case, Connection with Integrate and Fire Model | p. 151 |
| Introductory Remarks | p. 151 |
| Basic Equations Including Delay and Noise | p. 151 |
| Response of Dendritic Currents | p. 153 |
| The Phase-Locked State | p. 155 |
| Stability of the Phase-Locked State: Eigenvalue Equations | p. 156 |
| Example of the Solution of an Eigenvalue Equation of the Form of (8.59) | p. 159 |
| Stability of Phase-Locked State I: The Eigenvalues of the Lighthouse Model with ′ &neq; 0 | p. 161 |
| Stability of Phase-Locked State II: The Eigenvalues of the Integrate and Fire Model | p. 162 |
| Generalization to Several Delay Times | p. 165 |
| Time-Dependent Sensory Inputs | p. 166 |
| Impact of Noise and Delay | p. 167 |
| Partial Phase Locking | p. 167 |
| Derivation of Pulse-Averaged Equations | p. 168 |
| Evaluation of (8.35) | p. 173 |
| Fractal Derivatives | p. 177 |
| Pattern Recognition Versus Synchronization: Pattern Recognition | p. 181 |
| Introduction | p. 181 |
| Basic Equations | p. 182 |
| A Reminder of Pattern Recognition by the Synergetic Computer and an Alternative Approach | p. 185 |
| Properties of the Synergetic Computer of Type II | p. 188 |
| Limit of Dense Pulses | p. 193 |
| Pulse Rates Are Positive | p. 198 |
| Chopped Signals. Quasi-Attractors | p. 200 |
| Appendix to Sect. 9.5 | p. 203 |
| Pattern Recognition Versus Synchronization: Synchronization and Phase Locking | p. 207 |
| The Synchronized State | p. 207 |
| Stability of the Synchronized State | p. 212 |
| Stability Analysis Continued: Solution of the Stability Equations | p. 215 |
| Generalization to More Complicated Dendritic Responses* | p. 219 |
| Stability Analysis for the General Case of Dendritic Responses* | p. 223 |
| From Synchronization to Phase Locking | p. 227 |
| Conclusion to Chaps. 9 and 10: Two Pathways to Pattern Recognition | p. 234 |
| Phase Locking, Coordination and Spatio-Temporal Patterns | |
| Phase Locking via Sinusoidal Couplings | p. 239 |
| Coupling Between Two Neurons | p. 239 |
| A Chain of Coupled-Phase Oscillators | p. 242 |
| Coupled Finger Movements | p. 244 |
| Quadruped Motion | p. 247 |
| Populations of Neural Phase Oscillators | p. 249 |
| Synchronization Patterns | p. 249 |
| Pulse Stimulation | p. 249 |
| Periodic Stimulation | p. 250 |
| Pulse-Averaged Equations | p. 251 |
| Survey | p. 251 |
| The Wilson-Cowan Equations | p. 252 |
| A Simple Example | p. 253 |
| Cortical Dynamics Described by Wilson-Cowan Equations | p. 258 |
| Visual Hallucinations | p. 260 |
| Jirsa-Haken-Nunez Equations | p. 261 |
| An Application to Movement Control | p. 265 |
| The Kelso Experiment | p. 265 |
| The Sensory-Motor Feedback Loop | p. 267 |
| The Field Equation and Projection onto Modes | p. 268 |
| Some Conclusions | p. 269 |
| Conclusion | |
| The Single Neuron | p. 273 |
| Hodgkin-Huxley Equations | p. 273 |
| FitzHugh-Nagumo Equations | p. 276 |
| Some Generalizations of the Hodgkin-Huxley Equations | p. 280 |
| Dynamical Classes of Neurons | p. 281 |
| Some Conclusions on Network Models | p. 282 |
| Conclusion and Outlook | p. 283 |
| Solutions to Exercises | p. 287 |
| References | p. 317 |
| Index | p. 329 |
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