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