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Tables of Contents for Brain Dynamics
Chapter/Section Title
Page #
Page Count
Part I. Basic Experimental Facts and Theoretical Tools
Introduction
3
6
Goal
3
1
Brain: Structure and Functioning. A Brief Reminder
4
1
Network Models
5
1
How We Will Proceed
6
3
The Neuron -- Building Block of the Brain
9
8
Strcture and Basic Functions
9
1
Information Transmission in an Axon
10
2
Neural Code
12
1
Synapses - The Local Contacts
13
1
Naka-Rushton Relation
14
2
Learning and Memory
16
1
The Role of Dendrites
16
1
Neuronal Cooperativity
17
20
Structural Organization
17
6
Global Functional Studies. Location of Activity Centers
23
2
Interlude: A Minicourse on Correlations
25
6
Mesoscopic Neuronal Cooperativity
31
6
Spikes, Phases, Noise: How to Describe Them Mathematically? We Learn a Few Tricks and Some Important Concepts
37
40
The δ-Function and Its Properties
37
6
Perturbed Step Functions
43
3
Some More Technical Considerations
46
2
Kicks
48
3
Many Kicks
51
1
Random Kicks or a Look at Soccer Games
52
2
Noise Is Inevitable. Brownian Motion and the Langevin Equation
54
2
Noise in Active Systems
56
4
Introductory Remarks
56
1
Two-State Systems
57
1
Many Two-State Systems: Many Ion Channels
58
2
The Concept of Phase
60
8
Some Elementary Considerations
60
3
Regular Spike Trains
63
1
How to Determine Phases From Experimental Data? Hilbert Transform
64
4
Phase Noise
68
3
Origin of Phase Noise
71
6
Part II. Spiking in Neural Nets
The Lighthouse Model. Two Coupled Neurons
77
26
Formulation of the Model
77
3
Basic Equations for the Phases of Two Coupled Neurons
80
2
Two Neurons: Solution of the Phase-Locked State
82
4
Frequency Pulling and Mutual Activation of Two Neurons
86
3
Stability Equations
89
5
Phase Relaxation and the Impact of Noise
94
4
Delay Between Two Neurons
98
2
An Alternative Interpretation of the Lighthouse Model
100
3
The Lighthouse Model. Many Coupled Neurons
103
38
The Basic Equations
103
2
A Special Case. Equal Sensory Inputs. No Delay
105
2
A Further Special Case. Different Sensory Inputs, but No Delay and No Fluctuations
107
2
Associative Memory and Pattern Filter
109
4
Weak Associative Memory. General Case
113
3
The Phase-Locked State of N Neurons. Two Delay Times
116
2
Stability of the Phase-Locked State. Two Delay Times
118
5
Many Different Delay Times
123
1
Phase Waves in a Two-Dimensional Neural Sheet
124
1
Stability Limits of Phase-Locked State
125
1
Phase Noise
126
4
Strong Coupling Limit. The Nonsteady Phase-Locked State of Many Neurons
130
4
Fully Nonlinear Treatment of the Phase-Locked State
134
7
Integrate and Fire Models (IFM)
141
10
The General Equations of IFM
141
2
Peskin's Model
143
2
A Model with Long Relaxation Times of Synaptic and Dendritic Responses
145
6
Many Neurons, General Case, Connection with Integrate and Fire Model
151
32
Introductory Remarks
151
1
Basic Equations Including Delay and Noise
151
2
Response of Dendritic Currents
153
2
The Phase-Locked State
155
1
Stability of the Phase-Locked State: Eigenvalue Equations
156
3
Example of the Solution of an Eigenvalue Equation of the Form of (8.59)
159
2
Stability of Phase-Locked State I: The Eigenvalues of the Lighthouse Model with γ ≠ 0
161
1
Stability of Phase-Locked State II: The Eigenvalues of the Integrate and Fire Model
162
3
Generalization to Several Delay Times
165
1
Time-Dependent Sensory Inputs
166
1
Impact of Noise and Delay
167
1
Partial Phase Locking
167
1
Derivation of Pulse-Averaged Equations
168
15
Appendix 1 to Chap. 8: Evaluation of (8.35)
173
4
Appendix 2 to Chap. 8: Fractal Derivatives
177
6
Part III. Phase Locking, Coordination and Spatio-Temporal Patterns
Phase Locking via Sinusoidal Couplings
183
12
Coupling Between Two Neurons
183
3
A Chain of Coupled-Phase Oscillators
186
2
Coupled Finger Movements
188
3
Quadruped Motion
191
2
Populations of Neural Phase Oscillators
193
2
Synchronization Patterns
193
1
Pulse Stimulation
193
1
Periodic Stimulation
194
1
Pulse-Averaged Equations
195
22
Survey
195
1
The Wilson-Cowan Equations
196
1
A Simple Example
197
5
Cortical Dynamics Described by Wilson-Cowan Equations
202
2
Visual Hallucinations
204
1
Jirsa-Haken-Nunez Equations
205
4
An Application to Movement Control
209
8
The Kelso Experiment
209
2
The Sensory-Motor Feedback Loop
211
1
The Field Equation and Projection onto Modes
212
1
Some Conclusions
213
4
Part IV. Conclusion
The Single Neuron
217
8
Hodgkin-Huxley Equations
217
1
FitzHugh-Nagumo Equations
218
4
Some Generalizations of the Hodgkin-Huxley Equations
222
1
Dynamical Classes of Neurons
223
1
Some Conclusions on Network Models
224
1
Conclusion and Outlook
225
4
References
229
12
Index
241