Spiking Neuron Models

✍ Scribed by Wulfram Gerstner, Werner M. Kistler

• Publisher: Cambridge University Press
• Year: 2002
• Tongue: English
• Leaves: 496
• Edition: 1
• Category: Library

✦ Synopsis

very well written, easy to understand, walks you through the logic of each part of each equation. builds up more and more complex models based upon the previous models. You'll learn a lot of practical neurobiology stuff other than just modeling too.

✦ Table of Contents

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
Acknowledgments......Page 16
1.1 Elements of neuronal systems......Page 17
1.1.1 The ideal spiking neuron......Page 18
1.1.2 Spike trains......Page 19
1.2 Elements of neuronal dynamics......Page 20
1.2.2 Firing threshold and action potential......Page 22
1.3.1 Definition of the model SRM......Page 23
(i) Adaptation, bursting, and inhibitory rebound......Page 25
(ii) Saturating excitation and shunting inhibition......Page 26
1.4 The problem of neuronal coding......Page 29
1.5.1 Rate as a spike count (average over time)......Page 31
1.5.2 Rate as a spike density (average over several runs)......Page 33
1.5.3 Rate as a population activity (average over several neurons)......Page 34
1.6.1 Time-to-first-spike......Page 36
1.6.2 Phase......Page 37
1.6.3 Correlations and synchrony......Page 38
1.6.4 Stimulus reconstruction and reverse correlation......Page 39
1.7 Discussion: spikes or rates?......Page 41
1.8 Summary......Page 43
Literature......Page 44
Part one Single neuron models......Page 45
2.1.1 Nernst potential......Page 47
2.1.2 Reversal potential......Page 49
2.2.1 Definition of the model......Page 50
2.2.2 Dynamics......Page 53
2.3.1 Sodium channels......Page 57
2.3.2 Potassium channels......Page 59
2.3.3 Low-threshold calcium current......Page 61
2.3.4 High-threshold calcium current and calcium-activated potassium channels......Page 63
2.3.5 Calcium dynamics......Page 66
2.4.1 Inhibitory synapses......Page 67
2.4.2 Excitatory synapses......Page 68
2.5 Spatial structure: the dendritic tree......Page 69
2.5.1 Derivation of the cable equation......Page 70
2.5.2 Green’s function ()......Page 73
2.5.3 Nonlinear extensions to the cable equation......Page 76
2.6 Compartmental models......Page 77
2.7 Summary......Page 82
Literature......Page 83
3.1 Reduction to two dimensions......Page 85
3.1.1 General approach......Page 86
3.1.2 Mathematical steps ()......Page 88
3.2.1 Nullclines......Page 90
3.2.2 Stability of fixed points......Page 91
3.2.3 Limit cycles......Page 93
Hopf bifurcation ()......Page 95
3.2.4 Type I and type II models......Page 96
3.3 Threshold and excitability......Page 98
3.3.1 Type I models......Page 100
3.3.2 Type II models......Page 101
3.3.3 Separation of time scales......Page 102
Trajectory during a pulse ()......Page 105
3.4 Summary......Page 106
Literature......Page 107
4.1 Integrate-and-fire model......Page 109
4.1.1 Leaky integrate-and-fire model......Page 110
4.1.2 Nonlinear integrate-and-fire model......Page 113
Rescaling and standard forms ()......Page 114
4.1.3 Stimulation by synaptic currents......Page 116
4.2.1 Definition of the SRM......Page 118
Interpretation......Page 119
Refractoriness......Page 121
Removing the dynamic threshold......Page 122
4.2.2 Mapping the integrate-and-fire model to the SRM......Page 124
Dynamic threshold interpretation......Page 127
Relation to the integrate-and-fire model......Page 128
Relation between the kernels Epsilon and Episilon ()......Page 131
4.3 From detailed models to formal spiking neurons......Page 132
4.3.1 Reduction of the Hodgkin–Huxley model......Page 133
The Kappa kernel......Page 134
The threshold Theta......Page 135
Input scenarios......Page 136
4.3.2 Reduction of a cortical neuron model......Page 139
Reduction to a nonlinear integrate-and-fire model......Page 140
Reduction to a Spike Response Model......Page 144
4.3.3 Limitations......Page 147
4.4.1 Definition of the model......Page 149
4.4.2 Relation to the model SRM......Page 151
4.4.3 Relation to the full Spike Response Model ()......Page 153
Time-to-first-spike......Page 155
Phase coding......Page 156
Correlation coding......Page 157
Decoding: synchronous versus asynchronous input......Page 158
Literature......Page 161
5 Noise in spiking neuron models......Page 163
5.1.1 Are neurons noisy?......Page 164
5.1.2 Noise sources......Page 165
5.2 Statistics of spike trains......Page 166
5.2.1 Input-dependent renewal systems......Page 167
5.2.2 Interval distribution......Page 168
5.2.3 Survivor function and hazard......Page 169
Mean firing rate......Page 174
Noise spectrum......Page 175
5.2.5 Autocorrelation of a stationary renewal process......Page 176
5.3 Escape noise......Page 179
5.3.1 Escape rate and hazard function......Page 180
5.3.2 Interval distribution and mean firing rate......Page 184
5.4 Slow noise in the parameters......Page 188
5.5.1 Stochastic spike arrival......Page 190
5.5.2 Diffusion limit ()......Page 194
5.5.3 Interval distribution......Page 198
5.6 The subthreshold regime......Page 200
5.6.1 Sub- and superthreshold stimulation......Page 201
5.6.2 Coefficient of variation C......Page 203
5.7 From diffusive noise to escape noise......Page 204
5.8 Stochastic resonance......Page 207
5.9.1 Analog neurons......Page 210
5.9.2 Stochastic rate model......Page 212
5.9.3 Population rate model......Page 213
5.10 Summary......Page 214
Literature......Page 215
Part two Population models......Page 217
6 Population equations......Page 219
6.1 Fully connected homogeneous network......Page 220
6.2.1 Integrate-and-fire neurons with stochastic spike arrival......Page 223
Diffusion approximation......Page 226
6.2.2 Spike Response Model neurons with escape noise......Page 230
Integrating the partial differential equation ()......Page 232
Numerical implementation ()......Page 233
6.2.3 Relation between the approaches......Page 234
From membrane potential densities to phase densities ()......Page 235
From membrane potential densities to refractory densities ()......Page 237
6.3 Integral equations for the population activity......Page 238
6.3.2 Integral equation for the dynamics......Page 239
Absolute refractoriness and the Wilson–Cowan integral equation......Page 242
Derivation of the Wilson–Cowan integral equation ()......Page 243
Quasi-stationary dynamics ()......Page 244
6.4.1 Stationary activity and mean firing rate......Page 247
6.4.2 Gain function and fixed points of the activity......Page 249
6.4.3 Low-connectivity networks......Page 251
6.5.1 Several populations......Page 256
6.5.2 Spatial continuum limit......Page 258
6.6 Limitations......Page 261
Literature......Page 262
7 Signal transmission and neuronal coding......Page 265
7.1 Linearized population equation......Page 266
7.1.1 Noise-free population dynamics ()......Page 268
Linearization......Page 270
7.1.2 Escape noise ()......Page 272
The kernel…(x ) for escape noise ()......Page 274
7.1.3 Noisy reset ()......Page 276
7.2 Transients......Page 277
7.2.1 Transients in a noise-free network......Page 278
7.2.2 Transients with noise......Page 280
7.3.1 Signal term......Page 284
7.4 The significance of a single spike......Page 289
7.4.1 The effect of an input spike......Page 290
7.4.2 Reverse correlation – the significance of an output spike......Page 294
7.5 Summary......Page 298
Literature......Page 299
8 Oscillations and synchrony......Page 301
8.1 Instability of the asynchronous state......Page 302
8.2.1 Locking in noise-free populations......Page 308
Derivation of the Locking Theorem ()......Page 312
8.2.2 Locking in SRM neurons with noisy reset ()......Page 314
Pulse width in the presence of noise ()......Page 315
8.2.3 Cluster states......Page 316
8.3 Oscillations in reverberating loops......Page 318
8.3.1 From oscillations with spiking neurons to binary neurons......Page 321
Purely excitatory projections......Page 322
8.3.3 Microscopic dynamics......Page 325
Quantifying the information content ()......Page 328
8.4 Summary......Page 329
Literature......Page 330
9 Spatially structured networks......Page 331
9.1 Stationary patterns of neuronal activity......Page 332
9.1.1 Homogeneous solutions......Page 334
9.1.2 Stability of homogeneous states......Page 335
9.1.3 “Blobs" of activity: inhomogeneous states......Page 340
9.2 Dynamic patterns of neuronal activity......Page 345
9.2.1 Oscillations......Page 346
9.2.2 Traveling waves......Page 348
9.3 Patterns of spike activity......Page 350
9.3.1 Traveling fronts and waves ()......Page 353
9.3.2 Stability ()......Page 354
9.4 Robust transmission of temporal information......Page 357
Derivation of the spike packet transfer function ()......Page 361
Literature......Page 364
Part three Models of synaptic plasticity......Page 365
10.1 Synaptic plasticity......Page 367
10.1.1 Long-term potentiation......Page 368
10.1.2 Temporal aspects......Page 370
10.2.1 A mathematical formulation of Hebb’s rule......Page 372
10.3.1 Phenomenological model......Page 378
10.3.2 Consolidation of synaptic efficacies......Page 381
10.3.3 General framework ()......Page 383
(i) Sharply peaked back propagating action potential......Page 385
10.4 Detailed models of synaptic plasticity......Page 386
10.4.1 A simple mechanistic model......Page 387
10.4.2 A kinetic model based on NMDA receptors......Page 390
10.4.3 A calcium-based model......Page 393
NMDA receptor as a coincidence detector......Page 394
The calcium control hypothesis......Page 396
Results......Page 397
10.5 Summary......Page 399
Literature......Page 400
11.1.1 Correlation matrix and principal components......Page 403
11.1.2 Evolution of synaptic weights......Page 405
Self-averaging (*)......Page 409
11.1.3 Weight normalization......Page 410
11.1.4 Receptive field development......Page 414
Model architecture......Page 415
Plasticity......Page 417
Simulation results......Page 418
11.2 Learning in spiking models......Page 419
11.2.1 Learning equation......Page 420
11.2.2 Spike–spike correlations......Page 422
11.2.3 Relation of spike-based to rate-based learning......Page 425
Stabilization of postsynaptic rates......Page 426
11.2.4 Static-pattern scenario......Page 427
11.2.5 Distribution of synaptic weights......Page 431
Literature......Page 434
12.1 Learning to be fast......Page 437
12.2.1 The model......Page 441
12.2.2 Firing time distribution......Page 443
12.2.3 Stationary synaptic weights......Page 444
12.2.4 The role of the firing threshold......Page 446
12.3 Sequence learning......Page 448
12.4.1 Electro-sensory system of Mormoryd electric fish......Page 453
Neuron model......Page 455
Synaptic plasticity......Page 456
12.5 Transmission of temporal codes......Page 457
12.5.1 Auditory pathway and sound source localization......Page 458
12.5.2 Phase locking and coincidence detection......Page 460
12.5.3 Tuning of delay lines......Page 463
Literature......Page 468
References......Page 471
Index......Page 493

✦ Subjects

Психологические дисциплины;Нейропсихология;