## Spiking Neuron Models: Single Neurons, Populations, PlasticityNeurons in the brain communicate by short electrical pulses, the so-called action potentials or spikes. How can we understand the process of spike generation? How can we understand information transmission by neurons? What happens if thousands of neurons are coupled together in a seemingly random network? How does the network connectivity determine the activity patterns? And, vice versa, how does the spike activity influence the connectivity pattern? These questions are addressed in this 2002 introduction to spiking neurons aimed at those taking courses in computational neuroscience, theoretical biology, biophysics, or neural networks. The approach will suit students of physics, mathematics, or computer science; it will also be useful for biologists who are interested in mathematical modelling. The text is enhanced by many worked examples and illustrations. There are no mathematical prerequisites beyond what the audience would meet as undergraduates: more advanced techniques are introduced in an elementary, concrete fashion when needed. |

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### Contents

Preface | xi |

Acknowledgments | xiv |

Introduction | 1 |

1 1 1 The ideal spiking neuron | 2 |

1 12 Spike trains | 3 |

113 Synapses | 4 |

121 Postsynaptic potentials | 6 |

13 A phenomenological neuron model | 7 |

Population equations | 203 |

61 Fully connected homogeneous network | 204 |

62 Density equations | 207 |

622 Spike Response Model neurons with escape noise | 214 |

623 Relation between the approaches | 218 |

63 Integral equations for the population activity | 222 |

631 Assumptions | 223 |

64 Asynchronous firing | 231 |

132 Limitations of the model | 9 |

14 The problem of neuronal coding | 13 |

15 Rate codes | 15 |

152 Rate as a spike density average over several runs | 17 |

153 Rate as a population activity average over several neurons | 18 |

16 Spike codes | 20 |

162 Phase | 21 |

163 Correlations and synchrony | 22 |

164 Stimulus reconstruction and reverse correlation | 23 |

spikes or rates? | 25 |

18 Summary | 27 |

Single neuron models | 29 |

2 Detailed neuron models | 31 |

212 Reversal potential | 33 |

22 HodgkinHuxley model | 34 |

222 Dynamics | 37 |

23 The zoo of ion channels | 41 |

232 Potassium channels | 43 |

233 Lowthreshold calcium current | 45 |

234 Highthreshold calcium current and calciumactivated potassium | 47 |

235 Calcium dynamics | 50 |

24 Synapses | 51 |

242 Excitatory synapses | 52 |

the dendritic tree | 53 |

251 Derivation of the cable equation | 54 |

252 Greens function | 57 |

253 Nonlinear extensions to the cable equation | 60 |

26 Compartmental models | 61 |

27 Summary | 66 |

Twodimensional neuron models | 69 |

311 General approach | 70 |

312 Mathematical steps | 72 |

32 Phase plane analysis | 74 |

322 Stability of fixed points | 75 |

323 Limit cycles | 77 |

324 Type I and type II models | 80 |

33 Threshold and excitability | 82 |

331 Type I models | 84 |

332 Type II models | 85 |

333 Separation of time scales | 86 |

34 Summary | 90 |

Formal spiking neuron models | 93 |

411 Leaky integrateandfire model | 94 |

412 Nonlinear integrateandfire model | 97 |

413 Stimulation by synaptic currents | 100 |

42 Spike Response Model SRM | 102 |

422 Mapping the integrateandfire model to the SRM | 108 |

423 Simplified model SRMₒ | 111 |

43 From detailed models to formal spiking neurons | 116 |

431 Reduction of the HodgkinHuxley model | 117 |

432 Reduction of a cortical neuron model | 123 |

433 Limitations | 131 |

44 Multicompartment integrateandfire model | 133 |

442 Relation to the model SRMₒ | 135 |

443 Relation to the full Spike Response Model | 137 |

coding by spikes | 139 |

46 Summary | 145 |

Noise in spiking neuron models | 147 |

51 Spike train variability | 148 |

572 Noise sources | 149 |

52 Statistics of spike trains | 150 |

52 Inputdependent renewal systems | 151 |

522 Interval distribution | 152 |

523 Survivor function and hazard | 153 |

524 Stationary renewal theory and experiments | 158 |

525 Autocorrelation of a stationary renewal process | 160 |

53 Escape noise | 163 |

531 Escape rate and hazard function | 164 |

532 Interval distribution and mean firing rate | 168 |

54 Slow noise in the parameters | 172 |

55 Diffusive noise | 174 |

552 Diffusion limit | 178 |

553 Interval distribution | 182 |

56 The subthreshold regime | 184 |

561 Sub and superthreshold stimulation | 185 |

562 Coefficient of variation C𝘷 | 187 |

57 From diffusive noise to escape noise | 188 |

58 Stochastic resonance | 191 |

59 Stochastic firing and rate models | 194 |

592 Stochastic rate model | 196 |

593 Population rate model | 197 |

510 Summary | 198 |

Population models | 201 |

642 Gain function and fixed points of the activity | 233 |

643 Lowconnectivity networks | 235 |

65 Interacting populations and continuum models | 240 |

652 Spatial continuum limit | 242 |

66 Limitations | 245 |

67 Summary | 246 |

7 Signal transmission and neuronal coding | 249 |

71 Linearized population equation | 250 |

777 Noisefree population dynamics | 252 |

712 Escape noise | 256 |

713 Noisy reset | 260 |

72 Transients | 261 |

721 Transients in a noisefree network | 262 |

722 Transients with noise | 264 |

73 Transfer function | 268 |

732 Signaltonoise ratio | 273 |

741 The effect of an input spike | 274 |

742 Reverse correlation the significance of an output spike | 278 |

75 Summary | 282 |

Oscillations and synchrony | 285 |

81 Instability of the asynchronous state | 286 |

82 Synchronized oscillations and locking | 292 |

822 Locking in SRMₒ neurons with noisy reset | 298 |

823 Cluster states | 300 |

83 Oscillations in reverberating loops | 302 |

831 From oscillations with spiking neurons to binary neurons | 305 |

832 Mean field dynamics | 306 |

833 Microscopic dynamics | 309 |

84 Summary | 313 |

Spatially structured networks | 315 |

91 Stationary patterns of neuronal activity | 316 |

911 Homogeneous solutions | 318 |

912 Stability of homogeneous states | 319 |

inhomogeneous states | 324 |

92 Dynamic patterns of neuronal activity | 329 |

921 Oscillations | 330 |

922 Traveling waves | 332 |

93 Patterns of spike activity | 334 |

931 Traveling fronts and waves | 337 |

932 Stability | 338 |

94 Robust transmission of temporal information | 341 |

95 Summary | 348 |

Models of synaptic plasticity | 349 |

Hebbian models | 351 |

1011 Longterm potentiation | 352 |

1012 Temporal aspects | 354 |

102 Ratebased Hebbian learning | 356 |

103 Spiketimedependent plasticity | 362 |

1032 Consolidation of synaptic efficacies | 365 |

1033 General framework | 367 |

104 Detailed models of synaptic plasticity | 370 |

1041 A simple mechanistic model | 371 |

1042 A kinetic model based on NMDA receptors | 374 |

1043 A calciumbased model | 377 |

105 Summary | 383 |

Learning equations | 387 |

1112 Evolution of synaptic weights | 389 |

1113 Weight normalization | 394 |

1114 Receptive field development | 398 |

112 Learning in spiking models | 403 |

1121 Learning equation | 404 |

1122 Spikespike correlations | 406 |

1123 Relation of spikebased to ratebased learning | 409 |

1124 Staticpattern scenario | 411 |

1125 Distribution of synaptic weights | 415 |

113 Summary | 418 |

Plasticity and coding | 421 |

122 Learning to be precise | 425 |

1222 Firing time distribution | 427 |

1223 Stationary synoptic weights | 428 |

1224 The role of the firing threshold | 430 |

123 Sequence learning | 432 |

124 Subtraction of expectations | 437 |

1242 Sensory image cancellation | 439 |

125 Transmission of temporal codes | 441 |

1251 Auditory pathway and sound source localization | 442 |

1252 Phase locking and coincidence detection | 444 |

7253 Tuning of delay lines | 447 |

126 Summary | 452 |

455 | |

477 | |

### Common terms and phrases

Aabs absolute refractoriness action potential amplitude asynchronous firing calcium cell coding correlation current pulse dashed line dendritic density differential equation diffusive noise dynamics escape noise escape rate example excitatory firing threshold fixed point gain function Gerstner Hebbian learning Hodgkin-Huxley model inhibitory input current input potential input spike integral integrate-and-fire model integrate-and-fire neurons interspike interval interval distribution ion channels kernel learning rule learning window linear mean firing rate membrane potential neuron fires neuron models oscillation output spike parameters pattern phase Poisson process population activity population equation postsynaptic firing postsynaptic neuron postsynaptic potential postsynaptic spike presynaptic neurons presynaptic spike arrival receptive fields renewal theory reset resting potential reversal potential right-hand side Section side of Eq solid line solution Spike Response Model spike train spiking neuron models SRM0 SRMo neurons stable stationary stimulus synaptic efficacy synaptic plasticity synaptic weights temporal trajectory triggered variable voltage weight change

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