Spiking Neuron Models Single Neurons, Populations, Plasticity

A catalogue record of this book is available from the British Library ISBN 0 521 81384 0 hardback ISBN 0 521 89079 9 paperback Contents Preface page xi Acknowledgments xiv 1 Introduction 1 1.1 Elements of neuronal systems 1 1.1.1 The ideal spiking neuron 2 1.1.2 Spike trains 3 1.1.3 Synapses 4 1.2 Elements of neuronal dynamics 4 1.2.1 Postsynaptic potentials 6 1.2.2 Firing threshold and action potential 6 1.3 A phenomenological neuron model 7 1.3.1 Definition of the model SRM 0 7 1.3.2 Limitations of the model 9 1.4 The problem of neuronal coding 13 1.5 Rate codes 15 1.5.1 Rate as a spike count (average over time) 15 1.5.2 Rate as a spike density (average over several runs) 17 1.5.3 Rate as a population activity (average over several neurons) 18 1.6 Spike codes 20 1.6.1 Time-to-first-spike 20 1.6.2 Phase 21 1.6.3 Correlations and synchrony 22 1.6.4 Stimulus reconstruction and reverse correlation 23 1.7 Discussion: spikes or rates? 25 1.8 Summary 27 Part one: Single neuron models 29 v vi Contents 2 Detailed neuron models 31 2.1 Equilibrium potential 31 2.1.1 Nernst potential 31 2.1.2 Reversal potential 33 2.2 Hodgkin–Huxley model 34 2.2.1 Definition of the model 34 2.2.2 Dynamics 37 2.3 The zoo of ion channels 41 2.3.1 Sodium channels 41 2.3.2 Potassium channels 43 2.3.3 Low-threshold calcium current 45 2.3.4 High-threshold calcium current and calcium-activated potassium channels 47 2.3.5 Calcium dynamics 50 2.4 Synapses 51 2.4.1 Inhibitory synapses 51 2.4.2 Excitatory synapses 52 2.5 Spatial structure: the dendritic tree 53 2.5.1 Derivation of the cable equation 54 2.5.2 Green's function (*) 57 2.5.3 Nonlinear extensions to the cable equation 60 2.6 Compartmental models 61 2.7 Summary 66 3 Two-dimensional neuron models 69 3.1 Reduction to two dimensions 69 3. Contents vii 4 Formal spiking neuron models 93 4.1 Integrate-and-fire model 93 4.1.1 Leaky integrate-and-fire model 94 4.1.2 Nonlinear integrate-and-fire model 97 4.1.3 Stimulation by synaptic currents 100 4.2 Spike Response Model (SRM) 102 4.2.1 Definition of the SRM 102 4.2.2 Mapping the integrate-and-fire model to the SRM 108 4.2.3 Simplified model SRM 0 111 4.3 From detailed models to formal spiking neurons 116 4.3.1 Reduction of the Hodgkin–Huxley model 117 4.3.2 Reduction of a cortical neuron model 123 4.3.3 Limitations 131 4.4 Multicompartment integrate-and-fire model 133 4.4.1 Definition of the model 133 4.4.2 Relation to the model SRM 0 135 4.4.3 Relation to the full Spike Response …