We wrote this book as an introduction to spiking neuron models for advanced undergraduate or graduate students. It can be used either as the main text for a course that focuses on neuronal dynamics; or as part of a larger course in Computational Neuroscience, theoretical biology, neuronal modeling, biophysics, or neural networks. For a one-semester course on neuronal modeling, we usually teach one chapter per week focusing on the first sections of each chapter for lectures and give the remainder as reading assignment. Many of the examples can be reformulated as exercises. While writing the book we had in mind students of physics, mathematics, or computer science with an interest in biology; but it might also be useful for students of biology who are interested in mathematical modeling. All the necessary mathematical concepts are introduced in an elementary fashion. No prior knowledge beyond undergraduate mathematics should be necessary to read the book. An asterisk (*) marks those sections that have a more mathematical focus. These sections can be skipped at a first reading.

We have also tried to keep the book self-contained with respect to the underlying Neurobiology. The fundamentals of neuronal excitation and synaptic signal transmission are briefly introduced in Chapter 1 together with an outlook on the principal topics of the book, viz., formal spiking neuron models and the problem of neuronal coding. In Chapter 2 we review biophysical models of neuronal dynamics such as the Hodgkin-Huxley model and models of dendritic integration based on the cable equation. These models are the starting point for a systematic reduction to neuron models with a reduced complexity that are open to an analytical treatment. Whereas Chapter 3 is dedicated to two-dimensional differential equations as a description of neuronal dynamics, Chapter 4 introduces formal spiking neuron models, namely the integrate-and-fire model and the Spike Response Model. These formal neuron models are the foundation for all the following chapters. Part I on ``Single Neuron Models'' is completed by Chapter 5 which gives an overview of spike-train statistics and illustrates how noise can be implemented in spiking neuron models.

The step from single neuron models to networks of neurons is taken in Chapter 6 where equations for the macroscopic dynamics of large populations of neurons are derived. Based on these equations phenomena like signal transmission and coding (Chapter 7), oscillations and synchrony (Chapter 8), and pattern formation in spatially structured networks (Chapter 9) are investigated. So far, only networks with a fixed synaptic connectivity have been discussed. The third part of the book, finally, deals with synaptic plasticity and its role for development, learning, and memory. In Chapter 10, principles of Hebbian plasticity are presented and various models of synaptic plasticity are described that are more or less directly inspired by neurbiological findings. Equations that relate the synaptic weight dynamics to statistical properties of the neuronal spike activity are derived in Chapter 11. Last but not least, Chapter 12 presents an -- admittedly personal -- choice of illustrative applications of spike-timing dependent synaptic plasticity to fundamental problems of neuronal coding.

While the book contains material which is now considered as standard for courses in Computational Neuroscience, neuronal modeling, or neural networks, it also provides a bridge to current research which has developed over the last few years. In most chapters, the reader will find some sections which either report recent results or shed new light on well-known models. The viewpoint taken in the presentation of the material is of course highly subjective and a bias towards our own research is obvious. Nevertheless, we hope that the book will find the interest of students and researchers in the field.

Werner M. Kistler and W. Gerstner

Lausanne, November 2001