Editor’s summary: This book is a collection of articles containing known theorems and proofs on the dynamical behaviour of deterministic pulse-coupled neuron networks. The mathematical models assume pulse-coupled systems (neuronal networks) of individual simple dynamical units (neurons) governed by impulsive differential equations. The main purpose of most articles of the book is to study the asymptotic dynamics of such global systems. The proofs of the theorems are rigourous analytic. Each article is self-contained and occupies a different chapter of the book. In most theorems, the (finite) number \(N\) of neurons of the network is as large as wanted, and the neurons may be mutually different. For instance, for excitatory networks (positive coupling), the authors provide mathematical sufficient conditions for the full synchronization of the network, discussing on different structures of the graph of interactions (Chapter 3). For inhibitory networks (negative coupling), only networks with a complete graph of interactions are studied. Nevertheless, rather general results are obtained: for instance in Chapter 4, the authors prove that the return map to a Poincaré section of an inhibitory neuron network is piecewise contractive in a (\(N-1\))-dimensional compact space. Finally, in Chapter 5, the author studies networks with complete graphs of interactions, but with any sign of them. He proves, under additional assumptions, that the attractor is composed by a finite number of limit cycles.
The articles of this volume will not be indexed individually.