Collective Dynamics in Networks of Pulse-Coupled Oscillators

Abstract

Pulse-coupled oscillators constitute a paradigmatic class of dynamical systems interacting on networks. They model a variety of natural systems including earthquakes, flashing fireflies and chirping crickets as well as pacemaker cells of the heart and neural networks. In this thesis, we consider two main topics which arise in the theory of networks of pulse-coupled oscillators. These are motivated by biological networks of neurons: The first topic is the presence of interaction delays, the second the complexity of the network structure. The thesis is organized as follows. After an introduction to the subject, we briefly present the class of models of pulse-coupled oscillators that is studied throughout this thesis and discuss its advantages for analytical as well as numerical studies. We then analyze networks of oscillators with delayed, global pulse-coupling. Such a network constitutes the first example of a dynamical system that naturally exhibits periodic orbits that are simultaneously attracting and unstable. They are enclosed by basins of attraction of other attractors such that arbitrarily small noise leads to a switching among attractors. For a wide range of parameters, these unstable attractors become prevalent with increasing network size. Next, we consider networks exhibiting a complex structure. We first present an exact stability analysis for synchronous states in networks of arbitrary connectivity. As opposed to conventional stability analysis, here stability is determined by a multitude of linear operators. We treat this multioperator problem exactly and show that for inhibitory coupling the synchronous state is stable, independent of the parameters and the network connectivity. Furthermore, we present the first theoretical demonstration that, in randomly connected networks with strong interactions this synchronous state, displaying regular dynamics, coexists with a balanced state exhibiting irregular dynamics. We suggest simple mechanisms for switching between these qualitatively different states. The question arises whether the stability results are valid for a more general class of models that are obtained from the original by a structural perturbation of the dynamics. Finally, we analyze this question in detail identifying a subclass of models: Within this subclass, all multiple operators are degenerate and we are thus confronted with a standard, one operator stability problem. We present numerical evidence that random matrix theory provides useful estimates to the stability problem of synchronization on random networks. These estimates suggest that the stable synchronous state found for inhibitory coupling is robust against structural perturbations of the original model dynamics.