Wave phenomena in neuronal networks

Abstract

We study traveling wave solutions of a system of integro-differential equations which describe the activity of large-scale networks of excitatory neurons on spatially extended domains. The independent variables are the activity level, u, of a population of excitatory neurons which have long-range connections and a recovery variable, v. There is a critical value of the parameter β (β* > 0) that appears in the equation for v, at which the eigenvalues of the linearization of the system around the rest state (u, v)=(0,0) change from real to complex. In contrast to previous studies which analyzed properties of traveling waves when the eigenvalues are real, we examine the range β > β* >, where the eigenvalues are complex. In this case, our numerical experiments indicate that there is a range of parameters over which families of wave fronts and solitary and multi-bump waves can coexist as stable solutions. In two-space dimensions, we show how single-bump, double-bump and multi-ring waves form in response to a Gaussian-shaped stimulus. We also show how a stable, one-armed rotating spiral wave can form and fill the entire domain. All of these phenomena can be initiated at any point in the medium, as they are not driven by an underlying time-dependent periodic pacemaker, and they do not depend on the presence of a persistent external input. © 2008 Springer-Verlag Berlin Heidelberg.