From invasion to extinction in heterogeneous neural fields

In this paper, we analyze the invasion and extinction of activity in heterogeneous neural fields. We first consider the effects of spatial heterogeneities on the propagation of an invasive activity front. In contrast to previous studies of front propagation in neural media, we assume that the front propagates into an unstable rather than a metastable zero-activity state. For sufficiently localized initial conditions, the asymptotic velocity of the resulting pulled front is given by the linear spreading velocity, which is determined by linearizing about the unstable state within the leading edge of the front. One of the characteristic features of these so-called pulled fronts is their sensitivity to perturbations inside the leading edge. This means that standard perturbation methods for studying the effects of spatial heterogeneities or external noise fluctuations break down. We show how to extend a partial differential equation method for analyzing pulled fronts in slowly modulated environments to the case of neural fields with slowly modulated synaptic weights. The basic idea is to rescale space and time so that the front becomes a sharp interface whose location can be determined by solving a corresponding local Hamilton-Jacobi equation. We use steepest descents to derive the Hamilton-Jacobi equation from the original nonlocal neural field equation. In the case of weak synaptic heterogenities, we then use perturbation theory to solve the corresponding Hamilton equations and thus determine the time-dependent wave speed. In the second part of the paper, we investigate how time-dependent heterogenities in the form of extrinsic multiplicative noise can induce rare noise-driven transitions to the zero-activity state, which now acts as an absorbing state signaling the extinction of all activity. In this case, the most probable path to extinction can be obtained by solving the classical equations of motion that dominate a path integral representation of the stochastic neural field in the weak noise limit. These equations take the form of nonlocal Hamilton equations in an infinite-dimensional phase space.