Stability of the stationary solutions of neural field equations with propagation delays

Neural fields equations first appeared as a spatial-continuous extension of Hopfield networks with the seminal works of Wilson and Cowan, Amari [1,2]. These networks describe the mean activity of neural populations by nonlinear integral equations and play an important role in the modeling of various cortical areas including the visual cortex. They have been modified to take into account several relevant biological mechanisms like spike-frequency adaptation [3,4], the tuning properties of some populations [5] or the spatial organization of the populations of neurons [6]. In this work we focus on the role of the delays coming from the finite-velocity of signals in axons, dendrites or the time of synaptic transmission [7,8]. It turns out that delayed neural fields equations feature some interesting mathematical difficulties. The main question we address in the sequel is that of determining, once the stationary states of a non-delayed neural field equation are well-understood, what changes, if any, are caused by the introduction of propagation delays? We think this question is important since non-delayed neural field equations are pretty well understood by now, at least in terms of their stationary solutions, but the same is not true for their delayed versions which in many cases are better models closer to experimental findings. A lot of work has been done concerning the role of delays in waves propagation or in the linear stability of stationary states but except in [9] the method used reduces to the computation of the eigenvalues (which we call characteristic values) of the linearized equation in some analytically convenient cases (see [10]). Some results are known in the case of a finite number of neurons [11,12] and in the case of a few number of distinct delays [13,14]: the dynamical portrait is highly intricated even in the case of two neurons with delayed connections.The purpose of this article is to propose a solid mathematical framework to characterize the dyn