Interface dynamics in planar neural field models

Abstract

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.

Tutorial on Neural Field Theory

“…Hence, the travelling wave front for this example is neutrally stable. For a recent extension of the Amari interface dynamics to planar neural field models we refer the reader to [24].…”

“…More explicit progress has been possible for the case of Heaviside firing rate functions, especially as regards the stability of solutions using Evans functions [22]. The extension of results from one to two spatial dimensions has increased greatly in recent years [24,37,56,60,61,70,85] (see Chap. 7).…”

“…More recent work that tackles heterogeneity (primarily using simulations) can be found in [11] (also in Chap. 8), whilst perturbation theory and homogenisation techniques are developed in [13,24,80], and functional analytic results in [36]. The treatment of stochastic neural field models is a very new area, and we refer the reader to the recent review by Bressloff [14] and to Chaps.…”

Spatiotemporal canards in neural field equations

“…The interfacial description [30] applies in the case f (u) = Θ(u), where Θ(u) is the Heaviside step function. As customary, we consider localized regions of activity −ξ(t) ≤ x ≤ ξ(t), where the interfaces (or threshold crossings) x = ±ξ(t) are defined by the level set conditions u(±ξ(t), t) = h(t) with ∂ x u(±ξ(t), t) ≶ 0, for all t ∈ R + , and take their width 2ξ(t) as a measure of the spatial extent of the solution (see for instance Fig.…”

Spatiotemporal Pattern Formation in Neural Fields with Linear Adaptation

“…As is usually found in lateral inhibitory neural fields, the wide bump (C) is stable and the narrow bump ( ) is unstable in the limit of vanishing adaptation (ˇ! 0) [1,4,12,40]. At a criticalˇ, the wide bump undergoes a drift instability leading to a traveling bump.…”