Existence and properties of solutions for neural field equations

Abstract

The first goal of this work is to study solvability of the neural field equationwhich is an integro-differential equation in m+1 dimensions. In particular, we show the existence of global solutions for smooth activation functions f with values in [0, 1] and L 1 kernels w via the Banach fixpoint theorem. For a Heaviside type activation function f we show that the above approach fails. However, with slightly more regularity on the kernel function w (we use Hölder continuity with respect to the argument x) we can employ compactness arguments, integral equation techniques and the results for smooth nonlinearity functions to obtain a global existence result in a weaker space.Finally, general estimates on the speed and durability of waves are derived. We show that compactly supported waves with directed kernels (i.e. w(x, y) ≤ 0 for x ≤ y) decay exponentially after a finite time and that the field has a well defined finite speed.

Homogenization of a Wilson–Cowan model for neural fields

“…Thus by the Banach fixed point theorem, there exists a unique local solution u ε ∈ X to (3.1). Next, arguing exactly as in the proof of Theorem 2.7 in [19], we get the global existence of the solution to (3.1). Now, we need to check that estimate (3.3) holds true.…”

On the properties of nonlinear nonlocal operators arising in neural field models

“…[6,7]. These methods have been applied to the particular type of neural field model by various authors; see [8][9][10][11]. Dealing with the unit step function however leads to the discontinuity in the integral operator involved in (1.1), which makes it impossible to apply the classical theory.…”

“…While this conjecture is supported by numerical simulations (see for example [13]) there are few and far between works addressing this problem in a rigorous mathematical way. Namely, Potthast and Beim Graben provided a rigorous approach to study global existence of solutions to the Wilson-Cowan type of the model with the smooth firing rate function as well as with the unit step function, [11]. They demonstrated that the latter case requires more restrictions on the choice of a functional space as well as some extra assumptions on ω.…”

Amari Model

“…Most of the above work uses homogeneous kernels w(x À y). More recent work that tackles heterogeneity (primarily using simulations) can be found in Brackley and Turner (2007), Bressloff (2001), Schmidt et al (2009), andCoombes et al (2012) and functional analytic results in Faugeras et al (2008), and Potthast and beim Graben (2010). The inverse problems perspective for either homogeneous or nonhomogeneous kernels has been investigated by Potthast and beim Graben (2009) and beim .…”