Neural firing rate model with a steep firing rate function

Abstract

In this study we justify rigorously the approximation of the steep firing rate functions with a unit step function in a two-population neural firing rate model with steep firing rate functions. We do this justification by exploiting the theory of switching dynamical systems. It has been demonstrated that switching dynamics offer a possibility of simplifying the dynamical system and getting approximations of the solution of the system for any specific choice of parameters. In this approach the phase space of the system is divided into regular and singular domains, where the limit dynamics can be written down explicitly. The advantages of this method are illustrated by number of numerical examples for different cases of the singular domains (i.e. for black, white and transparent walls) and for specific choices of parameters involved. General conditions have been formulated on these parameters to give black, white and transparent walls. Further, the existence and stability of regular and singular stationary points have been investigated. It has been shown that the regular stationary points (i.e. stationary points inside the regular domains) are always stable and this property is preserved for smooth and sufficiently steep activation functions. In the most technical part of the paper we have provided conditions on the existence and stability of singular stationary points (i.e. those belonging to the singular domains). For the * Corresponding Author: Tel. +47 64965489; fax.+47 64965401.Email existence results, the implicit function theorem has been used, whereas the stability of singular stationary points is addressed by applying the singular perturbation analysis and the Tikhonov theorem.

Effect of localized input on bump solutions in a two-population neural-field model

“…While this conjecture is supported by numerical simulations (see for example [25]), there are few and far between the works addressing this problem in a rigorous mathematical way. We believe, however, that this problem can be dealt with methods of nonlinear functional analysis and degree theory in a way analogous to Oleynik et al [36] and singular perturbation analysis in a similar way as in Yousaf et al [37]. We do not pursue this problem here, however, but will return to it in future investigations.…”

A one-population Amari model with periodic microstructure

“…We will also compare the outcome of numerical simulations of the homogenized model ( 5) with the heterogeneous model (4). Finally, but not least we will like to address the problem of justification of the unit step approximation of firing rate functions in the homogenized model using nonlinear functional analysis and degree theory in a way similar to Oleynik et al [31] and singular perturbation analysis as in Yousaf et al [32]. Since the firing rate functions are assumed to be convex functions in the derivation of homogenized problem, there is also a need to justify the usage of non-convex functions such as sigmoidal-shaped firing rate functions.…”

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

“…Piecewise-linear systems of differential equations play an important role in many applications including hybrid dynamical systems [5], neural networks [16], genetic networks [4] and many others. Typically, such a system can be represented as a family of linear differential systems ·x = A α x + f α where the system no.…”

“…In some models (see e.g. [16]) the individual linear systems are not diagonal, but could be simultaneously transformed to a diagonal form.…”