Wandering bumps in a stochastic neural field: a variational approach
type: Item
instance of: scholarly article
title: Wandering bumps in a stochastic neural field: a variational approach
zbMATH Open document ID: 1493.92005
published in: Physica D
publication date: 2022-03-17T00:00:00Z
review text: The aim in this article is to develop a generalized variational method for analyzing wandering bumps in a stochastic ring attractor model. The second section is devoted to a short review of results on stationary bumps in the deterministic ring model, following mainly the presentation in the paper of \textit{Z. P. Kilpatrick} and \textit{B. Ermentrout} [SIAM J. Appl. Dyn. Syst. 12, No. 1, 61--94 (2013; Zbl 1278.92007)]. In the third section one introduces the stochastic version of the ring model and develops the variational method. In a compact form the considered model writes \[ du_t=[-u_t+\mathcal{F}(u_t)+\theta h]dt + \sqrt{\epsilon}\Phi(u_t)dW_t, t>0 \] where \[ \mathcal{F}(u_t)(\theta)=\int\limits_{-\pi}^{\pi}J(\theta - {\theta}') f(u_t(\theta'))d\theta' \] with notations $u(.,t)=u_t$, $dW_t=dW(\theta,t)$, $u(\theta,t)$ denoting the stochastic activity at time $t$ of a local population of cells with direction of preference $\theta$ and $W(\theta,t)$ a $Q$-Wiener process. One decomposes the solution $u_t$ into two components, a phase-shifted deterministic bump solution and an error term. \[ u_t =U_{\beta_{t}}+\sqrt{\epsilon}v_t. \] For the phase shift $\beta_t$ one obtains an exact stochastic differential equation. In the fourth section one derives an explicit stochastic differential equation for the phase in the weak noise limit. Finally in the fifth section, rigorous bounds on the error term are shown. A short part ``Discussion'' and forty References end the article.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 7492622
zbMATH Keywords: stochastic neural fields
zbMATH Keywords: stationary bumps
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: spatially extended noise
zbMATH Keywords: neural variability
zbMATH Keywords: variational principle
author: James Maclaurin
author: Paul C. Bressloff
MaRDI profile type: Publication
full work available at URL: j.physd.2020.132403
OpenAlex ID: W3006543920
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DOI: 10.1016/J.PHYSD.2020.132403
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description: scientific article; zbMATH DE number 7492622