Wandering bumps in a stochastic neural field: a variational approach
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.
scientific article; zbMATH DE number 7492622
scientific article