Simulation of SPDEs for Excitable Media Using Finite Elements

Abstract

In this paper, we address the question of the discretization of Stochastic Partial Differential Equations (SPDE's) for excitable media. Working with SPDE's driven by colored noise, we consider a numerical scheme based on finite differences in time (EulerMaruyama) and finite elements in space. Motivated by biological considerations, we study numerically the emergence of reentrant patterns in excitable systems such as the Barkley or Mitchell-Schaeffer models.

Numerical solutions of some stochastic hyperbolic wave equations including sine-Gordon equation

“…There have been many recent articles involving the applications of stochastic partial differential equations. These include parabolic equations which have enjoyed widespread attention in neurobiological applications (Boulakia et al, 2014;Dörsek et al, 2103;Faugeras and MacLaurin, 2014;Khoshnevisan and Kim, 2015;Petterson et al, 2014;Stannat, 2013;Tuckwell, 2013aTuckwell, , 2013b and to a lesser extent hyperbolic equations (Hajek, 1982). These applications have often employed two-parameter Wiener processes, or Brownian motion, {W (x, t), x ∈ X, t ∈ T }, with mean zero and covariance Cov[W (x, s), W (y, t)] = min(x, y) min(s, t) where X and T are sub-intervals of R or R + (or their formal derivatives, space-time white noise {w(x, t)}).…”