2014/02/19 by Martin Sauer, Wilhelm Stannat · 1 citation
Mathematics · #math.PR #msc:60H15 #msc:60H35 #msc:35R60 #msc:65C30 #msc:92C20
paper · pdf · doi:10.1090/mcom/3068
published as Math. Comp. 85 (2016), no. 301, 2457-2481 · 25 pages
arxiv created 2014/02/19 · arxiv updated 2020/01/16
We consider spatially extended conductance based neuronal models with noise described by a stochastic reaction diffusion equation with additive noise coupled to a control variable with multiplicative noise but no diffusion. We only assume a local Lipschitz condition on the nonlinearities together with a certain physiologically reasonable monotonicity to derive crucial L^∞-bounds for the solution. These play an essential role in both the proof of existence and uniqueness of solutions as well as the error analysis of the finite difference approximation in space. We derive explicit error estimates, in particular a pathwise convergence rate of √((1)/(n))- and a strong convergence rate of \frac1n in special cases. As applications, the Hodgkin-Huxley and FitzHugh-Nagumo systems with noise are considered.