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Analysis and approximation of stochastic nerve axon equations

Authors: Martin Sauer and Wilhelm Stannat
Journal: Math. Comp. 85 (2016), 2457-2481
MSC (2010): Primary 60H15, 60H35; Secondary 35R60, 65C30, 92C20
Published electronically: January 8, 2016
MathSciNet review: 3511288
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Abstract: 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 non-linearities together with a certain physiologically reasonable monotonicity to derive crucial $ L^\infty $-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 $ \sqrt {1/n}-$ and a strong convergence rate of $ 1/n$ in special cases. As applications, the Hodgkin-Huxley and FitzHugh-Nagumo systems with noise are considered.

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Additional Information

Martin Sauer
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany

Wilhelm Stannat
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany and Bernstein Center for Computational Neuroscience, Philippstr. 13, D-10115 Berlin, Germany

Keywords: Stochastic reaction diffusion equations, finite difference approximation, Hodgkin-Huxley equations, FitzHugh-Nagumo equations, conductance based neuronal models
Received by editor(s): February 19, 2014
Received by editor(s) in revised form: April 13, 2015
Published electronically: January 8, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society