A general framework for stochastic traveling waves and patterns, with application to neural field equations

James Inglis 1, 2 James Maclaurin 1
1 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
2 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : In this paper we present a general framework in which to rigorously study the effect of spatio-temporal noise on traveling waves and stationary patterns. In particular the framework can incorporate versions of the stochastic neural field equation that may exhibit traveling fronts, pulses or stationary patterns. To do this, we first formulate a local SDE that describes the position of the stochastic wave up until a discontinuity time, at which point the position of the wave may jump. We then study the local stability of this stochastic front, obtaining a result that recovers a well-known deterministic result in the small-noise limit. We finish with a study of the long-time behavior of the stochastic wave.
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Submitted on : Tuesday, June 30, 2015 - 9:40:50 AM
Last modification on : Friday, April 19, 2019 - 1:26:04 PM

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James Inglis, James Maclaurin. A general framework for stochastic traveling waves and patterns, with application to neural field equations. SIAM Journal on Applied Dynamical Systems, Society for Industrial and Applied Mathematics, 2016, 15 (1), pp.195-234. ⟨10.1137/15M102856X⟩. ⟨hal-01169697⟩

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