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Section: New Results

Neural fields theory

ERRATUM: A Center Manifold Result for Delayed Neural Fields Equations

Participants : Romain Veltz, Olivier Faugeras.

Lemma C.1 in [95] is wrong. This lemma is used in the proof of the existence of a smooth center manifold, Theorem 4.4 in that paper. An additional assumption is required to prove this existence. We spell out this assumption, correct the proofs, and show that the assumption is satisfied for a large class of delay functions τ. We also weaken the general assumptions on τ.

This work has been published in SIAM J. Math. Anal. and is available as [28] .

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

Participants : James Inglis, James Maclaurin.

In this work 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.

This work has appeared in SIAM J. on Applied Dynamical Systems (SIADS) [49] .