A gradient flow formulation for the stochastic Amari neural field model

Abstract

We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed. Date: November 27, 2018. 2010 Mathematics Subject Classification. Primary: 60H15; 92C20; Secondary: 34B10; 35B65; Key words and phrases. Gradient flow in Hilbert space; stochastic Amari neural field equation; nonlocal Hilbert space; spatially correlated additive noise; space-time regularity of solutions; nonnegative kernel; unique invariant measure; ergodic Feller semigroup; mathematical neuroscience. CK acknowledges financial support by a Lichtenberg Professorship. JMT would like to thank Dirk Blömker for some useful comments. Both authors are indebted to the anonymous reviewers for several helpful remarks.

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