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arXiv · 1807.02575

A gradient flow formulation for the stochastic Amari neural field model

Christian Kuehn·Jonas M. Tölle
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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.

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BibTeXRIS

Christian Kuehn, Jonas M. Tölle. 2018-07-06. A gradient flow formulation for the stochastic Amari neural field model. https://doi.org/10.1007/s00285-019-01393-w

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