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A gradient flow formulation for the stochastic Amari neural field model. (arXiv:1807.02575v1 [math.AP])
来源于:arXiv
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 and ergodic properties for the associated Feller
semigroups are also discussed. 查看全文>>