A gradient flow formulation for the stochastic Amari neural field model

Christian Kuehn, Jonas M. Tölle

Research output: Contribution to journalArticleScientificpeer-review


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.
Original languageEnglish
Pages (from-to)1227-1252
Number of pages26
JournalJournal of Mathematical Biology
Issue number4
Publication statusPublished - 2019
MoE publication typeA1 Journal article-refereed


  • J
  • 111 Mathematics


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