A gradient flow formulation for the stochastic Amari neural field model

  • Christian KuehnEmail author
  • Jonas M. Tölle


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.


Gradient flow in nonlocal Hilbert space Stochastic Amari neural field equation Spatially correlated additive noise Space-time regularity of solutions Nonnegative kernel Unique invariant measure of the ergodic Feller semigroup 

Mathematics Subject Classification

Primary 60H15 92C20 Secondary 34B10 35B65 



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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Research Unit “Multiscale and Stochastic Dynamics”, Faculty of MathematicsTechnical University of MunichGarching bei MünchenGermany
  2. 2.Institut für MathematikUniversität AugsburgAugsburgGermany

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