2018/07/31 by Christian Kuehn, Jonas M. Tölle · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Advanced Thermodynamics and Statistical Mechanics #Applied mathematics #Balanced flow #Ergodic theory #Field (mathematics) #Flow (mathematics) #Geometry #Hilbert space #Invariant (physics) #Mathematical Biology Tumor Growth #Mathematical analysis #Mathematical physics #Mathematics #Pure mathematics #Uniqueness #math.AP #math.DS #math.FA #math.PR #msc:34B10 #msc:35B65 #msc:60H15 #msc:92C20 #q-bio.NC
paper · pdf · doi:10.1007/s00285-019-01393-w
published as J. Math. Biol. 79 (2019), no. 4, 1227--1252 · 21 pages, 1 figure, 75 references
arxiv created 2018/11/26 · openalex publication_date 2019/06/18 · arxiv updated 2019/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.