Ergodicity for stochastic neural field equations

2025/05/20 by Anna-Mariya Otsetova, Otsetova, Anna-Mariya, Jonas M. Tölle +1 · 1 citation
Neuroscience · Physics and Astronomy · #35K58 #35R60 #37A25 #37L40 #45K05 #47D07 #60H20 #92B20 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Neural dynamics and brain function #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2505.14012

openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the well-posedness and long-time behavior of a general continuum neural field model with Gaussian noise on possibly unbounded domains. In particular, we give conditions for the existence of invariant probability measures by restricting the solution flow to an invariant subspace with a nonlocal metric. Under the assumption of a sufficiently large decay parameter relative to the noise intensity, the growth of the connectivity kernel, and the Lipschitz regularity of the activation function, we establish exponential ergodicity and exponential mixing of the associated Markovian Feller semigroup and the uniqueness of the invariant measure with second moments.

Citations

• Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling
• Fluctuations for Spatially Extended Hawkes Processes
• Conditional propagation of chaos for mean field systems of interacting neurons
• Exponential ergodicity for diffusions with jumps driven by a Hawkes process
• Travelling Waves in Monostable and Bistable Stochastic Partial\n Differential Equations
• Mean field limits for nonlinear spatially extended hawkes processes with exponential memory kernels
• A general framework for stochastic traveling waves and patterns, with\n application to neural field equations
• Microscopic approach of a time elapsed neural model
• Stochastic motion of bumps in planar neural fields
• Nonlinear Langevin equations for wandering patterns in stochastic neural\n fields
• Stochastic neural field equations: A rigorous footing
• Wandering bumps in stochastic neural fields
• Dynamics of pattern formation in lateral-inhibition type neural fields
• The chemical basis of morphogenesis

Cited by

• Neural Field Equations with random data

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