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Open Access Research

Laws of Large Numbers and Langevin Approximations for Stochastic Neural Field Equations

Martin G Riedler* and Evelyn Buckwar

Author Affiliations

Institute for Stochastics, Johannes Kepler University, Linz, Austria

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The Journal of Mathematical Neuroscience 2013, 3:1  doi:10.1186/2190-8567-3-1

Published: 23 January 2013

Abstract

In this study, we consider limit theorems for microscopic stochastic models of neural fields. We show that the Wilson–Cowan equation can be obtained as the limit in uniform convergence on compacts in probability for a sequence of microscopic models when the number of neuron populations distributed in space and the number of neurons per population tend to infinity. This result also allows to obtain limits for qualitatively different stochastic convergence concepts, e.g., convergence in the mean. Further, we present a central limit theorem for the martingale part of the microscopic models which, suitably re-scaled, converges to a centred Gaussian process with independent increments. These two results provide the basis for presenting the neural field Langevin equation, a stochastic differential equation taking values in a Hilbert space, which is the infinite-dimensional analogue of the chemical Langevin equation in the present setting. On a technical level, we apply recently developed law of large numbers and central limit theorems for piecewise deterministic processes taking values in Hilbert spaces to a master equation formulation of stochastic neuronal network models. These theorems are valid for processes taking values in Hilbert spaces, and by this are able to incorporate spatial structures of the underlying model.

Mathematics Subject Classification (2000): 60F05, 60J25, 60J75, 92C20.

Keywords:
Stochastic neural field equation; Wilson–Cowan model; Piecewise deterministic Markov process; Stochastic processes in infinite dimensions; Law of large numbers; Martingale central limit theorem; Chemical Langevin equation