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In this study we consider limit theorems for microscopic stochastic models ofneural fields. We show that the Wilson-Cowan equation can be obtained as thelimit in probability on compacts for a sequence of microscopic models when thenumber of neuron populations distributed in space and the number of neurons perpopulation tend to infinity. Though the latter divergence is not necessary.This result also allows to obtain limits for qualitatively different stochasticconvergence concepts, e.g., convergence in the mean. Further, we present acentral limit theorem for the martingale part of the microscopic models which,suitably rescaled, converges to a centered Gaussian process with independentincrements. These two results provide the basis for presenting the neural fieldLangevin equation, a stochastic differential equation taking values in aHilbert space, which is the infinite-dimensional analogue of the ChemicalLangevin Equation in the present setting. On a technical level we applyrecently developed law of large numbers and central limit theorems forpiecewise deterministic processes taking values in Hilbert spaces to a masterequation formulation of stochastic neuronal network models. These theorems arevalid for processes taking values in Hilbert spaces and by this are able toincorporate spatial structures of the underlying model.

Laws of large numbers and Langevin approximations for stochastic neural field equations