Stochastic neural field equations: a rigorous footing

Abstract

We here consider a stochastic version of the classical neural field equation that is currently actively studied in the mathematical neuroscience community. Our goal is to present a well-known rigorous probabilistic framework in which to study these equations in a way that is accessible to practitioners currently working in the area, and thus to bridge some of the cultural/scientific gaps between probability theory and mathematical biology. In this way, the paper is intended to act as a reference that collects together relevant rigorous results about notions of solutions and well-posedness, which although may be straightforward to experts from SPDEs, are largely unknown in the neuroscientific community, and difficult to find in a very large body of literature. Moreover, in the course of our study we provide some new specific conditions on the parameters appearing in the equation (in particular on the neural field kernel) that guarantee the existence of a solution.

Front Propagation in Stochastic Neural Fields: A Rigorous Mathematical Framework

“…Here, ξ will be chosen as Q-Wiener process on the Hilbert space L 2 (R). In contrast to the more direct rigorous approach recently suggested in [11], where a stochastic neural field is interpreted as SDE on a suitable weighted L 2 -space, our approach via decomposition has the advantage of providing information on the precise structure of solutions and even more, allows us to prove new stability results.…”

Noise sharing and Mexican-hat coupling in a stochastic neural field

“…Using the calculations in Appendix A, we may write B 0 (t) = LR(0)C 0 (t) and B k (t) = (L/2)R(2πk/L)C k (t) for k ≥ 1, where the processes C k are as stated in the proposition. For k ≥ 1 the equation (9) can then be rewritten as (7) and (8).…”

“…For each pair of parameters (c, η), which will determine the coupling strength and the width of the noise smoothing, defined in section III B, we are able to evaluate a functional of the distribution of the process Y c,η that is maximal at that value of (c, η) for which k c,η is the dominant mode. This functional, the expected squared amplitude of the kth mode of Y c,η , can be computed in terms of the drift and diffusion coefficients, λ k , σ k , of the process a k (t) defined by (7), (8).…”

A Multiscale Analysis of Traveling Waves in Stochastic Neural Fields

“…Set C m 0 (t) = t 0 c m 0 (s)ds and ϕ m 0 (t) = ct + ǫC m 0 (t). Formally identifying the highest order terms in (14) we define v m 0 to be the unique strong solution to…”