Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations

Neural field equations are used to describe the spatiotemporal evolution of the activity in a network of synaptically coupled populations of neurons in the continuum limit. Their heuristic derivation involves two approximation steps. Under the assumption that each population in the network is large, the activity is described in terms of a population average. The discrete network is then approximated by a continuum. In this article we make the two approximation steps explicit. Extending a model by Bressloff and Newby, we describe the evolution of the activity in a discrete network of finite populations by a Markov chain. In order to determine finite-size effects - deviations from the mean field limit due to the finite size of the populations in the network - we analyze the fluctuations of this Markov chain and set up an approximating system of diffusion processes. We show that a well-posed stochastic neural field equation with a noise term accounting for finite-size effects on traveling wave solutions is obtained as the strong continuum limit

Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction

We consider a system of $N$ disordered mean-field interacting diffusions within spatial constraints: each particle $\theta_i$ is attached to one site $x_i$ of a periodic lattice and the interaction between particles $\theta_i$ and $\theta_j$ decreases as $| x_i-x_j|^{-\alpha}$ for $\alpha\in[0,1)$. In a previous work, it was shown that the empirical measure of the particles converges in large population to the solution of a nonlinear partial differential equation of McKean-Vlasov type. The purpose of the present paper is to study the fluctuations associated to this convergence. We exhibit in particular a phase transition in the scaling and in the nature of the fluctuations: when $\alpha\in[0,\frac{1}{2})$, the fluctuations are Gaussian, governed by a linear SPDE, with scaling $\sqrt{N}$ whereas the fluctuations are deterministic with scaling $N^{1-\alpha}$ in the case $\alpha\in(\frac{1}{2},1)$.Comment: 56 page