The mean field limit of stochastic differential equation systems modelling grid cells

Abstract

Several differential equation models have been proposed to explain the formation of stationary activity patterns characteristic of the grid cell network. Understanding the robustness of these patterns with respect to noise is one of the key open questions in computational neuroscience. In the present work, we analyze a family of stochastic differential systems modelling grid cell networks. Furthermore, the well-posedness of the associated McKean-Vlasov and Fokker-Planck equations, describing the average behavior of the networks, is established. Finally, we rigorously prove the mean field limit of these systems and provide a sharp rate of convergence for their empirical measures.Remark 1.1. The way we choose the cloud of points x 1 , . . . x N P Q is not that important as far as we are concerned only with the discrete model for fixed M and N , and we may just consider them to be fixed a priori. However, to get a nice limiting behaviour as N, M Ñ 8, it is useful to choose these points to be independently uniformly distributed in Q and independent of the initial data and the subsequent stochastic evolution. Rigorously, we shall take N i.i.d. random variables X 1 , . . . , X N with uniform law in Q and we shall also take them to be independent of the initial data tu k px, 0qu kPN, xPQ and of the underlying white noise tW k px, tqu kPN, xPQ, tě0 .Remark 1.2. In this setting one should in general not expect the resulting initial data u k pX i , 0q to be independent if only one of i 1 ‰ i 2 or k 1 ‰ k 2 occurs. For example, from the point of view of modelling in neuroscience, u k px, 0q should be close to u k py, 0q for x close to y. Unfortunately, this will in turn have consequences on the rate of convergence towards the limiting behavior. The complete details are given in Section 5.As we let M, N Ñ 8 the limiting behaviour should be described by independent copies, in the column index k, of solutions to the associated mean-field McKean-Vlasov equation. Namely, the activity level of any neuron located at a point x P Q should satisfy the following equation:

On the fluctuations of an SDE system modelling grid cells

“…Understanding the effect of noise on these models is one of the key open questions in computational neuroscience. In the present work, we continue the analysis of the SDE system (1.1) initiated in [11]. We show that the fluctuations of the empirical measure associated to (1.1) around its mean field limit converge to the solution of a Langevin SPDE.…”

“…The present work is the continuation of [11], along the direction initiated in [12]. In [12] the authors studied a system of Fokker-Planck PDEs derived by adding noise to the attractor network models in [8,14] and formally taking the mean field limit.…”

“…The PDE has been further analyzed in [13]. The passage to the mean field limit has been rigorously proved in [11], where the authors derive the limit for a generalized model, covering diverse concrete models commonly proposed (cf. [2,8,9,14] and the review [6]).…”

“…, M , we have d v -dimensional Brownian motions pW β ik q β"1,...,dv ,which can also be correlated. The precise shape and properties of the coefficients b and σ are given in Section 2.1 and are deduced from those of the concrete models considered (see the introduction of [11]). We also denoted f N,M the empirical measure associated to these particles, that is f N,M pr, dy, duq "…”

Noise-driven bifurcations in a nonlinear Fokker-Planck system describing stochastic neural fields

“…In [25] understanding the effects of noise on networks of grid cells was emphasized as a challenge, and this sparked the development and initial study of (1.3) in [9]. The model (1.3), which is based on the ODE models of [10,4], has been rigorously shown to be the mean-field limit of a network of stochastic grid cells with Gaussian independent noise [7] by adapting Sznitman's coupling method [26]. Furthermore, novel evidence pointing towards a toroidal connectivity for grid cells has recently been provided in [15] by using topological data analysis tools.…”