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:

Several differential equation models have been proposed to explain the formation of patterns characteristic of the grid cell network. 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 interaction between different columns of neurons along the cortex prescribes a peculiar scaling regime.

We establish pathwise existence of solutions for stochastic porous media and fast diffusion equations of type (1.1), in the full regime m P p0, 8q and for any initial data u 0 P L 2 pQq. Moreover, if the initial data is positive, solutions are pathwise unique. In turn, the solution map to (1.1) is a continuous function of the driving noise and it generates an associated random dynamical system. Finally, in the regime m P t1u Y p2, 8q, all the aforementioned results also hold for signed initial data.