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:

We prove that a class of monotone, W 1 -contractive schemes for scalar conservation laws converge at a rate of Δx 2 in the Wasserstein distance (W 1 -distance), whenever the initial data are decreasing and consist of a finite number of piecewise constants. It is shown that the LaxFriedrichs, Enquist-Osher, and Godunov schemes are W 1 -contractive. Numerical experiments are presented to illustrate the main result. To the best of our knowledge, this is the first proof of secondorder convergence of any numerical method for discontinuous solutions of nonlinear conservation laws.

In the article a convergent numerical method for conservative solutions of the Hunter–Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction than the common CFL condition for conservation laws.

The activity generated by an ensemble of neurons is affected by various noise sources. It is a well-recognised challenge to understand the effects of noise on the stability of such networks. We demonstrate that the patterns of activity generated by networks of grid cells emerge from the instability of homogeneous activity for small levels of noise. This is carried out by upscaling a noisy grid cell model to a system of partial differential equations in order to analyse the robustness of network activity patterns with respect to noise. Inhomogeneous network patterns are numerically understood as branches bifurcating from unstable homogeneous states for small noise levels. We prove that there is a phase transition occurring as the level of noise decreases. Our numerical study also indicates the presence of hysteresis phenomena close to the precise critical noise value.

We consider linear hyperbolic systems with a stable rank 1 relaxation term and establish that the characteristic polynomial for the individual Fourier components of the solution can be written as a convex combination of the characteristic polynomials for the formal stiff and non-stiff limits. This allows us to provide a direct and elementary proof of the equivalence between linear stability and the subcharacteristic condition. In a similar vein, a maximum principle follows: The velocity of each individual Fourier component is bounded by the minimum and maximum eigenvalues of the non-stiff limit system.