Section: New Results

Macroscopic limits of stochastic neural networks and neural fields

We have studied in [26] the mean-field limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal assemblies. Our model takes into account explicitly the intrinsic randomness of firing times, contrasting with the classical integrate-and-fire model. The ergodicity properties of the Markov process associated with finite networks have been investigated. We have derived the limit in distribution of the sample path of the state of a neuron of the network when its size gets large. The invariant distributions of this limiting stochastic process have been analyzed as well as their stability properties. We have shown that the system undergoes transitions as a function of the averaged connectivity parameter, and can support trivial states (where the network activity dies out, which is also the unique stationary state of finite networks in some cases) and self-sustained activity when connectivity level is sufficiently large, both being possibly stable.

We have investigated in [23] existence and uniqueness of solutions of a McKean-Vlasov evolution PDE representing the macroscopic behavior of interacting Fitzhugh-Nagumo neurons. This equation is hypoelliptic, nonlocal and has unbounded coefficients. We have proven the existence of a solution to the evolution equation and non trivial stationary solutions. Moreover, we have demonstrated the uniqueness of the stationary solution in the weakly nonlinear regime. Eventually, using a semigroup factorisation method, we have shown exponential nonlinear stability in the small connectivity regime.

We have considered in [20] the ensemble of Real Ginibre matrices with a positive fraction $\alpha >0$ of real eigenvalues. We have demonstrated a large deviation principle for the joint eigenvalue density of such matrices and we have introduced a two phase log-gas whose stationary distribution coincides with the spectral measure of the ensemble. Using these tools we have provided an asymptotic expansion for the probability $p_{\alpha n}^n$ that an $n\times n$ Ginibre matrix has $k=\alpha n$ real eigenvalues and we have characterized the spectral measures of these matrices.