Qualitative Properties of Solutions for the Noisy Integrate & Fire model in Computational Neuroscience

International audienceThe Noisy Integrate-and-Fire equation is a standard non-linear Fokker-Planck Equation used to describe the activity of a homogeneous neural network characterized by its connectivity b (each neuron connected to all others through synaptic weights); b > 0 describes excitatory networks and b < 0 inhibitory networks. In the excitatory case, it was proved that, once the proportion of neurons that are close to their action potential V F is too high, solutions cannot exist for all times. In this paper, we show a priori uniform bounds in time on the firing rate to discard the scenario of blow-up, and, for small connectivity, we prove qualitative properties on the long time behavior of solutions. The methods are based on the one hand on relative entropy and Poincaré inequalities leading to L 2 estimates and on the other hand, on the notion of 'universal super-solution' and parabolic regularizing effects to obtain L ∞ bounds