Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience

The 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 ${{V}_{\text{F}}}$ 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 L² estimates and on the other hand, on the notion of 'universal super-solution' and parabolic regularizing effects to obtain ${{L}^{\infty}}$ bounds.