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

 Title Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience Publication Type Journal Article Year of Publication 2015 Authors Carrillo, JAntonio, Perthame, B, Salort, D, Smets, D Journal Nonlinearity Volume 28 Pagination 3365 Abstract 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 ##IMG## [http://ej.iop.org/images/0951-7715/28/9/3365/non518219ieqn001.gif] {${{V}_{\text{F}}}$} 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 L 2 estimates and on the other hand, on the notion of ‘universal super-solution’ and parabolic regularizing effects to obtain ##IMG## [http://ej.iop.org/images/0951-7715/28/9/3365/non518219ieqn002.gif] {${{L}^ınfty}}$} bounds. URL http://stacks.iop.org/0951-7715/28/i=9/a=3365