## Abstract

We analyse the behaviour of a mean-field partial differential equation
which describes the dynamics of a leaky integrate-and-fire neural
network. We are interested in the existence, (non-)uniqueness, and
stability of the steady states depending on the connectivity of the
network. In particular we prove the exponential relaxation to an
asynchronous state in the case of weak connectivity. The method relies
on a Doeblin contraction argument for the linear equation, which
corresponds to a population of uncoupled neurons, and a perturbation
argument for the extension to small nonlinearities.
It is a joint work with GrĂ©gory Dumont.