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 Grégory Dumont.