  We continue the study of a stochastic system of interacting neurons
introduced in De Masi-Galves-L\"ocherbach-Presutti (2014). The system consists
of N neurons, each spiking randomly with rate depending on its membrane
potential. At its spiking time, the neuron potential is reset to 0 and all
other neurons receive an additional amount 1/N of potential. Moreover,
electrical synapses induce a deterministic drift of the system towards its
center of mass. We prove propagation of chaos of the system, as N tends to
infinity, to a limit nonlinear jumping stochastic differential equation. We
consequently improve on the results of De Masi-Galves-L\"ocherbach-Presutti
(2014), since (i) we remove the compact support condition on the initial datum,
(ii) we get a rate of convergence in $1/\sqrt N$. Finally, we study the limit
equation: we describe the shape of its time-marginals, we prove the existence
of a unique non-trivial invariant distribution, we show that the trivial
invariant distribution is not attractive, and in a special case, we establish
the convergence to equilibrium.
