We investigate a stochastic network composed of Integrate-and-Fire spiking neurons, focusing on its mean-field asymptotic behavior. We consider an invariant probability measure of the McKean-Vlasov equation and establish an explicit sufficient condition for ensuring the local stability of this invariant distribution. Furthermore, we provide a proof of a conjecture originally proposed by P. Robert and J. Touboul regarding the bistable nature of a specific instance of this neuronal model.