  The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to
describe the dynamics of neural networks after a diffusive approximation of the
mean-field limit of a stochastic differential equation system. When the total
activity of the network has an instantaneous effect on the network, in the
average-excitatory case, a blow-up phenomenon occurs. This article is devoted
to the theoretical study of the NNLIF model in the case where a delay in the
effect of the total activity on the neurons is added. We first prove
global-in-time existence and uniqueness of classical solutions, independently
of the sign of the connectivity parameter, that is, for both cases: excitatory
and inhibitory. Secondly, we prove some qualitative properties of solutions:
asymptotic convergence to the stationary state for weak interconnections and a
non-existence result for periodic solutions if the connectivity parameter is
large enough. The proofs are mainly based on an appropriate change of variables
to rewrite the NNLIF equation as a Stefan-like free boundary problem,
constructions of universal super-solutions, the entropy dissipation method and
Poincar\'e's inequality.
