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

Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay