Global-in-time solutions and qualitative properties for the NNLIF neuron model with synaptic delay Cáceres Granados, María Josefa Roux, Pierre Salort, Delphine Schneider, Ricarda Leaky integrate and fire models Blow-up phenomena. Relaxation to steady state Neural networks Delay Global existence Stefan problem 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 “strong” 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 Poincaré’s inequality. 2025-01-23T08:42:53Z 2025-01-23T08:42:53Z 2019-07 journal article https://hdl.handle.net/10481/100074 10.1080/03605302.2019.1639732 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional TAYLOR & FRANCIS INC