@misc{10481/100074,
year = {2019},
month = {7},
url = {https://hdl.handle.net/10481/100074},
abstract = {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.},
publisher = {TAYLOR & FRANCIS INC},
keywords = {Leaky integrate and fire models},
keywords = {Blow-up phenomena.},
keywords = {Relaxation to steady state},
keywords = {Neural networks},
keywords = {Delay},
keywords = {Global existence},
keywords = {Stefan problem},
title = {Global-in-time solutions and qualitative properties for the NNLIF neuron model with synaptic delay},
doi = {10.1080/03605302.2019.1639732},
author = {Cáceres Granados, María Josefa and Roux, Pierre and Salort, Delphine and Schneider, Ricarda},
}