Alejandro Ramos Lora: Global behaviour of the NNLIF model with large delay

Speaker
Alejandro Ramos Lora, University of Granada

Abstract
Global behaviour of the NNLIF model with large delay Alejandro Ramos Lora. University of Granada. Abstract The Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model [1] describes a neuron population from the perspective of the membrane potential. Existence of solution and asymptotic behaviour of the NNLIF model have been the subject of several research papers in the field of PDEs over the last decades [3, 5]. In this talk we define a set of states, which we call “pseudo-equilibria” [2], and give evidence of their defining role in the behaviour of the NNLIF model when a significant synaptic delay is considered. These states are related to the NNLIF nonlinear system only by the systems’s parameter, and its behaviour can be studied through the analysis of a discrete sequence arising from a recurrence equation. Our work shows that the behaviour of this sequence of states completely determines the long-term behaviour of the non-linear system with large delay. To verify this, we provide ample numerical evidence. We also give a rigorous proof that the non-linear system converges exponentially to equilibrium for cases where the connectivity parameter is sufficiently small, following the argument that the non-linear system follows the behaviour of the sequence of pseudo-equilibria. This result establishes improvements on similar results previously known [4].