Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay

Abstract

There is a wide range of mathematical models that describe populations of large numbers of neurons. In this article, we focus on nonlinear noisy leaky integrate-and-fire (NNLIF) models that describe neuronal activity at the level of the membrane potential. We introduce a sequence of states, which we call pseudoequilibria, and give evidence of their defining role in the behavior of the NNLIF system when a significant synaptic delay is considered. The advantage is that these states are determined solely by the system's parameters and are derived from a sequence of firing rates that result from solving a recurrence equation. We propose a strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudoequilibria. Unlike direct entropy dissipation methods, this technique allows us to see how a large delay favors convergence. We present a detailed numerical study to support our results. This study helps us understand, among other phenomena, the appearance of periodic solutions in strongly inhibitory networks.