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 novel states, which we call pseudo-equilibria, and give evidence of their defining role in the behaviour 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 new strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudo-equilibria. Unlike direct entropy dissipation methods, this technique allows us to see how a large delay favours convergence. We present a detailed numerical study to support our results. This study helps understand, among other phenomena, the appearance of periodic solutions in strongly inhibitory networks.