Abstract

There is a wide range of mathematical models that describe populations of large numbers of neurons. In this talk, we focus on nonlinear noisy leaky integrate and fire (NNLIF) models that describe neuronal activity at the level of the membrane potential of neurons. 

We introduce a set 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. We also present a detailed numerical study to support our results. This study explores the overall behaviour of the NNLIF system and helps us understand, among other phenomena, periodic solutions in strongly inhibitory networks.
Furthermore, we show how the behaviour of the pseudo-equilibria sequence is related to the local stability/instability of the nonlinear system equilibrium depending on the connectivity parameter and the transmission delay.