An elapsed time model for strongly coupled inhibitory and excitatory neural networks

TL;DR

This paper analyzes an age-structured neural network model, proving convergence to steady states in inhibitory and weakly excitatory cases, and demonstrating periodic solutions with jumps in strongly excitatory networks, supported by numerical simulations.

Contribution

It introduces a reduction of the elapsed time model to delay equations and establishes convergence and periodic solutions in different network regimes.

Findings

Convergence to stationary states in inhibitory and weakly excitatory networks.

Existence of periodic solutions with jump discontinuities in strongly excitatory networks.

Numerical simulations confirm theoretical predictions.

Abstract

The elapsed time model has been widely studied in the context of mathematical neuroscience with many open questions left. The model consists of an age-structured equation that describes the dynamics of interacting neurons structured by the elapsed time since their last discharge. Our interest lies in highly connected networks leading to strong nonlinearities where perturbation methods do not apply. To deal with this problem, we choose a particular case which can be reduced to delay equations. We prove a general convergence result to a stationary state in the inhibitory and the weakly excitatory cases. Moreover, we prove the existence of particular periodic solutions with jump discontinuities in the strongly excitatory case. Finally, we present some numerical simulations which ilustrate various behaviors, which are consistent with the theoretical results.