Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models

TL;DR

This paper investigates the blow-up, steady states, and long-term dynamics of excitatory-inhibitory nonlinear neuron models, extending previous work on purely excitatory or inhibitory networks with analytical and numerical methods.

Contribution

It extends the analysis of neural network models to mixed excitatory-inhibitory systems, proving potential blow-up and characterizing steady states and long-term behavior.

Findings

Mixed excitatory-inhibitory networks can blow-up in finite time.

Steady states of the combined system are characterized.

Numerical simulations support analytical results.

Abstract

Excitatory and inhibitory nonlinear noisy leaky integrate and fire models are often used to describe neural networks. Recently, new mathematical results have provided a better understanding of them. It has been proved that a fully excitatory network can blow-up in finite time, while a fully inhibitory network has a global in time solution for any initial data. A general description of the steady states of a purely excitatory or inhibitory network has been also given. We extend this study to the system composed of an excitatory population and an inhibitory one. We prove that this system can also blow-up in finite time and analyse its steady states and long time behaviour. Besides, we illustrate our analytical description with some numerical results. The main tools used to reach our aims are: the control of an exponential moment for the blow-up results, a more complicate strategy than that considered in C\'aceres, Carrillo and Perthame (2011), for studying the number of steady states, entropy methods combined with Poincar\'e inequalities for the long time behaviour and, finally, high order numerical schemes together with parallel computation techniques in order to obtain our numerical results.