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

María J Cáceres,Ricarda Schneider

doi:10.3934/krm.2017024

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 [ 5 ] for studying the number of steady states, entropy methods combined with Poincare inequalities for the long time behaviour and, finally, high order numerical schemes together with parallel computation techniques in order to obtain our numerical results.

Highlights

One of the simplest self-contained mean field models for neural networks is the Network of Noisy Leaky Integrate and Fire (NNLIF) model
Some works have pointed to this direction; in [9] some existence results have been proven: For a fully inhibitory network there is global in time existence, while for a purely excitatory network there is a global in time solution only if the firing rate is finite for every time
These results are consistent with the fact that in the excitatory case, solutions can blow-up in finite time if the value of the connectivity parameter of the network is large enough or if the initial datum is concentrated close enough to the threshold potential, [5, 7, 14, 13, 12]

Summary

Introduction

One of the simplest self-contained mean field models for neural networks is the Network of Noisy Leaky Integrate and Fire (NNLIF) model. We analyze the set of stationary states, which is more complicate than in the case of purely excitatory or inhibitory networks, and prove exponential convergence to the unique steady state when all the connectivity parameters are small. The study developed in [3], together with the results in the present paper, show that this simple model (excitatory-inhibitory coupled NNLIF) could describe phenomena well known in neurophysiology: Synchronous and asynchronous states. On the other hand, when the model does not have stable steady states, there are syncronous states In this sense, the blow-up phenomenon could be understood as a synchronization of part of the network, because the firing rate diverges for a finite time.

The model and the definition of solution

Finite time blow-up

Steady states

Long time behaviour

Numerical results