Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models (1602.05604v2)

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.