Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods

Abstract

The network of noisy leaky integrate and fire (NNLIF) model is one of the simplest self-contained mean-field models considered to describe the behavior of neural networks. Even so, in studying its mathematical properties some simplifications are required [Cáceres and Perthame, J. Theor. Biol. 350 (2014) 81–89; Cáceres and Schneider, Kinet. Relat. Model. 10 (2017) 587–612; Cáceres, Carrillo and Perthame, J. Math. Neurosci. 1 (2011) 7] which disregard crucial phenomena. In this work we deal with the general NNLIF model without simplifications. It involves a network with two populations (excitatory and inhibitory), with transmission delays between the neurons and where the neurons remain in a refractory state for a certain time. In this paper we study the number of steady states in terms of the model parameters, the long time behaviour via the entropy method and Poincaré’s inequality, blow-up phenomena, and the importance of transmission delays between excitatory neurons to prevent blow-up and to give rise to synchronous solutions. Besides analytical results, we present a numerical solver, based on high order flux-splitting WENO schemes and an explicit third order TVD Runge-Kutta method, in order to describe the wide range of phenomena exhibited by the network: blow-up, asynchronous/synchronous solutions and instability/stability of the steady states. The solver also allows us to observe the time evolution of the firing rates, refractory states and the probability distributions of the excitatory and inhibitory populations.