Towards a realistic NNLIF model: Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods

The Network of Noisy Leaky Integrate and Fire (NNLIF) model describes the behavior of a neural network at mesoscopic level. It is one of the simplest self-contained mean-field models considered for that purpose. Even so, to study the mathematical properties of the model some simplifications were necessary C\'aceres-Carrillo-Perthame(2011), C\'aceres-Perthame(2014), C\'aceres-Schneider(2017), 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. We have studied the number of steady states in terms of the model parameters, the long time behaviour via the entropy method and Poincar\'e'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 have presented a numerical resolutor for this model, 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.