Well-posedness and numerical analysis of an elapsed time model with strongly coupled neural networks

Abstract

The elapsed time equation is an age-structured model that describes the dynamics of interconnected spiking neurons through the elapsed time since the last discharge, leading to many interesting questions on the evolution of the system from a mathematical and biological point of view. In this work, we deal with the case when the transmission after a spike is instantaneous and the case with a distributed delay that depends on the previous history of the system, which is a more realistic assumption. Since the instantaneous transmission case is known to be ill-posed due to non-uniqueness or jump discontinuities, we establish a criterion for well-posedness to determine when the solution remains continuous in time, through an invertibility condition that improves the existence theory under more relaxed hypothesis on the nonlinearity, including the strongly excitatory case. Inspired in the existence theory, we adapt the classical explicit upwind scheme through a robust fixed-point approach and we prove that the approximation given by this scheme converges to the solution of the nonlinear problem through BV-estimates and we extend the idea to the case with distributed delay. We also show some numerical simulations to compare the behavior of the system in the case of instantaneous transmission with the case of distributed delay under different parameters, leading to solutions with different asymptotic profiles.