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

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 first deal with the case when transmission after a spike is instantaneous and the case when there exists a distributed delay that depends on the previous history of the system, which is a more realistic assumption. Then we revisit the well-posedness in order to make a numerical analysis by adapting the classical upwind scheme through a fixed-point approach. We improve the previous results on well-posedness by relaxing some hypotheses on the non-linearity for instantaneous transmission, including the strongly excitatory case, while for the numerical analysis we prove that the approximation given by the explicit upwind scheme converges to the solution of the non-linear problem through BV-estimates. 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.