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The elapsed time equation is an age-structured model that describes dynamicsof interconnected spiking neurons through the elapsed time since the lastdischarge, leading to many interesting questions on the evolution of the systemfrom a mathematical and biological point of view. In this work, we first dealwith the case when transmission after a spike is instantaneous and the casewhen there exists a distributed delay that depends on previous history of thesystem, which is a more realistic assumption. Then we study the well-posednessand the numerical analysis of the elapsed time models. For existence anduniqueness we improve the previous works by relaxing some hypothesis on thenonlinearity, including the strongly excitatory case, while for the numericalanalysis we prove that the approximation given by the explicit upwind schemeconverges to the solution of the non-linear problem. We also show somenumerical simulations to compare the behavior of the system in the case ofinstantaneous transmission with the case of distributed delay under differentparameters, leading to solutions with different asymptotic profiles.

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