@misc{10481/107704,
year = {2026},
month = {1},
url = {https://hdl.handle.net/10481/107704},
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.},
organization = {ANID, Chile - (ECOS200018, general project funding; FB210005; Fondecyt-1220869)},
organization = {MCIN/AEI/10.13039/501100011033, European Union NextGenerationEU/PRTR - (Juan de la Cierva FJC2021-046894-I)},
publisher = {Elsevier},
keywords = {Structured equations},
keywords = {Mathematical neuroscience},
keywords = {Delay differential equations},
title = {Well-posedness and numerical analysis of an elapsed time model with strongly coupled neural networks},
doi = {10.1016/j.cnsns.2025.109144},
author = {Sepúlveda, Mauricio and Torres, Nicolás and Villada, Luis Miguel},
}