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.

翻译：历经时间方程是一种年龄结构模型，它通过自上次放电以来的历经时间描述相互关联的发放神经元的动力学特性，从数学和生物学角度引发了关于系统演化的诸多有趣问题。在本研究中，我们首先处理脉冲后瞬时传输的情形，以及存在依赖于系统历史信息的分布时滞的情形——后者是更符合实际的假设。随后我们重新审视适定性问题，以便通过不动点方法改进经典迎风格式进行数值分析。对于瞬时传输情形，我们通过放宽非线性项的部分假设（包括强兴奋性情形）改进了先前关于适定性的结果；对于数值分析，我们证明了显式迎风格式给出的近似解通过BV估计收敛于非线性问题的解。我们还展示了若干数值模拟结果，以比较不同参数下瞬时传输与分布时滞情形中系统的行为特征，这些情形会形成具有不同渐近轮廓的解。