The elapsed time equation is an age-structured model that describes 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 previous history of the system, which is a more realistic assumption. Then we study the well-posedness and the numerical analysis of the elapsed time models. For existence and uniqueness we improve the previous works by relaxing some hypothesis on the nonlinearity, 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. 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：时间流逝方程是一种年龄结构模型，通过自上次放电以来的时间流逝描述互连尖峰神经元的动态，从数学和生物学角度引出了系统演化的诸多有趣问题。本文首先处理尖峰后瞬时传输的情况，以及存在依赖于系统先前历史分布延迟的情况（后者是更现实的假设），进而研究时间流逝模型的适定性与数值分析。在存在唯一性方面，我们通过放宽非线性项的某些假设（包括强兴奋性情形）改进了先前的成果；在数值分析方面，我们证明了显式迎风格式给出的近似解收敛于非线性问题的真解。我们还通过数值模拟比较了不同参数下瞬时传输与分布延迟两种情况下系统的行为差异，得到了具有不同渐近轮廓的解。