The elapsed time model has been widely studied in the context of mathematical neuroscience with many open questions left. The model consists of an age-structured equation that describes the dynamics of interacting neurons structured by the elapsed time since their last discharge. Our interest lies in highly connected networks leading to strong nonlinearities where perturbation methods do not apply. To deal with this problem, we choose a particular case which can be reduced to delay equations.

We prove a general convergence result to a stationary state in the inhibitory and the weakly excitatory cases. Moreover, we prove the existence of particular periodic solutions with jump discontinuities in the strongly excitatory case. Finally, we present some numerical simulations which illustrate various behaviors, which are consistent with the theoretical results.

经过时间模型在数学神经科学的背景下得到了广泛的研究，还有许多悬而未决的问题。该模型由一个年龄结构方程组成，该方程描述了自上次放电以来经过的时间构成的相互作用神经元的动力学。我们的兴趣在于高度连接的网络，导致强非线性，其中扰动方法不适用。为了解决这个问题，我们选择一个可以简化为延迟方程的特殊情况。

我们证明了在抑制性和弱兴奋性情况下对静止状态的一般收敛结果。此外，我们证明了在强激励情况下具有跳跃不连续性的特定周期解的存在。最后，我们提出了一些数值模拟，说明了与理论结果一致的各种行为。