There is a wide range of mathematical models that describe populations of large numbers of neurons. In this article, we focus on nonlinear noisy leaky integrate-and-fire (NNLIF) models that describe neuronal activity at the level of the membrane potential. We introduce a sequence of states, which we call pseudoequilibria, and give evidence of their defining role in the behavior of the NNLIF system when a significant synaptic delay is considered. The advantage is that these states are determined solely by the system's parameters and are derived from a sequence of firing rates that result from solving a recurrence equation. We propose a strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudoequilibria. Unlike direct entropy dissipation methods, this technique allows us to see how a large delay favors convergence. We present a detailed numerical study to support our results. This study helps us understand, among other phenomena, the appearance of periodic solutions in strongly inhibitory networks.

翻译：存在大量描述大规模神经元群体的数学模型。本文聚焦于在膜电位层面描述神经元活动的非线性噪声泄漏积分发放（NNLIF）模型。我们引入一系列称为伪平衡的状态序列，并证明当考虑显著突触延迟时，这些状态在NNLIF系统行为中具有决定性作用。其优势在于这些状态完全由系统参数决定，且可通过求解递推方程得到的发放率序列推导得出。针对具有大传输延迟的弱连接系统，我们提出一种基于追踪伪平衡序列的策略来证明其向平衡态的收敛。与直接的熵耗散方法不同，该技术使我们能够观察大延迟如何促进收敛。我们通过详细的数值研究来支持结论。这项研究有助于理解强抑制性网络中周期解的出现等多种现象。