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 of neurons. We introduce a set of novel states, which we call "pseudo-equilibria", and give evidence of their defining role in the behaviour 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 new strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudo-equilibria. Unlike with the direct entropy dissipation method, this technique allows us to see how a large delay favours convergence. We also present a detailed numerical study to support our results. This study explores the overall behaviour of the NNLIF system and helps us understand, among other phenomena, 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。在本文中，我们专注于描述神经元膜电位层面活动的非线性噪声泄漏积分发放（NNLIF）模型。我们引入了一组新颖的状态，称之为“伪平衡”，并证明了当考虑显著的突触延迟时，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，并且源自求解递推方程所得的一系列发放率序列。针对具有大传输延迟的弱连接系统，我们提出了一种基于追踪伪平衡序列的新策略来展示其向平衡态的收敛性。与直接的熵耗散方法不同，该技术使我们能够理解大延迟如何促进收敛。我们还提供了详细的数值研究以支持我们的结果。该研究探讨了NNLIF系统的整体行为，并有助于我们理解强抑制性网络中的周期解等现象。