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：存在一系列描述大量神经元群体的数学模型。本文聚焦于描述神经元膜电位水平的非线性噪声泄漏积分发放（NNLIF）模型。我们引入一组称为"伪平衡"的新型状态，并证明在考虑显著突触延迟时，这些状态对NNLIF系统行为具有决定性作用。其优势在于，这些状态完全由系统参数决定，且源于通过求解递归方程得到的发放率序列。我们提出一种新策略，通过追踪伪平衡序列来证明具有大传输延迟的弱连接系统收敛到平衡态。与直接熵耗散方法不同，该技术使我们能够观察到大延迟如何促进收敛。我们还进行了详细的数值研究来支持结论。该研究探索了NNLIF系统的整体行为，有助于理解强抑制网络中周期解等现象。