伪平衡序列描述具有大延迟的非线性噪声泄漏积分点火模型的长时间行为
Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay
AI总结：
本文针对带大突触延迟的非线性噪声泄漏积分点火神经元模型，引入伪平衡序列概念，提出追踪该序列证明弱连接系统平衡收敛的策略，结合数值研究揭示大延迟对收敛的作用及强抑制网络周期解的成因。
摘要：
目前已有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点火（NNLIF，nonlinear noisy leaky integrate-and-fire）模型。我们引入了一系列被称为伪平衡的状态，并证明了当存在显著突触延迟时，这些状态对NNLIF系统的行为具有决定性作用。这类状态的优势在于它们仅由系统参数决定，且源自求解一个递推方程得到的放电率序列。我们提出了一种策略，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会收敛到平衡态。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。我们还开展了详细的数值研究以支撑所得结果。该研究有助于我们理解强抑制网络中周期解的出现等多种现象。
英文摘要：
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.
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