AI 中文总结
研究具有短期突触抑制的大规模积分发放神经元网络平均场极限，通过引入辅助线性化马尔可夫过程等方法，确定保证系统不变概率测度局部稳定性的解析准则，并通过数值例子展示了二维框架比一维模型有更丰富的长期动力学。
AI 中文摘要
本工作研究了受短期突触抑制（STD）影响的相互作用随机泄漏积分发放（LIF）神经元大规模网络的平均场极限。该系统的宏观动力学由一个二维非线性麦克凯恩 - 弗拉索夫方程控制，该方程将神经元膜电位的演化与一个突触抑制变量耦合。我们研究这个极限系统的长期行为。为此，通过将相互作用非线性冻结为常数引入一个辅助线性化马尔可夫过程。利用尖峰时刻膜电位的再生，我们能够在这个二维线性过程的不变测度下，明确计算出突触抑制变量在电位值条件下的条件期望。这是研究其时间相关局部扰动的关键要素。结果我们能够确定一个解析准则来保证完全非线性系统任何不变概率测度的局部稳定性。这个稳定性准则是根据特定线性响应函数的拉普拉斯变换的零点来制定的。最后，我们提供了数值例子表明二维框架比纯一维模型诱导出更丰富的长期动力学谱。例如，突触抑制可导致围绕一个唯一的、不稳定的不变测度的低频振荡，其振荡比神经元的发放率慢得多。
英文摘要
This work studies the mean-field limit of large networks of interacting stochastic leaky integrate-and-fire (LIF) neurons subject to short-term synaptic depression (STD). The macroscopic dynamics of this system is governed by a two-dimensional, non-linear McKean-Vlasov equation that couples the evolution of the neurons' membrane potentials with a synaptic depression variable. We investigate the long-time behavior of this limit system. To this end, we introduce an auxiliary linearized Markov process by freezing the interaction non-linearity to a constant. By exploiting the regeneration of the membrane potential at spike times, we are able to explicitly compute the conditional expectation of the synaptic depression variable, conditionally on the potential value, under the invariant measure of this two-dimensional linear process. This is a crucial ingredient to study time-dependent local perturbations thereof. As a consequence we are able to identify an analytic criterion guaranteeing the local stability of any invariant probability measure of the fully non-linear system. This stability criterion is formulated in terms of the zeros of the Laplace transform of a specific linear response function. Finally, we provide numerical examples demonstrating that the two-dimensional framework induces a richer spectrum of long-time dynamics than purely one-dimensional models. For example, synaptic depression can lead to low-frequency oscillations around a unique, unstable invariant measure where the oscillations are much slower than the neurons' firing rates.
