AI 中文总结
研究受共同噪声影响的紧密连接兴奋性积分发放神经元系统，证明神经元数量趋于无穷时弱收敛到随机福克 - 普朗克方程控制的膜电位密度，通过能量估计获方程唯一性，还给出膜电位密度的条件麦克凯恩 - 弗拉索夫表示。
AI 中文摘要
我们研究了受共同噪声影响的紧密连接的兴奋性积分发放神经元系统。该系统包含动作电位的渐进传输，并通过随机不应期捕捉复极化和超极化阶段，随后重置到低于静息电位的随机水平。当神经元数量趋于无穷时，我们证明存在弱收敛到由具有明确尖峰传输率的随机福克 - 普朗克方程控制的唯一膜电位密度。后者由平均累积尖峰计数驱动，其不可微但满足广义通量条件。我们从一阶索伯列夫空间对偶中的能量估计获得福克 - 普朗克方程的唯一性。最后，我们给出了膜电位密度的条件麦克凯恩 - 弗拉索夫表示，即给定共同噪声时代表性神经元的分布律。
英文摘要
We study a densely connected system of excitatory integrate-and-fire neurons which are subject to common noise. The system incorporates a gradual transmission of action potentials and captures the re- and hyperpolarization phases through a random refractory period followed by a reset to a randomized level below the rest potential. As the number of neurons tends to infinity, we show that there is weak convergence to a unique membrane potential density governed by a stochastic Fokker--Planck equation with a well-defined spike transmission rate. The latter is driven by the mean cumulative spike count, which is non-differentiable but shown to satisfy a generalized flux condition. We obtain the uniqueness of the Fokker--Planck equation from energy estimates in the dual of the first Sobolev space. Finally, we give a conditional McKean--Vlasov representation of the membrane potential density as the law of a representative neuron given the common noise.
Comments43 pages
