论文标题
尖峰神经元相互作用系统的平均场限制的波动
Fluctuations for mean field limits of interacting systems of spiking neurons
论文作者
论文摘要
我们考虑一个$ n $神经元的系统，每个系统都会根据其膜潜力随机峰值。当神经元尖峰时，其潜力将重置为$ 0 $，并且所有其他神经元将获得额外的$ h/n $潜力，其中$ h> 0 $是一些固定参数。在连续的尖峰之间，每个神经元的电位都以恒定速率$α的泄漏发生。 $虽然该系统的混乱繁殖为$ n \ to \ infty $，但已在一系列论文中已经建立了非线性跳跃随机微分方程，但请参见De Masi等。 （2015年）以及Fournier andLöcherbach（2016），本文专门介绍了相关的中央限制定理。更确切地说，我们研究了膜电位经验度量的$ n^{ - 1/2} $的度量值，膜电位的经验度量$ n^{ - 1/2} $以相关极限为中心。我们表明，这种波动过程被解释为在适当的加权Sobolev空间中采用值的Càdlàg过程，在法律上收敛到以高斯白噪声驱动的随机微分方程系统为特征的极限过程。我们通过研究波动，以$ n^{ - 1/2}的规模来完成这张图片，这是固定数量的膜电位过程围绕其相关极限数量的$，从而导致了膜电位的中镜近似，这些膜电位考虑了有限系统内的相关性。
We consider a system of $N$ neurons, each spiking randomly with rate depending on its membrane potential. When a neuron spikes, its potential is reset to $0$ and all other neurons receive an additional amount $h/N$ of potential, where $ h > 0$ is some fixed parameter. In between successive spikes, each neuron's potential undergoes some leakage at constant rate $ α. $ While the propagation of chaos of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation has already been established in a series of papers, see De Masi et al. (2015) and Fournier and Löcherbach (2016), the present paper is devoted to the associated central limit theorem. More precisely we study the measure valued process of fluctuations at scale $ N^{-1/2}$ of the empirical measures of the membrane potentials, centered around the associated limit. We show that this fluctuation process, interpreted as càdlàg process taking values in a suitable weighted Sobolev space, converges in law to a limit process characterized by a system of stochastic differential equations driven by Gaussian white noise. We complete this picture by studying the fluctuations, at scale $ N^{-1/2}, $ of a fixed number of membrane potential processes around their associated limit quantities, giving rise to a mesoscopic approximation of the membrane potentials that take into account the correlations within the finite system.
