标题： 平衡兴奋与抑制的神经网络的流体动力学极限 显示英文标题

标题： The Hydrodynamic Limit of Neural Networks with Balanced Excitation and Inhibition

摘要： 平衡神经网络理论是对大脑活动高度变异性和随机性的非常流行的解释。 我们确定了2n个神经元的平衡全连接网络在渐近大n情况下的流体力学极限方程。神经元被分为两类（兴奋性和抑制性）。 每个兴奋性神经元刺激其他所有神经元，每个抑制性神经元抑制其他所有神经元。 该模型具有随机混合性质，使得每个神经元的突触响应由常微分方程控制。 神经元j对神经元k的影响由一个尖峰泊松过程决定，其强度由神经元j的突触增强的逻辑函数给出。 相互作用按n^{-1/2}进行缩放，这比经典相互作用粒子系统的n^{-1}缩放要强得多。 我们证明，在适当条件下，当n趋于无穷时系统不会爆炸，因为网络活动在兴奋性和抑制性输入之间是平衡的。 极限种群动力学被证明是高斯的：均值由兴奋和抑制之间的平衡决定，方差由非齐次泊松过程的中心极限定理决定。 因此，极限方程可以表示为均值和方差的自主常微分方程。 显示英文摘要

摘要： The theory of `Balanced Neural Networks' is a very popular explanation for the high degree of variability and stochasticity in the brain's activity. We determine equations for the hydrodynamic limit of a balanced all-to-all network of 2n neurons for asymptotically large n. The neurons are divided into two classes (excitatory and inhibitory). Each excitatory neuron excites every other neuron, and each inhibitory neuron inhibits all of the other neurons. The model is of a stochastic hybrid nature, such that the synaptic response of each neuron is governed by an ordinary differential equation. The effect of neuron j on neuron k is dictated by a spiking Poisson Process, with intensity given by a sigmoidal function of the synaptic potentiation of neuron j. The interactions are scaled by n^{-1/2} , which is much stronger than the n^{-1} scaling of classical interacting particle systems. We demonstrate that, under suitable conditions, the system does not blow up as n asymptotes to infinity because the network activity is balanced between excitatory and inhibitory inputs. The limiting population dynamics is proved to be Gaussian: with the mean determined by the balanced between excitation and inhibition, and the variance determined by the Central Limit Theorem for inhomogeneous Poisson Processes. The limiting equations can thus be expressed as autonomous Ordinary Differential Equations for the means and variances.