标题： 关于随机神经网络的平稳分布 显示英文标题

标题： On stationary distributions of stochastic neural networks

摘要： 本文讨论具有有界记忆动态的非线性泊松神经元网络模型，这些模型可以包含赫布学习机制和不应期。 网络的状态由其神经元在突触后转移核记忆跨度内发射以来经过的时间以及突触连接的当前强度来描述，我们模型的状态空间是有限维组件的层次结构。 我们建立了描述网络行为的随机过程的遍历性，并证明了状态空间组件上存在连续可微的平稳分布密度（相对于相应维度的勒贝格测度），并找到了它们的上界。 对于密度组件，我们推导出一个仅在几个最简单情况下可以求解的微分方程组。 讨论了近似计算平稳密度的方法。 一种方法是通过修改网络，使得每个神经元在其在突触后转移核记忆跨度内发出的脉冲数达到给定阈值时不能发射，从而降低问题的维数。 我们证明了这个“截断”网络的平稳分布随着阈值的增加收敛到无限制网络的平稳分布，并且收敛速度是超指数级的。 一种互补方法是使用离散马尔可夫链来逼近网络过程。 我们推导了这些马尔可夫链的平稳分布的线性系统，并证明这些分布弱收敛到原始过程的平稳分布。 显示英文摘要

摘要： The paper deals with non-linear Poisson neuron network models with bounded memory dynamics, that can include both Hebbian learning mechanisms and refractory periods. The state of a network is described by the times elapsed since its neurons fired within the post-synaptic transfer kernel memory span, and the current strengths of synaptic connections, the state spaces of our models being hierarchies of finite-dimensional components. We establish ergodicity of the stochastic processes describing the behaviour of the networks and prove the existence of continuously differentiable stationary distribution densities (with respect to the Lebesgue measures of corresponding dimensionality) on the components of the state space and find upper bounds for them. For the density components, we derive a system of differential equations that can be solved in a few simplest cases only. Approaches to approximate computation of the stationary density are discussed. One is to reduce the dimensionality of the problem by modifying the network so that each neuron cannot fire if the number of spikes it emitted within the post-synaptic transfer kernel memory span reaches a given threshold. We show that the stationary distribution of this `truncated' network converges to that of the unrestricted one as the threshold increases, and that the convergence is at a super-exponential rate. A complementary approach uses discrete Markov chain approximations to the network process. We derive linear systems for the stationary distributions of these Markov chains and prove that these distributions converge weakly to the stationary laws for the original processes.