arXiv 1206.44892012-06-20math.PR

随机神经网络的平稳分布

On stationary distributions of stochastic neural networks

Konstantin Borovkov, Geoffrey Decrouez, Matthieu Gilson

备注：: 25 pages, 0 figures

摘要：

本文研究了具有有界记忆动力学的非线性Poisson神经元网络模型，该模型可以同时包含Hebbian学习机制和不应期。网络的状态由神经元在突触后传递内核记忆跨度内被激发的时间和突触连接的当前强度来描述，我们的模型的状态空间是有限维分量的层次结构。我们建立遍历性的随机过程描述的网络的行为，并证明存在连续可微的平稳分布密度（相对于相应的维数的勒贝格测度）的状态空间的组件，并找到他们的上界。对于密度分量，我们推导出一个系统的微分方程，可以解决只有在几个最简单的情况下。讨论了稳态密度的近似计算方法。一种是通过修改网络来降低问题的维度，这样如果每个神经元在突触后传递内核内存跨度内发出的尖峰数量达到给定阈值，它就不能被激发。我们发现，这个“截断”网络的平稳分布收敛到不受限制的阈值增加，并且收敛是在一个超指数的速度。一种互补的方法使用离散的马尔可夫链近似的网络过程。我们得到了这些马尔可夫链的平稳分布的线性系，并证明了这些分布弱收敛于原过程的平稳律。

英文摘要：

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.

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摘要：

英文摘要：

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