论文标题
平均可塑性模型的原则
Averaging Principles for Markovian Models of Plasticity
论文作者
论文摘要
生物神经网络的数学模型与丰富而复杂的随机过程相关。在本文中，我们考虑了一个简单的{\ em塑料}神经网络，其{\ em连通性/突触强度} $（w（t））$取决于一组与活动相关的过程，以模拟{\ em synaptic塑性}，这是一种神经科学的良好机制。在\ cite {robert_mathematical_2020}中引入了一类随机模型，以研究随机过程$（w（t））$。已经从实验上观察到，其动力学发生在比主要细胞过程慢得多的时间表上。本文的目的是建立在神经元过程的快速时间范围内限制$（w（w（t））$的定理。 本文的中心结果是随机过程$（w（t））$的平均原理。从数学上讲，关键变量是在神经元尖峰的瞬间出现跳跃的点过程。在缓慢的限制中，可以实现对其几个无限添加功能的彻底分析。此外，开发和使用有关相互作用射击过程的技术结果，并将其用于平均原理的一般证明。
Mathematical models of biological neural networks are associated to a rich and complex class of stochastic processes. In this paper, we consider a simple {\em plastic} neural network whose {\em connectivity/synaptic strength} $(W(t))$ depends on a set of activity-dependent processes to model {\em synaptic plasticity}, a well-studied mechanism from neuroscience. A general class of stochastic models has been introduced in \cite{robert_mathematical_2020} to study the stochastic process $(W(t))$. It has been observed experimentally that its dynamics occur on much slower timescale than that of the main cellular processes. The purpose of this paper is to establish limit theorems for the distribution of $(W(t))$ with respect to the fast timescale of neuronal processes. The central result of the paper is an averaging principle for the stochastic process $(W(t))$. Mathematically, the key variable is the point process whose jumps occur at the instants of neuronal spikes. A thorough analysis of several of its unbounded additive functionals is achieved in the slow-fast limit. Additionally, technical results on interacting shot-noise processes are developed and used in the general proof of the averaging principle.
