In this paper we consider a stochastic system with two connected nodes, whose unidirectional connection is variable and depends on point processes associated to each node. The input node is represented by an homogeneous Poisson process, whereas the output node jumps with an intensity that depends on the jumps of the input nodes and the connection intensity. We study a scaling regime when the rate of both point processes is large compared to the dynamics of the connection. In neuroscience, this system corresponds to a neural network composed by two neurons, connected by a single synapse. The strength of this synapse depends on the past activity of both neurons, the notion of synaptic plasticity refers to the associated mechanism. A general class of such stochastic models has been introduced in Robert and Vignoud (Stochastic models of synaptic plasticity in neural networks, 2020, arxiv: 2010.08195) to describe most of the models of long-term synaptic plasticity investigated in the literature. The scaling regime corresponds to a classical assumption in computational neuroscience that cellular processes evolve much more rapidly than the synaptic strength. The central result of the paper is an averaging principle for the time evolution of the connection intensity. Mathematically, the key variable is the point process, associated to the output node, whose intensity depends on the past activity of the system. The proof of the result involves a detailed analysis of several of its unbounded additive functionals in the slow-fast limit, and technical results on interacting shot-noise processes.

在本文中，我们考虑具有两个连接节点的随机系统，其单向连接是可变的并且取决于与每个节点关联的点过程。所述输入节点是由一个齐次泊松过程表示，而输出节点与依赖于输入节点和连接强度的跳跃的强度跳跃。当两个点过程的速率与连接的动态相比较大时，我们研究了缩放机制。在神经科学中，该系统对应于由两个神经元组成的神经网络，由单个突触连接。这种突触的强度取决于两个神经元过去的活动，即突触可塑性的概念指相关机制。Robert 和 Vignoud（神经网络中突触可塑性的随机模型，2020，arxiv：2010.08195）引入了一类通用的此类随机模型，以描述文献中研究的大多数长期突触可塑性模型。缩放机制对应于计算神经科学中的一个经典假设，即细胞过程的进化比突触强度快得多。该论文的核心结果是连接强度时间演化的平均原理。在数学上，关键变量是点过程，与输出节点相关联，其强度取决于系统过去的活动。结果的证明涉及对其在慢-快极限下的几个无界加性泛函的详细分析，