Abstract We consider a continuous-time stochastic model of spiking neurons originally introduced by Ferrari et al. in Ferrari et al. (2018). In this model, we have a finite or countable number of neurons which are vertices in some graph G where the edges indicate the synaptic connection between them. We focus on metastability, understood as the property for the time of extinction of the network to be asymptotically memory-less, and we prove that this model exhibits two different behaviors depending on the nature of the specific underlying graph of interaction G that is chosen. In this model the spiking activity of any given neuron is represented by a point process, whose rate fluctuates between 1 and 0 over time depending on whether the membrane potential is positive or null. The membrane potential of each neuron evolves in time by integrating all the spikes of its adjacent neurons up to the last spike of the said neuron, so that when a neuron spikes, its membrane potential is reset to 0 while the membrane potential of each of its adjacent neurons is increased by one unit. Moreover, each neuron is exposed to a leakage effect, modeled as an abrupt loss of membrane potential which occurs at random times driven by a Poisson process of some fixed rate γ . It was previously proven that when the graph G is the infinite one-dimensional lattice, this model presents a phase transition with respect to the parameter γ . It was also proven that, when γ is small enough, the renormalized time of extinction (the first time at which all neurons have a null membrane potential) of a finite version of the system converges in law toward an exponential random variable when the number of neurons goes to infinity. The present article is divided into two parts. First we prove that, in the finite one-dimensional lattice, this last result does not hold anymore if γ is large enough, and in fact we prove that for γ > 1 the renormalized time of extinction is asymptotically deterministic. Then we prove that conversely, if G is the complete graph, the result of metastability holds for any positive γ .

中文翻译：

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图连通性对尖峰神经元随机系统中亚稳态的影响

摘要 我们考虑了最初由 Ferrari 等人引入的尖峰神经元的连续时间随机模型。在法拉利等人。(2018)。在这个模型中，我们有有限或可数数量的神经元，它们是某些图 G 中的顶点，其中边表示它们之间的突触连接。我们专注于亚稳态，将其理解为网络消亡时间渐近无记忆的特性，并且我们证明该模型表现出两种不同的行为，具体取决于所选择的交互 G 的特定底层图的性质。在该模型中，任何给定神经元的尖峰活动由一个点过程表示，其速率随时间在 1 和 0 之间波动，具体取决于膜电位是正值还是零值。每个神经元的膜电位通过将其相邻神经元的所有尖峰积分到该神经元的最后一个尖峰来及时演变，这样当一个神经元尖峰时，其膜电位重置为0，而其每个的膜电位相邻的神经元增加一个单位。此外，每个神经元都受到泄漏效应的影响，模拟为膜电位的突然损失，这种损失在随机时间发生，由某个固定速率 γ 的泊松过程驱动。先前已经证明，当图 G 是无限一维晶格时，该模型呈现相对于参数 γ 的相变。还证明，当 γ 足够小时，当神经元数量趋于无穷大时，系统的有限版本的重新归一化灭绝时间（所有神经元第一次具有零膜电位）在法律上收敛于指数随机变量。本文分为两部分。首先我们证明，在有限的一维晶格中，如果 γ 足够大，则最后一个结果不再成立，事实上，我们证明对于 γ > 1，重新归一化的消光时间是渐近确定的。然后我们反过来证明，如果 G 是完整图，则亚稳态的结果对于任何正 γ 都成立。如果 γ 足够大，这最后一个结果不再成立，事实上我们证明，对于 γ > 1，重新归一化的灭绝时间是渐近确定的。然后我们反过来证明，如果 G 是完整图，则亚稳态的结果对于任何正 γ 都成立。如果 γ 足够大，这最后一个结果不再成立，事实上我们证明，对于 γ > 1，重新归一化的灭绝时间是渐近确定的。然后我们反过来证明，如果 G 是完整图，则亚稳态的结果对于任何正 γ 都成立。