On a toy model of interacting neurons
On a toy model of interacting neurons
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关于相互作用神经元的玩具模型
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Eva Locherbach
中科院分区：
文献类型：
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作者：
N. Fournier;Eva Locherbach
我们继续研究De Masi-Galves-L“ocherbach-Presutti（2014）中介绍的相互作用神经元的随机系统。该系统由N个神经元组成，每个神经元随机地以取决于其膜电位的速率尖峰。在其尖峰时刻，神经元电位被重置为0，并且所有其他神经元接收额外量的1/N电位。此外，电突触诱导系统向其质心的确定性漂移。证明了当N趋于无穷大时，系统的混沌传播为一个极限非线性跳跃随机微分方程。因此，我们改进了De Masi-Galves-L“ocherbach-Presutti（2014）的结果，因为（i）我们删除了初始数据上的紧支撑条件，（ii）我们得到了1/sqrt N$的收敛速度。最后，我们研究了极限方程：我们描述了它的时间边缘的形状，我们证明了唯一的非平凡不变分布的存在性，我们证明了平凡不变分布是不吸引的，并且在一个特殊的情况下，我们建立了收敛到平衡点。
We continue the study of a stochastic system of interacting neurons introduced in De Masi-Galves-L"ocherbach-Presutti (2014). The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to 0 and all other neurons receive an additional amount 1/N of potential. Moreover, electrical synapses induce a deterministic drift of the system towards its center of mass. We prove propagation of chaos of the system, as N tends to infinity, to a limit nonlinear jumping stochastic differential equation. We consequently improve on the results of De Masi-Galves-L"ocherbach-Presutti (2014), since (i) we remove the compact support condition on the initial datum, (ii) we get a rate of convergence in $1/sqrt N$. Finally, we study the limit equation: we describe the shape of its time-marginals, we prove the existence of a unique non-trivial invariant distribution, we show that the trivial invariant distribution is not attractive, and in a special case, we establish the convergence to equilibrium.