稀疏整合放电神经元网络的平均场极限
The mean-field Limit of sparse networks of integrate and fire neurons
AI总结：
研究基于随机IF动力学的稀疏生物神经元网络平均场极限，提出新可观测概念扩展至非相同/不可交换系统，通过定量稳定性估计在BBGKY层级中取极限，解决稀疏图与泊松跳奇异交互挑战。
摘要：
我们研究了基于所谓随机整合放电（IF）动力学的生物神经元网络模型的平均场极限。我们的方法允许推导出系统宏观行为（即单粒子分布）的连续极限，适用于大量神经元且对连接图无需在广义平均场缩放之外施加结构假设。我们提出了一种新的可观测量的概念，自然地将边缘分布的概念扩展到具有非相同或不可交换智能体的系统中。我们的新可观测值满足一个复杂的近似层级结构，本质上是经典BBGKY层级的树索引扩展。通过在某种适配的弱范数中的新型定量稳定性估计，我们能够在神经元数量增加时在该层级中取极限。虽然我们要求非零扩散，但该方法显著地解决了稀疏交互图/矩阵和来自泊松跳的奇异交互的挑战，且对初始分布无需额外正则性。
英文摘要：
We study the mean-field limit of a model of biological neuron networks based on the so-called stochastic integrate-and-fire (IF) dynamics. Our approach allows to derive a continuous limit for the macroscopic behavior of the system, the 1-particle distribution, for a large number of neurons with no structural assumptions on the connection map outside of a generalized mean-field scaling. We propose a novel notion of observables that naturally extends the notion of marginals to systems with non-identical or non-exchangeable agents. Our new observables satisfy a complex approximate hierarchy, essentially a tree-indexed extension of the classical BBGKY hierarchy. We are able to pass to the limit in this hierarchy as the number of neurons increases through novel quantitative stability estimates in some adapted weak norm. While we require non-vanishing diffusion, this approach notably addresses the challenges of sparse interacting graphs/matrices and singular interactions from Poisson jumps, and requires no additional regularity on the initial distribution.
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