We investigate the asymptotic behaviour of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctutations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resolvent are explicit, our analysis in principle also allows to derive approximation errors in terms of driving martingales. We also discuss extensions of our results to multi-class systems.

我们研究了相互作用的非线性霍克斯过程网络的渐近行为，这些过程在大种群限制中对同质神经元种群进行建模。特别是，我们证明了平均尖峰活动的功能中心极限定理，从而根据随机沃尔泰拉积分方程表征渐近波动。我们的方法与以前的方法不同，它利用相关的解析器将波动表示为弱收敛鞅的 Skorokhod 连续映射。由于解析器的 Lipschitz 特性是明确的，我们的分析原则上也允许根据驱动鞅得出近似误差。我们还讨论了将我们的结果扩展到多类系统。