SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 828-859, January 2020.
The present paper provides exact mathematical expressions for the high-order moments of spiking activity in a recurrently connected network of linear Hawkes processes. It extends previous studies that have explored the case of a (linear) Hawkes network driven by deterministic intensity functions to the case of a stimulation by external inputs (rate functions or spike trains) with arbitrary correlation structure. Our approach describes the spatio-temporal filtering induced by the afferent and recurrent connectivities (with arbitrary synaptic response kernels) using operators acting on the input moments. This algebraic viewpoint provides intuition about how the network ingredients shape the input-output mapping for moments, as well as cumulants. We also show using numerical simulation that our results hold for neurons with refractoriness implemented by self-inhibition, provided the corresponding negative feedback for each neuron only mildly alters its mean firing probability.

中文翻译：

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线性霍克斯网络中尖峰矩的传播

SIAM应用动力系统杂志，第19卷，第2期，第828-859页，2020年1月。
本文为线性霍克斯过程的递归连接网络中的尖峰活动的高阶矩提供了精确的数学表达式。它扩展了以前的研究，该研究探索了由确定性强度函数驱动的（线性）Hawkes网络的情况，到具有任意相关结构的外部输入（速率函数或峰值序列）刺激的情况。我们的方法描述了使用作用于输入力矩的算子，由传入和递归连接性（具有任意突触响应内核）引起的时空滤波。这种代数观点提供了关于网络要素如何塑造矩以及累积量的输入-输出映射的直觉。