We consider a dynamic system that is driven by an intensity-modulated Poisson process with intensity Λ(t)=λ(t)+ϵν(t). We derive an exact relation between the input-output cross-correlation in the spontaneous state (ϵ=0) and the linear response to the modulation (ϵ>0). If ϵ is sufficiently small, linear-response theory captures the full response. The relation can be regarded as a variant of the Furutsu-Novikov theorem for the case of shot noise. As we show, the relation is still valid in the presence of additional independent noise. Furthermore, we derive an extension to Cox-process input, which provides an instance of colored shot noise. We discuss applications to particle detection and to neuroscience. Using the new relation, we obtain a fluctuation-response relation for a leaky integrate-and-fire neuron. We also show how the new relation can be used in a remote control problem in a recurrent neural network. The relations are numerically tested for both stationary and nonstationary dynamics. Lastly, extensions to marked Poisson processes and to higher-order statistics are presented. Published by the American Physical Society 2024

中文翻译：

---

散粒噪声驱动系统的类互相关-响应关系


我们考虑一个动态系统，它由强度调制的泊松过程驱动，强度为 Λ（t）=λ（t）+εν（t）。我们推导出自发状态下的输入-输出互相关 （ε=0） 与对调制的线性响应 （ε>0） 之间的精确关系。如果 ε 足够小，则线性响应理论捕获完整响应。该关系可以看作是散粒噪声情况下的 Furutsu-Novikov 定理的变体。正如我们所展示的，在存在额外的独立噪声的情况下，这种关系仍然有效。此外，我们推导出了 Cox 进程输入的扩展，它提供了一个彩色散粒噪声的实例。我们讨论了粒子检测和神经科学的应用。使用新的关系，我们获得了泄漏的 integrate-and-fire 神经元的波动-响应关系。我们还展示了如何将新关系用于递归神经网络中的远程控制问题。这些关系针对稳态和非稳态动力学进行了数值检验。最后，介绍了标记 Poisson 过程和高阶统计量的扩展。 美国物理学会 2024 年出版