Furutsu-Novikov--like cross-correlation--response relations for systems driven by shot noise

TL;DR

This paper derives an exact relation between input-output cross-correlation and linear response for systems driven by shot noise, extending the Furutsu-Novikov theorem, with applications in neuroscience and particle detection.

Contribution

It introduces a novel Furutsu-Novikov-like relation for shot noise systems, including colored and Cox-process inputs, applicable to neuroscience and particle detection.

Findings

Derived an exact cross-correlation-response relation for shot noise systems

Extended the relation to colored shot noise and Cox processes

Demonstrated applications in neural and particle detection models

Abstract

We consider a dynamic system that is driven by an intensity-modulated Poisson process with intensity $\Lambda(t)=\lambda(t)+\varepsilon\nu(t)$. We derive an exact relation between the input-output cross-correlation in the spontaneous state ($\varepsilon=0$) and the linear response to the modulation ($\varepsilon>0$). If $\varepsilon$ 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 non-stationary dynamics. Lastly, extensions to marked Poisson processes and to higher-order statistics are presented.