Fluctuation-response relations for integrate-and-fire models with an absolute refractory period

We study the problem of relating the spontaneous fluctuations of a stochastic integrate-and-fire (IF) model to the response of the instantaneous firing rate to time-dependent stimulation if the IF model is endowed with a non-vanishing refractory period and a finite (stereotypical) spike shape. This seemingly harmless addition to the model is shown to complicate the analysis put forward by Lindner Phys. Rev. Lett. (2022), i.e., the incorporation of the reset into the model equation, the Rice-like averaging of the stochastic differential equation, and the application of the Furutsu-Novikov theorem. We derive a still exact (although more complicated) fluctuation-response relation (FRR) for an IF model with refractory state and a white Gaussian background noise. We also briefly discuss an approximation for the case of a colored Gaussian noise and conclude with a summary and outlook on open problems.