Fluctuation-response relations and response-response relations for membrane voltage and spike train of stochastic integrate-and-fire neurons

Neurons display spontaneous spiking (in the absence of stimulus signals) as well as a characteristic response to time-dependent external stimuli. In a simple but important class of stochastic neuron models, the integrate-and-fire model with Gaussian current noise, both aspects can mathematically be related via fluctuation-response relations (FRRs) as has been shown recently [B. Lindner, Phys. Rev. Lett. 129, 198101 (2022)0031-900710.1103/PhysRevLett.129.198101]. Here we extend the class of FRRs to include the susceptibilities of the membrane voltage and subthreshold voltage nonlinearity as well as the power spectrum of the membrane voltage. For a simple but often considered IF model, the leaky IF model with white Gaussian noise, we exploit the FRRs and derive explicit expressions for the power spectrum and susceptibility of the subthreshold membrane voltage. We also put forward a relation between the response functions of the spike train and membrane voltage, a response-response relation (RRR) that holds true for a more general setting than considered in most parts of the paper. For the generalized IF model with an adaptation current and colored Gaussian noise, we derive an FRR and an RRR. We briefly discuss useful applications of the derived FRRs and RRRs.