The spike-triggered average of the integrate-and-fire cell driven by gaussian white noise

We compute the exact spike-triggered average (STA) of the voltage for the nonleaky integrate-and-fire (IF) cell in continuous time, driven by gaussian white noise. The computation is based on techniques from the theory of renewal processes and continuous-time hidden Markov processes (e.g., the backward and forward Fokker-Planck partial differential equations associated with first-passage time densities). From the STA voltage, it is straightforward to derive the STA input current. The theory also gives an explicit asymptotic approximation for the STA of the leaky IF cell, valid in the low-noise regime sigma --> 0. We consider both the STA and the conditional average voltage given an observed spike "doublet" event, that is, two spikes separated by some fixed period of silence. In each case, we find that the STA as a function of time-preceding-spike, tau, has a square root singularity as tau approaches zero from below and scales linearly with the scale of injected noise current. We close by briefly examining the discrete-time case, where similar phenomena are observed.