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

Neural Computation 18: 2592-2616.

We compute the exact spike-triggered average (STA) of the voltage for
the nonleaky 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 $\sigma \to 0$. We
consider both the STA and the conditional average voltage given an
observed spike ``doublet'' event, i.e. 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.

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