Stimulus-correlated spike-triggered averaging

For a single linear filter followed by a rate nonlinearity and a Gaussian stimulus-history vector $s$ with covariance $C$, Gaussian integration by parts gives $\mathbb{E}[sg(k^{T}s)]\propto Ck$. Thus a spike-triggered average estimates $Ck$, rather than $k$ directly, when the stimulus is correlated. Covariance correction, with regularization if necessary, can recover the filter direction. For an even nonlinearity the proportionality factor can vanish; then a one-filter average is insufficient.