Spike-triggered covariance

Let ${\displaystyle \mathbf {x_{i}} }$ denote the spatio-temporal stimulus vector preceding the ${\displaystyle i}$'th time bin, and ${\displaystyle y_{i}}$ the spike count in that bin. The stimuli can be assumed to have zero mean (i.e., ${\displaystyle E[\mathbf {x} ]=0}$). If not, it can be transformed to have zero-mean by subtracting the mean stimulus from each vector. The spike-triggered covariance (STC) is given by

${\displaystyle \operatorname {STC} ={\frac {1}{n_{s}-1}}\sum _{i=1}^{T}y_{i}(\mathbf {x_{i}} -STA)(\mathbf {x_{i}} -STA)^{T},}$

where ${\displaystyle n_{s}=\sum y_{i}}$ is the total number of spikes, and STA is the spike-triggered average. The covariance of the stimulus is given by

${\displaystyle \mathrm {C} ={\frac {1}{n_{p}-1}}\sum _{i=1}^{T}\mathbf {x_{i}} \mathbf {x_{i}^{T}} ,}$

where ${\displaystyle n_{p}}$ is the number of stimuli ${\displaystyle \mathbf {x_{i}} }$ used during the experiment. The eigenvectors of ${\displaystyle (STC-C)}$ associated to significantly positive eigenvalues correspond to excitatory vectors, whereas eigenvectors associated to significantly negative eigenvalues are inhibitory eigenvectors.[4]