Spike-triggered covariance (STC) analysis is a tool for characterizing a neuron's response properties using the

covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

of stimuli that elicit spikes from a neuron. STC is related to the

spike-triggered average The spike-triggered averaging (STA) is a tool for characterizing the response properties of a neuron using the spikes emitted in response to a time-varying stimulus. The STA provides an estimate of a neuron's linear receptive field. It is a useful ...

(STA), and provides a complementary tool for estimating linear filters in a linear-nonlinear-Poisson (LNP) cascade model. Unlike STA, the STC can be used to identify a multi-dimensional feature space in which a neuron computes its response. STC analysis identifies the stimulus features affecting a neuron's response via an eigenvector decomposition of the spike-triggered

covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...

.Brenner, N., Bialek, W., & de Ruyter van Steveninck, R.R. (2000). Schwartz, O., Chichilnisky, E. J., & Simoncelli, E. P. (2002). Bialek, W. & de Ruyter van Steveninck, R. (2005). Arxiv preprint q-bio/0505003.Schwartz O., Pillow J. W., Rust N. C., & Simoncelli E. P. (2006). Spike-triggered neural characterization. ''Journal of Vision'' 6:484-507 Eigenvectors with eigenvalues significantly larger or smaller than the eigenvalues of the raw stimulus covariance correspond to stimulus axes along which the neural response is enhanced or suppressed. STC analysis is similar to

principal components analysis Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...

(PCA), although it differs in that the eigenvectors corresponding to largest ''and'' smallest eigenvalues are used for identifying the feature space. The STC matrix is also known as the 2nd-order

Volterra Volterra (; Latin: ''Volaterrae'') is a walled mountaintop town in the Tuscany region of Italy. Its history dates from before the 8th century BC and it has substantial structures from the Etruscan, Roman, and Medieval periods. History Volter ...

or Wiener kernel.

Mathematical definition

Standard STC

Let

\mathbf

denote the spatio-temporal stimulus vector preceding the

i

'th time bin, and

y_i

the spike count in that bin. The stimuli can be assumed to have zero mean (i.e.,

E mathbf 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 :

\operatorname = \frac\sum_^T y_i (\mathbf-STA)(\mathbf-STA)^T,

where

n_s = \sum y_i

is the total number of spikes, and STA is the

. The covariance of the stimulus is given by :

\mathrm = \frac 1 \sum_^T \mathbf\mathbf,

where

n_p

is the number of stimuli

\mathbf

used during the experiment. The eigenvectors of

(STC-C)

associated to significantly positive eigenvalues correspond to excitatory vectors, whereas eigenvectors associated to significantly negative eigenvalues are inhibitory eigenvectors.

References

{{reflist

External links

Matlab code for STA/STC analysis of neural data
Computational neuroscience Covariance and correlation