Standard STA

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 STA is given

${\displaystyle \mathrm {STA} ={\tfrac {1}{n_{sp}}}\sum _{i=1}^{T}y_{i}\mathbf {x_{i}} ,}$

where ${\displaystyle n_{sp}=\sum y_{i}}$, the total number of spikes.

This equation is more easily expressed in matrix notation: let ${\displaystyle X}$ denote a matrix whose ${\displaystyle i}$'th row is the stimulus vector ${\displaystyle \mathbf {x_{i}^{T}} }$ and let ${\displaystyle \mathbf {y} }$ denote a column vector whose ${\displaystyle i}$th element is ${\displaystyle y_{i}}$. Then the STA can be written

${\displaystyle \mathrm {STA} ={\tfrac {1}{n_{sp}}}X^{T}\mathbf {y} .}$

Whitened STA

If the stimulus is not white noise, but instead has non-zero correlation across space or time, the standard STA provides a biased estimate of the linear receptive field.^[5] It may therefore be appropriate to whiten the STA by the inverse of the stimulus covariance matrix. This resolves the spatial dependency issue, however we still assume the stimulus is temporally independent. The resulting estimator is known as the whitened STA, which is given by

${\displaystyle \mathrm {STA} _{w}=\left({\tfrac {1}{T}}\sum _{i=1}^{T}\mathbf {x_{i}} \mathbf {x_{i}} ^{T}\right)^{-1}\left({\tfrac {1}{n_{sp}}}\sum _{i=1}^{T}y_{i}\mathbf {x_{i}} \right),}$

where the first term is the inverse covariance matrix of the raw stimuli and the second is the standard STA. In matrix notation, this can be written

${\displaystyle \mathrm {STA} _{w}={\tfrac {T}{n_{sp}}}\left(X^{T}X\right)^{-1}X^{T}\mathbf {y} .}$

The whitened STA is unbiased only if the stimulus distribution can be described by a correlated Gaussian distribution ^[6] (correlated Gaussian distributions are elliptically symmetric, i.e. can be made spherically symmetric by a linear transformation, but not all elliptically symmetric distributions are Gaussian). This is a weaker condition than spherical symmetry.

The whitened STA is equivalent to linear least-squares regression of the stimulus against the spike train.

Regularized STA

In practice, it may be necessary to regularize the whitened STA, since whitening amplifies noise along stimulus dimensions that are poorly explored by the stimulus (i.e., axes along which the stimulus has low variance). A common approach to this problem is ridge regression. The regularized STA, computed using ridge regression, can be written

${\displaystyle \mathrm {STA} _{ridge}={\tfrac {T}{n_{sp}}}\left(X^{T}X+\lambda I\right)^{-1}X^{T}\mathbf {y} ,}$

where ${\displaystyle I}$ denotes the identity matrix and ${\displaystyle \lambda }$ is the ridge parameter controlling the amount of regularization. This procedure has a simple Bayesian interpretation: ridge regression is equivalent to placing a prior on the STA elements that says they are drawn i.i.d. from a zero-mean Gaussian prior with covariance proportional to the identity matrix. The ridge parameter sets the inverse variance of this prior, and is usually fit by cross-validation or empirical Bayes.

Consistency

The whitened STA is a consistent estimator, i.e., it converges to the true linear subspace, if

1. The stimulus distribution ${\displaystyle P(\mathbf {x} )}$ is elliptically symmetric, e.g., Gaussian. (Bussgang's theorem)
2. The expected STA is not zero, i.e., nonlinearity induces a shift in the spike-triggered stimuli.^[5]

Optimality

The whitened STA is an asymptotically efficient estimator if

1. The stimulus distribution ${\displaystyle P(\mathbf {x} )}$ is Gaussian
2. The neuron's nonlinear response function is the exponential, ${\displaystyle exp(x)}$.^[5]

For arbitrary stimuli, the STA is generally not consistent or efficient. For such cases, maximum likelihood and information-based estimators ^[5][6][13] have been developed that are both consistent and efficient.