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O. Schwartz, E. P. Simoncelli; A spike-triggered covariance method for characterizing divisive normalization models. Journal of Vision 2001;1(3):450. https://doi.org/10.1167/1.3.450.
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Spike-triggered average (reverse correlation) techniques are effective for linear characterization of neural responses. But cortical neurons exhibit striking nonlinear behaviors that are not captured by such analyses. Many of these nonlinear behaviors are consistent with a gain control (divisive normalization) model. We develop a spike-triggered covariance method for recovering the parameters of such a model. We assume a specific form of normalization, in which spike rate is determined by the half wave-rectified and squared response of a linear kernel divided by the weighted sum of squared responses of linear kernels at different positions, orientations, and spatial frequencies. The method proceeds in two steps. First, the linear kernel of the numerator is estimated using traditional spike-triggered averaging. Second, we measure responses with the exCitation of the numerator kernel held constant (this is accomplished by stimulus design, or during data analysis) but with random exCitation along all other axes. We construct a covariance matrix of the stimuli eliciting a spike, and perform a principal components decomposition of this matrix. The principal axes (eigenvectors) correspond to the directions in which the response of the neuron is modulated divisively. The variance along each axis (eigenvalue) is monotonically decreasing as a function of strength of suppression along that axis. The kernels and weights of an equivalent normalization model may be estimated from these eigenvalues and eigenvectors. We demonstrate through simulation that the technique yields a good estimate of the model parameters, and we examine accuracy as a function of the number of spikes. This method provides an opportunity to test a normalization model experimentally, by first estimating model parameters for an individual neuron, and then examining the ability of the resulting model to account for responses of that neuron to a variety of other stimuli.
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