Spike-triggered covariance

Last updated February 23, 2019

Spike-triggered covariance (STC) analysis is a tool for characterizing a neuron's response properties using the covariance of stimuli that elicit spikes from a neuron. STC is related to the spike-triggered average (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.

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 lesser values,, the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other,, the covariance is negative. The sign of the covariance therefore shows the tendency in the linear relationship between the variables. The magnitude of the covariance is not easy to interpret because it is not normalized and hence depends on the magnitudes of the variables. The normalized version of the covariance, the correlation coefficient, however, shows by its magnitude the strength of the linear relation.

The spike-triggered average (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 technique for the analysis of electrophysiological data.

The linear-nonlinear-Poisson (LNP) cascade model is a simplified functional model of neural spike responses. It has been successfully used to describe the response characteristics of neurons in early sensory pathways, especially the visual system. The LNP model is generally implicit when using reverse correlation or the spike-triggered average to characterize neural responses with white-noise stimuli.

STC analysis identifies the stimulus features affecting a neuron's response via an eigenvector decomposition of the spike-triggered covariance matrix. [1] [2] [3] [4] 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.

In probability theory and statistics, a covariance matrix, also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix, is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has either a finite number of observed empirical values or a finite or infinite number of potential values. The potential values are specified by a theoretical joint probability distribution.

STC analysis is similar to principal components analysis (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 or Wiener kernel.

In mathematics, the Wiener series originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to the Volterra series and having the same relation to it as an orthogonal Hermite polynomial expansion has to a power series. For this reason it is also known as the Wiener-Hermite expansion. The analogue of the coefficients are referred to as Wiener kernels. The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of white noise. This property allows the terms to be identified in applications by the Lee-Schetzen method.

Mathematical definition

Standard STC

Let $\mathbf {x_{i}}$ 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., $E[\mathbf {x} ]=0$ ). 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

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

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

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

where $n_{p}$ is the number of stimuli $\mathbf {x_{i}}$ 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. [4]

Related Research Articles

Principal component analysis conversion of a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components

Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. If there are $observations with variables, then the number of distinct principal components is . This transformation is defined in such a way that the first principal component has the largest possible variance, and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. The resulting vectors are an uncorrelated orthogonal basis set. PCA is sensitive to the relative scaling of the original variables.$

Eigenface set of eigenvectors used in the computer vision problem of human face recognition

Eigenfaces is the name given to a set of eigenvectors when they are used in the computer vision problem of human face recognition. The approach of using eigenfaces for recognition was developed by Sirovich and Kirby (1987) and used by Matthew Turk and Alex Pentland in face classification. The eigenvectors are derived from the covariance matrix of the probability distribution over the high-dimensional vector space of face images. The eigenfaces themselves form a basis set of all images used to construct the covariance matrix. This produces dimension reduction by allowing the smaller set of basis images to represent the original training images. Classification can be achieved by comparing how faces are represented by the basis set.

In linear algebra, a generalized eigenvector of an n × n matrix $is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.$

FastICA is an efficient and popular algorithm for independent component analysis invented by Aapo Hyvärinen at Helsinki University of Technology. Like most ICA algorithms, FastICA seeks an orthogonal rotation of prewhitened data, through a fixed-point iteration scheme, that maximizes a measure of non-Gaussianity of the rotated components. Non-gaussianity serves as a proxy for statistical independence, which is a very strong condition and requires infinite data to verify. FastICA can also be alternatively derived as an approximative Newton iteration.

In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say, or etc.. It is a two-dimensional case of the general n-dimensional phase space.

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. More formally, if $T$ is a linear transformation from a vector space $V$ over a field $F$ into itself and $v$ is a vector in $V$ that is not the zero vector, then $v$ is an eigenvector of $T$ if $T (v)$ is a scalar multiple of $v$ . This condition can be written as the equation

In graph theory, eigenvector centrality is a measure of the influence of a node in a network. Relative scores are assigned to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes. A high eigenvector score means that a node is connected to many nodes who themselves have high scores.

In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method.

In the field of multivariate statistics, kernel principal component analysis is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space.

Neural coding is a neuroscience field concerned with characterising the hypothetical relationship between the stimulus and the individual or ensemble neuronal responses and the relationship among the electrical activity of the neurons in the ensemble. Based on the theory that sensory and other information is represented in the brain by networks of neurons, it is thought that neurons can encode both digital and analog information.

Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja, is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons.

The point distribution model is a model for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes.

In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way.

In statistics, the projection matrix $, sometimes also called the influence matrix or hat matrix, maps the vector of response values to the vector of fitted values. It describes the influence each response value has on each fitted value. The diagonal elements of the projection matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation.$

In statistics, principal component regression (PCR) is a regression analysis technique that is based on principal component analysis (PCA). Typically, it considers regressing the outcome on a set of covariates based on a standard linear regression model, but uses PCA for estimating the unknown regression coefficients in the model.

Common spatial pattern (CSP) is a mathematical procedure used in signal processing for separating a multivariate signal into additive subcomponents which have maximum differences in variance between two windows.

Maximally informative dimensions is a dimensionality reduction technique used in the statistical analyses of neural responses. Specifically, it is a way of projecting a stimulus onto a low-dimensional subspace so that as much information as possible about the stimulus is preserved in the neural response. It is motivated by the fact that natural stimuli are typically confined by their statistics to a lower-dimensional space than that spanned by white noise. Within this subspace, however, stimulus-response functions may be either linear or nonlinear. The idea was originally developed by Tatyana Sharpee, Nicole Rust, and William Bialek in 2003.

References

↑ 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.
1 2 Schwartz O., Pillow J. W., Rust N. C., & Simoncelli E. P. (2006). Spike-triggered neural characterization. Journal of Vision 6:484-507

External links

Matlab code for STA/STC analysis of neural data

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[brenner00-1] Brenner, N., Bialek, W., & de Ruyter van Steveninck, R.R. (2000).

[2] Schwartz, O., Chichilnisky, E. J., & Simoncelli, E. P. (2002).

[bialek05-3] Bialek, W. & de Ruyter van Steveninck, R. (2005). Arxiv preprint q-bio/0505003.

[schwartz06-4] 1 2 Schwartz O., Pillow J. W., Rust N. C., & Simoncelli E. P. (2006). Spike-triggered neural characterization. Journal of Vision 6:484-507