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'''Maximally informative dimensions''' is a [[dimensionality reduction]] technique used in the statistical analyses of [[neural coding|neural responses]]. Specifically, it is a way of projecting a stimulus onto a low dimensional [[subspace]] so that as much [[mutual information|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 [[linear span|spanned]] by [[white noise]].<ref>D.J. Field. "Relations between the statistics of natural images and the response properties of cortical cells." J. Opt. Soc. am. A 4:2479-2394, 1987.</ref> Within this subspace, however, [[stimulus-response model|stimulus-response functions]] may be either [[linear]] or [[nonlinear]]. The idea was originally developed by Tatyana Sharpee, Nicole Rust, and [[William Bialek]] in 2003.<ref>Sharpee, Tatyana, Nicole C. Rust, and William Bialek. "Maximally informative dimensions: analyzing neural responses to natural signals." Advances in Neural Information Processing Systems (2003): 277-284.</ref>
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==Mathematical formulation==
Neural stimulus-response functions are typically given as the probability of a [[neuron]] generating an [[action potential]], or spike, in response to a stimulus <math>\mathbf{s}</math>. The goal of maximally informative dimensions is to find a small relevant subspace of the much larger stimulus space that accurately captures the salient features of <math>\mathbf{s}</math>. Let <math>D</math> denote the dimensionality of the entire stimulus space and <math>K</math> denote the dimensionality of the relevant subspace, such that <math>K \ll D</math>. We let <math>\{\mathbf{v}^K\}</math> denote the basis of the relevant subspace, and <math>\mathbf{s}^K</math> the [[Projection (mathematics)|projection]] of <math>\mathbf{s}</math> onto <math>\{ \mathbf{v}^K \}</math>. Using [[Bayes' theorem]] we can write out the probability of a spike given a stimulus:
 
: <math>P(spike|\mathbf{s}^K) = P(spike)f(\mathbf{s}^K)</math>
 
where
 
: <math>f(\mathbf{s}^K) = \frac{P(\mathbf{s}^K|spike)}{P(\mathbf{s}^K)}</math>
 
is some nonlinear function of the projected stimulus.
 
In order to choose the optimal <math>\{ \mathbf{v}^K \}</math>, we compare the prior stimulus distribution <math>P(\mathbf{s})</math> with the spike-triggered stimulus distribution <math>P(\mathbf{s}|spike)</math> using the [[mutual information|Shannon information]]. The [[mean|average]] information (averaged across all presented stimuli) per spike is given by
 
: <math>I_{spike} = \sum_{\mathbf{s}} P(\mathbf{s}|spike) log_2 [P(\mathbf{s}|spike)/P(\mathbf{s})]</math>.<ref>N. Brenner, S. P. Strong, R. Koberle, W. Bialek, and R. R. de Ruyter van Steveninck. "Synergy in a neural code. Neural Comp., 12:1531-1552, 2000.</ref>
 
Now consider a <math>K = 1</math> dimensional subspace defined by a single direction <math>\mathbf{v}</math>. The average information conveyed by a single spike about the projection <math>x = \mathbf{s} \cdot \mathbf{v}</math> is
 
: <math>I(\mathbf{v}) = \int dx P_{\mathbf{v}}(x|spike)log2[P_{\mathbf{v}}(x|spike)/P_{\mathbf{v}}(x)]</math>,
 
where the probability distributions are approximated by a measured data set via <math>P_{\mathbf{v}}(x|spike) = \langle \delta(x - \mathbf{s} \cdot \mathbf{v}) |spike \rangle_{\mathbf{s}}</math> and <math>P_{\mathbf{v}}(x) = \langle \delta(x - \mathbf{s} \cdot \mathbf{v})\rangle_{\mathbf{s}}</math>, i.e., each presented stimulus is represented by a scaled [[Dirac delta function]] and the probability distributions are created by averaging over all spike-eliciting stimuli, in the former case, or the entire presented stimulus set, in the latter case. For a given dataset, the average information is a function only of the direction <math>\mathbf{v}</math>. Under this formulation, the relevant subspace of dimension <math>K = 1</math> would be defined by the direction <math>\mathbf{v}</math> that maximizes the average information <math>I(\mathbf{v})</math>.
 
This procedure can readily be extended to a relevant subspace of dimension <math>K > 1</math> by defining
 
: <math>P_{\mathbf{v}^K}(\mathbf{x}|spike) = \langle \prod_{i=1}^K \delta(x_i - \mathbf{s} \cdot \mathbf{v}_i) |spike \rangle_{\mathbf{s}}</math>
and
: <math>P_{\mathbf{v}^K}(\mathbf{x}) = \langle \prod_{i=1}^K \delta(x_i - \mathbf{s} \cdot \mathbf{v}_i) \rangle_{\mathbf{s}}</math>
 
and maximizing <math>I({\mathbf{v}^K})</math>.
 
==Importance==
 
Maximally informative dimensions does not make any assumptions about the [[normal distribution|Gaussianity]] of the stimulus set, which is important, because naturalistic stimuli tend to have non-Gaussian statistics. In this way the technique is more robust than other dimensionality reduction techniques such as [[spike-triggered covariance]] analyses.
 
==References==
{{Reflist}}
 
[[Category:Neurology]]

Latest revision as of 18:15, 30 August 2014

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