101:2186-2193, 2009. First published Jan 7, 2009; doi:10.1152/jn.90727.2008 J NeurophysiolSam Behseta, Tamara Berdyyeva, Carl R. Olson and Robert E. Kass You might find this additional information useful...15 articles, 6 of which you can access free at: This article cites http://jn.physiology.org/cgi/content/full/101/4/2186#BIBLincluding high-resolution figures, can be found at: Updated information and services http://jn.physiology.org/cgi/content/full/101/4/2186 can be found at: Journal of Neurophysiologyabout Additional material and information http://www.the-aps.org/publications/jnThis information is current as of March 10, 2010 . http://www.the-aps.org/.American Physiological Society. ISSN: 0022-3077, ESSN: 1522-1598. Visit our website at (monthly) by the American Physiological Society, 9650 Rockville Pike, Bethesda MD 20814-3991. Copyright © 2005 by the publishes original articles on the function of the nervous system. It is published 12 times a yearJournal of Neurophysiology on March 10, 2010 jn.physiology.orgDownloaded fromInnovative MethodologyBayesian Correction for Attenuation of Correlation in Multi-Trial SpikeCount DataSam Behseta,1Tamara Berdyyeva,2Carl R. Olson,2and Robert E. Kass2,31Department of Mathematics, California State University, Fullerton, California; and2Center for the Neural Basis of Cognitionand3Department of Statistics, Carnegie Mellon University, Pittsburgh, PennsylvaniaSubmitted 2 July 2008; accepted in final form 20 December 2008Behseta S, Berdyyeva T, Olson CR, Kass RE. Bayesian correctionfor attenuation of correlation in multi-trial spike count data. J Neu-rophysiol 101: 2186 –2193, 2009. First published January 7, 2009;doi:10.1152/jn.90727.2008. When correlation is measured in the pres-ence of noise, its value is decreased. In single-neuron recordingexperiments, for example, the correlation of selectivity indices in apair of tasks may be assessed across neurons, but, because the numberof trials is limited, the measured index values for each neuron will benoisy. This attenuates the correlation. A correction for such attenua-tion was proposed by Spearman more than 100 yr ago, and morerecent work has shown how confidence intervals may be constructedto supplement the correction. In this paper, we propose an alternativeBayesian correction. A simulation study shows that this approach can befar superior to Spearman’s, both in accuracy of the correction and incoverage of the resulting confidence intervals. We demonstrate theusefulness of this technology by applying it to a set of data obtained fromthe frontal cortex of a macaque monkey while performing serial order andvariable reward saccade tasks. There the correction results in a substantialincrease in the correlation across neurons in the two tasks.INTRODUCTIONA central theme in the statistical analysis of neuronal data isthe appropriate accounting for uncertainty. This often involvesthe inclusion of sources of variability that might otherwise beomitted. Although in some cases taking into account additionalsources of variability may decrease the magnitude of an effect(Behseta et al. 2005), in other cases, the effect of interest mayactually increase. An important example of this second situa-tion involves estimation of correlation in the presence of noise.Suppose␪and␰are random variables having a positivecorrelation␳␪␰and ␧ and␦are independent “noise” randomvariables that corrupt the measurement of␪and␰producingX ⫽␪⫹ ␧ and Y ⫽␰⫹␦. A simple mathematical argument(see APPENDIX) shows that␳XY⬍␳␪␰where␳XYis the correlation between X and Y. In words, thepresence of noise decreases the correlation. Thus in many cir-cumstances, a measured correlation will underestimate thestrength of the actual correlation between two variables. However,if the likely magnitude of the noise is known, it becomes possibleto correct the estimate. The purpose of this note is to provide aBayesian correction and to show that it can produce good resultswhen examining correlations derived from multi-trial spikecounts.We apply the method in the context of single-neuronalrecording experiments. Broadly speaking, it is sometimes nec-essary to compare the selectivity of a neuron for a particularvariable across two task contexts. For example, one might wishto compare shape selectivity across blocks of trials in which theshape has different colors (Edwards et al. 2003) or compareselectivity for saccade direction across blocks of trials in whichthe saccade is selected according to different rules (Olson et al.2000). It is also sometimes necessary to compare selectivity fortwo different variables as measured in separate task contexts.For example, we might wish to compare selectivity for thedirection of motion of a visual stimulus viewed during passivefixation with selectivity for saccade direction in a task requir-ing eye movements (Horwitz and Newsome 2001). The stan-dard approach to making such comparisons is to compute, formultiple neurons, index 1 in context 1 and index 2 in context2 and then to compute the correlation between the two indicesacross neurons. The correlation may be statistically significantbut smaller than one might expect, which raises the question: isthe small correlation due to a genuine discordance between thetwo forms of selectivity, or is it due to noise arising fromrandom trial-to-trial variability in the neuronal firing rates?This is the kind of question the methods of this article aredesigned to answer.The idea of introducing a “correction for attenuation” of thecorrelation goes back at least to Spearman (1904). He did notat that time, however, have the technology to provide confi-dence intervals associated with his proposed technique. Frostand Thompson (2000) reviewed some solutions to the problemof constructing confidence intervals for the slope of a noise-corrupted regression line, and Charles (2005) gave proceduresfor obtaining confidence intervals for the correlation based onSpearman’s formula. We performed a computer simulationstudy to compare the Bayesian correction with Spearman’scorrection and the Bayesian confidence intervals with thosebased on Spearman’s correction. We found the Bayesianmethod to be far superior. We then applied the method to datafrom the frontal cortex of a macaque monkey recorded whilethe monkey was performing serial order and variable rewardsaccade tasks.METHODSNotationLet Xi⫽␪i⫹ ␧i, and Yi⫽␰i⫹␦i,