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Guilherme Wood, Moritz Mahr, H. Nuerk (2005)
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FMRI data observed under a given experimental condition may be decomposed into two parts: the average effect and the deviation of single replications from this average effect. The average effect is represented by the mean activation over a specific condition. The deviation from this average effect may be decomposed into two components as well: systematic variation due to known empirical factors and pure measurement error. In most fMRI designs deviations from mean activation may be treated as measurement error. Nevertheless, often deviation from the average also may contain systematic variation that can be distinguished from simple measurement error. In these cases, the average fMRI signal may provide only a coarse picture of real brain activation. The larger the variation within-condition, the coarser the average effect and the more relevant is the impact of deviations from it. Systematic deviation from the mean activation may be examined by defining a set of parametric regressors. Here, the applicability of parametric methods to refine the evaluation of fMRI studies is discussed with special emphasis on (i) examination of the impact of continuous predictors on the fMRI signal, (ii) control for variation within each experimental condition and (iii) isolation of specific contributions by different features of a single complex stimulus, especially in the case of a sampled stimulus. The usefulness and applicability of this method are discussed and an example with real data is presented. use of parametric designs in fMRI research, discuss shortly Background We present an update about the use of a special type of its mathematical background and applicability, and parametric designs in fMRI research that can be very useful present an empirical example where parametric regressors in investigations involving natural and multi-featured carry the most relevant modulation of the fMRI signal. stimuli such as pictures or words. This method has already been developed by Büchel and colleagues [1] but unfortu- In several fields of neurocognition, stimuli can be nately it has not been used as frequently as it deserves. For assigned to experimental conditions so that they (i) are this reason, we present a summary of the logic behind the homogeneous within each cell of an experimental design Page 1 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 and (ii) differ only with regard to a single aspect across the Overview of the method different levels of an experimental factor. Each statistical Activation Y in a particular voxel can be described as in ij contrast unequivocally isolates therefore one and only (1) for each replication i (for every i from 1 to p) of an one neurocognitive process. However, in the case of natu- experimental condition j (for every j from 1 to q): ral stimuli, such as pictures (e.g. kitchen utensils vs. garage tools) or written words (e.g. varying in length, number of Y = α + β X + ε (1) ij j j ij syllables, frequency, neighbourhoods, regularities, con- sistencies etc.), the task of matching different groups of having a as the intercept, X as a (continuous) parameter items for their attributes is particularly challenging, β as the describing the present experimental design, because there is often only a finite a number of stimuli to regression coefficient for the parameter X and ε as resid- j ij fit into each of the different cells of the experimental ual error. The coefficient β describes the event- or block- design that vary simultaneously in more than one feature. specific expected BOLD-response under a given experi- In these cases, different dimensions of stimuli can only be mental condition j assuming that within an experimental matched on average. Words, for instance, can vary in the condition the BOLD-response induced by event- or block- number of letters, the frequency of occurrence, the specific stimulation will be a constant. A corollary of this number of lexical neighbours as well as the frequency of assumption is that variation in the BOLD response occur- occurrence of orthographic or phonological sub-lexical ring within an experimental condition will be considered units. Different words may have for example 1, 2 or 8 dif- residual error. ferent lexical neighbours. Therefore, for each stimulus dimension (i.e. frequency of occurrence, number of lexi- When stimuli in an experimental condition are sampled cal neighbours, etc.) there is a non-zero distance between from a universe of natural items, some variation among each single item and the average for each of the different items will always be present. An artificial increase of resid- dimensions, characterizing the amount of variation ual error ε due to variation in the BOLD-response pro- within condition. duced by variation within condition contributes negatively to the sensitivity of the fMRI design. Impor- Due to variation within condition, the statistics for the tantly, when the variance among items not only repre- size of fMRI signal elicited by the different items pertain- sents a confounding factor but genuine scaling properties ing to the very same condition may vary considerably. of stimulus features, it is mandatory to deal with them Consequently, the type-II error for detecting a difference appropriately by modelling this variance within condi- in fMRI signal between two different conditions may be tions. inflated. The main problem for the interpretation of the results of such an experiment is whether it is acceptable to Parametric modelling always allows for the description of consider the variation within condition as measurement variation in the event- or block specific BOLD-response, error or not. If the variation within condition is small in when the source (or sources) of variation is known a priori comparison with the variation between different condi- and can be specified numerically as parametric regressor. tions, treating it as measurement error is not problematic. Importantly, the variation within conditions may be due However, if the variation within a cell of the experimental not only to one single stimulus feature, but rather be due design increases due to systematic variation in known to two or more features. In this case, for each of the dimensions of multi-featured stimuli, the validity of the dimensions of multi-featured stimuli a regressor can be whole study may be questioned. defined, which absorbs the contribution of that dimen- sion for the variation within a given experimental condi- In the present paper we examine a method proposed by tion (but see the section on the limitations of this Büchel and colleagues [1] for dealing with systematic var- approach in the discussion, below). The specification of iation between items. The method involves the definition parametric regressors is given as follows: the parameter X of parametric regressors representing each of the several described by a canonical hemodynamic function in com- dimensions of complex stimuli. These parametric regres- mon fMRI designs, which has exactly the same form sors absorb the systematic variation inherent in different across all replications i of a given experimental condition dimensions of complex stimuli such as words, sentences j, can be expressed as the average effect X of a predictor X or arithmetic problems, and allow for separating it from on brain activation. Moreover, in parametric designs a sec- genuine measurement error. In the following, we will ond set of predictors may be complemented by a set of k present the method, discuss its main applications, and dimensions (for every k from 1 to r) which are nested present an example in which the variation between items under each condition j and which absorb the variation (and their exact scaling properties) was the most relevant within each condition. The full model presented in (2) aspect of the experimental design. contains a predictor representing the average effect of experimental condition j plus an additional parameter for Page 2 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 each parametric regressor k considered. X , X , ... X ... normally differ only with respect to one out of a set of par- ij1 ij2 ijk X contain the variation in each of k different stimulus ametric regressors. With this second type of application, ijr β are hierarchically dimensions. Note that parameters we are able to statistically test hypotheses about the exact jk bound to the average parameter β and that the number of form of variation in fMRI activation. parameters β associated with each average parameter β jk j may differ. Therefore, (1) can be generalized by assuming In the following example, we compare the model fit a set of r > 1 dimensions: obtained for different numerical compressions of the pre- dictors employed (i.e. logarithmic vs. linear scale). Results Y = α + β X + β X + ... + β X + ... +β X + ε of these comparisons may help to determine the exact ij j j j1 ij1 jk ijk jr ijr ij form of variation and the underlying rate of neuronal (2) response to each of the different stimulus dimensions examined. By entering parametric regressors in the fMRI design, the proportion of variance which can be accounted for by the An empirical example variation within conditions is separated from the residual Numerical cognition provides a straightforward example error ε . This extension of the model presented in (1) has ij for the usefulness of parametric regression. Numbers do two consequences: (i) the statistical test on the signifi- naturally differ in their parametric properties, such as, for cance of null-order parameter β will not be biased by var- instance, their magnitude [2-4]. Since no number shares iation within conditions, which can be explained by the same magnitude with another, naturally there is vari- predictors β to β . (ii) Furthermore, the relevance of j1 jr ation in this dimension within every experimental condi- regression coefficients β to β may be assessed. j1 jr tion in which different numbers are used. Number magnitude is assumed to be represented in the cortex The definition of parametric regressors with the single around the intraparietal sulcus (IPS) [2,3]. Behavioral purpose of isolating variation within conditions as a con- studies [5] and a neural network model [6] have indicated founding factor is trivial and has been employed regularly that numerical distance is logarithmically compressed. in fMRI research. The sole purpose of this application is to Some recent single-cell recording studies reported that control for the impact of undesired sources of variance cells in pre-frontal and parietal cortex are tuned to specific affecting statistics about the effects of interest. In this case, magnitudes [7-11]; their signal is best described by a log- variation within conditions can be considered an effect of arithmically compressed scale [10,11]. Similar results non-interest, the impact of which on the statistics can be have been obtained in fMRI studies [12,13]. Furthermore, partialled out from residual error. studies on two-digit number processing have shown that participants may not be able to compare the magnitude of Nevertheless, parametric regressors also may be defined decade digits while ignoring the unit digits, even when the with the aim of directly investigating theoretical predic- units are totally irrelevant for the task at hand [see tions with respect to the fMRI activation observed. In the [5,14,15] for a review, [16,17]]. following, we will concentrate on the advantages and lim- itations of such an application. In fact, parametric regres- Given this theoretical background, we ask two empirical sors make possible an investigation of the direction and questions about the fMRI signal that can be investigated actual scaling properties of variation of fMRI activation. more precisely by means of parametric than by conven- Examination of the impact of quantitative regressors on tional categorical methods. The first question is whether the fMRI activation has been presented by Büchel and col- the fMRI signal in the intraparietal cortex is better leagues [1]. In that study the authors defined one single accounted for by the overall distance when participants parametric regressor and applied polynomial expansions are asked to choose the larger from two two-digit Arabic (i.e. quadratic, cubic, etc.) to investigate non-linear rela- numbers or by the distance between decade digits. Since tionships between the BOLD-response and this single there are no two-digit numbers "without" a unit digit to experimental parameter. Here we use the method [1] for serve as stimuli for a control condition, the only way to two purposes: (i) instead of examining the impact of pol- examine this problem empirically is to compare the ynomial expansion of a single parametric predictor on BOLD-response evoked by overall distance with that fMRI activation, defining a set of predictors which, accord- evoked by decade distance (decade digit – dec- ing to some theoretical expectation, may account for a sig- larger number ade digit ). If the statistical fit for overall dis- nificant amount of variability among trials produced by smaller number tance is better than for decade distance, one may infer that known and quantifiable properties of stimuli. Further- the fMRI activation in the intraparietal cortex due to the more, the method is useful for (ii) assessing the signifi- contrast (overall distance > decade distance) is associated cance of each single parameter for brain activation (i.e. with the processing of the overall magnitude of numbers. one-sample t-tests) to the comparison between different models (i.e. statistics for two or more samples), which Page 3 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 A second empirical question is whether the fMRI signal in fMRI acquisition intraparietal cortex is better accounted for by the loga- Two functional imaging runs sensitive to blood oxygena- rithm of the distance than by the linear distance between tion level-dependent (BOLD) contrast were recorded for two-digit numbers. This question has been investigated each participant with a Philips 1.5T Gyroscan MRI system first in an fMRI study by Pinel and colleagues [4]. These (T2*-weighted echo-planar sequence, TR = 2800 ms; TE = authors found that in six out of seven regions of interest 50 ms; flip angle = 90°; FOV = 220 mm, 64 × 64 matrix; the percent signal change dropped in accord with the log- 30 slices, voxel size = 3.4 × 3.4 × 4 mm). In each run, 316 arithm of the distance between numbers rather than with scans + 5 dummy scans were acquired. In a rapid event- the linear distance. Nevertheless, the authors examined related design, a total of 576 trials (480 experimental trials the effect of logarithmic scaling on fMRI signal by splitting + 96 null events) were presented at a rate of 3 seconds. The the range of distances into three arbitrary categories (i.e. fMRI time series was corrected for movement and small, medium and large distances) instead of treating unwarped in SPM2. Images were resampled every 4-mm distance as a continuum. This approach presents disad- and normalized to a standard EPI template using the sinc vantages in comparison with the modelling with paramet- interpolation method. Moreover, functional images were ric regressors: The method employed by Pinel and co-registered with the normalized anatomical pictures. colleagues [4] may fail to distinguish between the impact Finally, functional images were smoothed with an 8-mm of decade distance and overall distance on fMRI signal Gaussian kernel. (i.e. the first empirical question examined in the present Parametric design example). This may have affected the determination of the exact spatial distribution of the neurons responding more We convolved brain activity for all experimental trials strongly to the logarithmically compressed magnitude of with the canonical hemodynamic response function numbers. (HRF) in a single experimental condition and defined three parametric regressors representing overall distance, dec- In the following, we describe the results of the parametric ade distance, and problem size. The correlations between analysis of an fMRI experiment examining the two empir- the different parameters and the in-line correlations (i.e. ical questions stated above. the correlations obtained after convolution with the HRF function) between the parametric regressors and the aver- Procedure age hemodynamic response are shown in Table 1 and Fourteen male right-handed volunteers (mean age = 27, Table 2, respectively. In order to scale the estimated regres- range 21–38 years) took part in the study after giving their sion parameters uniformly, the parametric regressors rep- written consent to the imaging protocol which has been resenting overall distance, decade distance, and problem approved by the local Ethics Committee of the Medical size were standardized to a mean of 0 and a standard devi- Faculty and is in compliance with the Helsinki Declara- ation of 1. tion. Participants had to select the larger number of a pair of two-digit Arabic numbers (range: 21–98) and press a In order to examine whether the fMRI signal in the intra- key [for further details on the design of experiment and parietal cortex can be better accounted for by the overall characteristics of the task as well as on behavioural data, distance than by decade distance alone, we estimated two see [16], including supplementary material]. Overall dis- separate models. In one model, overall distance and prob- tance, decade distance, unit distance and problem size lem size were entered as parametric regressors and, in a were matched both absolutely and logarithmically separate model, decade distance and problem size were between stimulus categories [16]. The four digits chosen as units and decades of the two-digit number pair were Table 1: Means and correlation matrix for the parametric always different. Furthermore, in the present study unit regressors (n = 240 items, variances in the main diagonal) numbers were totally irrelevant for magnitude compari- dist10 logdist10 dist logdist size son since no within decade comparisons were included. dist10 391 MRI acquisition logdist10 0.97 0.09 For each participant, a high-resolution T1-weighted ana- dist 0.98 0.95 377 tomical scan was acquired with a Philips 1.5T Gyroscan logdist 0.95 0.96 0.97 0.08 size -0.03 -0.05 -0.03 -0.05 652 MRI system (TR = 30 ms, matrix = 256 × 256 mm, 170 slices, voxel size = 0.86 × 0.86 × 2 mm; FOV = 220 mm, mean 36.63 1.48 36.72 1.49 118.43 TE = 4.6 ms; flip angle = 30°). The anatomical scans were normalized using the standard T1 template of SPM2. dist10: decade distance, logdist10: base-10 logarithm of decade distance, dist: overall distance, logdist: base-10 logarithm of overall distance, size: problem size Page 4 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 Table 2: In line correlation matrix for the parametric regressors parietal cortex centered at Talairach coordinates x = -40, y = -34, z = 41 (t(13) = 5.54; p < .001) and x = 45, y = -40, z Model 1 Model 2 = 51 (t(13) = 4.21; p = .001; Figure 2). The contrast "dec- "overall distance" "decade distance" ade distance > overall distance" revealed no activation in intraparietal cortex but only a slight deactivation in the Overall Problem Decade Problem distance size distance size left angular gyrus. ROI analysis revealed a significant deac- tivation for logarithmic decade distance in comparison Average BOLD function -0.05 -0.05 with logarithmic overall distance in the left angular gyrus Overall distance -0.03 -0.03 -0.03 -0.02 (x = -44, y = -62, z = 31; t(13) = -4.63; p < .001; Figure 2). entered as parametric regressors. A summary of the proce- To examine whether the fMRI signal in the intraparietal dure for definition, estimation, and statistical assessment cortex could be better accounted for by the logarithm of of the different parametric models is presented in Table 3. the distance than by the linear overall distance between the two two-digit numbers, we estimated two separate ROI analysis models: In one model, overall distance was entered as a To avoid the problem of multiple comparisons typical for parametric regressor; in a separate model, the logarithm of whole brain analysis when assessing the empirical overall distance was employed as a parametric regressor. hypotheses about the amount of signal captured by para- Both linear and logarithmic overall distance were signifi- metric predictors, small volume analysis was carried out cant predictors of activation in intraparietal cortex. Inter- in specific sub-regions of parietal cortex. For the analysis estingly, the logarithmic overall distance was a better of the regions of interest (ROI), 6 mm-spheres in the left predictor of fMRI activation in a broad network of brain and right parietal cortex were extracted from the brain regions including the right and left posterior intraparietal images using the toolbox MARSBAR. Selection of these cortex, left anterior intraparietal cortex, left extrastriate ROIs was based on regions showing significant differences cortex, left premotor cortex, right frontal operculum, right in the experimental contrasts in the whole brain analysis. SMA, right ventrolateral prefrontal cortex, right premotor cortex and the right orbitofrontal cortex (Table 4). In the Results contrast "linear overall distance > logarithmic overall dis- One-sample t-contrasts revealed that overall- and decade tance" no activation was observed at the threshold of p = distance as well as problem size predicted activation in .001, uncorrected, k = 10. parietal cortex – especially in the cortical regions in the vicinity of the intraparietal sulcus – as well as in occipital, To examine the differential impact of logarithmic overall premotor, and prefrontal cortices (Figure 1). Interestingly, distance on the MRI activation in IPS, four ROIs were the paired two-sample contrast "overall distance > decade extracted for the contrast "logarithmic overall distance > distance" revealed strong fMRI activation in the intrapari- linear overall distance". Posterior left (x = -32, y = -53, z = etal cortex (Table 4). ROI analyses pointed out that signif- 44, t(13) = 7.15; p < .001) and right intraparietal cortex (x icantly more activation in response to overall distance = 37, y = -56, z = 49; t(13) = 8.10; p < .001) as well as left than for decade distance was found in the left and right (x = -52, y = -46, z = 38; t(13) = 5.97; p < .001) and right Table 3: Summary of model definition, estimation and statistical comparison using parametric predictors Model 1 Model 2 Model name "overall distance" "decade distance" First level predictors "overall distance" "decade distance" (coefficient estimation) , "problem size" , "problem size" Second level "overall distance" > "decade distance" Paired two-sample t-tests "overall distance" < "decade distance" Model name "log-overall distance" "overall distance" First level predictors "log overall distance", "overall distance", (coefficient estimation) "problem size" "problem size" Second level "log-overall distance" > "overall distance" Paired two-sample t-tests "log-overall distance" < "overall distance" Page 5 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 Table 4: Brain areas activated more by overall distance or decade distance, respectively Overall distance > decade distance Region Talairach t-value BA Cluster coordinates x, y, df = 13 size k z§ Left extrastriate cortex -24, -93, 8 10.10** 19 356 Right extrastriate cortex 32, -86, -2 10.02** 19 - Left anterior intraparietal cortex -51, -33, 38 8.39** 40 32 Left striate cortex -16, -66, 7 5.87** 40 15 Right anterior intraparietal cortex 44, -37, 42 5.15** 40 20 Left superior parietal lobule -4, -60, 51 6.07** 7 18 Right fusiform gyrus 36, -13, -20 5.17** 19 10 Decade distance > overall distance Left angular gyrus -44, -60, 33 -5.38* 39 10 Logarithmic overall distance > overall distance Left posterior intraparietal cortex -33, -52, 44 7.42** 7 75 Left anterior intraparietal cortex -32, -30, 41 5.61** 40 Left extrastriate cortex -28, -84, 18 6.18** 26 Left premotor cortex -24, -5, 48 7.49** 26 Left premotor cortex -43, 1, 44 7.07** 46 Right posterior intraparietal cortex 37, -56, 50 8.45** 7 234 Right anterior intraparietal cortex 40, -33, 41 7.65** -a 40 Right frontal operculum 34, 18, 10 7.21** 21 Right SMA 4, 7, 51 6.41** 29 Right ventrolateral prefrontal cortex 45, 16, -2 6.06** 21 Right premotor cortex 44, -1, 49 6.05** 34 Right orbitofrontal cortex 31, 39, -9 5.80** 16 Overall distance > logarithmic overall distance No suprathreshold clusters § transformed from the MNI coordinates with the SPM tool mni2tal; ** p-value at the cluster level < .05, corrected; a: local maximum in the same cluster as the right posterior intraparietal cortex anterior intraparietal cortex (x = 40, y = -33, z = 41; t(13) not able to ignore the magnitude of unit digits when com- = 7.10; p < .001) were activated more strongly by logarith- paring two-digit numbers: unit magnitudes are processed mic than by linear overall distance. – behaviorally as well as neurofunctionally – even if they are irrelevant for the comparison [4,5,14-17]. Further- Discussion more, by examining the influence of decades and units on In the present paper we have examined the applicability of brain activation, we found that the left angular gyrus was the parametric methods presented in [1] in two ways: (i) deactivated more in response to decade distances than to the specific impact of each one out of a set of parametric overall distances. Commonly, deactivation of the angular regressors on fMRI activation (Figure 1) and (ii) the com- gyrus is interpreted as the product of an enhancement of parison of different quantitative models of the scaling visuospatial attention [18]. In the present case we tenta- properties of fMRI activation (Figures 2 and 3). Using par- tively interpret the stronger deactivation in response to ametric modelling of fMRI data, we have shown that the decade distances as a product of the effort implied in the hemodynamic response in the intraparietal cortex, bilater- selection of just the decade distances for comparison. ally, is sensitive to the overall magnitude of two-digit Since the correct result for the magnitude comparison numbers (decade + unit distances), since the parametric could be reached in the experimental task by comparing model containing overall distance predicted fMRI activa- the decade digits alone, participants may have tried to tion significantly better than the model containing decade process their magnitude in more detail than unit magni- distance only. These results indicate that participants are tudes. For doing so, they need more visuospatial attention Page 6 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 A 10 Figure 1 ctivation produced by voxels) (A) logarithm of overall distance, (B) overall distance and (C) problem size (p < .001, uncorrected, k = Activation produced by (A) logarithm of overall distance, (B) overall distance and (C) problem size (p < .001, uncorrected, k = 10 voxels). to select decade digits in the visual display. To our knowl- plex stimuli, especially in the case of stimuli sampled edge this is the first report of that effect, which should be from a pool of natural items. In the final section we will investigated in further studies. discuss some limitations of the parametric method. Continuous predictors of fMRI signal Together, results also support the view that number mag- nitude is represented in the intraparietal cortex in a loga- The most important feature of the parametric method is rithmically compressed fashion, that irrelevant unit the modelling of variation in different dimensions of nat- magnitudes determine fMRI activation, and that partici- ural stimuli in a natural way that is not constrained by the pants may engage visuospatial attention in order to select necessity of generating (sometimes arbitrary) categories of decade digits for processing. In general, the present data stimuli in order to look at average differences between are in line with an extensive behavioural and fMRI litera- these categories. Therefore, no categorization of distances ture [[2,4,5,12,14], and [3] for a review]. is necessary when comparing for instance linear and loga- rithmic scales, since the distances themselves are entered In the following, we will discuss the relevance of the in the model as predictors of activation. present results as an illustration of the advantages of mod- elling the fMRI signal with parametric regressors. Specifi- One could argue that instead of employing parametric cally, three points will be emphazised: (i) the impact of modelling, the experimenter could mask the decade or continuous predictors on fMRI signal, (ii) the control of unit digit of different numbers in order to isolate the variation within experimental conditions due to known effects of overall magnitude and decades in the different features of complex stimuli and (iii) the isolation of spe- experimental conditions. This approach is, however, cific contributions by quantifiable features of single com- problematic, since it is not clear whether the single digits Page 7 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 stronger activation for th Figure 2 Voxels showing stronger a e de ctivation for the overall distance cade distance than for overall distance are than for decade coloured red ( distance are colo p < .001, ured blu uncor erected, k = 10 voxels) while voxels showing Voxels showing stronger activation for the overall distance than for decade distance are coloured blue while voxels showing stronger activation for the decade distance than for overall distance are coloured red (p < .001, uncorrected, k = 10 voxels). While overall distance activated large portions of the anterior intraparietal cortex bilaterally in comparison with decade distance only, as well as in the extrastriate cortex, decade distance only deactivated voxels in the left angular gyrus relative to overall distance. ROI analyses revealed stronger activation in the intraparietal cortex bilaterally in response to overall distance compared to decade distance as well as a slight deactivation in the left angular gyrus. at the decade and unit positions in two-digit numbers also signal. Parametric regressors were not entered in the have their own magnitude representation. For this reason, model only in order to control for known sources of inho- presenting digits of a two-digit number separately from mogeneity, but they were the actual experimental factor. each other would possibly lead to activation of separate magnitudes [17,19] and cannot be considered as a valid As already pointed out in [1], the experimenter may iso- test about two-digit number processing. With parametric late linear and non-linear contributions of the same pre- regressors, the selective contribution of one stimulus fea- dictor to the BOLD-response. In this sense, parametric ture can be assessed in experimental settings in which models may be able to better approximate scaling proper- other stimulus features cannot be held constant without ties of activity in very specialized groups of neurons. The destroying their usual perceptual or semantic structure. empirical hypotheses tested in the present study involved Among these features, which cannot be held constant, one predominantly the scaling of the fMRI signal (linear vs. may differentiate between those which are of experimen- logarithmic distances). Comparing the relative fit tal interest and those which are simply confounds without obtained by modelling data with linear predictors vs. log- any special theoretical meaning. This latter aspect of sam- arithmically compressed regressors was made possible by pled stimuli will be examined in the next section. employing parametric models. The aim of modelling data with linear vs. logarithmic predictors was to compare the Variation of interest and of no-interest within an relative fit obtained by modelling fMRI signal either with experimental condition linear overall distance or with the logarithm of overall dis- As mentioned in the overview about this method, para- tance. As illustrated by the present results, the parametric metric regressors may represent the only source of vari- method allows for detecting even very subtle details of the ance of interest in an experiment. In the empirical scaling properties of the BOLD response. Inspection of example, parametric regressors were shown to be suited Table 1 reveals that the parametric regressors representing for testing non-trivial empirical hypotheses about fMRI overall distance and decade distance are highly correlated Page 8 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 RO Figure 3 I analysis of the contrast logarithmic overall distance > linear overall distance ROI analysis of the contrast logarithmic overall distance > linear overall distance. Both, in the anterior and poste- rior intraparietal cortex, bilaterally, more activation was found for the logarithm of overall distance. (r(240) = .96, p < .05). This means that the advantage of predictors should be orthogonal to each other. If this con- overall distance in predicting fMRI activation in the intra- dition cannot be reached in a specific case, the correla- parietal cortex bilaterally is due to fine-grained differences tions between the different predictors should be held low in the scaling properties of regressors representing overall (see the section on the limitations of parametric models, distance and decade distance, which can only be captured below). Furthermore, the number of predictors entering a in a parametric model. Voxels significantly better tuned to parametric model should be much smaller than the the logarithm of overall distance than to the overall dis- number of scans in the individual time series from which tance itself could be observed in the intraparietal cortex. the beta coefficients for each condition are estimated. In Similar arguments can be put forward for comparing the general this is not problematic, since the number of scans relative response of voxels in the angular gyrus to decade acquired for reach participant is very large in comparison distance and overall distance. Our analyses suggest that with the number of experimental conditions of interested. overall distance rather than decade distance is generally more closely related to IPS activation while decade dis- In the example presented above, the activation produced tance led to deactivation of the left angular gyrus. by numerical distance and by problem size in a given voxel could be disentangled using the parametric model: As mentioned before, in parametric analyses, predictors Both overall distance and problem size activated highly may represent different features of stimuli, which, even overlapping regions in the intraparietal cortex bilaterally. being theoretically different constructs, may selectively Nevertheless, there is no doubt that overall distance and contribute to the activation in a single voxel. Importantly, problem size remain different features of two-digit num- the covariance structure between different predictors bers. We also have demonstrated a selective increase in should be taken into consideration when designing a par- fMRI signal in the intraparietal cortex which was better ametric fMRI study. In the optimal case the parametric explained by the overall distance than by decade distance Page 9 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 alone. Moreover, we also found a small cluster of voxels Parametric designs are not a substitute for the careful in the left angular gyrus, which responded more strongly selection of items for an empirical investigation. The cor- to decade distance alone rather than to overall distance relation matrix of the different properties of stimuli is the (Figure 2). Only with parametric regressors it was possible most adequate index for assessing the suitability of para- to "extract" and assess the selective contribution of the metric modelling of fMRI data. When the correlations decades for the fMRI signal to a complex stimulus, which between different predictors are moderate or high, their by commonly activates different perceptual, motor and interpretability as separate conceptual entities is compro- cognitive representations [3]. mised. For this reason, the selection of adequate predic- tors for a parametric study may turn out to be a non-trivial Isolating the contribution made by different features of a task, since the correlations between the different paramet- single complex stimulus ric properties of items, which are controlled for in a given Natural stimuli are complex objects which carry different empirical investigation, should be held small or even perceptual as well as different conceptual features. These non-significant. In the present study, we were able to iso- features may present variation across the different objects late the impact of overall distance and problem size on selected to form a category of objects in an empirical fMRI signal because the correlation between these two investigation. Ollinger and colleagues [20] presented a properties of items could be kept very low (r(240) = -.03, method for separating processes within a trial in event- n.s.). Accordingly, in the study by Wood and colleagues related fMRI designs. The authors have shown that percep- [16] the selective impact of decade and unit distance tual, cognitive, and motor processes may be confounded could be well separated in a parametric analysis since the in complex tasks. They presented a method for isolating correlation between the parametric regressors represent- the relative impact of each one of these single processes by ing them was low, too (r(240) = .18, p < .05). means of a manipulation of the sequence of trials. One important precondition for employing this method is that The only case in which parametric modelling with highly the different processes activated within a trial can be correlated predictors is informative is the comparison assessed separately. Unfortunately, this assumption is not between the relative fit of two different models, one con- valid for all experimental designs. In the domain of taining one of the two predictors and the other model number processing, examination of the relative impact of containing the other one. A paired two-sample t-test may decade and unit digits on brain activation cannot be reveal whether one of these predictors explains more var- investigated in a natural way without presenting decade iance of fMRI signal (see the section on linear vs. logarith- and unit digits in every trial, for there is no two-digit mic scaling of overall distance, above). In this case, the number without a decade or a unit digit. In parametric only difference between the two models should be pro- models the impact of different features of a complex stim- duced by the scaling of the parametric predictor. Neurons ulus can be modelled simultaneously, their specific effects in intraparietal cortex bilaterally respond to the magni- can be isolated from the effects of other features, and most tude of two-digit numbers in a logarithmically com- importantly, the specific effect of each feature on the acti- pressed fashion. This piece of evidence is in line with vation observed in a given voxel can be assessed statisti- current theories of number magnitude processing [3] and cally. In the example presented above, the amount of with evidence from behavioural [14], fMRI [4,12] as well signal produced by problem size was controlled for in the as single-cell recording studies [7-11]. Moreover, a discus- different analyses, since problem size was entered as a sec- sion on the compression of magnitude representation has ond predictor in every model examined. In the present been put forward by Dehaene [6], who argues that the case, the correlation between regressors representing magnitude representation may assume a more linear scal- those numerical distances and that representing problem ing with training in arithmetical tasks. The question size was not different from 0 (Table 1). Therefore, the whether the neural response in the intraparietal cortex effect of problem size on fMRI activation did not interfere also changes from a logarithmically compressed scaling to substantially with the impact of overall and decade dis- a more linear one can be directly assessed with parametric tances on fMRI activation. models, indicating that the parametric method represents not only an alternative method for data analysis but also However, in specific applications the correlation between a tool for testing specific empirical hypotheses with more different stimulus dimensions may differ from 0. In such precision. cases it is imperative to define the different stimulus Some limitations dimensions in the same model, in order to isolate the spe- cific contribution of each dimension to the fMRI activa- Multi-collinearity tion and to obtain the correct statistics regarding each of As in every implementation of the general linear model, these correlated dimensions. only the orthogonal part of the variance of a parametric regressor has its impact on fMRI signal tested for statistical Page 10 of 12 (page number not for citation purposes) Behavioral and Brain Functions 2008, 4:38 http://www.behavioralandbrainfunctions.com/content/4/1/38 significance [21]. When the different parametric regres- Authors' contributions sors are highly correlated, the orthogonal part of each par- GW Participated in the design of the study and carried out ametric regressor may become very small and lose behavioral and fMRI measures and statistical analyses, DS empirical relevance. For this reason, the interpretation of Carried out the region-of-interest analyses, HCN and KW parametric designs in the presence of multi-collinearity is participated in the conceptual formulation of the research problematic. Before carrying out a parametric fMRI study, question and the design of the study. All authors read and a careful selection of items should be conducted in order approved the final version of the manuscript. to avoid large correlations between the predictors of inter- est. In any case, the correlations between the different pre- Acknowledgements None. dictors in an fMRI design should be inspected and reported in the manuscript. References 1. Büchel C, Holmes AP, Rees G, Friston KJ: Characterizing stimu- "Deactivation" and inverted contrasts lus-response functions using nonlinear regressors in para- Since the parametric regressors represent the deviation of metric fMRI experiments. Neuroimage 1998, 8:140-148. 2. Cohen Kadosh R, Lammertyn J, Izard V: Are numbers special? An single items from the average of the experimental condi- overview of chronometric, neuroimaging, developmental tion and not the average activation itself, the notion of and comparative studies of magnitude representation. Prog "deactivation" must be viewed differently in parametric Neurobiol 2008, 84:132-147. 3. Hubbard EM, Piazza M, Pinel P, Dehaene S: Interactions between models. "Deactivation" in a paramentric regressor means number and space in parietal cortex. Nat Rev Neurosci 2005, that the direction of the association between variation 6:435-448. 4. Pinel P, Dehaene S, Riviére D, LeBihan D: Modulation of parietal within condition and fMRI signal is inverted. activation by magnitude distance in a number comparison task. Neuroimage 2001, 14:1013-1026. Conclusion 5. Dehaene S, Dupoux E, Mehler J: Is numerical comparison digital? Analogical and symbolic effects in two-digit number compar- The parametric method can be very useful for investiga- ison. J Exp Psychol Hum Percept Perform 1990, 16:626-641. tions involving complex stimuli characterised by several 6. Dehaene S: Symbols and quantities in parietal cortex: Ele- ments of a mathematical theory of number representation different features. The more complex the tasks (e.g. com- and manipulation. In Attention & Performance XXII. Sensori-motor plex arithmetic tasks or reading words from very specific foundations of higher cognition Edited by: Haggard P, Rossetti Y. Cam- word classes), the more adequate is the parametric mod- bridge, Mass.: Harvard University Press; 2007. 7. Nieder A, Diester I, Tudusciuc O: Temporal and spatial enumer- elling of stimulus features, since in many occasions ation processes in the primate parietal cortex. Science 2006, authentic variation in stimulus properties cannot be 313:1431-1435. matched exactly between different conditions, but only 8. Nieder A, Freedman DJ, Miller EK: Representation of the quan- tity of visual items in the primate prefrontal cortex. Science on average. In these cases genuine variance within condi- 2002, 297:1708-1711. tions will be present and should be treated as such and not 9. Nieder A, Miller EK: Coding of cognitive magnitude: Com- pressed scaling of numerical information in the primate pre- as measurement error. Modelling fMRI data using para- frontal cortex. Neuron 2003, 37:149-157. metric regressors allows for the simultaneous quantifica- 10. Nieder A, Miller EK: Analog numerical representations in Rhe- tion of variation in many stimulus dimensions and may sus monkeys: Evidence for parallel processing. J Cogn Neurosci 2004, 16:889-901. be very useful for simultaneously isolating and statisti- 11. Nieder A, Miller EK: A parieto-frontal network for visual cally assessing the contribution of variation in different numerical information in the monkey. Proc Natl Acad Sci USA 2004, 101:7457-7462. dimensions. However, the careful choice of items in each 12. Piazza M, Izard V, Pinel P, Le Bihan D, Dehaene S: Tuning curves for experimental condition cannot be substituted by adding approximate numerosity in the human intraparietal sulcus. parametric regressors to the statistical models at the high Neuron 2004, 44:547-55. 13. Piazza M, Pinel P, Le Bihan D, Dehaene S: A magnitude code com- cost of interpretability of results. Moreover, the correla- mon to numerosities and number symbols in human intrapa- tions between the different parametric predictors entering rietal cortex. Neuron 2007, 53:293-305. the statistical model should be zero or close to zero. The 14. Nuerk H-C, Weger U, Willmes K: Decade breaks in the mental number line? Putting the tens and units back in different only exception from this rule is the comparison between bins. Cognition 2001, 82:B25-B33. linear and non-linear transformation of the same para- 15. Nuerk H-C, Willmes K: On the magnitude representation of two-digit numbers. Psych Sci 2005, 47:52-72. metric predictor. Finally, when the (statistical) assump- 16. Wood G, Nuerk H-C, Willmes K: Neural representations of tions for their use are fulfilled, parametric models two-digit numbers: A parametric fMRI study. Neuroimage represent a very useful tool for assessing empirical 2006, 29:358-367. 17. Wood G, Nuerk H-C, Moeller K, Geppert B, Schnitker R, Weber J, hypotheses in fMRI studies more precisely. Willmes K: All for one but not one for all: How multiple number representations are recruited in one numerical task. Brain Res 2008, 1187:154-166. Competing interests 18. 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Behavioral and Brain Functions – Springer Journals
Published: Aug 15, 2008
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