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a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m

w w w. e l s e v i e r. c o m / l o c a t e / b r a i n r e s

Research Report

Parametric effects of numerical distance on the intraparietal sulcus during passive viewing of rapid numerosity changes Daniel Ansari ⁎, Bibek Dhital, Soon Chun Siong Numerical Cognition Laboratory, Department of Education, Dartmouth College, 3 Maynard Street, Raven House, Hanover, NH 03755, USA

A R T I C LE I N FO

AB S T R A C T

Article history:

A number of functional neuroimaging studies have revealed that regions in and around the

Accepted 5 October 2005

intraparietal sulcus (IPS) are parametrically modulated by numerical distance, whereby

Available online 15 December 2005

there is an inverse relationship between distance and levels of activation. These areas are thus thought to contain the internal representation of numerical magnitude. Nevertheless,

Theme:

it has also been suggested that the IPS is involved in response selection during number

Neural basis of behavior

comparison tasks rather than the representation of numerical magnitude per se. In order to

Topic:

test the independence of the effect of distance on cortical regions, we employed a passive

Cognition

viewing paradigm. Sixteen right-handed male participants viewed rapidly changing slides containing arrays of squares. By varying the distance between the numerosity presented in

Keywords:

separate blocks (8 vs. 8, 8 vs. 12, and 8 vs. 16), we examined which regions exhibit a

Functional MRI

parametric effect of numerical distance. This analysis revealed such effects in the superior

Numerical cognition

part of the IPS bilaterally as well as the superior parietal lobule and the supramarginal gyrus.

Numerical magnitude

In contrast, slides rapidly changing in area but not number (Area constant, Area × 1, and

Distance effect

Area × 2) did not yield a parametric effect of distance in these regions. Instead, a reverse

Intraparietal sulcus

effect of area was found in a region of the calcarine sulcus. These findings suggest that areas in and around the IPS are involved in numerical magnitude discrimination in the absence of an explicit task and response requirements. © 2005 Elsevier B.V. All rights reserved.

1.

Introduction

How does the brain enable the representation of numerical magnitudes? Much research into the neural basis of numerical magnitude processing today is related to an influential paper by Moyer and Landauer (1967). In this paper, the authors reported that reaction time is inversely related to the distance between numbers when adults perform relative magnitude comparisons. This so-called “numerical distance effect” has since been studied in young children, infants, and animals (Brannon and Terrace, 1998; Feigenson et al., 2004; Huntley-

Fenner and Cannon, 2000; Sekuler and Mierkiewicz, 1977; Xu and Spelke, 2000). The distance effect is well replicated and is thought to reveal important characteristics of the semantic organization of numerical magnitudes. The fact that it can be measured in both animals and preverbal infants suggests that it represents a fundamental property of the way in which numerical stimuli are processed. In this vein, it has been hypothesized that numerical magnitudes are represented on a “number line,” where magnitudes close to each other share more variance in representational signal than those relatively far apart and are therefore harder to discriminate.

⁎ Corresponding author. Fax: +1 603 646 3968. E-mail address: [email protected] (D. Ansari). URL: http://www.dartmouth.edu/~numcog (D. Ansari). 0006-8993/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.brainres.2005.10.083

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In addition to the behavioral findings, functional neuroimaging studies have revealed the neural correlates of the numerical distance effect. In a series of studies using both functional Magnetic Resonance Imaging (fMRI) and eventrelated brain potentials (ERPs), Pinel et al. (1999, 2001) isolated the neural correlates of different stages of processing during number comparison of both number words and Arabic numerals. While identification of Arabic numerals and number words led to activation of different brain areas, the activation converged on areas in and around the intraparietal sulcus (IPS) and precuneus during the numerical magnitude comparison stage. Moreover, activation of these parietal sites was found to be correlated with numerical distance. In other words, greater parietal activation was observed for close compared with far numerals. Such parametric effects of numerical distance on parietal cortex suggest that the IPS contains the internal representation of numerical magnitude and is involved in the discrimination of numerical stimuli. In more recent work, Pinel et al. (2004) have directly contrasted the neural correlates of different comparative judgments. Participants were asked to compare stimuli for size, luminance and number. While the effect of distance on brain activation resulted in different local peaks for each comparison condition, substantial overlap in activation along the IPS between conditions emerged. This may suggest that neurons in and around the IPS contain a distributed and overlapping code for continuous quantity, with an activation peak in the horizontal segment of the right IPS for number comparisons. Thus, while the IPS appears to house the semantic representation of numerical magnitude, its features may overlap with those necessary to perform discrimination of other continuous stimulus variables. In related work, it has been shown that effects of number discrimination on the IPS are not restricted to symbolic stimuli such as Arabic numerals and number words, but extend to nonsymbolic stimuli such as judgments of angle width and line lengths (Fias et al., 2003). Such findings have led to the contention that the IPS contains a stimulus-independent representation of numerical magnitude. Moreover, in a task requiring no explicit attention to numerosity, numerical stimuli were found to lead to greater bilateral activation of the IPS than color words or letters (Eger et al., 2003). Importantly, the response of the IPS was greater for numerical stimuli regardless of whether these were presented as visual or auditory stimuli. Thus, the representation of numerical magnitude in the IPS appears to be both stimulus-independent and amodal. Recent data from single-cell physiology also support the involvement of the IPS in numerosity representations (Nieder and Miller, 2004). However, the IPS is also activated when participants select a response, independently of whether such a response is made in the context of number-related tasks (Culham and Kanwisher, 2001; Jiang and Kanwisher, 2003). This calls into question whether the IPS modulates numerical magnitude processing over and above response selection (Gobel and Rushworth, 2004). In a recent experiment, Gobel et al. (2004) compared activation of the IPS during number comparison with a control task in which participants had to judge whether a vertical line was absent or present. While the contrast of

number comparison against rest revealed activation in the IPS, the contrast between the number comparison and control task revealed no significant activation in the IPS. These results suggest that activation in the IPS during number comparison may be related to response selection and task difficulty, rather than semantic processing of numerical magnitude or numberspecific discrimination. In other words, greater activation of the IPS for close vs. far numerical distances may reflect nothing more than the increased level of difficulty in response selection. One way to dissociate processes related to task difficulty and response selection from those involved in the semantic processing of numerical magnitude is to utilize a passive design in which participants are not required to make any active responses. If the IPS contains the semantic representation of numerical magnitude, effects of numerical distance on this cortical network should be observed even when no active response is required. To assess this possibility, participants were presented with alternating slides of 8 vs. 8 (distance 0), 8 vs. 12 (distance 4), and 8 vs. 16 (distance 8) squares in separate blocks. We predicted that if distance modulates activation of the IPS in the absence of active decision making and response requirements, a negative, parametric relationship between numerical distance and BOLD signal should emerge in the IPS, whereby rapid changes between 8 and 8 would lead to the greatest responses and stimulus changes between 8 and 16 would lead to the smallest responses. To ascertain whether any effect of distance on neural activation patterns was specific to numerosity, we also presented participants with square stimuli which remained constant in number but varied in area. To match the physical magnitude of change in the area condition with that of the number condition, participants were presented with displays of 8 squares, where each individual square increased in area either by a factor of 1, 1.5, or 2. We predicted that if the presentation of stimuli invoked generalized, higher-level visual discrimination of continuous stimulus attributes, then distance in both the number and area conditions should have similar effects on the neural response. If, however, the rapid changes in numerical stimuli lead to passive discrimination of numerosity, we would expect the effects of numerical distance to differ from those for area.

2.

Results

2.1.

Number change effect

Bilateral regions in the intraparietal sulcus and the superior parietal lobule were sensitive to the number effect at our selected threshold. Additional activation was observed in the left supramarginal gyrus. See Fig. 1.

2.2.

Area change effect

The only cortical region exhibiting the predicted effect (greater activation for smaller difference in area change) was the right precentral gyrus. In addition, an effect in the reverse direction (greater activation for larger differences in area change) was found in the left calcarine sulcus. See Fig. 2.

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183

Fig. 1 – Bilateral parietal regions showing significant (P b 0.0001, uncorrected) parametric decrease of activation with increasing distance in number change in a random effects GLM. Red lines indicate the effects of Number Change, and green dotted lines indicate the effects of Area Change. Lines were calculated by estimating the three data points from the predictor betas and the weights for each condition. To illustrate the match between the parametric predictors and individual conditions, the z scores and standard errors for each condition were also calculated. These estimates for each condition are shown in the blue bars. Error bars indicate the standard error of the mean.

2.3.

Common activations

Activations common to all tasks were found in a large network of areas comprising bilateral occipital and parietal, as well as left prefrontal areas. We verified that there were no specific effects of number change and area change in these regions. We present a subset of these regions in Fig. 3.

3.

Discussion

We report an effect of numerical distance on regions in and around bilateral intraparietal sulci, as well as in the superior parietal lobules. It is thought that the numerical distance effect on reaction time and accuracy provides important insights into the features of numerical magnitude representations. The distance effect suggests that the representational features of numerical magnitudes close to each other overlap more than those far apart. A number of studies have revealed that areas in and around the IPS are sensitive to numerical distance (Dehaene, 1996; Pinel et al., 1999, 2001, 2004). Such findings and the general observation that the IPS is involved in number processing and calculation have led to the suggestion that the IPS contains the internal representation of numerical magnitude. Yet, to date, paradigms exploring the numerical

distance effect have required participants to make active responses (such as judging whether a number on a screen is larger or smaller than a reference). This has thus far left open the possibility that activation in the IPS and other parietal areas during number discrimination is attributable to response-selection rather than representation and discrimination of numerical magnitude (Gobel et al., 2004; Gobel and Rushworth, 2004). The results reported above throw new light on this debate by revealing a parametric effect of numerical distance on the bilateral IPS and the superior parietal lobules in the absence of any active response. This suggests that the IPS is involved in numerical magnitude discrimination over and above response-related activation. The areas found to exhibit a significant distance effect overlap closely with those reported to exhibit parametric modulation of distance in active tasks involving symbolic stimuli (Pinel et al., 1999, 2001, 2004). Moreover, the bilateral inferior parietal sites found here are close to the horizontal segment of the intraparietal sulcus (HIPS) which, it has been argued, contains the core representation of numerical quantity (for a review, see Dehaene et al., 2003). In addition to areas close to the HIPS, we found superior parietal activations bilaterally. The role of the superior parietal lobule in number processing is poorly understood, though it has been suggested that its involvement reflects spatial attentional resources both related to and independent of number processing (Dehaene et al., 2003). It is plausible that a

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Fig. 2 – Left calcarine region showing significant (P b 0.0001, uncorrected) parametric increase of activation with increasing distance in area change in a random effects GLM using parametric predictors. Red lines indicate the effects of Number Change, and green dotted lines indicate the effects of Area Change. Lines were calculated by estimating the three data points from the predictor betas and the weights for each condition. To illustrate the match between the parametric predictors and individual conditions, the z scores and standard errors for each condition were also calculated. These estimates for each condition are shown in the blue bars. Error bars indicate the standard error of the mean.

Fig. 3 – Activation profiles from some regions that showed significant Common Activations across all conditions (P b 0.0001, uncorrected) in a random effects GLM, but did not show any modulation with Number Change or Area Change. Red lines indicate the effects of Number Change, and green dotted lines indicate the effects of Area Change. Lines were calculated by estimating the three data points from the predictor betas and the weights for each condition. To illustrate the match between the parametric predictors and individual conditions, the z scores and standard errors for each condition were also calculated. These estimates for each condition are shown in the blue bars. Error bars indicate the standard error of the mean.

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superior parietal system for visuo-spatial processing directs attention to numerically relevant dimensions of stimuli and tracks their transformations. In the context of the present findings, it might be plausible that greater attentional resources for discrimination of numerical stimulus characteristics are required when the numerical distance is small. However, these effects cannot be solely attributed to differences in global attention across conditions. If the effect of numerical distance on the parietal areas was merely due to attentional effects independent of number processing, then a similar effect should have been observed for the conditions in which the total number of stimuli remained constant while the area occupied changed. However, no significant “area distance” effect on parietal regions was observed. An “area distance” effect on parietal regions revealed for the “number distance” effect was also absent at a more generous statistical threshold (P b 0.001, uncorrected). The only region which was found to exhibit such an effect was the right precentral gyrus. Interestingly, the reverse of the predicted distance effect (i.e., greater activation for large compared with small differences in area) was found in a region of the left calcarine sulcus. The calcarine sulcus is typically involved in basic visual processing. It therefore seems likely that the greater activation of this region for greater area differences is related to the greater visual discriminability of larger differences in the area occupied by squares. Moreover, many cortical regions were activated to similar extents for all conditions, and were not modulated by number or area change in either the predicted or reverse directions. This included regions in the bilateral calcarine sulcus and the inferior parietal lobe which were close to those exhibiting Number Change and Area Change effects, and the dorso-lateral prefrontal cortex which is often associated with executive functions, cognitive load, and attention. Together, these results suggest that the parametric effect of distance on activation in the IPS and superior parietal lobes cannot be explained by global attentional effects. Instead, the data reveal a degree of specificity in response to numerosity changes.

3.1. Passive activation of numerosity representation and discrimination The present results show that bilateral parietal regions are involved in the processing and discrimination of nonsymbolic numerosity stimuli even during passive viewing. This suggests that previously reported involvement of the IPS in numerosity processing during active tasks was not just due to task difficulty and response selection. This may also explain why Shuman and Kanwisher (2004) did not find any significant difference between number and color discrimination tasks in various parietal regions. Against the background of their results, Shuman and Kanwisher challenge the notion that the IPS is the locus of a domain specific system for number representation. However, in order to equate stimulus properties across the tasks, numerosity changes occurred similarly in all conditions. Given our present findings, it is possible that the same numerosity representation and discrimination mechanisms in the IPS were automatically activated in both number and color tasks, resulting in the lack of task differences in activation.

3.2.

185

fMRI adaptation for numerosity

In the same publication, Shuman and Kanwisher reported another experiment, part of which involved passive viewing of nonsymbolic numerosity stimuli. They failed to find a significant fMRI adaptation effect of number in the IPS when the same nonsymbolic numerosity was presented repeatedly. Interestingly, they found slightly greater activation in a condition in which numerosity was held constant across the block compared to one in which numerosity was varied within the block. The absence of adaptation of the BOLD signal in the IPS to the repetition of numerosity is taken by the authors to indicate a lack of representational specificity for number. However, these findings are actually somewhat comparable to the results reported here, as there was slightly greater activation when numerosity was repeated compared to when it was varied. In contrast, at the same time as Shuman and Kanwisher reported their negative findings, Piazza et al. (2004) revealed fMRI adaptation effects for numerosity repetition in bilateral regions of the IPS when participants passively viewed a stream of stimuli with repeated numerosity (habituation) interspersed irregularly with stimuli of deviant numerosity. Piazza et al. found greater activation during the presentation of the deviant compared with the habituated numerosity stimuli in bilateral regions of the IPS. Furthermore, it was found that the response to the deviant numerosity increased with the degree of deviance. Again, the apparent discrepancy in results may be explained by differences in experimental paradigm. In Shuman and Kanwisher's study, the independent variable was simply the number of repetitions in a block. In contrast, Piazza et al. modeled the deviant trials as events of interest. In such a design, the dependent measure may be conceived as dishabituation or change detection. The greater the degree of difference in numerosity in the deviant trial, the less variance it shares with the habituated numerosity, the more salient the change. In agreement with Piazza et al.'s findings, our data suggest that the IPS is indeed involved in the internal representation and discrimination of numerical magnitude. Yet on closer examination, the two sets of results may be perceived as contradictory. While the present results suggest that areas in and around the IPS respond more when the numerical difference between two nonsymbolic numerosities is small, Piazza et al.'s results suggest the opposite—greater activation of the IPS for more numerically deviant numerosities. However, it should be noted that while Piazza et al. optimized their experimental design to obtain fMRI adaptation effects (Grill-Spector and Malach, 2001), the present paradigm was not optimized for adaptation and was not designed to test this effect. Each number or area change block contained 40 repetitions of each type of stimulus, while the Constant × 1.0 block had 80 repetitions of one type of stimulus. Thus, fMRI adaptation for numerosity should also be expected even in the number and area change conditions, and it is plausible that the Constant × 1.0 block did not lead to a significantly greater extent of adaptation than other blocks. This difference in design offered the opportunity to explore whether the rapid presentation of changing nonsymbolic numerosities leads to greater activation of numerosity

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discrimination processes, exhibiting the distance effect typically observed during active tasks. While representing a well-documented effect, especially for visual processing, the specific mechanisms and dynamics underlying fMRI adaptation are not clear. It remains for future experiments to establish the conditions under which numerosity adaptation can occur. Moreover, the distance effect and fMRI adaptation appear to generate opposite predictions with regard to functional activation, and there may be interesting interactions between these two effects. The resultant activation profile may be specifically dependent on the task and temporal specifics of stimulus presentation. The above considerations make it likely that the present data and those reported by Piazza et al. are complementary rather than contradictory. While Piazza et al.'s results reveal the degree of overlap in the representation of numerical magnitude, the present findings suggest the existence of neuronal mechanisms for numerosity discrimination that exist in the absence of response requirements.

purpose of the study and are not required to select a response, the IPS is parametrically modulated by numerical distance. In contrast, no such parametric effect on the IPS was found when area instead of number was varied. Thus, our findings suggest a degree of specificity for the involvement of the IPS in numerical magnitude discrimination.

5.

Experimental procedures

5.1.

Participants

Sixteen healthy, right-handed, male adults (mean age = 20 years, 5 months) participated in this experiment. The procedure was approved by the Committee for the Protection of Human Subjects at Dartmouth College and all participants signed informed consent. Subjects were not told the purpose of the experiment and were simply instructed to fixate on a small white crosshair in the center of the screen. 5.2.

4.

Conclusion

The numerical distance effect has provided significant insights into the nature of numerical magnitude representations. A number of functional neuroimaging studies have implicated areas in and around the IPS in numerical magnitude processing and have revealed that numerical distance has parametric effects on the activation levels in the IPS. Despite this, there has been controversy over the neural correlates of the numerical distance effect. It has been contended that activation of the IPS during magnitude processing reflects response selection rather than magnitude representation per se. Results from our study suggest that even when participants are not made aware of the

Procedure and task design

In each run, subjects were presented with blocks of rapidly changing slides of white squares against a black background (see Fig. 4). Stimuli were produced using Adobe Photoshop CS™ and were presented with E-Prime (Psychological Software Tools, Pittsburgh, USA). All the squares within a slide were of the same size and randomly arranged within the constraint that there was no spatial overlap between squares. There were five different types of stimuli blocks: Number × 2.0, Number × 1.5, Constant × 1.0, Area × 1.5, Area × 2.0. Each block contained a total of 80 slides. Each number change (Number × 1.5 and Number × 2.0) and area change (Area × 1.5 and Area × 2.0) block contained two types of slides, 40 of each type. For the number change blocks, there were 40 slides of each number, i.e., 40 slides of 8 squares and 40 slides of 12 squares in each Number × 1.5 block; 40 slides of 8 squares and 40 slides of 16 squares in each Number × 2.0 block. Similarly, for each area

Fig. 4 – Examples of all stimulus conditions and timings. Every 20 s block consisted of 80 randomly presented stimuli, each presented for 250 ms.

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change block (Area × 1.5 and Area × 2.0), there were 40 slides for each square size. In the Constant × 1.0 block, participants viewed 80 slides of 8 dots with area and number held constant between slides. To ensure that changes in numerosity were disconfounded from changes in total area occupied by the squares, the total area occupied by the squares in the numerosity conditions was equated within and between blocks. To enable a comparison between area changes, the total area occupied by squares was equated between conditions. In other words, the total area occupied by the combination of all slides in the Area × 1.5 condition was equivalent to the total area taken up by summing the slides of squares in the Area × 2.0 condition. Moreover, the total area occupied in each of the area conditions was equated with the total area occupied by squares in the number conditions. Thus, for all conditions, the total area was the same. This enabled comparison between all conditions. Each slide was presented for a total duration of 250 ms with no fixation ISI between slides. The order of presentation of the slides within each block was randomized by the stimulus presentation software. Blocks were separated by 20 s of rest, during which participants fixated on the centrally located white crosshair. Participants saw a total of 7 repetitions of each block type. The order of the block types was completely randomized for each run. Each run of stimulation began and ended with 30 s of rest. Between runs, participants were reminded by the investigators to concentrate on the white crosshair and were given a short break. 5.3.

Data acquisition

Functional images were acquired in a 1.5 T General Electric whole body MRI scanner. A standard birdcage head coil was used and head movements were restricted through the use of a foam pillow. Using a fast spin echo sequence, 25 T1 weighted structural slices were acquired in the axial plane. Co-planar to the T1 weighted structural images, functional images were acquired using a gradient echo-planar T2*-sequence sensitive to blood-oxygenation level-dependent (BOLD) contrast. Image volumes consisted of 25 noncontiguous slices (4.5 mm thickness, 1 mm gap, 64 by 64 matrix, repetition time = 2.5 s, TE = 40 ms, flip angle = 90°, field of view = 24 by 24 cm) covering the whole brain. Each run of functional imaging consisted of the acquisition of 96 volumes. Three-dimensional whole-brain high-resolution (0.94 × 0.94 1.2) T1 weighted images were acquired in the saggital plane using a standard GE SPGR 3-D sequence.

5.4.

187

Data analysis

Structural and functional images were analyzed using Brain Voyager QX 1.2.6 (Brain Innovation, Maastricht, Holland). Functional images were corrected for slice time acquisition differences, head motion, and linear trend. Functional images were aligned to the T-1 weighted co-planar images and subsequently to the threedimensional high-resolution images. The realigned data set was then transformed into Tailarach space (Tailarach and Tournoux, 1988). Following the numerical distance effect, we investigated which areas of the brain exhibited parametric effects of distance (area or number) on BOLD signal. We therefore constructed a parametric general linear model (GLM) design matrix with two predictors: (1) Number Change effect, (2) Area Change effect, and a nonparametric predictor (3) Common Activations (see Fig. 5). Based on behavioral data and previous neuroimaging studies of the distance effect, we expected smaller signal change for those conditions with greater changes in distance (area or number), and we weighted the Number Change and Area Change predictors accordingly, e.g., for the Number Change predictor, the smallest weight was given to the Number × 2.0 blocks. The Common Activations predictor captures generic task-related activation that is not modulated by changes in number or area. Our parametric design matrix was then convolved with the expected BOLD signal. Following Boynton et al. (1996), the expected BOLD signal change was modeled using a gamma function (tau of 2.5 s and a delta of 1.5). Random-effects analyses were performed to examine the effects of all three predictors in this parametric GLM. Voxels were considered to be significantly activated when they passed a threshold of P b 0.0001, uncorrected. Using our parametric predictors, we identified regions sensitive to (1) Number Change and (2) Area Change. To verify that the parametric predictors in the parametric GLM accurately modeled the signal changes in these regions-of-interest (ROI), we ran a separate, nonparametric, GLM analysis on each ROI, modeling each condition with a separate predictor. The parameter estimate of each predictor from this GLM would yield a more direct estimate of the activation level for each condition. To illustrate graphically how well these two GLMs matched, the results obtained from the parametric GLM was then plotted together with the results from the nonparametric GLM (Figs. 3–5).

Fig. 5 – Schematic of the parametric predictors used in the GLM. Based on the distance effect, greater BOLD signal change is expected for smaller changes in number or area. The Common Activations predictor captures generic task-related variance that is not modulated by changes in number or area.

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Acknowledgments We would like to thank the Dartmouth Brain Imaging Center. This work was supported by grants from Dartmouth College and the Dickey Center for International Understanding at Dartmouth College to DA. We would also like to thank Stanislas Dehaene, Scott T. Grafton, and Jonathan Fugelsang for comments on an earlier draft.

REFERENCES

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Parametric effects of numerical distance on the intraparietal sulcus ...

Dec 15, 2005 - reported that reaction time is inversely related to the distance. between numbers when adults perform relative magnitude. comparisons. This so-called “numerical distance effect” has. since been studied in young children, infants, and animals. (Brannon and Terrace, 1998; Feigenson et al., 2004; Huntley- ...

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Oct 3, 2012 - V.W., and M.F.S.R. analyzed data; I.C.G., A.C.N., and M.F.S.R. wrote the paper. This work was .... EEG recording, preprocessing, and spectral analysis. ...... Oostenveld R, Fries P, Maris E, Schoffelen JM (2011) Fieldtrip: open source s

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There was a problem previewing this document. ... numerical magnitudes onto symbols-the numeri ... vidual differences in childrens math achievement.pdf.

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Multimedia systems are more and more used in distance learning. Since these systems are often structured as a hypertext, they pose additional problems to the ...

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Oct 3, 2012 - tice trials during which TMS was delivered on 50% of trials, and then 360 .... All topographies depict a birds-eye view of the scalp, with anterior ...

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In the present study Akaike's Information Criterion (AIC). (Akaike 1973) was used as the quantitative method for model selection. The adequacy of the selected ...

The effects of size distance and suppression
coding and comparison). On the .... Procedure. The lists of numbers were presented in the centre of a Nec computer screen, using e-Prime ..... 101–135. Hasher, L., Zacks, R. T., & May, C. P. (1999). Inhibitory control, circadian arousal, and age.

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kicking leg movements in Taekwondo roundhouse kick. Twelve male ..... To facilitate data analysis a group of meaningful events were defined: Start, Toeoff,.

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The effect of mathematics anxiety on the processing of numerical magnitude.pdf. The effect of mathematics anxiety on the processing of numerical magnitude.pdf.

The effect of mathematics anxiety on the processing of numerical ...
The effect of mathematics anxiety on the processing of numerical magnitude.pdf. The effect of mathematics anxiety on the processing of numerical magnitude.pdf.

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Apr 21, 2008 - recognition, signal detection and estimation, and communications. ..... [1] J.G. Proakis and D.G. Manolakis, Digital Signal Processing: Principles ...

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Aug 9, 2006 - future needs and by their ability to offer solutions to those needs that are superior to rivals' offerings (Slater and Narver 2000). Firms that are able to .... components, such as customer orientation (e.g., Voss and Voss, 2000) or int

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stitute I thank Theo Geisel and Fred Wolf for hosting me under their scientific ...... IN vdS + ˆ. Ω. svdV. Here, σ,Φ,v represent the conductivity, potential and test ...

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