Abstract

We examined the effects of number magnitude (within vs. outside the subitizable range) and notation (symbolic vs. nonsymbolic number) on neural responses to visual displays in the human brain using fMRI at 7T. We found that the right temporoparietal junction (rTPJ) responded more strongly to small than to larger numbers (2, 4 > 6, 8), while there was greater activity bilaterally within and around the intraparietal sulcus (IPS) as number magnitude increased (6, 8 > 2, 4). The effects of number magnitude were greatest for nonsymbolic stimuli. In addition, there was striking overlap between rTPJ regions responding to small numbers and those most strongly activated by symbolic stimuli, and between IPS regions responding to large numbers and those most activated by nonsymbolic stimuli. The results are consistent with distinct neural processes recruited for the processing of small- and large-number magnitudes. Contributions due to differences in representing exact number (small nonsymbolic arrays and all symbolic numbers, in rTPJ) and overall magnitude (particularly with large nonsymbolic arrays, in IPS), and the associated theoretical implications of the findings, are discussed.

Introduction

The ability to apprehend the number of objects present in the environment may reflect a basic cognitive capacity found not only in literate adults but also young infants and nonhuman primates (Feigenson et al. 2004; Brannon 2006; Nieder and Dehaene 2009). This ability seems also to be represented by specialist neural regions in posterior parietal cortex (Nieder 2005; Piazza and Izard 2009). What remains controversial is whether 1) once stimuli have been individuated, brain regions respond differentially to small numbers (usually up to four, referred to as “subitizing,” first coined by (Kaufman et al. 1949), for the rapid and accurate enumeration of objects in a small set) and large numbers (usually >4), and 2) brain regions respond in an “abstract” manner (equivalent across different number notations) or in a manner specific to properties of the stimulus (e.g., differentially to visual patterns as opposed to symbols). The evidence on these points is controversial. For example, brain imaging studies have shown evidence both for distinct neural responses to small and large numbers (Sathian et al. 1999; Piazza et al. 2003; Ansari et al. 2007; Hyde and Spelke 2009; Vetter et al. 2010) and also for an undifferentiated response that solely reflects stimulus magnitude (Izard et al. 2008; Jacob and Nieder 2009; Notebaert et al. 2010). Likewise, there are data supporting both the argument for an abstract response to number, that is, common across notations (Naccache and Dehaene 2001; Pinel et al. 2001; Libertus et al. 2007), and for there being selective responses to symbolic and nonsymbolic number representations (Cohen Kadosh et al. 2007; Cohen Kadosh and Walsh 2009). Here, we present the first ever study of number processing where the brain is imaged at 7T to generate both higher signal-to-noise ratio (SNR) and greater sensitivity for blood oxygen-level–dependent (BOLD) contrast, in order to assess whether the neural responses to small and large numbers, and to symbolic and nonsymbolic stimuli, can be separated. Participants performed a task of numerical comparison for 2 consecutive displays while being scanned. First, we modeled the neural response for the 2 displays individually but, to avoid effects linked to comparisons between the stimuli, we analyzed only the signal for enumerating the initial display as a function of the notation and magnitude conditions. Further, to investigate the notation effect between the two displays within a trial, we modeled the whole trial as a function of the different combinations of notations for the 2 compared stimuli. The evidence we present indicates that there are distinct responses in different neural areas to symbolic and nonsymbolic numbers, and to small and large numbers, which emerge most clearly when magnitude is conveyed by nonsymbolic forms.

Materials and Methods

Subjects

Twenty healthy human adults (12 males and 8 females, mean aged 21 years) participated. All were right-handed, reported normal or corrected-to-normal vision, and had no known neurological disorders. They gave written, informed consent in accordance with procedures and protocols approved by Institutional Review Board of Beijing MRI Center for Brain Research.

Procedures, Stimuli, and Designs

The stimuli were projected on a rear-projection screen placed 70 cm from the participants' eyes and seen through an angled mirror attached to the head coil. Trials started with brief presentation (200 ms) of a sample stimulus, followed by a memory delay which varied from 3.8 to 6.8 s. Next, the second display, serving as a match, appeared for 200 ms, followed by a response delay varying from 3.8 to 6.8 s (Fig. 1A). Subjects were asked to judge which 1 of the 2 sequential displays, the first or the second, contained a larger numerical value. After each response, the green fixation cross turned red, signaling the start of next trial. Each trial lasted 11 s in total.

All visual displays were projected on gray squares subtending ∼5°of visual angle and presented against a black background. On each trial, the sample and match stimuli could be, respectively, dot patterns or Arabic numerals, either sharing or not sharing the same numerical notation. Each dot pattern contained a variable number of randomly distributed black dots of circular shape with a diameter depending on the dot number, controlling for the overall luminance across four levels of numerosities (2, 4, 6, and 8). Each digit pattern contained a black Arabic numeral subtending ∼0.7 × 1.2° of visual angle, of a random position within the pattern and of variable numerical values (2, 4, 6, and 8).

The experiment consisted of 8 blocks, each with 32 trials. Stimuli with different numerical values represented by different notations were displayed on equal numbers of trials. The order of the numerical values was randomized within each block. On half the trials, the sample was numerically smaller than the match stimulus (2 vs. 4, 2 vs. 6, 4 vs. 8, and 6 vs. 8) and on the other half, the sample was numerically larger than the match stimulus (4 vs. 2, 6 vs. 4, 8 vs. 2, and 8 vs. 6). On an equal proportion of trials, sample and match stimuli had the same numerical notations (dot pattern followed by dot pattern, Arabic numeral followed by Arabic numeral) or they had different notations (dot pattern followed by Arabic numeral, Arabic numeral followed by dot pattern).

Figure 1.

Illustration of the experimental procedure and design. (A) Participants were presented with 2 sequential displays and judged which was numerically larger. (B) Sample and match stimuli could vary in their numerical magnitude (4 levels: 2, 4, 6, and 8) and notation (either dot patterns with matched luminance or Arabic numbers) (C) A mid-sagittal slice of one of the participants. The yellow shade represents the area that was covered during the functional scanning, which included the occipital and parietal lobes along with part of the temporal lobe.

Figure 1.

Illustration of the experimental procedure and design. (A) Participants were presented with 2 sequential displays and judged which was numerically larger. (B) Sample and match stimuli could vary in their numerical magnitude (4 levels: 2, 4, 6, and 8) and notation (either dot patterns with matched luminance or Arabic numbers) (C) A mid-sagittal slice of one of the participants. The yellow shade represents the area that was covered during the functional scanning, which included the occipital and parietal lobes along with part of the temporal lobe.

fMRI Scanning and Analysis

Structural and functional images were acquired with a 7T scanner (Siemens, Erlangen, Germany) with a Nova medical volume transmit and 24-channel receiver coil. Functional images were acquired with 1.4-mm isotropic voxels by using single-shot echo-planar imaging sequence optimized for 7T (32 oblique transverse slices covering parietal and occipital lobes; FOV = 224 × 224 mm2; acquisition matrix = 160 × 160; TR = 2000 ms; TE = 23 ms; flip angle = 90°; 6/8 partial Fourier, GRAPPA factor = 3, sinusoidal readout gradient). For all functional images, distortion was corrected by performing online reconstruction based on a reference measurement of the local point spread function (Zaitsev et al. 2004). The higher resolution T1-weighted structural images were acquired by 3D-MPRAGE (0.7-mm isotropic resolution, TI = 1050 ms, TR = 2200 ms, flip angle 7°) for each participant.

The data were analyzed with SPM8 (http://www.fil.ion.ucl.ac.uk/spm). Standard preprocessing included realignment, slice timing, spatial normalization, and smoothing with a 6-mm Gaussian kernel. For each subject, activations were modeled by a linear combination of regressors for the 8 sample stimulus conditions and 8 match stimulus conditions (2 stimuli type (sample/match) × 4 magnitudes (2, 4, 6, and 8) × 2 notations (dot/digit)) with a standard hemodynamic response function. The resulting 8 independent estimates of fMRI signal change (beta values) for each sample condition were submitted to a second level random-effect analysis, with a 2-way factorial design. For the group analysis, the data are reported for the overall contrasts of large > small and nonsymbolic > symbolic, and contrasts of small > large and symbolic > nonsymbolic, at P < 0.05 (both were FWE corrected, >10 consecutive voxels). Subsequently, we performed separate ROI analyses to assess activation across the step from small to larger numbers within regions showing either a stronger response to the largest relative to the smallest magnitude (activation for number 8> activation for number 2) or those showing a stronger response to the extreme small relative to large numbers (activation for number 2> activation for number 8). To avoid the nonindependence problem from using a within-subject functional localizer, we used a leave-1-subject-out approach in which 20 general linear models (GLMs) were run with 1 subject left out in each and with each GLM defining the ROIs for the subject that was left out (Esterman et al. 2010). Specifically, for iterations in this method, 1 subject was left out. For each of the 20 subjects, the ROIs were defined by the group analysis based on the other 19 subjects in the contrast of number 8> number 2 and the reverse contrast under a threshold of P < 0.001 uncorrected. The resulting ROIs were used to extract beta values from the one remaining subject for subsequent multifactor ANOVAs with the factors being activation area, numerical magnitude, and notation. Further, to investigate the notation effect in a comparison trial, we remodeled the data with 4 conditions, depending on the relation between the notations of the 2 consecutive stimuli on a trial (the sample and the match), 1) dot number followed by dot number (dot–dot), 2) dot number followed by Arabic number (dot–digit), 3) Arabic number followed by dot number (digit–dot), and 4) Arabic number followed by Arabic number (digit–digit). Note that the duration (the time from the onset of the sample to the offset of the match) for each condition may vary from trial to trial, because of the variable delay after the sample.

Results

Activation patterns evoked by sample stimuli were analyzed as a function of 1) the numerical magnitude present in each display, and 2) the notation. Further, to investigate the notation effect when 2 consecutive stimuli were compared, we analyzed the data modeling, the 2 compared stimuli in a single trial with 4 conditions, dot–dot, dot–digit, digit–dot, and digit–digit.

Comparison Between Small and Large Numbers

Whole-Volume Group Analysis

The contrast between large and small numbers (including nonsymbolic and symbolic numbers) revealed greater activation for processing the large numbers (6 and 8) compared with processing small numbers (2 and 4) in the left and right IPS at MNI coordinates [−23 −62 51; 27 −60 51], and an area in the right postcentral gyrus [51 −34 49] (P < 0.05, FWE corrected, cluster size >10, see results of group analysis in Fig. 2A and Table 1). In contrast, enumerating small numbers relative to large numbers activated the right temporoparietal junction (TPJ) at MNI coordinates [51 -34 49] (P < 0.05, FWE corrected, cluster size >10; the results of the group analysis are presented in Fig. 2B and Table 2).

Table 1

Regions showing stronger activations for large > small

X Y Z Cluster size T values Cortical region 
−23 −62 51 1898 8.34 Left superior IPS 
−30 −49 45   Left anterior IPS 
−26 −67 35   Left inferior IPS 
27 −60 51 1848 8.25 Right superior IPS 
28 −67 38   Right inferior IPS 
51 −34 49 254 6.61 Right postcentral gyrus 
X Y Z Cluster size T values Cortical region 
−23 −62 51 1898 8.34 Left superior IPS 
−30 −49 45   Left anterior IPS 
−26 −67 35   Left inferior IPS 
27 −60 51 1848 8.25 Right superior IPS 
28 −67 38   Right inferior IPS 
51 −34 49 254 6.61 Right postcentral gyrus 
Table 2

Regions showing stronger activations for small > large

X Y Z Cluster size T values Cortical region 
49 −63 28 66 5.11 Right TPJ 
X Y Z Cluster size T values Cortical region 
49 −63 28 66 5.11 Right TPJ 
Figure 2.

Brain regions activated in the contrast large > small (A) and the contrast small > large (B), and results of the ROI analysis for different numerical notations and magnitudes in the bilateral IPS for the contrast number 8 > 2 (C), and in the right TPJ for the contrast number 2 > 8 (D).

Figure 2.

Brain regions activated in the contrast large > small (A) and the contrast small > large (B), and results of the ROI analysis for different numerical notations and magnitudes in the bilateral IPS for the contrast number 8 > 2 (C), and in the right TPJ for the contrast number 2 > 8 (D).

ROI Analysis

The contrast between small and larger numerical values typically translates into a contrast between processing magnitudes of ≤4 and magnitudes >4 (Trick and Pylyshyn 1994). To evaluate whether regions responding to magnitude showed a shift across this critical contrast, we examined activity in regions of interest defined by contrasting the most extreme magnitudes (the smallest number 2 vs. the largest number 8). To avoid any possibility of “double dipping” where the same data are used both to define a ROI and to contrast between the effects of related conditions within the ROI (see Kriegeskorte et al. 2009), we defined the ROIs by using a leave-1-subject-out approach (Esterman et al. 2010). Note that we present a conservative treatment of our data given that we make critical contrasts within each ROI between the relatively close numbers 4 (small) and 6 (large), for which neural responses are less likely to differ than to the numbers 2 and 8. For greater responses to large over small numbers, ROIs in bilateral parietal cortex were identified (showing a reliable profile of activity for the contrast 8 > 2). For greater responses to small over large numbers, ROIs were extracted in the right TPJ (showing a reliable effect of the contrast 2 > 8).

Contrast of Number 8> Number 2

In areas showing stronger activation for the largest relative to the smallest number, ROIs were initially defined on a subject-by-subject basis and beta values were extracted for the different magnitude and notation conditions. An analysis of variance (ANOVA) was conducted with the factors being brain region (left and right IPS), numerical magnitude, and notation. There were reliable main effects of region (F1,19= 7.24, P < 0.014), numerical magnitude (F3,57 = 39.50, P < 0.001), and notation (F1,19 = 175.63, P < 0.001), and also a significant interaction between magnitude and notation (F3,57 = 81.57, P < 0.001) and a significant interaction between region, magnitude and notation (F3,57 = 3.96, P < 0.02) (see Fig. 2C). We then analyzed data for the left and right IPS separately. For both the left and right IPS, there were reliable effects of numerical magnitude (F3,57 = 33.61, P < 0.001, for left IPS; F3,57 = 41.81, P < 0.001, for right IPS) and notation (F1,19 = 132.34, P < 0.001, for left IPS; F1,19 = 166.87, P < 0.001, for right IPS), and interactions between magnitude and notation (F3,57 = 64.07, P < 0.001, for left IPS; F3,57 = 78.32, P < 0.001, for right IPS). For nonsymbolic numbers, activation increased with numerical magnitude for both the left and right IPS (F3,57 = 65.52, P < 0.001, for left IPS; F3,57 = 85.61, P < 0.001, for right IPS); this was found for symbolic numbers in the left IPS (F3,57 = 4.65, P < 0.022) but not in the right IPS (F3,57 = 1.94, P > 0.159). This result suggests a greater contrast between large and small magnitudes for nonsymbolic numbers, particularly, for the right IPS. Crucially, the activation profile exhibited a step change for large and small numerosities around the number 4. For both the left and right IPS, there was a reliable difference between numerosities 4 and 6 (within the ROI defined by the contrast 8 > 2) (post hoc paired t-test P < 0.001, Bonferroni corrected for multiple comparisons, for both left and right IPS). No significant difference was observed within either the small number range (P > 0.9, for left IPS; P > 0.94, for right IPS) or the large-number range (P > 0.079, for left IPS; P > 0.064, for right IPS).

Contrast of Number 2> Number 8

We obtained ROIs from the rTPJ for each subject and submitted the beta values to an ANOVA with the factors being numerical magnitude and notation. This revealed significant main effects of numerical magnitude (F3,57= 10.16, P < 0.001) and notation (F1,19 = 25.04, P < 0.001), along with a reliable interaction between magnitude and notation (F3,57 = 8.58, P < 0.001) (see Fig. 2D). Post hoc analysis showed that for nonsymbolic numbers, there was an overall decrease with increasing numerical magnitude in the rTPJ (F3,57 = 15.21, P < 0.001), but not for symbolic numbers (F3,57 = 1.35, P > 0.27). Critically, the greater activation for small relative to larger nonsymbolic magnitudes held for the contrast between numerosities 4 and 6 (P < 0.004, Bonferroni corrected for multiple comparisons). This reflected a step function from small to large number in nonsymbolic displays, as there were no differences within the small or within the larger number ranges (2 vs. 4, P > 0.157, and 6 vs. 8, P > 0.9) (see Fig. 2D). Therefore, the activation profile showing a stronger response for smaller numbers in rTPJ was specific to nonsymbolic (random dot) displays.

Comparison Between Nonsymbolic and Symbolic Numbers

The contrast for nonsymbolic numerosities > symbolic Arabic digits revealed a distributed network of dorsal regions including the bilateral IPS, posterior parietal, and occipital cortices (P < 0.05, FWE corrected, cluster size >10, see results of group analysis in Fig. 3A and Table 3). Greater activation for the reverse contrast (symbolic > nonsymbolic) was found in the right TPJ (P < 0.05, FWE corrected, cluster size >10, see Fig. 3B and Table 4). Note that the activation profiles for nonsymbolic > symbolic and symbolic > nonsymbolic were, respectively, similar to the activation profiles observed for large (6, 8) > small (2, 4) and small (2, 4) > large (6, 8) magnitudes.

Table 3

Regions showing stronger activation for nonsymbolic > symbolic stimuli

X Y Z Cluster size T values Cortical region 
26 −60 51 3866 11.14 Right superior IPS 
31 −70 28   Right inferior IPS 
35 −81 19   Right dorsomedial occipital 
−22 −62 51 3179 10.23 Left superior IPS 
−33 −41 41   Left anterior IPS 
−25 −69 31   Left inferior IPS 
−35 −87 14   Left dorsomedial occipital 
51 −34 52 156 6.78 Right postcentral gyrus 
X Y Z Cluster size T values Cortical region 
26 −60 51 3866 11.14 Right superior IPS 
31 −70 28   Right inferior IPS 
35 −81 19   Right dorsomedial occipital 
−22 −62 51 3179 10.23 Left superior IPS 
−33 −41 41   Left anterior IPS 
−25 −69 31   Left inferior IPS 
−35 −87 14   Left dorsomedial occipital 
51 −34 52 156 6.78 Right postcentral gyrus 
Table 4

Regions showing stronger activation for symbolic > nonsymbolic stimuli

X Y Z Cluster size T values Cortical region 
55 −56 26 47 4.78 Right TPJ 
48 −63 30    
X Y Z Cluster size T values Cortical region 
55 −56 26 47 4.78 Right TPJ 
48 −63 30    
Figure 3.

Brain regions activated in the contrast nonsymbolic > symbolic (A), and in the contrast symbolic > nonsymbolic (B).

Figure 3.

Brain regions activated in the contrast nonsymbolic > symbolic (A), and in the contrast symbolic > nonsymbolic (B).

We next investigated the notation effect by analyzing data following the modeling of 4 conditions involved in comparisons of the sample and match stimuli: dot–dot, dot–digit, digit–dot, and digit–digit combinations. First, we examined the difference between within-notation (including dot–dot and digit–digit conditions) and cross-notation conditions (including dot–digit and digit–dot conditions). No significant activation was found, throughout the whole volume, for the contrast of within-notation > cross-notation or its reverse, even at a reduced threshold level (up to P = 0.005, uncorrected). These results suggest that the notation relation between the 2 comparison stimuli had little impact on brain activity, possibly because we used a relatively long interstimulus interval to separate out the neural signals to the comparison and match stimuli. Considering the distinct neural response to symbolic and nonsymbolic numbers observed for the sample stimuli (see Fig. 3), we then performed the comparison between the within- and cross-notation trials by separating the 2 within-notation conditions (dot–dot and digit–digit), with balanced contrast weights for each comparison (i.e., for both contrasts dot–dot > cross-notation and digit–digit > cross-notation, we used [2 −1 −1]). There was stronger activation bilaterally in the IPS for dot–dot over cross-notation conditions under a threshold of P < 0.05 (FWE corrected) (see Fig. 4A and Table 5). No significant activation was found for the contrast of digit–digit trials over cross-notation trials at the same threshold, though there was evidence for activation in the rTPJ when we lowered the threshold (P < 0.001, uncorrected) (see Fig. 4B and Table 6). In contrast, we observed no significant activations for cross-notation over the dot–dot conditions even at a reduced level of threshold (P < 0.001, uncorrected), while there was a robust activation in the right IPS for the contrast of cross-notation trials > digit–digit trials (P < 0.05, FWE corrected) (see Fig. 4C and Table 7).

Table 5

Regions showing stronger activation for dot–dot > cross-notation trials

X Y Z Cluster size T values Cortical region 
31 −69 30 169 5.96 Right inferior IPS 
−22 −62 44 197 5.69 Left superior IPS 
31 −57 52 384 5.67 Right superior IPS 
−30 −41 40 31 5.13 Left anterior IPS 
X Y Z Cluster size T values Cortical region 
31 −69 30 169 5.96 Right inferior IPS 
−22 −62 44 197 5.69 Left superior IPS 
31 −57 52 384 5.67 Right superior IPS 
−30 −41 40 31 5.13 Left anterior IPS 
Table 6

Regions showing stronger activation for digit–digit > cross-notation trials

X Y Z Cluster size T values Cortical region 
51 −53 27 15 3.37 Right TPJ 
X Y Z Cluster size T values Cortical region 
51 −53 27 15 3.37 Right TPJ 
Table 7

Regions showing stronger activation for cross-notation > digit–digit trials

X Y Z Cluster size T values Cortical region 
30 −55 44 49 5.07 Right superior IPS 
X Y Z Cluster size T values Cortical region 
30 −55 44 49 5.07 Right superior IPS 
Figure 4.

Effects of cross- and within-notation for the 2 consecutive stimuli on a trial, with 2 within-notation conditions (dot–dot and digit–digit) analyzed separately. (A) Brain regions activated for the contrast dot–dot > cross-notation (left panel), and for the contrast digit–digit > cross-notation (right panel). (B) Brain regions activated in the contrast cross-notation > digit–digit.

Figure 4.

Effects of cross- and within-notation for the 2 consecutive stimuli on a trial, with 2 within-notation conditions (dot–dot and digit–digit) analyzed separately. (A) Brain regions activated for the contrast dot–dot > cross-notation (left panel), and for the contrast digit–digit > cross-notation (right panel). (B) Brain regions activated in the contrast cross-notation > digit–digit.

The results reflecting the effect of notation across sample and match trials are mostly consistent with those in analysis of individual sample stimuli. In both cases, there was stronger activation in the rTPJ for symbolic over nonsymbolic stimuli, and stronger bilateral activation in occipito-parietal networks for nonsymbolic over symbolic stimuli. Note that the analysis across individual sample stimuli seems more sensitive than the analysis across both the sample and match stimuli because significant activation was observed at a threshold of P < 0.05 (FWE corrected) in the former but not the latter method (where reliable activity was found at a lower threshold, P < 0.001, uncorrected). The difference was also mirrored by the results from the contrast of digit–digit > digit–dot and of digit–digit > dot–digit. For the conditions with the same notation for the sample stimuli, no stronger activation was found for symbolic over nonsymbolic numbers in the match display, while for conditions with the same notation of the match stimuli, stronger activation was found for symbolic over nonsymbolic sample displays in the rTPJ and bilateral precuneus (both contrasts P = 0.001 uncorrected) (see Supplementary Fig. S1A). The difference in 2 analysis methods, with one focusing on sample data only and the other combining sample and match events as a single trial, may reflect confounding contributions from distinctive cognitive processes of comparison and response selection that may smear or even eliminate the effects of interest (Pollmann et al. 2003; Gobel et al. 2004). Indeed, there was a difference in the activation maps for the sample and match stimuli. For instance, we found a significant activation in bilateral occipito-parietal networks for dot–dot > dot–digit (P < 0.05, FWE corrected, see Supplementary Fig. S2B), but not for dot–dot > digit–dot. Similarly, right occipito-parietal network and left occipital area were activated for digit–dot > digit–digit (P < 0.05, FWE corrected, see Supplementary Fig. S1C), but not for dot–digit > digit–digit using the same threshold.

The Relations Between Notation and Magnitude

To examine the relations between the notation-sensitive and magnitude-sensitive areas, we performed conjunction analyses of the neural response to sample stimuli respectively for the contrasts 1) nonsymbolic > symbolic and large > small, and 2) symbolic > nonsymbolic and small > large. Overlaps for the contrasts of nonsymbolic > symbolic and large > small were found in the bilateral IPS and the right postcentral gyrus, and overlaps for symbolic > nonsymbolic and small > large were found in the right TPJ (both conjunction results were reported at a threshold of P < 0.001 uncorrected at peak level, FWE corrected at cluster level). (See overlaps for individual comparisons in Fig. 5A,B, and central coordinates in Tables 8 and 9.).

Table 8

Regions of overlap for the contrast of nonsymbolic > symbolic and large > small displays

X Y Z Cluster size T values Cortical region 
−23 −62 51 3371 8.34 Left superior IPS 
−33 −41 41   Left anterior IPS 
−26 −67 35   Left inferior IPS 
27 −60 51 3297 8.25 Right superior IPS 
28 −67 38   Right inferior IPS 
51 −34 51 299 6.53 Right postcentral gyrus 
X Y Z Cluster size T values Cortical region 
−23 −62 51 3371 8.34 Left superior IPS 
−33 −41 41   Left anterior IPS 
−26 −67 35   Left inferior IPS 
27 −60 51 3297 8.25 Right superior IPS 
28 −67 38   Right inferior IPS 
51 −34 51 299 6.53 Right postcentral gyrus 
Table 9

Regions of overlap for the contrast of symbolic > nonsymbolic and small > large trials

X Y Z Cluster size T values Cortical region 
48 −63 30 85 4.51 Right TPJ 
49 −57 29    
X Y Z Cluster size T values Cortical region 
48 −63 30 85 4.51 Right TPJ 
49 −57 29    
Figure 5.

Overlaps for the contrasts (A) nonsymbolic > symbolic and large > small, and (B) symbolic > nonsymbolic and small > large, using conjunction analysis at a threshold of P < 0.001 uncorrected at peak level, FWE corrected at cluster level.

Figure 5.

Overlaps for the contrasts (A) nonsymbolic > symbolic and large > small, and (B) symbolic > nonsymbolic and small > large, using conjunction analysis at a threshold of P < 0.001 uncorrected at peak level, FWE corrected at cluster level.

Discussion

The present study showed distinct neural responses recorded at 7T associated with the processing of large and small magnitudes, for both whole-volume group analyses and subject-by-subject ROI analyses. Specifically, the rTPJ showed stronger activation for small numbers compared with larger numbers, and this region displayed lower activity with increasing magnitudes, with a step decrease in activity when large (6+) relative to small magnitude displays (<4) were presented. In contrast, the bilateral IPS exhibited an activation profile in which its overall signal increased with larger numerical magnitudes, with a step increase in activity when large (6+) relative to small magnitude displays (<4) were presented. Importantly, the dissociations for small and large numbers observed in the rTPJ and bilateral IPS (most notably the step change in the functions between the magnitudes 4 and 6) were specific to nonsymbolic number stimuli.

In addition to the contrasts between small and large magnitude displays, there were also differential responses as a function of the notation of the stimuli. Symbolic numbers generated stronger activation in the rTPJ compared with nonsymbolic numbers, while nonsymbolic numbers activated the IPS bilaterally (though particularly on the right IPS) more than was the case for symbolic numbers. It is notable that there was close correspondence at the voxel level between the brain regions that responded most strongly to small number displays and those responding to symbolic number representations. In contrast, the regions activated by large relative to small magnitudes were also activated more strongly by nonsymbolic representations relative to symbolic number representations. These results have a number of important implications for understanding number processing in the brain.

Processing Magnitude

Our data demonstrate for the first time a double dissociation between the processing of small and large nonsymbolic number displays within a single fMRI experiment. This is consistent with evidence from both brain imaging (Sathian et al. 1999; Piazza et al. 2003; Ansari et al. 2007; Vetter et al. 2010) and neuropsychology (Dehaene and Cohen 1994; Demeyere et al. 2010) for distinct processes contributing to the apprehension of small and larger numbers, once stimuli have been parsed as distinct perceptual units. Similar to earlier imaging evidence, the rTPJ was strongly activated by small number displays and the bilateral IPS by large-number displays (with the IPS showing a nonlinear effect of magnitude opposite to that present in the rTPJ for nonsymbolic stimuli)—but the effects have not been established before within a single experiment. There may be several reasons for our finding of the double dissociation here: The rTPJ has been associated with a number of cognitive processes, including the bottom-up engagement of attention (Corbetta and Shulman 2002) and linking perceptual and response selection (Pollmann et al. 2003). Here, we suggest that the rTPJ activation to small rather than large numbers reflects the stronger bottom-up engagement of attention to a small set of stimuli and/or the stronger linkage between displays with few items and the exact number presented, which participants may derive from the sample display to match against the subsequent comparison number. We return to these points after considering the effects of stimulus notation.

  1. Using a pure numerical task. We employed a single task of numerical comparison and thus avoided potentially confounding contributions from distinctive cognitive processes of search, selection, and comparison that may affect tasks such as enumeration (Sathian et al. 1999; Gobel et al. 2004). These additional processes might smear or even eliminate effects of numerical magnitude and notation on fMRI.

  2. Improved resolution, SNR, and BOLD contrast. Our event-related 7T fMRI procedure used both a higher resolution protocol (1.4 × 1.4 × 1.4 mm3) compared with previous studies (Piazza et al. 2002, 2003; Ansari et al. 2007; Vetter et al. 2010), a SNR over 1.6 times greater than the SNR at 4T (Vaughan et al. 2001) and a higher sensitivity of BOLD contrast (1.6 times of that for 3T and 2 times of that for 1.5T) (van der Zwaag et al. 2009).

The greater activation of the bilateral IPS for large- rather than small-number displays may come about because neurons in this brain region encode numerical magnitude, for example, through a monotonically increasing response function (summation coding) (Verguts and Fias 2004), or a compressed (e.g., logarithmic) code (Nieder 2005). This last possibility is supported by current single-neuron evidence from the monkey which shows compressed scaling of numerical quantity (Nieder and Miller 2003). On the other hand, other single-cell–recording studies show a decrease in the number of neurons tuned to a preferred numerosity as number magnitude increases, with a marked drop from numbers 2 to 8 (as used here). In this case, a weaker overall response may be found for larger magnitudes (Nieder and Merten 2007). An alternative view is that the IPS represents exact quantity and this is more difficult to compute for larger than for smaller magnitude displays, and this would be required in the comparison task used here. The net result is that the IPS is exercised more in representing larger numbers, generating the activation profiles we observed. This may be particularly the case for nonsymbolic number displays, where different magnitudes may be more difficult to compute than with symbolic numbers. One other possible view is that processing large numbers might involve goal-directed serial scanning of attention; in contrast, small numbers might be apprehended in a bottom-up manner without demanding attention shifts (although some attentional resources may still be required). Evidence shows that the IPS is part of the attentional network-supporting top-down, goal-directed selection of stimuli (Corbetta and Shulman 2002). Enumerating numerosities beyond the subitizing range might involve serial attentional “tagging” of counted items (Trick and Pylyshyn 1994) and recursive assimilation of smaller numbers, which consequently generates a stronger fMRI signaling regions sensitive to these processes (i.e., the IPS). This could also explain the stronger activation in the IPS for nonsymbolic compared with symbolic numbers—extracting the numerical values from symbolic numbers does not necessarily involve serial shifts of attention to the stimuli.

It might seem that the differential role of attention in processing small and large numbers, implicated here, is inconsistent with previous studies suggesting that subitizing is an attention-demanding process, while estimating larger numbers is not. For instance, in the task of rapid numerosity judgment, enumerating small numbers is modulated by attentional load while estimating large numbers is relatively unaffected (Burr et al. 2010). This is supported by evidence indicating that there is attention-modulated activity in rTPJ for displays in the subitizing range but not for large numbers (Vetter et al. 2010). The divergence may be due to the different experimental protocol used in the current study compared with previous studies. In contrast to manipulating the attentional resource for the numerosity task, we asked participants to attentively compare the 2 consecutive stimuli, so that a relatively precise rather than estimated number was required. Also, our fMRI analysis focused on the process of enumerating the initial comparison stimulus—a process that could take place across the long interstimulus interval (3.8–6.8 s). This more exact, enumeration may require more goal-directed attention, unlike estimation. Thus, the more demanding the process for enumeration, the stronger the activation in IPS (see also (Piazza et al. 2003)) and the longer the associated reaction time (P < 0.001). Interestingly, there was no effect of sample notation was observed on reaction times (P > 0.65).

Effects of Working Memory

There is evidence showing that activity in the IPS is sensitive to number of items held in visual short-term memory (VSTM). Accordingly, it is possible to argue that the present findings represent effects of working memory rather than numerosity per se. Indeed, there is a tight correlation between activity in IPS and the amount of information that can be stored in VSTM. Specifically, activation in IPS increases with the number of objects before reaching a participant's capacity (i.e., 3 or 4), after which it levels off (i.e., across 4, 6, and 8) (Todd and Marois 2004). Capacity is modulated by object complexity and tracked particularly by the superior IPS, which is dissociated from the inferior IPS which encodes a fixed number of objects regardless of object complexity (Xu and Chun 2006). However, working memory processes cannot adequately explain the data we observed. First, for the nonsymbolic numbers, the percent signal change steadily increased with the numbers of objects from 2 to 8, with a step function between small (2 and 4) and large (6 and 8) numbers. This is inconsistent with the activation pattern for working memory which increases up to 4 maximally and reaches a plateau thereafter. As a further test, we analyzed data in superior and inferior IPS regions separately and found similar patterns of increasing signal with numerical magnitudes for nonsymbolic numbers in both the superior and inferior IPS regions in both hemispheres. Thirdly, we reasoned that, if the activation reflects the number of objects kept in working memory, there should be more activation for the nonsymbolic number 2 (when 2 objects are present) compared with symbolic numbers (where only 1 perceptual object is present). However, we found no activations, throughout the whole volume, for either the contrast dot2 > digits, with different numbers of perceptual objects, or for the contrast dot2 > digit2, with the numerical magnitude controlled but different numbers of perceptual objects, even at a reduced threshold (P = 0.005 uncorrected). Therefore, these results indicate a distinct pattern of activation for the enumeration task used here compared with prior results on working memory.

Effects of Notation

As well as contrasting the effects of different number magnitudes, we also compared brain activity for symbolic (Arabic numerals) and nonsymbolic (dot pattern) stimuli. Recent fMRI studies have examined the relations between the coding of symbolic and nonsymbolic number using visual adaptation, with conflicting results. Evidence for transfer from number words to Arabic numerals (Naccache and Dehaene 2001) or from dot numerosities to Arabic numerals (Piazza et al. 2007) suggests an abstract response across notation. On the other hand, there is also evidence for selective adaptation effects across different notations (e.g., no effects of numerical distance on adaptation in the left IPS when nonsymbolic numbers follow symbolic numbers vs. effects of numerical distance when both habituation and probe stimuli are nonsymbolic (Piazza et al. 2007)), suggesting distinct representations for different notations. Studies using multivoxel pattern classification to decode neural representations highlight both distinct and common representations of different notations (Eger et al. 2009). Our evidence, using high-resolution imaging with high SNR and sensitive BOLD contrast indicates both a close overlap in the neural response to symbolic and nonsymbolic number and some differences. The overlap was interesting, however. The regions responding more strongly to large relative to small magnitudes (the IPS, bilaterally) were also more activated by nonsymbolic than symbolic representations. On the other hand, the region showing a strong response to small, exact number (rTPJ) was also differentially activated by symbolic compared with nonsymbolic numbers. The common rTPJ activity might result from the representation of exact magnitudes which, in our paradigm, can be extracted from engaging stimulus-driven attention for small sets of objects and in preparation for the response to the forthcoming comparison display. With nonsymbolic displays, exact number may be accessed more directly from small magnitude arrays by attending individual objects in the display, contrasting the process for large magnitudes in which attention to individual elements may not be an optimal strategy for making an estimate of numerical quantity, and hence effects may be stronger for small magnitudes. With symbolic displays, though, exact magnitudes may be computed directly for small and larger numbers alike (for all stimuli, there is only 1 “perceptual object” in the display), eliciting a stronger response across the board to symbolic displays. According to this account, the rTPJ may be involved in representing exact number and in attending to stimuli in a bottom-up fashion, and perhaps also for subsequent response matching and selection. The IPS, however, is activated by magnitude and, as this is more difficult to compute (perhaps due to limitations in visuospatial attention) for nonsymbolic compared with symbolic stimuli—indeed IPS activity, especially the right hemisphere (Fig. 4C and Supplementary Fig. S1C) was overall greatest for nonsymbolic displays. Note that the differential activation of the rTPJ for exact compared with large nonsymbolic numbers cannot be adequately explained by general task effort. If the rTPJ is significantly modulated by task difficulty across the conditions, it should be significantly correlated with response latencies in all cases. However, there was a significant reverse correlation between the neural response to match stimuli in the rTPJ and reaction times in the comparison task for symbolic numbers (r = −0.541, P < 0.014; P > 0.12 for nonsymbolic numbers) (See Fig. S2 and detailed analysis in Supplementary Material). This reverse correlation may be expected if this region is involved in representing small exact numbers, but not if it is modulated by task difficulty. Another alternative interpretation for the differential responses of rTPJ to exact and large nonsymbolic numbers is that these effects are driven by deactivation during saccades across space to accumulate quantity information about nonsymbolic dots. However, there are several reasons that this interpretation cannot hold. First, Ansari et al. (2007) observed a relative low activation or suppression for large numerosities compared with that for small numerosities, and importantly, larger magnitudes of suppression were associated with smaller reaction times (Ansari et al. 2007). We reason that, if serial saccades result in neural suppression, larger magnitudes should be associated with a larger number of saccades and thus with longer not shorter reaction times. Second, in our current study, subjects were required to judge which of the 2 sequential displays contained the larger number, so exact counting was not necessary for the numerical comparison. Third, in our experimental design, the sample stimuli were presented for 200 ms, which should be sufficiently brief to exclude eye movements across elements in the visual display (Carpenter 1988; Gottlieb et al. 2005).

As well as the common activation patterns, there were cases where different activation patterns arose with symbolic and nonsymbolic stimuli. With nonsymbolic stimuli in particular, there were examples of a nonlinear response to magnitude, with the rTPJ and bilateral IPS showing a step-change inactivity between small (≤4) and large-number displays. With symbolic stimuli, there was an overall stronger activation for the left IPS compared with the rIPS (P < 0.002). Contrasts of nonsymbolic and symbolic numbers revealed a specific role of the rTPJ for symbolic over nonsymbolic numbers (Fig. 3B, right panel of Fig. 4A, and Supplementary Fig. S1A), while there was a distributed network of bilateral, but more right-laterized, dorsal regions including the IPS, posterior parietal and occipital cortices for nonsymbolic relative to symbolic numbers (Fig. 3A, left panel of Fig. 4A,B, and Supplementary Fig. S1B,C). We did not observe significant activation of the left TPJ for symbolic over nonsymbolic numbers as did some previous studies (Grabner et al. 2007; Holloway et al. 2010; Price and Ansari 2011). The divergence may be due to the different experimental paradigm we used. First, we found activity in the rTPJ for symbolic over nonsymbolic numbers when sample stimuli were enumerated. This procedure avoids possible effects from the processes of comparison and response selection once the match stimulus is presented. Indeed, different activation maps were observed for contrasts of sample and match stimuli (e.g., rTPJ was activated in the contrast digit–digit > digit–dot but not in the contrast digit–digit > dot–digit under the same threshold of P < 0.001 uncorrected, see Supplementary Fig. S1A). Second, the relatively long delay after the sample may provide enough time for subjects to enumerate both large and small numerosities, and thus to map nonsymbolic numbers to symbolic numbers, which may consequently activate the left TPJ (important for symbolic number identification). This could cancel out the effect of signal change in left TPJ for symbolic and nonsymbolic numbers. The left TPJ may play an important role in language-mediated processes such as the conceptual identification of symbolic numbers, the exact computation of arithmetic facts, and automatic fact retrieval, while the right TPJ is more involved in bottom-up attentional mechanisms engaged in symbolic and small nonsymbolic number processing.

The evidence for differential involvement of the left IPS for symbolic numbers here links to the results of Notebaert et al. (2010) who reported adaptation effects for symbolic stimuli in the left but not in the right IPS. The results also support the idea of Ansari et al. (2007) that the left IPS is specialized for encoding symbols of numerical magnitude. Importantly, the representation of nonsymbolic small numbers also correlated with stronger activation in the left than in the right IPS (P < 0.022 for nonsymbolic small numbers, but P > 0.6 for nonsymbolic large numbers), suggesting an important role of left IPS for exact numerical judgments including both symbolic and small nonsymbolic numbers. The striking similarity between symbolic and small nonsymbolic numbers in neural response may make contributions to theories of mathematical cognition. For example, the current data are consistent with symbolic numbers being mapped onto small individuated sets of objects (Le Corre and Carey 2007; Carey 2009), while they challenge the hypothesis that symbolic numbers derived their meaning by association with analog magnitudes (e.g., the approximate numerical meaning of large nonsymbolic numbers) (Dehaene 1997; Wynn 1998). The evidence for differential involvement of the right IPS for large nonsymbolic numbers also links to the results of Izard et al. (2008) who reported adaptation effects for nonsymbolic stimuli in the right but not in the left IPS in infants. What is not clear from our data is whether this differential activation stems from specialization of the right hemisphere for approximate estimation (see Andres et al. 2005; Burr et al. 2010), or from the greater difficulty in computing magnitude from large nonsymbolic relative to symbolic stimuli and stimuli in small nonsymbolic displays. Additional work incorporating adaptation as well as stimulus comparison procedures might be useful here to prise-apart effects of approximate number representation and the difficulty in computing exact larger numbers.

Supplementary Material

Supplementary material can be found at: http://www.cercor.oxfordjournals.org/

Funding

This work was supported by the Ministry of Science and Technology of China grants (2012CB825500, 2012IM030100, 2010IM030800), National Nature Science Foundation of China grants (91132302, 90820307, 31100813), a grant from the Chinese Academy of Sciences which supported a Visiting Professorial post for Glyn Humphreys.

Notes

We thank 3 anonymous reviewers for valuable comments on the manuscript. Conflict of Interest: None declared.

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