Edited by: Seppe Santens, Ghent University, Belgium
Reviewed by: Lars Nyberg, Umeå University, Sweden; Roland Grabner, Swiss Federal Institute of Technology Zurich, Switzerland; Samuel Shaki, Ariel University Center, Israel
*Correspondence: Kristen P. Blair, Stanford University School of Education, Wallenberg Hall, Building 160, 450 Serra Mall, Stanford, CA 94305, USA. email:
^{†}Kristen P. Blair and Miriam RosenbergLee have contributed equally to this work.
This is an openaccess article distributed under the terms of the
Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a
How are abstract mathematical concepts represented in the brain? Negative integers are among the earliest abstract concepts encountered in mathematics curricula. Unlike positive numbers, negative numbers have no obvious perceptual referents, and therefore, children can struggle when learning about them (Liebeck,
A standard paradigm for investigating mental representations of positive numbers has participants quickly select which of two numbers is quantitatively larger or smaller (Moyer and Landauer,
Two general accounts of negative number processing have been proposed in the behavioral literature (Varma and Schwartz,
The second,
In sum, the behavioral research has been inconclusive with respect to the representation of negative numbers, with several studies supporting each of the leading models. One explanation of this inconsistency is that adults have multiple ways of interacting with negative numbers dependent on the task at hand (GanorStern et al.,
The current study explores the neural correlates of negative number processing in the context of a symbolic magnitude comparison task where positive and negative trials are intermixed. The intraparietal sulcus (IPS) within the dorsal aspects of the posterior parietal cortex has been implicated in numerical processing of positive numbers. IPS activation has also been found more generally for tasks requiring spatial attention and serialization (Majerus et al.,
Very little is currently known about “neural distance effects” for negative numbers in the IPS and or other brain areas like the prefrontal cortex, which is sensitive to task difficulty and rulebased processing. To our knowledge no previous neuroimaging study has explicitly manipulated numerical distance on comparisons with negative numbers. Based on the positive integer literature, we expect negative number processing to rely on the IPS. An important question is whether negative number processing engages this area differentially from the positive numbers. One imaging study to date has examined the representation and processing of negative numbers. Chassy and Grodd (
In the current study, we used eventrelated fMRI to investigate the processing and representation of negative and positive numbers. A factorial design crossed number type (positive vs. negative vs. mixed) and distance (near vs. far). A univariate analysis explored the neural correlates of number type and distance, and we discuss the results in relation to the two competing models of integer representation. A complication in interpreting the results of signal level differences between negative and positive comparisons is that negative comparisons take longer than positives, and signal level differences could be due to longer processing time. Thus, it is critical to address the extent to which activation differences reflect neural responses specific to negative number processing, as opposed to general task difficulty.
We complemented traditional univariate analyses of signal change with a representational similarity analysis (RSA), a multivoxel approach for examining stimulusrelated brain responses (Kriegeskorte et al.,
To further investigate number representation in the IPS in an anatomically unbiased manner, we used cytoarchitectonically defined maps to quantify both the overall level of activity (Wu et al.,
Twentytwo righthanded individuals (16 females), mean age 24.2 (SD 6.8), participated in the study. Three additional participants were excluded due to technical problems during data collection (two participants), or failure to complete the study (one participant). Participants were drawn from a paid subjects pool, and were compensated for their participation. All participants provided written informed consent in compliance with Stanford University’s Human Research Protection Program.
The study design was modeled on behavioral research by Varma and Schwartz (
Positive  Negative  Mixed  

Near  [(1, 3); (1, 4); (2, 4) (6, 8); (6, 9); (7, 9)]^{a}  [(−1, −3); (−1, −4); (−2, −4) (−6, −8); (−6, −9); (−7, −9)]^{a}  [(1, −2); (−1, 2); (−1, 1)]^{b} 
Far  [(1, 9); (1, 8); (2, 9)]^{b}  [(−1, −9); (−1, −8); (−2, −9)]^{b}  [(1, −6); (−1, 6); (1, −7) (−1, 7); (2, −6); (−2, 7)]^{a} 
Stimuli were presented in four runs using a fast eventrelated design. There were 72 trials in each run, yielding 288 total trials (48 per condition). Within each run, participants saw an equal number of trials from all six conditions, presented in a random order. Left/right configuration of the digits (e.g., 2, 7 vs. 7, 2) was counterbalanced within each run. For the far trials, each problem was repeated twice within a run, in each configuration, because there are fewer possible problems of distance seven or eight among the single digits.
Stimuli were displayed using Eprime presentation software (Psychological Software Tools, Pittsburgh, PA, USA), and were projected onto a screen at the head of the scanner bore. Participants viewed the screen through a mirror directly in their line of vision. The two digits were presented in green on a black background, equidistant from the center of the screen. Participants held a button box in their right hand and indicated which number was greater (or lesser) by pressing their index finger to choose the number on the left, and their middle finger to choose the number on the right. Before each stimulus was presented, participants saw a blank screen, jittered between 0.5 and 5.5 s, in 100 ms increments. Participants then saw a center fixation cross for 500 ms, followed by the stimulus, which was present for 1500 ms (see Figure
Prior to entering the scanner, participants completed a brief 20 problem training session. At the beginning of each run in the scanner, two short instruction screens reminded the participants of the task and informed them whether they were making a greater or lesser judgment for the run. Participants then completed two unrecorded practice trials. Data collection began after the practice trials and lasted 5 min and 38 s. Thus, the time between the instructions and the first trial was approximately 20 s, including 14 s for signal equilibration and the initial jitter period.
Images were acquired on a 3T GE Signa scanner using a standard GE 8channel head coil (software Lx 8.3). Head movement was minimized during scanning with small cushions fit between the head and the coil. A total of 30 axial slices (4.0 mm thickness, 0.5 mm spacing) parallel to the AC–PC line and covering the whole brain were imaged using a T2* weighted gradient echo spiral in/out pulse sequence (TR = 2000 ms, TE = 30 ms, flip angle = 80°; Glover and Lai,
The first seven volumes were discarded to allow for signal equilibration effects. A linear shim correction was applied separately for each slice during reconstruction using a magnetic field map acquired automatically by the pulse sequence at the beginning of the scan (Glover and Lai,
Images were realigned to the first scan to correct for motion and slice acquisition timing. Images were spatially normalized to standard MNI space using the echoplanar imaging template provided with SPM8, resampled every 2 mm using trilinear sinc interpolation, and smoothed with a 6mm fullwidth halfmaximum Gaussian kernel to decrease spatial noise prior to statistical analysis. Translational movement in millimeters (
Statistical analysis was performed on individual and group data using the general linear model implemented in SPM8. Taskrelated regressors were modeled as boxcar functions corresponding to each condition. There were six regressors (three number types × two distances) for the correct trials, with one additional regressor for all incorrect trials. Additionally, the six movement parameters generated from the realignment procedure were included as regressors of no interest. Regressors of interest were convolved with a hemodynamic response function and a time derivative to account for voxelwise latency differences in hemodynamic response. Lowfrequency drifts at each voxel were removed using a highpass filter (0.5 cycles/min) and serial correlations were accounted for by modeling the fMRI time series as a first degree autoregressive process (Friston et al.,
Group analysis was performed using a randomeffects model that incorporated a twostage hierarchical procedure (Holmes and Friston,
In each iteration of the Monte Carlo procedure, a 3D image with the same resolution and dimensions as the fMRI scan was randomly generated and smoothed with a 6mm FWHM Gaussian kernel for consistency with the inclusive mask used to report the results of the general linear model analysis. A gray matter mask was then applied to this image. The maximum cluster size at a given height threshold was recorded for each iteration, and 10,000 iterations were performed. At a height threshold of
Functionally defined regions of interest were compared to cytoarchitectonic maps of parietal cortex, using the Anatomy Toolbox in SPM8 (Eickhoff et al.,
Two sets of ROIs were identified: (1) Functional ROIs were constructed using 10 mm spheres centered at the peaks of significant activation in the pairedsample
Representational similarity analysis (RSA) considers the voxelwise similarity between the activation patterns of task conditions within an ROI (Kriegeskorte et al.,
Across all cells of the design, average accuracy was above 90%. Accuracy data were analyzed using a three Number Type (positive, negative, mixed) × 2 Distance (near, far) repeated measures ANOVA. There was a main effect of number type [
For each participant, the mean RT and SD on correct trials were computed for each number type and distance. RTs more than 2.5 SD from the individual mean were removed. A three Number Type (positive, negative, mixed) × 2 Distance (near, far) repeated measures ANOVA revealed a significant main effect of number type, [
Mixed trials are not considered further in the main text because they failed to show the classic indicator of magnitude processing, differential RTs for near and far comparisons. These trials may have been solved with the strategy of identifying a negative sign to find the smaller number without considering magnitude. Here we focus on pure negative and positive comparisons, which did show robust distance effects. GLM results comparing mixed vs. positive and negative trials are presented in the Appendix (Table
Compared to positive numbers, negative numbers elicited greater activity in the bilateral middle frontal gyrus (MFG), presupplementary motor area (Figure
Brain region  Peak MNI coordinates 
Peak 
No. of voxels  



Bilat PCC  0  −42  46  4.65  87 


Left ITG  −40  −56  −6  5.64  237 
Right IPS  30  −64  48  5.43  396 
Bilat preSMA  2  16  48  5.36  147 
Bilat LG  −16  −78  2  5.27  1621 
Right inferior LOC  40  −88  −8  4.67  215 
Left IPS  −30  −52  38  4.65  89 
Left posterior IPS  −24  −68  42  4.46  84 
Right MFG  54  32  24  4.43  35 
Left MFG  −32  14  30  4.4  40 
Left inferior LOC  −46  −80  −10  4.16  49 
Left SPL/IPS  −38  −54  54  4.11  44 
Because RTs are longer for negative than positive trials, greater activity for negative numbers could be driven by task difficulty. To test this possibility, in each brain region that showed differences between negative and positive numbers (Table
Collapsing across negative and positive numbers, near trials showed increased activation over far trials in the left premotor cortex and bilateral somatosensory cortex (SC) extending posteriorly into the superior parietal lobe (SPL; Table
Brain region  Peak MNI coordinates 
Peak 
No. of voxels  

Near > far  
Left premotor cortex  −28  −18  72  4.76  49 
Left SPL/somatosensory  −34  −42  70  4.68  71 
Right SPL/somatosensory  36  −38  70  4.46  39 
Far > near  
No significant clusters  
Negative (near–far) > positive (near–far)  
Right TOF  28  −48  −12  4.71  40 
Positive (near–far) > negative (near–far)  
No significant clusters 
For the interaction between number type and distance, only the right temporal–occipital fusiform cortex (TOF) was statistically significant, but this effect was driven by differential levels of deactivation rather than greater activation during number comparison (Figure
We examined differential responses in six anatomically defined IPS subdivisions (left and right hIP1, hIP2, hIP3) derived from previous cytoarchitectonic mapping studies (Choi et al.,
We used RSA to examine similarity of IPS response patterns to near and far trials. RSA between these two trial types was computed separately for positive and negative numbers.
We first examined RSA in four functional ROIs defined as 10mm spheres around activation peaks of the IPS regions that showed greater activation to negative vs. positive numbers. The left IPS ROI centered at (−30, −52, 38) showed a greater near–far similarity for negative numbers than for positive numbers [
Additional analyses were conducted using the six (three in each hemisphere) cytoarchitectonically defined IPS ROIs described in the previous section. A repeated measures ANOVA with number type (negative, positive), ROI (hIP1, hIP2, hIP3), and hemisphere (L, R) as within participant factors revealed a significant effect of ROI [
While a large body of neuroscience research has addressed the representation of positive numbers, much less is known about the negative numbers. In this study, we examined neural responses and representations of negative integers using traditional univariate analyses and a novel multivariate analysis of representational similarity. To our knowledge, this is first brain imaging study to use a distance manipulation to investigate the representation of negative numbers. Compared to positives, negative number comparisons elicited greater activation in several parietal, frontal, and occipital regions, including bilateral IPS, bilateral MFG, and bilateral LOC. Univariate analyses failed to reveal strong neural distance effects in the IPS, but the multivariate RSA revealed a less differentiated representation for negative, compared to positive, numbers. Furthermore, neural representations were associated with individual differences in performance such that individuals with more distinct neural representations of negative magnitudes performed faster.
The IPS is crucial to positive number processing, and in the following sections, we focus on the role of the IPS in negative number processing, first in terms of overall signal levels and then in terms of multivoxel representations. Next, we consider the potential role of the prefrontal cortex in the rulebased processing of negative numbers. Finally, we discuss our findings in the context of the previous behavioral research on negative numbers and demonstrate how multivariate approaches can provide novel insights into abstract number representation.
Both positive and negative numbers elicited robust bilateral activity in the IPS. For negative numbers compared to positive numbers, whole brain analyses revealed greater activity in a distributed set of regions within the IPS, specifically three clusters in the left IPS and one cluster in the right (Figure
Negative numbers took longer to process, suggesting that task difficulty may drive differences in activation to negative and positive numbers. Indeed, after covarying out RT, we found no differences between negative and positive numbers, highlighting the difficulty of disentangling general task difficulty effects from polarityspecific processing. Similar findings likely apply in the domain of positive number comparisons, where near distance comparisons are known to have longer RT and elicit greater activation in the IPS compared to far number comparisons. Few studies have considered whether these effects are independent of RT differences (Gobel et al.,
Behavioral distance effects are thought to reflect an analog magnitude representation of positive numbers (Moyer and Landauer,
In contrast to univariate analysis, RSA revealed differences between neural representations of near and far comparisons across the two number types. Neural responses in the IPS were less differentiated for negative than positive numbers. Specifically, there was greater similarity between the multivoxel activity patterns for near and far negative number pairs than near and far positive pairs. That is, there were smaller neural representational distance effects for negative numbers than for positive numbers. Critically, increased representational differentiation between near and far negative numbers was associated with faster response times across subjects, consistent with a broader claim that greater differentiation in neural representation facilitates comparative processes. We suggest that greater experience with positive numbers leads to more distinct representations compared to negative numbers (RosenbergLee et al.,
Representational similarity analysis in both the functional and structural ROI implicated a midanterior region of the IPS, the hIP1, as a common locus of less differentiated representations for negative numbers. Resting state fMRI and diffusion tensor imaging analyses have shown that relative to the posteriormost IPS region hIP3, hIP1 has greater functional and structural connectivity with lateral prefrontal cortex, while hIP3 is more strongly connected to ventral visual areas (Uddin et al.,
In contrast to the IPS, the prefrontal cortex was robustly engaged only for the more demanding task of negative number comparison. The left and right MFG also showed greater activation for negative, compared to positive, numbers. However, we did not find greater prefrontal cortex activation over baseline for positive numbers. While lateral prefrontal cortex activity is often reported for arithmetic tasks, previous studies of number processing have not consistently found activation in this area (Arsalidou and Taylor,
Consistent with previous findings on integer comparison, we found that negative and positive number comparisons showed parallel distance effect slopes, but negatives took longer than positives. While prior behavioral studies have not reported accuracy differences (Tzelgov et al.,
Based on extant behavioral literature, two theoretical models have been proposed for negative number processing. The
The
The signal level differences found by univariate analyses in our study are consistent with either a
Representational similarity analysis provides a way to examine patterns of activation independent of overall signal level. RSA in both functionally and cytoarchitectonically defined ROIs showed that in subregions of the IPS, representations of near and far positive numbers were more differentiated than near and far negative numbers. If the differentiation of far and near in negatives is functionally important, then we should predict an effect on behavior. This is what we found – a greater degree of differentiation among negative numbers was correlated with faster RTs. These findings point to a unique, but less welldeveloped, magnitude representation for negative numbers.
Alternate explanations are possible. For example, the application of rules might produce more noise in the positive magnitude representations when used for negative trials, resulting in decreased differentiation. However, if this were the case, prefrontal cortex activation for the application of a constant rule for negatives should not have diminished when covarying out RT (although caution should be taken when interpreting null results).
Our preferred interpretation, based on the RSA, is that an
The field of cognitive neuroscience has focused considerable attention on how the natural numbers are represented in adults. During formal education, students are exposed to increasingly abstract quantitative relations, and mastery of these concepts forms a foundation for higher mathematics such as algebra and calculus. Yet little is known about how the brain enables and organizes abstract quantitative concepts. Examining negative numbers provides a first step toward a fuller understanding of the neural basis of these processes. The multivariate analysis technique used here reveals for the first time that negative numbers appear less well differentiated than positive numbers in the IPS, and that greater differentiation within negative number problems is associated with faster RT on negative problems. These findings support the proposal that people develop facility with negative numbers by creating a new representation that incorporates magnitude properties while remaining distinct from the natural numbers. Beyond the domain of negative numbers, our findings may reflect a general property of neural representation: that experience leads to greater differentiation between stimuli, even for abstract concepts.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
As shown in Table
Brain region  Peak 


No significant clusters  
Bilat caudate  −6  6  8  4.86  133 
Right precentral gyrus  46  4  40  4.44  33 
Right cerebellum  34  −62  −28  4.43  63 
No significant clusters  
Right cerebellum  30  −64  −28  7.37  1127 
Left cerebellum  −24  −70  −20  6.97  874 
Left IPS  −22  −60  46  6.21  1145 
Left precentral gyrus  −54  0  42  5.77  306 
Left MFG  −32  12  28  5.58  390 
Right MFG  58  24  28  5.39  536 
Bilat cerebellum  −6  −80  −22  4.35  306 
Bilat preSMA  0  12  58  5.27  382 
Right frontal operculum cortex  48  18  −4  5.08  85 
Left premotor cortex  −26  −4  70  5  106 
Right IPS  34  −58  42  4.98  302 
Right MFG  36  6  64  4.94  67 
Left SPL  2  −82  38  4.62  81 
Left caudate  −14  16  −10  4.46  30 
Left precentral gyrus  −54  8  20  4.28  62 
Right SPL  32  −42  48  4.07  42 
Region  Number of voxels in the region  % of cluster in region  % of region activated 

R hIP1  48  11.9  21.1 
R hIP3  45  11.2  14.8 
R area 2  24  5.8  2.5 
L hIP1  22  25.0  4.8 
L hIP1  22  48.9  4.7 
L hIP3  14  32.4  5.1 
L SPL 7PC  3  7.4  1.6 
L hIP2  2  4.5  0.9 
L SPL 7A  2  4.5  0.1 
L area 2  1  2.3  0.1 
No overlap with cytoarchitectonic areas 
We thank Dr. Sashank Varma for assistance with experimental design and Dr. Sarit Ashkenazi for useful discussions. This work was supported by the National Institutes of Health (HD047520 and HD045914); and the National Science Foundation (BCS/DRL 0449927 and DRL 0814768). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the granting agencies.