Numerosity processing is context driven even in the subitizing range: An fMRI study.

Numerical judgments are involved in almost every aspect of our daily life. They are carried out so efficiently that they are often considered to be automatic and innate. However, numerosity of non-symbolic stimuli is highly correlated with its continuous properties (e.g., density, area), and so it is hard to determine whether numerosity and continuous properties rely on the same mechanism. Here we examined the behavioral and neuronal mechanisms underlying such judgments. We scanned subjects' hemodynamic responses to a numerosity comparison task and to a surface area comparison task. In these tasks, numerical and continuous magnitudes could be either congruent or incongruent. Behaviorally, an interaction between the order of the tasks and the relevant dimension modulated the congruency effects. Continuous magnitudes always interfered with numerosity comparison. Numerosity, on the other hand, interfered with the surface area comparison only when participants began with the numerosity task. Hemodynamic activity showed that context (induced by task order) determined the neuronal pathways in which the dimensions were processed. Starting with the numerosity task led to enhanced activity in the right hemisphere, while starting with the continuous task led to enhanced left hemisphere activity. Continuous magnitudes processing relied on activation of the frontal eye field and the post-central gyrus. Processing of numerosities, on the other hand, relied on deactivation of these areas, suggesting active suppression of the continuous dimension. Accordingly, we suggest that numerosities, even in the subitizing range, are not always processed automatically; their processing depends on context and task demands.

Introduction

In daily life we rely profoundly on numerical judgments, for example, to choose the shortest checkout line in a store, or choosing the larger pile of candy. To study the cognitive mechanisms underlying numerical cognition, many studies used non-symbolic representations of numbers—arrays of items—as stimuli. In a typical non-symbolic numerosity comparison task, participants are presented with two arrays of items and asked to choose the array containing more dots (e.g., Cantlon et al., 2006; Piazza, 2010). These studies have suggested that processing of numerosities is innate and automatic (e.g., Cantlon et al., 2009; Coubart et al., 2014; Feigenson et al., 2004). Moreover, It was proposed that due to their importance, numerosities are processed by a dedicated brain circuitry (e.g., Dehaene et al., 2003; Harvey et al., 2013).

However, recent studies have demonstrated that it is very difficult to study the mechanisms underlying non-symbolic numerosity in isolation from continuous magnitudes. Indeed, numerosities and continuous magnitudes usually correlate. For example, more apples will fill more of a bag than fewer ones; when placed on the ground they would either occupy more area, or be more crowded. Therefore, whenever two to-be-compared arrays differ in numerosity, they also differ in their continuous magnitudes. These continuous magnitudes can potentially influence performance (for reviews see Leibovich and Henik (2013) and Mix et al. (2002)). Studies investigating the impact of continuous magnitudes in non-symbolic numerical comparison tasks have suggested that numerical abilities are not primary but depend on other cognitive abilities and do not rely on designated brain regions (e.g., Cantrell and Smith, 2013; Gebuis and Reynvoet, 2012a, Gebuis and Reynvoet, 2013; Leibovich and Henik, 2014). Accordingly, it is unclear whether numerosity and continuous magnitudes processing are separable and what behavioral and neuronal mechanisms they share.

Numerosity and continuous magnitudes—a mere correlation or inseparable processing?

The notion of numerosity as an innate and primary ability is based on studies with non-human animals (McComb et al., 1994, Nieder and Dehaene, 2009, Pisa and Agrillo, 2008), young babies and newborns (e.g., Cantlon et al., 2009; Coubart et al., 2014; Xu and Spelke, 2000) who exhibited a spontaneous ability to discriminate numerosities. These studies, however, suffered from an inherent confound. The numerosity in the presented arrays was correlated with continuous magnitudes (Mix et al., 2002). It is therefore possible that animals and newborns do not rely on numerosities to discriminate between arrays but instead they rely on continuous magnitudes (for an extended discussion see Leibovich and Henik (2013)). In 2002, Mix et al. showed that results of seminal developmental studies that used non-symbolic numerosities could be explained by discrimination of continuous magnitudes. For example, Xu and Spelke (2000) habituated 6-month-old babies to arrays containing dots of different sizes. The test display was different either in the number of the dots or in the overall area of the dots. Since the babies looked longer when the number of dots changed (compared to when the area changed), the authors concluded that babies were sensitive to changes in numerosities. However, Mix et al. noted that the contour length was positively correlated with numerosity. Hence, babies might have responded to changes in contour length rather than numerosity.

With the problem of correlation between numerosity and continuous magnitudes in mind, different studies employed different methods to control for a possible influence of continuous magnitudes in the experimental designs. Some manipulated only one continuous magnitude at a time (Mussolin et al., 2010), others assigned a random dot size to each array (Piazza and Izard, 2009), some used a single array containing two different colors of dots where participants had to indicate the color of the more numerous dots (Halberda et al., 2008), etc.

All these methods shared an inherent assumption that continuous magnitudes are not processed automatically and would not affect performance unless they reliably predicted numerosity. Recent studies, however, suggested that continuous magnitudes are processed even when they are irrelevant and are an unreliable cue of numerosity. Leibovich and Henik (2014) presented participants with pairs of dot arrays (5–25 dots per array) and asked them to choose the array containing more dots. Using stimuli generated by code provided by Gebuis and Reynvoet (2011), numerosity was minimally correlated with continuous magnitudes, so the magnitudes were neither a reliable cue of numerosity nor relevant to the task. Regression analysis with the numerosity ratio and the ratio of five continuous magnitudes as predictors (i.e., average diameter, total surface area (the sum of all dots area in the array), area extended (the smallest contour that included all of the dots, as if an elastic band was wrapped around the dots), density and total circumference) revealed that half of the explained variance in response times (RTs) was explained by continuous magnitudes.

The importance and utility of the continuous magnitudes in numerosity judgments was demonstrated recently in an event related potentials (ERP) experiment (Gebuis and Reynvoet, 2013). ERPs were recorded while participants passively viewed arrays of dots in which continuous magnitudes were carefully controlled. Results showed that ERP fluctuations were correlated with continuous magnitude changes but not with numerosity change, even when participants were told that the number of dots would change. The authors concluded that extraction of continuous magnitudes and not the extraction of numerosities from a visual scene is automatic. In a functional magnetic resonance imaging (fMRI) study, Leroux et al. (2009) presented a “Piaget-like” task for adults. In the experiment, same-size horizontal bars were presented in two rows. The spaces between the bars were manipulated and participants were asked to decide if the two lines contained the same number of bars or not. In congruent trials, in the row containing more bars, the bars were further apart from each other than in the row with the fewer bars (i.e., convex-hull positively correlated with numerosity), and vice-versa in incongruent trials (i.e., convex-hull negatively correlated with numerosity). Although success rates were high in both conditions, there was more activation of brain areas related to conflict monitoring and cognitive control in incongruent trials, suggesting that adults process continuous magnitudes, even when they are not consistently correlated with numerosity. Taken together, these studies show that continuous magnitudes are processed during numerical judgments, even when they are completely irrelevant to the task and do not consistently correlate with numerosity.

Brain areas related to processing of numerosity and continuous magnitudes

Behavioral evidence suggesting numerosity processing is basic and innate led to the notion that numerosity processing is executed by a dedicated brain circuitry (Dehaene, 1997, Dehaene and Changeux, 1993). Efforts to find such a brain region led to conflicting results. While some studies reported the existence of a dedicated neuronal substrate (e.g., Castelli et al., 2006; Chassy and Grodd, 2012; Dehaene et al., 2003; Harvey et al., 2013; Piazza et al., 2007), others proposed numerosities were processed by the same brain areas as other magnitudes (e.g., Fias et al., 2003; Piazza et al., 2007; Pinel et al., 2004). Naturally, isolating the neuronal mechanism underlying numerical cognition requires dissociating numerosity from continuous magnitudes; but, is that at all possible? In an fMRI study, Chassy and Grodd (2012) had participants compare either disk size (i.e., continuous task) or numerosity of dots (i.e., numerosity task). While some brain areas were active to the same extent for both tasks (e.g., the left middle occipital gyrus and the right supramarginal gyrus), some areas were more active in the numerosity task (e.g., bilateral primary visual cortices (BA17), right superior parietal lobule, and the bilateral middle occipital gyri). In this task, all the dots were presented in the same size. As a result, continuous magnitude and numerosity were correlated and the same strategy could have been employed for both tasks. This can explain the overlapping areas. Importantly, in the numerosity comparison task, one has to integrate items, that is, to sum the numerosity (or the total surface area) of each array before comparing. Such a step does not exist in the continuous comparison task. Hence, the areas that were more active during the numerosity comparison task might reflect item integration and not processing of numerosities.

In a more recent study aiming to find brain circuitry specific to numerosity processing, Harvey et al. (2013) asked whether processing of numerosity is organized topographically in the brain. Participants were presented with 1–7 dots. To control for the influence of continuous magnitudes, five different groups of 3 arrays of dots were used; in each different group, a different continuous magnitude was kept constant across all numerosities. For example, in the “constant area” group of stimuli, the total surface area of the 3 arrays (i.e., with 1, 4 and 7 dots) was identical; that is, the 1 dot in the 1-dot array was 4 times larger than the 4 dots in the 4-dot array, and at the same time the 1 dot was 7 times larger than the 7 dots in the 7-dot array (see Fig. 1 in Harvey et al. (2013)). The authors reported that areas in the right posterior parietal lobe showed a topographical representation, with small numerosities processed in more medial areas and larger numerosities in more lateral areas. However, as suggested by Gebuis et al. (2014), while one continuous magnitude was controlled for, all the other continuous magnitudes correlated with numerosity. Hence, the topographic map found “reflects a weighted response of neurons that encode different sensory cues rather than a pure numerosity estimate” (p. 1). Up until this point, it is unclear whether the automaticity associated with numerical processing is a consequence of continuous magnitudes processing, or whether it is a primary function. As such, it is unclear whether brain areas dedicated to numerosity processing exist.

Fig. 1

Example for congruent and incongruent trials.

The current study

The current fMRI study has two major aims: first, to investigate the reciprocal relations between numerosity and continuous magnitudes; and second, to investigate the corresponding brain mechanisms underlying these relations.

We chose to use numerosities between 2 and 4. This range of numerosities is known as the subitizing range (Revkin et al., 2008, Trick and Pylyshyn, 1994). The main reason for choosing this range is that estimation of numerosities in this range has been consistently found to be very fast and accurate. Some researchers even assert that subitizing is a perceptual rather than a numerical process (Hyde, 2011; but see also Burr et al. (2010)). Thus, we used the easiest numerical task available relative to continuous magnitudes to decrease salience differences between the tasks. In each trial, participants were presented with two arrays of gray dots and were asked to indicate where there were more dots (i.e., numerical task) or more gray area (i.e., continuous task), while their neural activity was measured using fMRI. Half of the trials were congruent; namely, the array containing more dots also contained more surface area, was denser, had larger dots, etc., and vice-versa for incongruent trials. The same stimuli were used for both tasks. Importantly, half of the participants started with the numerical task (i.e., the NC group) and half with the continuous task (i.e., the CN group), and the effect of order was tested at both the behavioral and functional levels.

To investigate the reciprocal relations at the behavioral level, we analyzed the difference (in RTs and error rates) between congruent and incongruent trials (i.e., the congruity effect) for each task and each group. A congruity effect in the numerical task would indicate that the irrelevant continuous magnitudes were processed automatically. A congruity effect in the continuous task would indicate that the irrelevant numerosity was processed automatically. An asymmetric congruity effect, namely a congruity effect in the numerical task but not in the continuous task (i.e., task×congruity interaction), would support the notion that continuous magnitudes underlie numerosity processing. If the size of the congruity effect is modulated not only by task, but also by tasks order (i.e., task×order interaction), it would suggest that the reciprocal relations between the dimensions are not constant, but depend on the context in which the task is being performed.

In order to understand the underlying neural mechanisms of the reciprocal relationship between numerosity and continuous magnitudes, we conducted a whole brain analysis corresponding to the behavioral ones. We contrasted the different groups, namely, the group that started with the numerical task and the group that started with the continuous task, in order to reveal brain areas where the activity was modulated by the context in which the task was performed.

We contrasted the brain activity during incongruent trials of the numerosity and continuous tasks, separately for the different groups, to test whether the brain resolves conflict differently when the conflict arises from different types of magnitudes. To test whether activity during conflicting trials is further modulated by context, we also contrasted incongruent trials from both groups (NC group: incongruent continuous task>incongruent numerical task)>(CN group: incongruent continuous task>incongruent numerical task).

To explore the brain areas activated when top-down attention is directed towards numerosity and continuous magnitudes, we contrasted the brain activity during congruent trials of the numerosity and continuous tasks separately for the different groups. If there are brain areas that are more active when attention is directed to numerosities, they would be more active in the congruent trials of the numerical task (for a similar idea see O’Craven et al., 1999). To test whether activity during congruent trials is further modulated by context, we also contrasted congruent trials from both groups (NC group: congruent continuous task>congruent numerical task)>(CN group: congruent continuous task>congruent numerical task).

Methods

Participants

Forty-eight students from Ben-Gurion University of the Negev participated in the experiment. The experimental procedures were approved by the local Helsinki committee. All participants were right-handed, monolingual native Hebrew speakers, with intact or corrected vision, and no reported learning disabilities or attention deficits. Participants were compensated for their participation in the experiment with a monetary reimbursement (100 NIS). Eight participants were excluded from the analysis; three subjects were removed due to excessive motion during scanning (we allowed no more than a 3 mm deviation from the first image collected and no more than a 1 mm deviation between one functional image to the next functional image); three others were excluded because they did not comply with the tasks instructions (e.g., responding to the numerical and not the continuous magnitudes or vice versa) and two were excluded due to technical problems during the experiment. For the remaining 40 participants, 20 (seven females) started with the numerosity task, and 20 (nine females) started with the continuous task. The average age of the participants was 24 years and 5 months (SD=2 years and 1 month). There were no significant age differences between the groups (t<1, ns).

Stimuli

Stimuli were composed of pairs of light gray dot arrays presented on a black background and separated by a vertical gray line. Stimuli were presented in the center of a 1024×768 pixel screen consisting of an area of 300×175 pixels. The dot array pairs were created with Matlab code detailed in Gebuis and Reynvoet (2011). This code records five different continuous magnitudes separately for each array in a pair. It recorded average diameter, total surface area (the sum of all dots area in the array), area extended (the smallest contour that included all of the dots, as if an elastic band was wrapped around the dots), density and total circumference. Each array contained 2, 3 or 4 dots. Pairs were always composed of two different-sized dot arrays. One dot was not used since terms such as density and convex-hull do not apply for one dot.

Congruity manipulation

Numerical magnitude was either congruent or incongruent with all five continuous properties. In congruent stimuli, in the array containing more dots, the average size of the dots, their total circumference, total surface area and the area occupied by the dots was larger and the dots were denser, compared with the array containing fewer dots. In incongruent stimuli, all five continuous properties were smaller compared to the array that contained the more numerous dots (see Fig. 1).

Controlling for the continuous magnitudes of the dot arrays

In order to assure that in our set of stimuli the influence of continuous magnitudes was consistent across different numerical ratios, the following analyses were conducted. The current set of stimuli had a numerical ratio of either 0.5 (i.e., 2 versus 4 dots), 0.67 (i.e., 2 versus 3 dots) or 0.75 (i.e., 3 versus 4 dots). To check if the ratio between the 5 continuous magnitudes mentioned above was different in the three numerical ratios, we performed a two-way analysis of variance (ANOVA) with each ratio as a group (i.e., 3 groups of numerical ratio; 0.5, 0.67 and 0.75), continuous magnitudes as the independent measure (5 levels for 5 different continuous magnitudes), and the ratio between the continuous magnitudes of each pair of dot arrays as the dependent measure. This analysis revealed no interaction between continuous properties and numerical ratio groups, F (8, 468)=1.12, p=0.35, ƞp=0.02, suggesting that the continuous ratios did not differ among the different groups of dots. Therefore, in the continuous task, the level of difficulty should have been similar across different numerical ratios. Moreover, in the numerical task, the level of interference from continuous magnitudes should have been similar across different numerical ratios.

Numerical ratio might affect performance even in the subitizing range (Hyde, 2011). Namely, the larger the ratio, the more difficult it is to decide which array contains more dots (Moyer and Landauer, 1967). This ratio effect might also explain some of the congruency effect. Barth (2008) showed that the ratio between the cumulative area in incongruent trials tends to be higher (i.e., the magnitudes are more similar and the ratio between them is closer to 1) than in congruent trials. In such cases, one might suggest that the effect is driven not by congruency of numerical and continuous magnitudes but instead it is driven by the higher ratio of incongruent compared to congruent trials. To avoid this confound we kept the ratio between average continuous properties in all our stimuli (both congruent and incongruent) within a constant range. The average continuous ratio for a pair of dots was calculated by averaging the ratio of all five continuous magnitudes for every pair of arrays. For example, for pair “x”, where the ratio between the densities is 0.5, between the total surface areas is 0.7, between average dot sizes is 0.3, between total circumference is 0.6 and between the area occupied by the arrays is 0.5, the average continuous ratio would be 0.52. The range of average continuous ratios for all our stimuli was between 0.34 and 0.41 (average=0.38, SD=0.01).

Tasks

Participants performed two tasks in two separate runs. The only difference between the two runs was the task instructions given at the beginning of each run. In the numerosity task, participants were asked to choose the array containing more dots. In the continuous task, participants were asked to choose the array containing more gray area. Each run contained 120 stimuli: 2(congruity)×6(pairs)×10(different variations for each pair). The side of the larger dot array was counterbalanced. All stimuli were repeated only once in each run. Thus, participants encountered the same stimuli twice: once in the numerical run and once in the continuous run.

Procedure

Before starting the scan, participants signed a consent form and were given general instructions. An event-related fMRI design was used to acquire functional imaging data. Each scan included an anatomical scan and two functional runs. Before each functional run, participants read instructions specific to the task. Each trial started with a black screen with a vertical gray line in its center. In the middle of the gray line a red fixation cross appeared. To achieve deconvolution of the blood oxygen level dependent (BOLD) response, a jitter interval between 4000 and 6000 ms (120 different intervals in total) was used before the fixation cross changed to green (for 250 ms) in order to alert the participant that a dot stimulus was soon to appear and thereby allowing participants to get ready to respond. After the green fixation cross disappeared, a black screen with a vertical gray line was presented for 700 ms. Thereafter, two dot arrays were presented for 700 ms and then were replaced by a black screen with a gray vertical line for 1100 ms. Participants were able to respond to the stimuli from the time the dot arrays were displayed on the screen until a new trial with a red cross started. The participants were instructed to respond with a button press with the hand corresponding to the side of the presentation of the array containing the larger area/numerosity. Between the first and second run, there was a four-minute break where participants watched a short video of nature scenes. This was done in an effort to prevent general habituation to the dot stimuli that could potentially have reduced brain activity during the second run. The procedure of a typical trial is depicted in Fig. 2.

Fig. 2

An example of a typical trial. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

fMRI data acquisition

A 3-Tesla Philips ingenia whole-body MRI scanner was used to collect the functional and structural data of this study. The brain anatomy of each participant was collected with high-resolution T1 weighted sequence (1×1x1 mm3, TR: 8200 ms, TE: 3.8 ms, flip angle: 8°). An echo planar (EPI SE) sequence was used to measure the BOLD brain signal of the functional run with a 32-channel Siemens head coil. The order of imaging acquisition was ascending – interleaved, covering the whole brain of participants. The acquisition resulted in 353 whole-brain images per functional run, with a total length of 10 min and 30 s per run. For each functional volume, 38 slices were collected resulting in a 3 mm isovoxel resolution over a 64×64 voxel matrix. The time of repetition (TR) was 2700 ms, the echo time (TE) was 52 ms and the flip angle was 78°.

Imaging analysis

Brain Voyager QX 2.8 (Brain Innovation, Maastricht, The Netherlands) was used to analyze the functional and structural data sets. Each individual data set was preprocessed according to the following steps. Functional imaging data were first corrected for slice scan time acquisition (ascending – interleaved; using a cubic-spline interpolation algorithm). A high-pass (GLM – Fourier) frequency filter with a cut off value of 2 sines/cosines cycles was applied to remove low frequency signals. Finally, a Trilinear/sinc interpolation approach was used to remove and to adjust head motions. In order to be included into the study, a participant's movement parameters had to stay within 3 mm of overall movement (maximum deviation from the first volume) and within 1 mm volume-to-volume movement (maximum deviation from one collected functional image to the next collected functional image).

An automatic alignment procedure (implemented in Brain Voyager) was used in order to spatially align the functional runs of each participant onto the corresponding anatomical scan. The quality of the alignment was checked visually and corrected manually if the automatic procedure did not reveal a sufficient alignment. Subsequently, the co-aligned images were transformed into Talairach space (Talairach and Tournoux, 1988). This was achieved in two consecutive steps: first, using the landmarks of the anterior commissure (AC) and the posterior commissure (PC), the anatomical image of each participant was transformed into the ACPC-plane position. Then the boundaries of the brain tissue were manually selected and transformed into the Talairach grid using a trilinear interpolation algorithm (Talairach and Tournoux, 1988).

Individual data sets were entered into a general linear model (GLM) for group-based analysis. All functional events of the two conditions (i.e., numerical and continuous) were convolved with a two-gamma hemodynamic response function (HRF) in order to predict the BOLD function (Friston et al., 1998). Congruent and incongruent trials across the two functional runs were modeled separately in order to investigate brain activation differences related to congruency. The statistical maps derived from brain activation contrasts (see below) were applied a threshold with an uncorrected p Value of 0.005 and subsequently cluster corrected in order to correct for multiple comparisons and to adjust the Type I error to a level of p<0.05. This was achieved by an iterative “Monte Carlo Simulation” (1000 iterations), which estimates the minimum size of a functional cluster to be significant on the basis of functional data from the present study (Forman et al., 1995).

Results

Behavioral analysis

An ANOVA with task and congruity as within-subject variables, and the order of task administration as a between-subjects variable, was performed twice: once with error rates and once with RT as dependent variables. Averages and standard errors (SEs) are depicted in Fig. 3. A main effect of order was marginally significant for error rates, F (1, 38)=3.06, p=0.09, ƞp=0.07, and only showed a pattern for RTs, F (1, 38)=1.6, p=0.2, ƞp=0.04. This pattern suggests that the group that started with the continuous task (i.e., the CN group) was slightly more accurate and faster than the NC group.

Fig. 3

Behavioral analysis. (A) Error rates in the different groups and conditions. (B) RTs in the different groups and conditions. All participants conducted both tasks. The NC group started with the numerosity task (i.e., where are there more dots?) and the CN group started with the continuous task (i.e., where is there more gray area?). C=congruent, IC=incongruent. * p<0.05.

Both RTs and error rates were affected by task demands, congruity and order. Participants were generally more accurate and faster in congruent than incongruent trials across tasks; F (1, 38)=24.36, p<0.001, ƞp=0.39 and F (1, 38)=72.85, p<0.001, ƞp=0.66, for accuracy and RT, respectively. Responses to the continuous task were generally more accurate and faster than to the numerosity task; F (1, 38)=33.65, p<0.001, ƞp=0.47 and F (1, 38)=131.91, p<0.001, ƞp=0.77, for accuracy and RT, respectively.

The interaction of task and order revealed that error rates in the numerosity task were modulated by task order; in the numerosity task, participants in the NC group were less accurate than participants in the CN group. This modulation was specific to the numerosity task; error rates in the continuous task were not modulated by task order, F (1, 38)=4.63, p<0.05, ƞp=0.11. For RTs, the interaction of task and order was marginally significant; F (1, 38)=3.69, p=0.06, ƞp=0.09. Further investigation of this interaction revealed that while order did not affect RTs in the numerosity task (F<1, ns), performance in the continuous task was slightly faster in the CN group, F (1, 38)=3.02, p=0.09, ƞp=0.07.

The interaction of task and congruity revealed that the difference in error rates and in RT between congruent and incongruent trials (i.e., the congruity effect) was greater in the numerosity task than in the continuous task, F (1, 38)=18.88, p<0.001, ƞp=0.33 and F (1, 38)=10.63, p<0.05, ƞp=0.22, for error rates and RT, respectively. For RT, there was also a triple interaction of task, congruity and order, F (1, 38)=6.09, p<0.05, ƞp=0.14; in the NC group the congruity effect was similar for both tasks (F<1, ns). In contrast, in the CN group there was no congruity effect in the continuous task, F (1, 38)=1.22, p=0.27, ƞp=0.03, whereas the size of the congruity effect in the numerosity task remained similar to the congruity effect in the numerosity task of the NC group (F<1, ns).

Functional analysis

To test our hypothesis that task order might affect the way magnitudes are processed and also in light of the effect of order on RT and accuracy, we conducted a whole-brain analysis for each group separately, in addition to the contrasts between the groups.

Main effect of order

In order to investigate how task order affected brain activation, we calculated a whole-brain t-test statistic, contrasting the brain signals of both groups (NC and CN) across all conditions. The results of this analysis (Table 1) revealed two regions of activation – the posterior medial aspect of the cingulate gyrus, and the right superior frontal gyrus (Fig. 4). These areas were more strongly activated in the CN group than in the NC group.

Table 1

Order of main effect.

Brain region	Coordinates
	x	y	z	t	Cluster size
Order main effect: CN>NC
R.SFG/MiFG (FEF)	11	37	36	4.9	1141
PoG (cingulate gyrus)	−1	−53	45	3.68	799

Note: p<0.005 (cluster corrected for multiple comparisons: p=0.05). Coordinates are in Talairach space. SFG=superior frontal gyrus; MiFG=middle frontal gyrus; FEF=frontal eye field; PoG=postcentral gyrus.

Fig. 4

Main effect for task. Transversal view; x=0, y=−51. The z coordinates are indicated in gray in the figure. R.SFG: right superior frontal gyrus; MiFG: middle frontal gyrus. PoG: postcentral gyrus.

Brain activation related to task effect for congruent trials

In order to investigate how the different tasks affected brain activation, we performed a whole-brain t-test, contrasting brain signals related to the different relevant dimensions according to task instructions (i.e., either numerosity or continuous magnitudes). This was done only for congruent trials; in these trials, participants in both tasks viewed the same stimuli and responded with the same hand. The only difference was task instructions, directing attention to different dimensions. This analysis was made for each group separately because of possible effects of order.

Brain activation for the different tasks in the CN group

The results of the CN group analysis revealed fronto-parietal areas in the left hemisphere that were more strongly activated during the continuous task; the superior frontal gyrus, precentral gyrus and the middle frontal gyrus. No areas were found to be more strongly activated during the numerosity task in this group (see Table 2 and Fig. 5 – yellow clusters).

Table 2

Task effect for congruent trials.

Brain region	Coordinates
	x	y	z	t	Cluster size
CN; congruent; Cont>Num
L.SFG	−4	55	33	5.05	642
L.PrG	−7	−38	66	5.5	1612
L.MiFG	−22	13	42	4.19	679
NC; congruent; Num>Cont
R.SMG/IPL	44	−32	48	5.27	1252
R.PrG	29	−23	63	4.6	864
R.SFG	5	−11	60	4.18	1140
L.PoG (Insula)	−52	−17	15	5.16	633
Congruent; [NC (Num>Cont)]>[CN (Cont>Num)]
R.ITG/OcG	50	−65	3	5.25	1396
R.SFG/MiFG	23	−17	63	4.13	802
R.SFG	5	−11	60	4.38	845
L.SFG (cingulate gyrus)	−7	−17	45	4.34	2304
L.PrG (Insula)	−43	−26	18	5.18	2774
L.IFG/OFG	−43	31	−9	−3.81	603

Note: p<0.005 (cluster corrected for multiple comparisons: p=0.05). Coordinates are in Talairach space. SFG=superior frontal gyrus; MiFG=middle frontal gyrus; PoG=postcentral gyrus; SMG=supramarginal gyrus; PrG=precentral gyrus; OcG=occipital gyrus; ITG=inferior temporal gyrus; IFG=inferior frontal gyrus; OFG=orbitofrontal gyrus; IPL=inferior parietal lobule.

Fig. 5

Brain activation related to the task effect for congruent trials. Transversal view; x=0, y=−40. z coordinates are indicated in gray in the figure.; MiFG=middle frontal gyrus; PoG=postcentral gyrus; SMG=supramarginal gyrus; PrG=precentral gyrus; OcG=occipital gyrus; ITG=inferior temporal gyrus; IFG=inferior frontal gyrus; OFG=orbitofrontal gyrus; IPL=inferior parietal lobule. AnG=angular gyrus. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Brain activation for the different tasks in the NC group

The results of the NC group analysis revealed fronto-parietal areas that were more strongly activated during the numerosity task; the right middle frontal gyrus and the right precentral gyrus in the frontal lobe, and the left postcentral gyrus (around the insula) and the left supramarginal gyrus (around the postcentral gyrus) in the parietal lobe. No areas were found to be more strongly activated during the continuous task in this group (see Table 2 and Fig. 5 – red clusters).

Differences in brain activation related to task and order

To investigate if the impact of task was further modulated by task order, we performed a whole-brain t-test, pitting the brain signals related to task for the congruent trials of the NC group to those of the CN group. The results of this analysis revealed activation in frontal, parietal and occipital areas; the right cingulate gyrus (in the area of the superior frontal gyrus), right middle frontal gyrus, left insula (in the area of the precentral gyrus), and the inferior temporal gyrus, on the border of the occipital gyrus. In the CN group, these areas were more strongly activated during the continuous task; in the NC group the pattern was reversed—the same areas were more strongly activated during the numerosity task. In the left inferior frontal gyrus on the boarder of the orbitofrontal gyrus, the activation was negative compared to baseline. Namely, in the CN group this area was more strongly deactivated in the continuous task and in the NC group, this area was more strongly deactivated during the numerosity task (see Table 2 and Fig. 5 – blue clusters).

Brain activation related to conflict in the different tasks

To investigate the impact of conflict from different sources (e.g., when the to-be-ignored dimension was different), we performed a whole-brain t-test, pitting the brain signals related to incongruent trials in the numerosity task with those of the continuous task. This was done separately for each group.

Brain activation related to conflict in the different tasks for the CN group

This analysis revealed parietal and occipital areas in the right hemisphere that were modulated by the different sources of conflict. Specifically, the right postcentral gyrus in the area of the supramarginal gyrus was more active during incongruent trials of the continuous task (where the irrelevant dimension was numerosity), and the right posterior cingulate and the right inferior occipital gyrus (the striate area) were more strongly activated during incongruent trials of the numerosity task (where the irrelevant dimension was continuous) (see Table 3 and Fig. 6 – yellow clusters).

Table 3

Task effect for incongruent trials.

Brain region	Coordinates
	x	y	z	t	Cluster size
CN; incongruent; Cont>Num
R.PoG/SMG (IPS)	60	−29	42	−4.99	690
CN; incongruent; Num>Cont
R.PoG	17	−62	12	4.92	1088
R.IOG	−10	−89	−12	4.45	860
NC; incongruent; Num>Cont
R.PoG/SMG (IPS)	56	−26	45	4.8	964
R.SFG/MiFG	26	−17	66	4.79	1731
SFG (cingulate gyrus)	−1	−14	45	4.73	2735
L.PrG	−16	−11	66	4.34	915
L.PoG	−43	−26	27	4.62	981
Incongruent; [NC (Num>Cont)]>[CN (Cont>Num)]
R.SMG/IPS	59	−29	42	4.21	1297
R.ITG/OcG	47	−65	3	5.27	810
R.SFG	20	−14	60	3.89	868
L.PrG	−16	−14	63	4.42	755
L.IPS (AnG)	−37	−50	48	4.05	797
L.Insula	−43	−29	21	4.71	949

Note: p<0.005 (cluster corrected for multiple comparisons: p=0.05). Coordinates are in Talairach space. PoG=postcentral gyrus; SMG=supramarginal gyrus; IOG=inferior orbital gyrus; SFG=superior frontal gyrus; PrG=precentral gyrus; MiFG=middle frontal gyrus; IPL=inferior parietal lobe; ITG=inferior temporal gyrus; OcG=occipital gyrus; IPS=intraparietal sulcus; AnG=angular gyrus.

Fig. 6

Brain activation related to conflict in the different tasks. Transversal view; x=0, y=−40. z coordinates are indicated in gray in the figure. PoG=postcentral gyrus; SMG=supramarginal gyrus; IOG=inferior orbital gyrus; SFG=superior frontal gyrus; PrG=precentral gyrus; MiFG=middle frontal gyrus; IPL=inferior parietal lobe; ITG=inferior temporal gyrus; OcG=occipital gyrus; IPS=intraparietal sulcus; AnG=angular gyrus. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Brain activation related to conflict in the different tasks for the NC group

This analysis revealed activation in frontal and parietal areas in the left and right hemispheres—along the right superior frontal gyrus, middle frontal gyrus and cingulate gyrus, and the left pre and postcentral gyri. These areas were more strongly activated in the NC group during the incongruent trials of the numerosity task—where the irrelevant dimension was continuous (see Table 3 and Fig. 6 – red clusters). There were no areas that were more active during the continuous task.

Differences in brain activation related to conflict and task order differences

To investigate if the impact of conflict from different sources was further modulated by task order, we performed a whole-brain t-test, pitting the brain signals related to task for the incongruent trials of the NC group to those of the CN group. The results of this analysis revealed activation in frontal, parietal and occipital areas in the left and right hemispheres; the right and left superior frontal gyri, the left insula, the right supramarginal gyrus in the area of the inferior parietal lobe and the intraparietal sulcus (IPS), and the right inferior temporal gyrus on the boarder of the occipital gyrus. In the CN group, the activity in these areas was similar. However, in the NC group these areas were more strongly activated in the numerosity task (see Table 3 and Fig. 6 – blue clusters).

Discussion

In the current work, we investigated the neuronal mechanisms underlying magnitude comparisons. Namely, we were interested in examining the effect of processing continuous magnitudes on numerosities in the subitizing range and vice versa. We chose the subitizing range because estimating numerosities in this range is fast and accurate (e.g., Knops et al., 2014; Piazza et al., 2011; Revkin et al., 2008; Trick and Pylyshyn, 1994). Furthermore, we chose this range in order to try to make the processing of numerosities as easy as possible. Despite these efforts, the behavioral and functional results suggest that numerosity comparison was more difficult than continuous magnitudes comparison. In addition, continuous magnitudes always influenced numerosity comparisons, whereas numerosities affected continuous magnitude comparisons only if the concept of numerosities was primed. Our results suggest that performance during a comparative judgment task is affected by context; namely, the relevance of the dimension and the order of the tasks.

Performance in a comparative judgment task is affected by order and relevance of the dimension even in the subitizing range

The behavioral results exhibited an asymmetrical pattern; continuous magnitudes interfered with numerical magnitude judgment regardless of task order, whereas numerosities affected continuous magnitude judgment only if the numerosity task preceded the continuous task. This pattern of results is in line with previous suggestions that continuous properties affect performance even when irrelevant to the task and not correlated with numerosities (Gebuis and Reynvoet, 2012a, Leibovich and Henik, 2014).

Similar context-dependency has been shown with a basic visual feature like color (Yee et al., 2012). Yee and colleagues' study included two tasks: a color-word Stroop task, where participants had to name the font color and avoid reading the word; and a priming task, in which participants were presented with a prime word and a target word, and asked to decide whether the target word was the name of an animal. The order of the tasks was counterbalanced. The results revealed that responses were faster when the prime and the target word represented the same color (e.g., emerald and cucumber), compared with target words in which there was no such connection. This, however, occurred only when the color-word Stroop task preceded the priming task, namely, only when the notion of color was primed. Accordingly, the authors suggested that the extent to which color information is activated depends not only on long-term factors but also on short-term, context-dependent factors.

Following the logic of Yee et al. (2012), our result suggests that the extent to which numerical information is processed is modulated by short-term, context-related factors as well; it is not automatically processed all the time. In contrast to numerosities, the processing of continuous magnitudes was not modulated by context; the same congruity effect appeared in the numerical task, regardless of tasks order.

The novel finding here is that continuous magnitudes affect performance even when the to-be-compared numerosities are in the subitizing range—where it is the easiest. Nonetheless, some researchers assume that in the subitizing range a different system is applied than for larger numerosities (i.e., the two-system view; Feigenson et al., 2004). Therefore, it should be tested whether the same pattern of results would generalize to larger numerosities as well.

Some studies suggest that subitizing is made possible by a parallel individuation process; each item is assigned an index token in parallel; the tokens are then mapped into number names. Since indexing and parallel individuation are limited by the capacity of visual working memory, this process is possible only with small numerosities (Knops et al., 2014, Trick and Pylyshyn, 1994). Dehaene and Changeux (1993) suggested that individuation does not take into account different continuous magnitudes. If such an individuation process that ignores continuous magnitudes (i.e., size, density) occurred during the numerosity comparison task, we would not expect continuous magnitudes to be processed when irrelevant. Therefore, we would not expect our resultant congruity effect. We argue that this congruity effect, in the numerosity task, suggests that even in the subitizing range, continuous magnitudes influence performance. This notion may support the one-system view, suggesting that all numerosities, within and outside of the subitizing range, are represented by the approximate number system (e.g., ANS).

There are, however, findings in the literature supporting the role of individuation in the subitizing range. Knops et al. (2014), for example, demonstrated that object enumeration and visual working memory share a neural mechanism. Namely, that the numerosity of objects in the subitizing range is represented via saliency maps; maps that topographically represent the saliency of items in a specific location (for a different view see Pagano et al., 2014). The authors also provide evidence for the flexibility of this mechanism and its ability to deal with changing environmental demands. Importantly, the findings of Knops et al. support the possibility that during individuation, the continuous magnitudes of the items (e.g., the individual item size) or of the array (e.g., the density of the items) are also encoded. Thus, it is possible that during the comparison of two numerosities in the subitizing range, an individuation process occurs that takes into account the continuous magnitudes of the stimuli. Moreover, because this system is flexible, the magnitude that would be encoded can be context-dependent, as demonstrated by the congruity×task×order interaction in our results. Further work should empirically test this notion.

The effect of task order on brain activity

The posterior cingulate was more active in the group that started with the continuous task. The posterior cingulate has a key role in the default mode network (Fransson and Marrelec, 2008). Activations of areas in the default mode network are inversely related to task difficulty; the more difficult a task, the less active the area. According to the behavioral results, the group that started with the continuous task was generally faster than the group that started with the numerosity task. This suggests that the tasks were easier and required fewer resources in the CN group, since the notion of numerosity was not prompted and hence, did not interfere with the continuous task.

The posterior cingulate also plays a role in shifts of visual attention (Small et al., 2003); it monitors eye movements and responds to sensory stimuli (Vogt et al., 1992). The differential activation in this area according to task order suggests that the two groups used different scanning strategies. Namely, the group that started with the numerosity task could have used less eye movements, since numerosities in the subitizing range may be grasped in parallel (e.g., Trick and Pylyshyn, 1994). Watson et al. (2007) have shown that there are significantly more eye movements during enumeration of numerosity beyond the subitizing range compared with numerosities within the subitizing range. The authors also reported that enumeration of up to four items was accurate and fast even when participants were not allowed to move their eyes. This strategy of individuation with less eye movements could have been carried over to the continuous task. In contrast, the group that started with the continuous task might have used more eye movements in order to assess and sum all surfaces across dots. This could have been manifested in higher activation of the posterior cingulate. It might appear counter-intuitive that the strategy of less eye movements would result in longer RTs, however, it should be noted that the initial visual processing is only one step in the comparison process. As suggested by Durgin (2008), processing numerosities requires more calculations than processing continuous properties. Thus, even though the visual processing might be faster for subitizing, the RT patterns show that the rest of the comparison process is probably more lengthy and complicated than that for continuous magnitudes. To our knowledge, there is no work comparing eye movement during estimation of numerosity and continuous properties. Subsequently, this suggestion should be examined empirically.

Different scanning strategies might also be the cause for the right middle frontal gyrus (MiFG) activation. The MiFG (BA8) that includes the frontal eye field (FEF) was activated in the group that started with the continuous task, and deactivated (compared to rest) in the group that started with the numerosity task. Traditionally, the FEF is known to have a role in controlling and monitoring eye movements. Importantly, single-cell recordings and fMRI studies converge to suggest that the human FEF region, although located in the prefrontal cortex, is part of a cortical network of low-level sensory areas. As such, the FEF is able to differentiate simple stimuli such as 2D shapes during active or passive viewing (Peng et al., 2008) and colors (Münch et al., 2014). ERP studies have revealed that the FEF response to stimuli can occur very fast (as soon as 50 ms after the onset of the visual stimuli). The time of FEF activation is modulated by the complexity level of the stimuli (Kirchner et al., 2009). Kirchner et al. also raised the possibility that the human brain may have kept, through species evolution, a specialized system to process some aspects of sensory information very quickly. In light of these findings, we argue that due to their evolutionary importance (Henik et al., 2012, Leibovich and Henik, 2013), the FEF supports the automatic processing of continuous properties. In order to voluntarily attend to numerosities, attention to continuous properties should be inhibited via deactivation of the FEF (for the role of the FEF in inhibiting reflexive saccades to exogenous signal see Henik et al., 1994). This pattern fits with the behavioral interaction of task, congruity and order (i.e., while the continuous dimension always interferes with performance, numerosity interferes only when being prompted), and with our suggestion that continuous properties are processed more automatically than numerosities, even in the subitizing range.

The effect of the relevant dimension on brain activity

We contrasted congruent trials in the different tasks for the CN and the NC groups separately. This contrast allowed us to track brain activity that was modulated by the allocation of attention to a specific dimension. Both contrasts revealed activity in the precentral and the superior frontal gyri, albeit with a different pattern. In the CN group, the left precentral and superior frontal gyri were more active in the continuous task than in the numerosity task. In contrast, in the NC group, the right precentral and superior frontal gyri were more active in the numerosity task. These areas, as well as other fronto-parietal areas that were found here (see Table 2), were previously found to be involved in numerical cognition tasks (for review see Ansari (2012)). The right precentral gyrus was found to be modulated by the distance between two to-be-compared symbolic numbers in adults (Ansari et al., 2005). The left precentral gyrus and the left superior frontal gyrus were found to be more active in 4-year-old children when they were adapted to numerosities compared with shapes. The right superior frontal gyrus was more active in response to shape dishabituation, which is also a basic visual feature (Cantlon et al., 2006).

Is there a lateralization of numerical abilities? In a variety of methods (lesion studies, transcranial magnetic stimuli (TMS), positron-emission tomography (PET), fMRI), it has been shown that for non-symbolic numerosities, both the right and left fronto-parietal cortices play a role. However, the left parietal lobe is more associated with symbolic number processing (Andres et al., 2005, Ashkenazi et al., 2008, Cantlon et al., 2006, Cohen Kadosh et al., 2005). In almost all studies that investigated non-symbolic numerical processing, the relevant dimension was numerosity, and participants were able to use continuous magnitudes to solve the task (for more details on such studies see Gebuis and Reynvoet, 2012b, Gebuis and Reynvoet, 2013; Leibovich and Henik, 2013). This brings to mind the possibility that while both lobes are engaged when processing non-symbolic numerosities, the left fronto-parietal region might process the continuous aspect and the right fronto-parietal region might process the numerosity of the non-symbolic stimuli. Since this is the first functional study that presented the same non-symbolic stimuli and asked subjects to voluntarily attend to numerosity or continuous properties, and considered the order of the tasks, more empirical evidence is required to confirm this hypothesis.

In addition to these areas, the right inferior parietal lobe (r.IPL) was found only in the NC group. The r.IPL was found to be anatomically different in a 22q11.2 deletion syndrome ― a syndrome in which patients have lower abilities in mathematics (Simon et al., 2005). It was also found to be involved in more complex arithmetical calculations (Fehr et al., 2007). Dehaene and Cohen (1997) described a patient with a right inferior parietal lesion who was able to read symbolic numerals and write them but could not do calculations. From such studies, it is clear that the r.IPL plays an important role in processing numerosities (maybe in order to manipulate them to be able to perform calculations). However, this region was only active when the notion of numerosity was prompted by being the first task. This finding is in line with our hypothesis that although some brain regions might be more specific to numerosities than to continuous properties, they are context-dependent and not automatic.

The effect of congruity on brain activity

Contrasting only incongruent trials was aimed at revealing areas that were involved in attending to and processing of the relevant dimension. Because we hypothesized that task order may affect performance and brain activity, we conducted this analysis for each group separately and also directly contrasted task and order.

Some areas that were found in the current contrast were also found in previous contrasts; the r.MiFG and the PoG were found to be related to the main effect of order (see Section 4.2), and involved in the general direction of attention according to the first task. The activity in the PrG and SFG (bilaterally) was modulated by the relevant dimension (see Section 4.3).

The left angular gyrus (l.AnG) activity was modulated by both task order and the relevant dimension: namely, in the NC group the activity was higher in the numerosity task, and in the CN group the activity was higher in the continuous task. The activity of this area was not modulated by task within each group. Instead, interaction stemmed from an opposite activity pattern in the two groups.

The anterior-lateral aspect of the right intraparietal sulcus (IPS) was modulated by the context determined according to the first task; in the NC group, the r.IPS was more active in the numerosity task, and in the CN group, the r.IPS was more active in the continuous task. Importantly, this area was not active in the main effect of order, or when comparing only congruent trials in the different tasks. This suggests that the activity of the r.IPS is not selective to a specific aspect of magnitude; instead, it has a role in directing attention according to the relevant dimension, and it is affected by task context (i.e., by the first task). Similar findings, where IPS activity was found to be modulated by cues that determined a task’s relevant dimension, were found in several studies (e.g., Ruge et al., 2009; Schultz and Lennert, 2009; Waskom et al., 2014).

Ventral areas ― the right inferior orbital gyrus (r.IOG), the right inferior temporal gyrus (r.ITG) and the right occipital gyrus (r.OCG) ― were also more active in the first task, regardless of the nature of the task. These areas were not significant in the previous contrasts, suggesting that they might be specific to the incongruent nature of the trials.

Conclusion

Our results indicated that the context of the task determines both behavioral performance and the neuronal routes in which the stimuli are processed. The pattern of brain activity was determined by the first relevant dimension and carried over to the next task. Continuous magnitudes processing relied on activation of the FEF and POG, associated with eye movements, and processing of basic visual features. Processing of numerosities, on the other hand, relied on deactivation of these areas, suggesting active suppression of the continuous dimension. Our experimental design allowed us to dissociate continuous magnitudes and numerosity processing; whereas previous studies usually reported bilateral activity associated with non-symbolic numerosity processing, we demonstrated lateralized activity that was associated with task context and the relevant dimension. Starting with the numerosity task led to enhanced activity in the right hemisphere, while starting with the continuous task led to enhanced left hemisphere activity. Further research is required to examine the mechanisms underlying this lateralization. Moreover, our experimental design allowed us to demonstrate that parietal activations that were previously associated with magnitude processing are in fact related to task context. This result does not support the existence of a dedicated brain circuitry for the processing of numerosities (e.g., Burr and Ross, 2008; Dehaene and Changeux, 1993; Harvey et al., 2013; Piazza, 2010; Piazza et al., 2013).