Individual differences in the activity of executive function brain regions during number comparison

1.1. Effects of format type

Different formats of numerical information (i.e., symbolic, nonsymbolic) may vary in the numerical and EF demands they place on an individual [40]. Across neuroimaging studies, typically developing individuals display activation in the right and left intraparietal sulci (IPS) during different types of number and math tasks including magnitude comparison and calculation [15,64]. However, the hemisphere of the IPS activation often varies and this may be due to the format of the number, with nonsymbolic processing typically associated with right hemisphere activation while symbolic processing relies more commonly on the left IPS [4]. However, little is known regarding the extent to which EF regions function differentially during symbolic vs nonsymbolic magnitude processing, and how those differences relate to behavior [44]. Accordingly, in this study we examine differences between neural activation during nonsymbolic and symbolic numerical processing in canonical EF and number processing brain regions.

1.2. Individual differences in math skills

One method to assess math skills is a magnitude comparison task, in which individuals are presented with two numerical representations and are asked to determine which numerical representation is greater. Centered on the Approximate Number System (ANS) which insinuates that numbers closer together in magnitude have overlapping neural representations [24], individuals tend to perform slower and less accurately when discriminating numerical representations closer together in magnitude than numerical representations that are further apart (for review see [61]. Consequently, the neural ratio between the numerical pairs can be modeled as a ratio effect, with a smaller ratio effect associated with increased ANS precision (e.g., [20,54]).

Individual differences in the function of numerical processing brain regions have been found to be associated with individual differences in math skills in a number of studies (e.g., [10,32,55]). However, the ways in which individual differences in the function of EF regions during numerical processing are associated with behavioral outcomes are less clear. Behavioral research suggests, for example, that math ability is associated with EF, such that individuals with low math ability tend to exhibit poor EF [11,22,40,50,51,60,73]. In addition, parallel to those with reading difficulties, individuals with math difficulties (MD) exhibit greater activation in executive functioning areas [19], suggesting increased effort. However, EF skills have been shown to act as a protective factor for students at risk for MD [56], once again highlighting the complex interplay between math and EF ability.

The relationship between math and EF may also differ depending on the format used to assess the processing of numerical magnitude. For instance, it has been shown in adults that nonsymbolic processing requires specific demands related to multistep strategies which in turn necessitate working memory and cognitive control skills [46]. Therefore, we expect brain activation in EF regions to be stronger during nonsymbolic than symbolic numerical processing, but for both formats, activation in EF and numerical brain regions will correlate positively with math skills.

1.3. Individual differences in EF

In the current study, we examine the association between individual differences in EF to brain activation in response to nonsymbolic and symbolic numerical processing in each of numerical processing and canonical EF brain regions. Previous research suggests that working memory and inhibitory control components of EF may be particularly relevant for different aspects of numerical and mathematical processing (e.g., [11,18,66,70]).

The Triple Code Model of numerical cognition suggests three number representations, including the Arabic numeral, auditory verbal word, and the nonsymbolic magnitude representation [23]. To determine which number magnitude is greater in a nonsymbolic and symbolic numerical processing, participants may transform Arabic numerals (i.e., symbolic number representations) into nonsymbolic magnitude representations, taxing the working memory system, which has a restricted capacity [7]. Thus, it is not surprising that working memory skills predict symbolic magnitude skills in young children [37]; yet, it remains unclear if this same pattern will be present in older children.

In contrast to working memory, much less is known about the relationship between math and inhibition. Similar to working memory, poor math ability is also associated with low inhibition skills [11]. Inhibition seems to play a particular role in suppressing incorrect solutions of inefficient strategies in solving calculation problems [9]. In nonsymbolic magnitude comparison tasks where the size of the dots is incongruent with the number of dots, inhibition can play a role in suppressing the distracting information (e.g., [16,27,31]); however, inhibition’s role in symbolic and controlled nonsymbolic magnitude comparison tasks remains unclear. Therefore, we investigated working memory and inhibitory control as distinct constructs in this study. We expect that individuals with higher scores in working memory and inhibitory control will show stronger activation of EF regions during both symbolic and nonsymbolic numerical comparison tasks.

1.4. Summary

In the current study, we examine the underlying EF neural relationships involved in numerical processing by assessing the association between individual differences in EF to brain activation in response to nonsymbolic and symbolic numerical processing in each of the numerical processing and canonical EF regions. We compare said activation between numerical formats, and examine the ways in which such activation is associated with individual differences in behavioral measures of math skills and EF. A richer characterization of the function of EF regions during numerical processing will contribute to our understanding of how the neural systems that underlie numerical magnitude processing interface with domain-general systems to produce individual differences in behavior.

2.1. Participants

Participants for the study were recruited from a previously described longitudinal twin sample [68] at the University of Colorado Boulder. Participants were 151 children and adolescents (Mean age = 166.23 months; SD = 29.70, age range 120–203 months, 53.6 % female). The sample included 35 monozygotic (MZ; i.e., identical) and 101 dizygotic (DZ; i.e., fraternal) twin pairs, as well as 14 twins whose co-twin was either ineligible for scanning, removed during quality control, or declined to participate. Twin status was not the focus of the current study, nor was the sample large enough to perform a well-powered twin analysis. As a result, family status (i.e., twin pairs) was statistically controlled for in analyses. All participants reported normal or corrected-to-normal vision and had no history of neurological disorders. The racial and ethnic composition of the sample is 80.13 % White (11.26 % Hispanic), 0.66 % Other race, 17.22 % reporting more than one race (3.31 % American Indian or Alaska Native, 3.31 % Asian, 1.32 % Native Hawaiian or Pacific Islander, 5.96 % African American, and 3.97 % Other) and 2.65 % not reporting. Informed consent or assent was obtained from each individual prior to participation and all study procedures were approved by the Institutional Review Board at the University of Colorado Boulder.

2.2. Procedures

Participants completed a 2.5-hour session that consisted of approximately 50 min of behavioral testing and 90 min of neuroimaging. Parents of participants also completed questionnaires about their child’s behavioral and socioemotional functioning. fMRI was used to assess brain activity patterns while participants completed numerical comparison, lexical decision, and working memory tasks. The current study did not include the analyses of the lexical decision and the working memory task. The results of the numerical comparison tasks that assessed participants’ processing of symbolic and nonsymbolic qualities are the sole focus of the current study. Participants were paid for their time and received images of their anatomical brain scans for participating.

2.3. Behavioral assessments

Mathematical ability and executive functioning were assessed in participants using the following measures. For math, the Calculation and Math Facts Fluency subtests from the Woodcock-Johnson III (WJ-III; [76]) were administered to estimate participants’ ability to perform basic mathematical operations (untimed) and apply calculation skills to single-digit numbers (timed). The z-scores for Calculation and Math Facts Fluency were used for the analyses so that all relevant measures were on the same scale. Two tasks were used to measure executive functioning: the Delis-Kaplan Executive Function Systems (D-KEFS) Color-Word Interference Test [26] was used to assess inhibition and the 2-back task on an N-back, modeled on that used in the Human Connectome Project [8], was used to assess working memory. For the D-KEFS inhibition measure, the standard scores from the inhibition minus the color naming trials were calculated. The 2-back consisted of 8 blocks where participants indicated by button press whether the current stimulus matched the item presented 2 trials prior. Each 2.5 s trial consisted of a 250 ms fixation cross, 2000 ms stimulus presentation, and 250 ms inter-trial interval. There were three types of stimuli used: letters (A, C, E, H, J, M, P, T, X, Z), Arabic numerals (0–9), and nonsense shapes adapted from Vanderplas and Garvin [67]. See Table 1 for participants’ behavioral performance on measures.

2.4. MRI acquisition parameters

All imaging data was collected on a 3-Tesla Siemens Prisma MRI scanner with 32-channel parallel imaging using multi-band sequences at the University of Colorado Boulder. To reduce motion, foam padding was placed around the participant’s head within the head coil and a weighted blanket was placed over the participant’s legs. The order of acquisitions was distortion correction scans (time = 44 s), two resting state scans (time = 6 m:07 s each), functional N-back scan (the behavioral results of which were used for the working memory measure), T1 MPRAGE sequence, the symbolic number comparison task, the nonsymbolic number comparison task, the circle comparison task (not of interest in current paper), four Diffusion Tensor Imagining (DTI) scans (3 m:16 s, 3 m:28 s, 3 m:8 s, 3 m:0 s), and a functional lexical decision scan (not of interest in current paper). The functional number comparison scans were collected with the following parameters: TR = 460 ms, TE = 27.2 ms, FOV = 248 mm; slices = 82; multiband acceleration factor = 8; voxel size = 3 × 3 × 3 mm; flip angle = 44°; time = 3 m:58 s each. The structural images were collected using a T1-weighted high-resolution 3D magnetization prepared rapid gradient echo (MPRAGE) sequence acquired in the sagittal plane with the following parameters: TR = 2400 ms, TE = 2.07 ms, FOV = 256 mm; slices = 320; iPAT GRAPPA acceleration factor = 2; voxel size = 0.8 × 0.8 × 0.8 mm; flip angle = 8°; time = 7 m:02 s).

2.5. fMRI task

Participants completed one run each of a (1) symbolic and (2) nonsymbolic number comparison task using an event-related fMRI design. For each trial, participants were asked to respond as quickly and as accurately as possible to indicate whether an Arabic digit (symbolic) or dot array (nonsymbolic) was greater or less than five by pressing a button with either their right index or left index finger, respectively. Notably, the stimuli do not require participants to suppress incongruent visual information such as surface area, density, or convex hull, such as other nonsymbolic tasks [30,71], which could account for a differential role of inhibition across the formats. For each task, 72 trials were presented within a run. Six stimulus numerosities —2, 3, 4, 6, 7, or 8—occurred with equal probability (i.e., 12 trials per numerosity) in an intermixed and pseudo-random order (i.e., no more than 2 and 4 consecutive trials were of the same numerosity and same response, respectively). These numerosities yielded ratios of 0.4, 0.6, 0.625, 0.714, 0.8, and 0.833 with respect to reference value of five. To minimize the use of non-numerical visual cues for nonsymbolic number comparison, we controlled for visual parameters of dot arrays across trials by generating the stimuli using the script by Dehaene et al. [25]. Specifically, half of the set of dot arrays was controlled for the total surface area across numerosities such that dot size decreased with numerosity, and half of the set was controlled for dot size such that the total surface area increased with numerosity. The font size of the Arabic digits for the symbolic number comparison trials was selected to match the mean luminance of the dot arrays. Each trial began with an initial fixation of 500 ms, stimulus presentation of 750 ms, and a mean inter-trial interval (ITI) of 1630 ms (ranging from 1230 to 2030 ms, randomized in equal probability within each run). Each run also began with 9.50 s of fixation and ended with 10.50 s of fixation to estimate baseline activity. The total duration of each run was 3.79 min.

Response times and accuracy rates were recorded during the fMRI tasks. A Repeated Measures 2 × 6 ANOVA, with Format (Symbolic, Nonsymbolic) and Ratio (0.4, 0.6, 0.63, 0.71, 0.80, 0.83) was conducted to examine participant performance (p) on the fMRI task for response time (ms) and percent accuracy, using Eq. (1), where RT is response time and ER is the error rate [43].

2.6. fMRI preprocessing

Image data pre-processing was implemented using the FSL package (analysis group, FMRIB, Oxford, UK, http://www.fmrib.ox.ac.uk/fsl/), using standard preprocessing procedures: for each functional image, brain extraction using Brain Extraction Tool (BET), motion correction, distortion correction, spatial smoothing, and resampling. Motion correction using a 12 degree of freedom model were performed. For distortion correction, fieldmap images were derived from B0 mapping image pairs for each functional series. To ensure intensity stabilization, the initial 10 volumes of each functional series were removed. Two stages were used for image registration to spatially transform all functional series to MNI152 standard space; first, using a 6 degree of freedom model between a single band reference image and high-resolution structural scan, then using a 12 degree of freedom model between the subject’s high-resolution structural scan and MNI152 high-resolution scan. Denoising was performed using AROMA, using nonaggressive noise removal. Quality assurance and inclusion were conducted on participants’ motion graphs, signal-to-noise ratios, and brain coverage parameters, using a 0–3 scale for motion and a 0–2 scale for signal-to-noise/coverage quality via visual and quantitative inspection of participant data.

2.7. fMRI analysis

3.1. Behavioral performance in MRI tasks

Overall p (response time adjusted for accuracy) was greater for nonsymbolic than symbolic number comparisons (F(1172) = 8.352, p < 0.01) and for larger (numerically closer) than smaller (numerically further) ratios (F(5860) = 116.843, p < 0.01), see Fig. 1. There was also a greater ratio effect for nonsymbolic than symbolic number comparisons (F(5860) = 49.786, p < 0.01). See Table 1 for descriptive statistics of participants’ behavioral performance. This pattern of effects is in line with past reports in the literature.

3.2. Overall ratio effect

As a preliminary analysis, we analyzed the comparisons of ratio effect > fixation for symbolic and nonsymbolic individually to verify that the number comparison task performed as expected for each format. Consistent with prior literature [17,21,36,44,64], the intraparietal sulcus was activated during both format types (Fig. 2; see also Table 3) in addition to activation in various pre-frontal regions [17,44,68]. Thus, the tasks each exhibited expected effects.

In the next sections, we discuss the relationship of individual differences in each of math and EF abilities with the neural ratio effects first for symbolic stimuli and then for non-symbolic stimuli.

3.3. Symbolic neural ratio effect

Findings revealed differential relations between individuals’ differences in math/EF ability and the neural ratio effect.

3.4. Nonsymbolic neural ratio effect

In contrast to the symbolic task, for the nonsymbolic number comparison task, math ability and EF ability each correlated with the activation associated with the ratio effect in brain regions associated with math, while EF ability only correlated with activation associated with the ratio effect in brain regions associated with EF (see Table 5 and Fig. 4).

4.1. Limitations and future directions

Limitations should be considered when interpreting the findings of the current study. The current study measured EF using two tasks: one for inhibition and one for working memory. Future studies should examine whether individual differences in the current findings correspond when measuring EF using different methods. Our nonsymbolic magnitude task also included small number values that can be subitized [42]; therefore, our findings may not be generalized to other ANS tasks using larger number values. Future studies with larger twin samples may also examine the heritability of the effects in the current study to determine if individual differences are likely due to environment or genetics.

4.2. Conclusion and implications

In sum, the current study provides new insights into the complex interplay between EF and numerical magnitude processing across number formats, showing that the relationship between EF systems and number systems varies across number formats in terms of their relevance for math skills development. Most notably, our findings indicated that the association between individual differences in EF differed across numerical formats. Specifically, while individuals’ inhibitory control positively correlated with the nonsymbolic neural ratio effect in the left IFJ, working memory positively correlated with the symbolic neural ratio effect in the left IPL, left IFJ, left precentral gyrus, and left posterior dorsolateral prefrontal cortex. These findings suggest that the format of numerical information influences the neural systems engaged and that engagement varies with individual differences in math and EF abilities.