Does the parietal cortex distinguish between "10," "ten," and ten dots?

A new ontology for numerical cognition: Integrating evolutionary, embodied, and data informatics approaches

Numerical cognition is a field that investigates the sociocultural, developmental, cognitive, and biological aspects of mathematical abilities. Recent findings in cognitive neuroscience suggest that cognitive skills are facilitated by distributed, transient, and dynamic networks in the brain, rather than isolated functional modules. Further, research on the bodily and evolutionary bases of cognition reveals that our cognitive skills harness capacities originally evolved for action and that cognition is best understood in conjunction with perceptuomotor capacities. Despite these insights, neural models of numerical cognition struggle to capture the relation between mathematical skills and perceptuomotor systems. One front to addressing this issue is to identify building block sensorimotor processes (BBPs) in the brain that support numerical skills and develop a new ontology connecting the sensorimotor system with mathematical cognition. BBPs here are identified as sensorimotor functions, associated with distributed networks in the brain, and are consistently identified as supporting different cognitive abilities. BBPs can be identified with new approaches to neuroimaging; by examining an array of sensorimotor and cognitive tasks in experimental designs, employing data-driven informatics approaches to identify sensorimotor networks supporting cognitive processes, and interpreting the results considering the evolutionary and bodily foundations of mathematical abilities. New empirical insights on the BBPs can eventually lead to a revamped embodied cognitive ontology in numerical cognition. Among other mathematical skills, numerical magnitude processing and its sensorimotor origins are discussed to substantiate the arguments presented. Additionally, an fMRI study design is provided to illustrate the application of the arguments presented in empirical research.

Inter-individual, hemispheric and sex variability of brain activations during numerosity processing

Numerosity perception is a fundamental and innate cognitive function shared by both humans and many animal species. Previous research has primarily focused on exploring the spatial and functional consistency of neural activations that were associated with the processing of numerosity information. However, the inter-individual variability of brain activations of numerosity perception remains unclear. In the present study, with a large-sample functional magnetic resonance imaging (fMRI) dataset (n = 460), we aimed to localize the functional regions related to numerosity perceptions and explore the inter-individual, hemispheric, and sex differences within these brain regions. Fifteen subject-specific activated regions, including the anterior intraparietal sulcus (aIPS), posterior intraparietal sulcus (pIPS), insula, inferior frontal gyrus (IFG), inferior temporal gyrus (ITG), premotor area (PM), middle occipital gyrus (MOG) and anterior cingulate cortex (ACC), were delineated in each individual and then used to create a functional probabilistic atlas to quantify individual variability in brain activations of numerosity processing. Though the activation percentages of most regions were higher than 60%, the intersections of most regions across individuals were considerably lower, falling below 50%, indicating substantial variations in brain activations related to numerosity processing among individuals. Furthermore, significant hemispheric and sex differences in activation location, extent, and magnitude were also found in these regions. Most activated regions in the right hemisphere had larger activation volumes and activation magnitudes, and were located more lateral and anterior than their counterparts in the left hemisphere. In addition, in most of these regions, males displayed stronger activations than females. Our findings demonstrate large inter-individual, hemispheric, and sex differences in brain activations related to numerosity processing, and our probabilistic atlas can serve as a robust functional and spatial reference for mapping the numerosity-related neural networks.

tDCS effects in basic symbolic number magnitude processing are not significantly lateralized

Functional lateralization was previously established for various cognitive domains-but not for number processing. Although numbers are considered to be bilaterally represented in the intraparietal sulcus (IPS), there are some indications of different functional roles of the left vs. right IPS in processing number pairs with small vs. large distance, respectively. This raises the question whether number size plays a distinct role in the lateralization within the IPS. In our preregistered study, we applied anodal transcranial direct current stimulation (tDCS) over the left vs. right IPS to investigate the effect of stimulation as compared to sham on small vs. large distance, in both single-digit and two-digit number comparison. We expected that anodal tDCS over the left IPS facilitates number comparison with small distance, while anodal tDCS over the right IPS facilitates number comparison with large distance. Results indicated no effect of stimulation; however, exploratory analyses revealed that tDCS over the right IPS slowed down single-digit number processing after controlling for the training effect. In conclusion, number magnitude processing might be bilaterally represented in the IPS, however, our exploratory analyses emphasise the need for further investigation on functional lateralization of number processing.

The neural correlates of interference effects of numerical Stroop task: An ALE meta-analysis and connectometry

Numerical skills are part of cognitive and formal education development, and low performance in math has been associated with adverse features such as low income and unemployment. The studies of functional magnetic resonance imaging (fMRI) activation in numerical Stroop interference had been accomplished to evidence neural correlates of numerical, automatic, and controlled processes. The aim of this research was to summarize the results of the neural correlates of a number-size congruity task through meta-analysis of fMRI, behavioral evidence, and connectometry. Our study includes 15 fMRI papers (total number of subjects n=155-302, the total number of foci=81-233). Meta-analyses used an activation likelihood estimation (ALE) logarithm. For connectometry, it was used the diffusion tensor image. We found that, for the attentional control numerical Stroop effect, the activated areas were the dorsolateral prefrontal cortex, the anterior cingulate gyrus, and the intraparietal sulcus. Consistent activation over both paradigms was found in five clusters, two in the frontal and three in the parietal lobe. The matrix of connectivity showed connections between insula and inferior parietal right with 587 fibers, cingulate gyrus, and inferior parietal right with 843 fibers. Both paradigms activate parietal areas but differ in the activation of regions correlated to attentional control. The results of these meta-analyses summarized results from fMRI studies that may contribute to current theories. The results of connectometry could be interpreted regarding the fibers connection between the clusters right inferior parietal with insula and cingulate gyrus that suggests the integration of information.

Magnitude comparison and automaticity in number processing in adolescents with prenatal alcohol exposure: An event-related potentials study

BACKGROUND

Individuals with fetal alcohol spectrum disorders may exhibit a distinct pattern of dysmorphic facial features, growth restriction, and cognitive deficits, particularly in arithmetic. Magnitude comparison, a fundamental element of numerical cognition, is modulated by the numerical distance effect, with numbers closer in value more difficult to compare than those further apart, and by the automaticity of the association of numerical values with their symbolic representations (Arabic numerals).

METHODS

We examined event-related potentials acquired during the Numerical Stroop numerical and physical tasks administered to 24 alcohol-exposed adolescents (eight fetal alcohol syndrome (FAS), eight partial FAS (PFAS), eight heavily exposed (HE) nonsyndromal) and 23 typically developing (TD), same- age controls. The distance effect was assessed on the numerical task to examine differences in reaction time (RT) and accuracy when two numbers are close in value (e.g., 1 vs. 2) compared to when the numbers are less close (e.g., 1 vs. 6). Automaticity was assessed in the physical task by examining the degree to which RT and accuracy are reduced when the relative physical size of two numerals is incongruent with their numerical values (e.g., 1 vs. 6).

RESULTS

Adolescents in all four groups performed behaviorally as expected on these relatively simple magnitude comparison tasks, but accuracy was poorer and RT was slower on both tasks in the FAS and PFAS than the HE and TD groups. At the neurophysiological level, in the numerical task, a higher level of prenatal alcohol exposure was associated with smaller P2p amplitude. In the physical task, only the TD and nonsyndromal HE groups exhibited the expected smaller P300 amplitude in the incongruent than the congruent condition.

CONCLUSIONS

These findings suggest that magnitude comparison in alcohol-exposed individuals may be mediated by recruitment of alternative neural pathways that are likely to be inefficient when number processing becomes more challenging.

Foundational Number Sense Training Gains Are Predicted by Hippocampal-Parietal Circuits

The development of mathematical skills in early childhood relies on number sense, the foundational ability to discriminate among quantities. Number sense in early childhood is predictive of academic and professional success, and deficits in number sense are thought to underlie lifelong impairments in mathematical abilities. Despite its importance, the brain circuit mechanisms that support number sense learning remain poorly understood. Here, we designed a theoretically motivated training program to determine brain circuit mechanisms underlying foundational number sense learning in female and male elementary school-age children (7-10 years). Our 4 week integrative number sense training program gradually strengthened the understanding of the relations between symbolic (Arabic numerals) and nonsymbolic (sets of items) representations of quantity. We found that our number sense training program improved symbolic quantity discrimination ability in children across a wide range of math abilities including children with learning difficulties. Crucially, the strength of pretraining functional connectivity between the hippocampus and intraparietal sulcus, brain regions implicated in associative learning and quantity discrimination, respectively, predicted individual differences in number sense learning across typically developing children and children with learning difficulties. Reverse meta-analysis of interregional coactivations across 14,371 fMRI studies and 89 cognitive functions confirmed a reliable role for hippocampal-intraparietal sulcus circuits in learning. Our study identifies a canonical hippocampal-parietal circuit for learning that plays a foundational role in children's cognitive skill acquisition. Findings provide important insights into neurobiological circuit markers of individual differences in children's learning and delineate a robust target for effective cognitive interventions.SIGNIFICANCE STATEMENT Mathematical skill development relies on number sense, the ability to discriminate among quantities. Here, we develop a theoretically motivated training program and investigate brain circuits that predict number sense learning in children during a period important for acquisition of foundational cognitive skills. Our integrated number sense training program was effective in children across a wide a range of math abilities, including children with learning difficulties. We identify hippocampal-parietal circuits that predict individual differences in learning gains. Our study identifies a brain circuit critical for the acquisition of foundational cognitive skills, which will be useful for developing effective interventions to remediate learning disabilities.

Cognitive and brain correlates of acquired number-colour synaesthetic-like associations

Synaesthesia is a condition in which one sensory dimension triggers another sensation. The exact contribution of genetic and environmental factors in synaesthesia is not yet fully understood. Most synaesthesia phenotypes involve associations in which the synaesthetic inducer constitutes some form of linguistic/conceptual information acquired during the course of development (e.g., digits, letters, and names of months). To study the role of learning in synaesthesia, we induced synaesthesia by training and examined the behavioural and brain correlates of number-colour associations. We took advantage of the well-known behavioural and neural signature of numbers and used number symbols as inducers. Short (2 weeks) and long (4 weeks) training protocols were conducted with two different groups. Task-related BOLD response was acquired while participants performed Stroop tasks requiring naming colours while ignoring the stimuli (i.e., number symbols, dots, words). If the arbitrary association involving number-colour is automatic, the irrelevant dimension (i.e., numbers) would interfere with the colour response. In addition, if number-colour associations are transferred to linguistic and non-symbolic representations, the passive viewing of stimuli (i.e., words and dots) would disrupt colour naming. Behavioural findings showed automatic associations as both training protocols elicited reliable congruency effects for all stimulus dimensions. Congruency effects following both training protocols produced reliable brain activations in various cortical sites involved in number and in cognitive control. The behavioural and brain patterns reported here support the role of learning in the brain correlates of developmental synaesthesia and provide the first evidence that automatic associations involving different magnitude dimensions can be acquired.

Electrophysiological Signatures of Numerosity Encoding in a Delayed Match-to-Sample Task

The number of elements in a small set of items is appraised in a fast and exact manner, a phenomenon called subitizing. In contrast, humans provide imprecise responses when comparing larger numerosities, with decreasing precision as the number of elements increases. Estimation is thought to rely on a dedicated system for the approximate representation of numerosity. While previous behavioral and neuroimaging studies associate subitizing to a domain-general system related to object tracking and identification, the nature of small numerosity processing is still debated. We investigated the neural processing of numerosity across subitizing and estimation ranges by examining electrophysiological activity during the memory retention period in a delayed numerical match-to-sample task. We also assessed potential differences in the neural signature of numerical magnitude in a fully non-symbolic or cross-format comparison. In line with behavioral performance, we observed modulation of parietal-occipital neural activity as a function of numerosity that differed in two ranges, with distinctive neural signatures of small numerosities showing clear similarities with those observed in visuospatial working memory tasks. We also found differences in neural activity related to numerical information in anticipation of single vs. cross-format comparison, suggesting a top-down modulation of numerical processing. Finally, behavioral results revealed enhanced performance in the mixed-format conditions and a significant correlation between task performance and symbolic mathematical skills. Overall, we provide evidence for distinct mechanisms related to small and large numerosity and differences in numerical encoding based on task demands.

Cultural factors weaken but do not reverse left-to-right spatial biases in numerosity processing: Data from Arabic and English monoliterates and Arabic-English biliterates

Directional response biases due to a conceptual link between space and number, such as a left-to-right hand bias for increasing numerical magnitude, are known as the SNARC (Spatial-Numerical Association of Response Codes) effect. We investigated how the SNARC effect for numerosities would be influenced by reading-writing direction, task instructions, and ambient visual environment in four literate populations exemplifying opposite reading-writing cultures-namely, Arabic (right-to-left script) and English (left-to-right script). Monoliterates and biliterates in Jordan and the U.S. completed a speeded numerosity comparison task to assess the directionality and magnitude of a SNARC effect in their numerosity processing. Monoliterates' results replicated previously documented effects of reading-writing direction and task instructions: the SNARC effect found in left-to-right readers was weakened in right-to-left readers, and the left-to-right group exhibited a task-dependency effect (SNARC effect in the smaller condition, reverse SNARC effect in the larger condition). Biliterates' results did not show a clear effect of environment; instead, both biliterate groups resembled English monoliterates in showing a left-to-right, task-dependent SNARC effect, albeit weaker than English monoliterates'. The absence of significant biases in all Arabic-reading groups (biliterates and Arabic monoliterates) points to a potential conflict between distinct spatial-numerical mapping codes. This view is explained in terms of the proposed Multiple Competing Codes Theory (MCCT), which posits three distinct spatial-numerical mapping codes (innate, cardinal, ordinal) during numerical processing-each involved at varying levels depending on individual and task factors.

Symbols Are Special: An fMRI Adaptation Study of Symbolic, Nonsymbolic, and Non-Numerical Magnitude Processing in the Human Brain

How are different formats of magnitudes represented in the human brain? We used functional magnetic resonance imaging adaptation to isolate representations of symbols, quantities, and physical size in 45 adults. Results indicate that the neural correlates supporting the passive processing of number symbols are largely dissociable from those supporting quantities and physical size, anatomically and representationally. Anatomically, passive processing of quantities and size correlate with activation in the right intraparietal sulcus, whereas symbolic number processing, compared with quantity processing, correlates with activation in the left inferior parietal lobule. Representationally, neural patterns of activation supporting symbols are dissimilar from neural activation patterns supporting quantity and size in the bilateral parietal lobes. These findings challenge the longstanding notion that the culturally acquired ability to conceptualize symbolic numbers is represented using entirely the same brain systems that support the evolutionarily ancient system used to process quantities. Moreover, these data reveal that regions that support numerical magnitude processing are also important for the processing of non-numerical magnitudes. This discovery compels future investigations of the neural consequences of acquiring knowledge of symbolic numbers.

Children's neural activity during number line estimations assessed by functional near-infrared spectroscopy (fNIRS)

Number line estimation (NLE) is an educational task in which children estimate the location of a value (e.g., 25) on a blank line that represents a numerical range (e.g., 0-100). NLE performance is a strong predictor of success in mathematics, and error patterns on this task help provide a glimpse into how children may represent number internally. However, a missing and fundamental element of this puzzle is the identification of neural correlates of NLE in children. That is, understanding possible neural signatures related to NLE performance will provide valuable insight into the cognitive processes that underlie children's development of NLE ability. Using functional near-infrared spectroscopy (fNIRS), we provide the first investigation of concurrent behavioral and cortical signatures of NLE performance in children. Specifically, our results highlight significant fronto-parietal changes in cortical activation in response to increases in NLE scale (e.g., 0-100 vs. 0-100,000). Furthermore, our results demonstrate that NLE performance feedback (auditory, visual, or audiovisual), as well as children's grade (2nd vs. 3rd) influence cortical responding during an NLE task.

Numerical cognition: A meta-analysis of neuroimaging, transcranial magnetic stimulation and brain-damaged patients studies

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We review neuroimaging, TMS, and patients studies on numerical cognition.

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We focused on the predictions derived from the Triple Code Model (TCM).

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Our findings generally agree with TCM predictions.

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Our results open avenues for the study of the neural bases of numerical cognition.

This article offers the first comprehensive review examining the neurocognitive bases of numerical cognition from neuroimaging, Transcranial Magnetic Stimulation (TMS) and brain-damaged patients studies. We focused on the predictions derived from the Triple Code Model (TCM), particularly the assumption that the representation of numerical quantities rests on a single format-independent representation (i.e., the analogical code) involving both intraparietal sulci (IPS). To do so, we conducted a meta-analysis based on 28 neuroimaging, 12 TMS and 12 brain-damaged patients studies, including arithmetic and magnitude tasks in symbolic and non-symbolic formats. Our findings generally agree with the TCM predictions indicating that both IPS are engaged in all tasks. Nonetheless, the results of brain-damaged patients studies conflicted with neuroimaging and TMS studies, suggesting a right hemisphere lateralization for non-symbolic formats. Our findings also led us to discuss the involvement of brain regions other than IPS in the processing of the analogical code as well as the neural substrate of other codes underlying numerical cognition (i.e., the auditory-verbal code).

Tactile enumeration: A case study of acalculia

Enumeration is one of the building blocks of arithmetic and fingers are used as a counting tool in early steps. Subitizing-fast and accurate enumeration of small quantities-has been vastly studied in the visual modality, but less in the tactile modality. We explored tactile enumeration using fingers, and gray matter (GM) changes using voxel-based morphometry (VBM), in acalculia. We examined JD, a 22-year-old female with acalculia following a stroke to the left inferior parietal cortex. JD and a neurologically healthy normal comparison (NC) group reported how many fingers were stimulated. JD was tested at several time points, including at acute and chronic phases. Using the sensory intact hand for tactile enumeration, JD showed deficit in the acute phase, compared to the NC group, and improvement in the chronic phase of (1) the RT slope of enumerating up to four stimuli, (2) enumerating neighboring fingers, and (3) arithmetic fluency performance. Moreover, VBM analysis showed a larger GM volume for JD relative to the NC group in the right middle occipital cortex, most profoundly in the chronic phase. JD's performance serves as a first glance of tactile enumeration in acalculia. Pattern-recognition-based results support the suggestion of subitizing being the enumeration process when using one hand. Moreover, the increase in GM in the occipital cortex lays the groundwork for studying the innate and primitive ability to perceive and evaluate sizes or amounts-"sense of magnitude"- as a multisensory magnitude area and as part of a recovery path for deficits in basic numerical abilities.

Numerical encoding in early visual cortex

The ability to estimate numerosity in a visual array arose early in evolution, develops early in human development, and is correlated with mathematical ability. Previous work with visually presented arrays indicates that the intraparietal sulcus (IPS) represents number. However, it is not clear if the number signal originates in IPS or is propagated from earlier visual areas. Previous work from our group has demonstrated a rapidly instantiated representation of number in low-level regions of visual cortex using the high temporal resolution of event-related electro-encephalography (EEG). Here, we use a rapid event-related functional magnetic resonance imaging (fMRI) paradigm and find convergent evidence for a number signal in low-level visual cortex (areas V1, V2, and V3). Employing a stringent set of stimulus controls, we demonstrate that this signal cannot be explained by the total extent of the array, the density of the items in the array, the aggregate visual area of the items, the size of individual items, the proportion of the array covered by items, nor the overall scale of the array and items. Our findings thus provide strong support for the hypothesis that number is rapidly and directly encoded early in the visual processing stream.

Prospective relations between resting-state connectivity of parietal subdivisions and arithmetic competence

The present study investigates the relation between resting-state functional connectivity (rsFC) of cytoarchitectonically defined subdivisions of the parietal cortex at the end of 1st grade and arithmetic performance at the end of 2nd grade. Results revealed a dissociable pattern of relations between rsFC and arithmetic competence among subdivisions of intraparietal sulcus (IPS) and angular gyrus (AG). rsFC between right hemisphere IPS subdivisions and contralateral IPS subdivisions positively correlated with arithmetic competence. In contrast, rsFC between the left hIP1 and the right medial temporal lobe, and rsFC between the left AG and left superior frontal gyrus, were negatively correlated with arithmetic competence. These results suggest that strong inter-hemispheric IPS connectivity is important for math development, reflecting either neurocognitive mechanisms specific to arithmetic processing, domain-general mechanisms that are particularly relevant to arithmetic competence, or structural 'cortical maturity'. Stronger connectivity between IPS, and AG, subdivisions and frontal and temporal cortices, however, appears to be negatively associated with math development, possibly reflecting the ability to disengage suboptimal problem-solving strategies during mathematical processing, or to flexibly reorient task-based networks. Importantly, the reported results pertain even when controlling for reading, spatial attention, and working memory, suggesting that the observed rsFC-behavior relations are specific to arithmetic competence.

Spatial Arrangement and Set Size Influence the Coding of Non-symbolic Quantities in the Intraparietal Sulcus

Performance in visual quantification tasks shows two characteristic patterns as a function of set size. A precise subitizing process for small sets (up to four) was contrasted with an approximate estimation process for larger sets. The spatial arrangement of elements in a set also influences visual quantification performance, with frequently perceived arrangements (e.g., dice patterns) being faster enumerated than random arrangements. Neuropsychological and imaging studies identified the intraparietal sulcus (IPS), as key brain area for quantification, both within and above the subitizing range. However, it is not yet clear if and how set size and spatial arrangement of elements in a set modulate IPS activity during quantification. In an fMRI study, participants enumerated briefly presented dot patterns with random, canonical or dice arrangement within and above the subitizing range. We evaluated how activity amplitude and pattern in the IPS were influenced by size and spatial arrangement of a set. We found a discontinuity in the amplitude of IPS response between subitizing and estimation range, with steep activity increase for sets exceeding four elements. In the estimation range, random dot arrangements elicited stronger IPS response than canonical arrangements which in turn elicited stronger response than dice arrangements. Furthermore, IPS activity patterns differed systematically between arrangements. We found a signature in the IPS response for a transition between subitizing and estimation processes during quantification. Differences in amplitude and pattern of IPS activity for different spatial arrangements indicated a more precise representation of non-symbolic numerical magnitude for dice and canonical than for random arrangements. These findings challenge the idea of an abstract coding of numerosity in the IPS even within a single notation.

Visual and Imagery Magnitude Comparisons Are Affected Following Left Parietal Lesion

We describe Jane Dow (JD), a young right-handed female with acalculia following a cerebral infarction in the left intraparietal sulcus. We investigated automatic processing of different types of magnitudes that were presented visually or through imagery. We employed the size congruity task and the mental clock task that differ in stimuli presentation and in working memory load. In the size congruity task, for physical comparisons, JD presented a lack of facilitation effect, suggesting a deficit in the automatic processing of numerical values. In the mental clock task, JD performed as accurate as controls did but much slower. In both tasks, JD presented a steeper distance effect compared to controls, suggesting a deficit in a domain-general comparison process. Our findings present an atypical pattern of magnitude processing following a left parietal lesion that appears not only for visually presented stimuli but also for imagery-based magnitudes. These finding support recent theories suggesting different types of magnitudes are interconnected with each other.

Measuring arithmetic: A psychometric approach to understanding formatting effects and domain specificity

Using multitrait, multimethod data, and confirmatory factor analysis, the current study examined the effects of arithmetic item formatting and the possibility that across formats, abilities other than arithmetic may contribute to children's answers. Measurement hypotheses were guided by several leading theories of arithmetic cognition. With a sample of 1314 3rd grade students (age M=103.24 months, SD=5.41 months), Abstract Code Theory, Encoding Complex Theory, Triple Code Theory, and the Exact versus Approximate Calculations Hypothesis were evaluated, using 11 measures of arithmetic with symbolic problem formats (e.g., Arabic numeral and language-based formats) and various problem demands (e.g., requiring both exact and approximate calculations). In general, results provided support for both Triple Code Theory and Encoding Complex Theory. As predicted by Triple Code Theory, arithmetic outcomes with language formatting, Arabic numeral formatting, and estimation demands (across formats) were related but distinct from one another. As predicted by Encoding Complex Theory, executive attention was a direct predictor of all arithmetic outcomes. Language was no longer a direct predictor of arithmetic outcomes when executive attention was accounted for in the model; however, a strong and enduring relationship between language and executive attention suggested that language may play a facilitative role in reasoning during numeric processing. These findings have important implications for assessing arithmetic in educational settings and suggest that in addition to arithmetic-focused interventions, interventions targeting executive attention, language, and/or the interplay between them (i.e., internal speech during problem-solving) may be a promising avenues of mathematical problem-solving intervention.

Common and distinct brain regions in both parietal and frontal cortex support symbolic and nonsymbolic number processing in humans: A functional neuroimaging meta-analysis

In recent years, there has been substantial growth in neuroimaging studies investigating neural correlates of symbolic (e.g. Arabic numerals) and non-symbolic (e.g. dot arrays) number processing. At present it remains contested whether number is represented abstractly, or if number representations in the brain are format-dependent. In order to quantitatively evaluate the available neuroimaging evidence, we used activation likelihood estimation (ALE) to conduct quantitative meta-analyses of the results reported in 57 neuroimaging papers. Consistent with the existence of an abstract representation of number in the brain, conjunction analyses revealed overlapping activation for symbolic and nonsymbolic numbers in frontal and parietal lobes. Consistent with the notion of format-dependent activation, contrast analyses demonstrated anatomically distinct fronto-parietal activation for symbolic and non-symbolic processing. Therefore, symbolic and non-symbolic numbers are subserved by format-dependent and abstract neural systems. Moreover, the present results suggest that regions across the parietal cortex, not just the intraparietal sulcus, are engaged in both symbolic and non-symbolic number processing, challenging the notion that the intraparietal sulcus is the key region for number processing. Additionally, our analyses indicate that regions in the frontal cortex subserve magnitude representations rather than non-numerical cognitive processes associated with number tasks, thereby highlighting the importance of considering both frontal and parietal regions as important for number processing.

Modality-independent representations of small quantities based on brain activation patterns

Machine learning or MVPA (Multi Voxel Pattern Analysis) studies have shown that the neural representation of quantities of objects can be decoded from fMRI patterns, in cases where the quantities were visually displayed. Here we apply these techniques to investigate whether neural representations of quantities depicted in one modality (say, visual) can be decoded from brain activation patterns evoked by quantities depicted in the other modality (say, auditory). The main finding demonstrated, for the first time, that quantities of dots were decodable by a classifier that was trained on the neural patterns evoked by quantities of auditory tones, and vice-versa. The representations that were common across modalities were mainly right-lateralized in frontal and parietal regions. A second finding was that the neural patterns in parietal cortex that represent quantities were common across participants. These findings demonstrate a common neuronal foundation for the representation of quantities across sensory modalities and participants and provide insight into the role of parietal cortex in the representation of quantity information.

Topology-defined units in numerosity perception

What is a number? The number sense hypothesis suggests that numerosity is "a primary visual property" like color, contrast, or orientation. However, exactly what attribute of a stimulus is the primary visual property and determines numbers in the number sense? To verify the invariant nature of numerosity perception, we manipulated the numbers of items connected/enclosed in arbitrary and irregular forms while controlling for low-level features (e.g., orientation, color, and size). Subjects performed discrimination, estimation, and equality judgment tasks in a wide range of presentation durations and across small and large numbers. Results consistently show that connecting/enclosing items led to robust numerosity underestimation, with the extent of underestimation increasing monotonically with the number of connected/enclosed items. In contrast, grouping based on color similarity had no effect on numerosity judgment. We propose that numbers or the primitive units counted in numerosity perception are influenced by topological invariants, such as connectivity and the inside/outside relationship. Beyond the behavioral measures, neural tuning curves to numerosity in the intraparietal sulcus were obtained using functional MRI adaptation, and the tuning curves showed that numbers represented in the intraparietal sulcus were strongly influenced by topology.

Visual Number Beats Abstract Numerical Magnitude: Format-dependent Representation of Arabic Digits and Dot Patterns in Human Parietal Cortex

In numerical cognition, there is a well-known but contested hypothesis that proposes an abstract representation of numerical magnitude in human intraparietal sulcus (IPS). On the other hand, researchers of object cognition have suggested another hypothesis for brain activity in IPS during the processing of number, namely that this activity simply correlates with the number of visual objects or units that are perceived. We contrasted these two accounts by analyzing multivoxel activity patterns elicited by dot patterns and Arabic digits of different magnitudes while participants were explicitly processing the represented numerical magnitude. The activity pattern elicited by the digit “8” was more similar to the activity pattern elicited by one dot (with which the digit shares the number of visual units but not the magnitude) compared to the activity pattern elicited by eight dots, with which the digit shares the represented abstract numerical magnitude. A multivoxel pattern classifier trained to differentiate one dot from eight dots classified all Arabic digits in the one-dot pattern category, irrespective of the numerical magnitude symbolized by the digit. These results were consistently obtained for different digits in IPS, its subregions, and many other brain regions. As predicted from object cognition theories, the number of presented visual units forms the link between the parietal activation elicited by symbolic and nonsymbolic numbers. The current study is difficult to reconcile with the hypothesis that parietal activation elicited by numbers would reflect a format-independent representation of number.

Developmental specialization of the left parietal cortex for the semantic representation of Arabic numerals: an fMR-adaptation study

The way the human brain constructs representations of numerical symbols is poorly understood. While increasing evidence from neuroimaging studies has indicated that the intraparietal sulcus (IPS) becomes increasingly specialized for symbolic numerical magnitude representation over developmental time, the extent to which these changes are associated with age-related differences in symbolic numerical magnitude representation or with developmental changes in non-numerical processes, such as response selection, remains to be uncovered. To address these outstanding questions we investigated developmental changes in the cortical representation of symbolic numerical magnitude in 6- to 14-year-old children using a passive functional magnetic resonance imaging adaptation design, thereby mitigating the influence of response selection. A single-digit Arabic numeral was repeatedly presented on a computer screen and interspersed with the presentation of novel digits deviating as a function of numerical ratio (smaller/larger number). Results demonstrated a correlation between age and numerical ratio in the left IPS, suggesting an age-related increase in the extent to which numerical symbols are represented in the left IPS. Brain activation of the right IPS was modulated by numerical ratio but did not correlate with age, indicating hemispheric differences in IPS engagement during the development of symbolic numerical representation.

Children's mappings between number words and the approximate number system

Humans can represent number either exactly--using their knowledge of exact numbers as supported by language, or approximately--using their approximate number system (ANS). Adults can map between these two systems--they can both translate from an approximate sense of the number of items in a brief visual display to a discrete number word estimate (i.e., ANS-to-Word), and can generate an approximation, for example by rapidly tapping, when provided with an exact verbal number (i.e., Word-to-ANS). Here we ask how these mappings are initially formed and whether one mapping direction may become functional before the other during development. In two experiments, we gave 2-5 year old children both an ANS-to-Word task, where they had to give a verbal number response to an approximate presentation (i.e., after seeing rapidly flashed dots, or watching rapid hand taps), and a Word-to-ANS task, where they had to generate an approximate response to a verbal number request (i.e., rapidly tapping after hearing a number word). Replicating previous results, children did not successfully generate numerically appropriate verbal responses in the ANS-to-Word task until after 4 years of age--well after they had acquired the Cardinality Principle of verbal counting. In contrast, children successfully generated numerically appropriate tapping sequences in the Word-to-ANS task before 4 years of age--well before many understood the Cardinality Principle. We further found that the accuracy of the mapping between the ANS and number words, as captured by error rates, continues to develop after this initial formation of the interface. These results suggest that the mapping between the ANS and verbal number representations is not functionally bidirectional in early development, and that the mapping direction from number representations to the ANS is established before the reverse.

Common neural substrates for ordinal representation in short-term memory, numerical and alphabetical cognition

The representation and maintenance of serial order information is one of the main functions of verbal short-term memory (STM) but its neural correlates remain poorly understood. We show here that the neural substrates allowing for coding of order information in STM are shared with those supporting ordinal processing in the numerical and alphabetical domains. We designed an fMRI experiment determining the neural substrates sensitive to ordinal distance effects in numerical judgment, alphabetical judgment and serial order STM tasks. Null conjunction analyses for parametric ordinal distance effects showed a common involvement of the horizontal segment of the left intraparietal sulcus over the three tasks; in addition, right intraparietal sulcus involvement was also observed for ordinal distance effects in the STM and numerical judgment tasks. These findings demonstrate that shared neural correlates in the intraparietal cortex support processing of order information in verbal STM, number and alphabetical domains, and suggest the existence of domain general, potentially ordinal, comparison processes supported by the left intraparietal sulcus.

Neural connectivity patterns underlying symbolic number processing indicate mathematical achievement in children

In early childhood, humans learn culturally specific symbols for number that allow them entry into the world of complex numerical thinking. Yet little is known about how the brain supports the development of the uniquely human symbolic number system. Here, we use functional magnetic resonance imaging along with an effective connectivity analysis to investigate the neural substrates for symbolic number processing in young children. We hypothesized that, as children solidify the mapping between symbols and underlying magnitudes, important developmental changes occur in the neural communication between the right parietal region, important for the representation of non-symbolic numerical magnitudes, and other brain regions known to be critical for processing numerical symbols. To test this hypothesis, we scanned children between 4 and 6 years of age while they performed a magnitude comparison task with Arabic numerals (numerical, symbolic), dot arrays (numerical, non-symbolic), and lines (non-numerical). We then identified the right parietal seed region that showed greater blood-oxygen-level-dependent signal in the numerical versus the non-numerical conditions. A psychophysiological interaction method was used to find patterns of effective connectivity arising from this parietal seed region specific to symbolic compared to non-symbolic number processing. Two brain regions, the left supramarginal gyrus and the right precentral gyrus, showed significant effective connectivity from the right parietal cortex. Moreover, the degree of this effective connectivity to the left supramarginal gyrus was correlated with age, and the degree of the connectivity to the right precentral gyrus predicted performance on a standardized symbolic math test. These findings suggest that effective connectivity underlying symbolic number processing may be critical as children master the associations between numerical symbols and magnitudes, and that these connectivity patterns may serve as an important indicator of mathematical achievement.

Understanding less than nothing: children's neural response to negative numbers shifts across age and accuracy

We examined the brain activity underlying the development of our understanding of negative numbers, which are amounts lacking direct physical counterparts. Children performed a paired comparison task with positive and negative numbers during an fMRI session. As previously shown in adults, both pre-instruction fifth-graders and post-instruction seventh-graders demonstrated typical behavioral and neural distance effects to negative numbers, where response times and parietal and frontal activity increased as comparison distance decreased. We then determined the factors impacting the distance effect in each age group. Behaviorally, the fifth-grader distance effect for negatives was significantly predicted only by positive comparison accuracy, indicating that children who were generally better at working with numbers were better at comparing negatives. In seventh-graders, negative number comparison accuracy significantly predicted their negative number distance effect, indicating that children who were better at working with negative numbers demonstrated a more typical distance effect. Across children, as age increased, the negative number distance effect increased in the bilateral IPS and decreased frontally, indicating a frontoparietal shift consistent with previous numerical development literature. In contrast, as negative comparison task accuracy increased, the parietal distance effect increased in the left IPS and decreased in the right, possibly indicating a change from an approximate understanding of negatives' values to a more exact, precise representation (particularly supported by the left IPS) with increasing expertise. These shifts separately indicate the effects of increasing maturity generally in numeric processing and specifically in negative number understanding.

Enhancing performance in numerical magnitude processing and mental arithmetic using transcranial Direct Current Stimulation (tDCS)

The ability to accurately process numerical magnitudes and solve mental arithmetic is of highest importance for schooling and professional career. Although impairments in these domains in disorders such as developmental dyscalculia (DD) are highly detrimental, remediation is still sparse. In recent years, transcranial brain stimulation methods such as transcranial Direct Current Stimulation (tDCS) have been suggested as a treatment for various neurologic and neuropsychiatric disorders. The posterior parietal cortex (PPC) is known to be crucially involved in numerical magnitude processing and mental arithmetic. In this study, we evaluated whether tDCS has a beneficial effect on numerical magnitude processing and mental arithmetic. Due to the unclear lateralization, we stimulated the left, right as well as both hemispheres simultaneously in two experiments. We found that left anodal tDCS significantly enhanced performance in a number comparison and a subtraction task, while bilateral and right anodal tDCS did not induce any improvements compared to sham. Our findings demonstrate that the left PPC is causally involved in numerical magnitude processing and mental arithmetic. Furthermore, we show that these cognitive functions can be enhanced by means of tDCS. These findings encourage to further investigate the beneficial effect of tDCS in the domain of mathematics in healthy and impaired humans.

Separate pathway characteristics of numerical surface form processing: evidence from operand-related error effects

Two perspectives compete to explain how the surface form of digits affects cognitive processing of numerical magnitude; one argues for a common pathway and the other for separate pathways. This study examined the operand-related error effect in simple multiplication operations using different combinations of visually presented Arabic digits and auditorily presented Mandarin number words. The study suggested two conclusions, both consistent with the separate pathway perspective. First, the numerical surface form (Arabic digits, spoken Mandarin number words) affected retrieval. That is, surface properties were maintained as specific codes throughout processing. Second, the phonological code activated by spoken Mandarin number words interfered with activation of answers during retrieval.

The role of the left intraparietal sulcus in the relationship between symbolic number processing and children's arithmetic competence

The neural foundations of arithmetic learning are not well understood. While behavioral studies have revealed relationships between symbolic number processing and individual differences in children's arithmetic performance, the neurocognitive mechanisms that bind symbolic number processing and arithmetic are unknown. The current fMRI study investigated the relationship between children's brain activation during symbolic number comparison (Arabic digits) and individual differences in arithmetic fluency. A significant correlation was found between the numerical ratio effect on reaction times and accuracy and children's arithmetic scores. Furthermore, children with a stronger neural ratio effect in the left intraparietal sulcus (IPS) during symbolic number processing exhibited higher arithmetic scores. Previous research has demonstrated that activation of the IPS during numerical magnitude processing increases over the course of development, and that the left IPS plays an important role in symbolic number processing. The present findings extend this knowledge to show that children with more mature response modulation of the IPS during symbolic number processing exhibit higher arithmetic competence. These results suggest that the left IPS is a key neural substrate for the relationship between the relative of precision of the representation of numerical magnitude and school-level arithmetic competence.

Decoding the representation of numerical values from brain activation patterns

Human neuroimaging studies have increasingly converged on the possibility that the neural representation of specific numbers may be decodable from brain activity, particularly in parietal cortex. Multivariate machine learning techniques have recently demonstrated that the neural representation of individual concrete nouns can be decoded from fMRI patterns, and that some patterns are general over people. Here we use these techniques to investigate whether the neural codes for quantities of objects can be accurately decoded. The pictorial mode (nonsymbolic) depicted a set of objects pictorially (e.g., a picture of three tomatoes), whereas the digit-object mode depicted quantities as combination of a digit (e.g., 3) with a picture of a single object. The study demonstrated that quantities of objects were decodable from neural activation patterns, in parietal regions. These brain activation patterns corresponding to a given quantity were common across objects and across participants in the pictorial mode. Other important findings included better identification of individual numbers in the pictorial mode, partial commonality of neural patterns across the two modes, and hemispheric asymmetry with pictorially-depicted numbers represented bilaterally and numbers in the digit-object mode represented primarily in the left parietal regions. The findings demonstrate the ability to identify individual quantities of objects based on neural patterns, indicating the presence of stable neural representations of numbers. Additionally, they indicate a predominance of neural representation of pictorially depicted numbers over the digit-object mode.

Meta-analyses of developmental fMRI studies investigating typical and atypical trajectories of number processing and calculation

The main aim of the present coordinate-based meta-analysis is to identify brain regions that are commonly activated in functional magnetic resonance imaging studies (fMRI) investigating number processing and calculation in children. Here, we include 19 developmental fMRI papers, five of which also examine children diagnosed with developmental dyscalculia and/or mathematical disability. Results reveal that children produce consistent fronto-parietal activation patterns in response to number processing and calculation. Importantly, activation patterns are clearly modulated by notation, task complexity, and competence level. Activation differences between children with and without dyscalculia are observable in number-relevant parietal regions as well as in (pre)frontal and occipital cortex.

Electrophysiological evidence for numerosity processing in infancy

In previous studies we established that similar to adults, 6-9-month-old infants' brain activity showed increased negativity when presented with incorrect solutions to simple arithmetic equations. Moreover, we demonstrated that in adults this frontal reaction was preceded by activity over right parietal areas, which are involved in evaluation of numerical value and its discrepancy from the expected correct solution to the equation. The current article suggests that similar to findings in adults, infants' frontal activity also seems to be preceded by activity in right parietal electrodes, plausibly related to the evaluation of numerosity.

Multi-representation of symbolic and nonsymbolic numerical magnitude in Chinese number processing

Numerical information can be conveyed by either symbolic or nonsymbolic representation. Some symbolic numerals can also be identified as nonsymbolic quantities defined by the number of lines (e.g., I, II, III in Roman and , , in Japanese Kanji and Chinese). Here we report that such multi-representation of magnitude can facilitate the processing of these numerals under certain circumstances. In a magnitude comparison task judging 1 to 9 (except 5) Chinese and Arabic numerals presented at the foveal (at the center) or parafoveal (3° left or right of the center) location, multi-representational small-value Chinese numerals showed a processing advantage over single-representational Arabic numerals and large-value Chinese numerals only in the parafoveal condition, demonstrated by lower error rates and faster reaction times. Further event-related potential (ERP) analysis showed that such a processing advantage was not reflected by traditional ERP components identified in previous studies of number processing, such as N1 or P2p. Instead, the difference was found much later in a N400 component between 300–550 msec over parietal regions, suggesting that those behavioral differences may not be due to early processing of visual identification, but later processing of subitizing or accessing mental number line when lacking attentional resources. These results suggest that there could be three stages of number processing represented separately by the N1, P2p and N400 ERP components. In addition, numerical information can be represented simultaneously by both symbolic and nonsymbolic systems, which will facilitate number processing in certain situations.

The Magnitude Representation of Small and Large Symbolic Numbers in the Left and Right Hemisphere: An Event-related fMRI Study

Numbers are known to be processed along the left and right intraparietal sulcus. The present study investigated hemispheric differences between the magnitude representation of small and large symbolic numbers. To this purpose, an fMRI adaptation paradigm was used, where the continuous presentation of a habituation number was interrupted by an occasional deviant number. The results presented a distance-dependent increase of activation: larger ratios of habituation and deviant number caused a larger recovery of activation. Similar activation patterns were observed for small and large symbolic numbers, which is in line with the idea of a more coarse magnitude representation for large numbers. Interestingly, this pattern of activation was only observed in the left parietal cortex, supporting the recently proposed idea of Ansari [Ansari, D. Does the parietal cortex distinguish between “10”, “Ten,” and Ten Dots? Neuron, 53, 165–167, 2007] that the left parietal cortex is specialized in the processing of encultured symbolically presented numbers.

An fMRI study of magnitude comparison and exact addition in children

By contrast to the adult literature, in which a consistent parietofrontal network for number processing has been identified, the data from studies of number processing in children have been less consistent, probably due to differences in study design and control conditions. Number processing was examined using functional magnetic resonance imaging in 18 right-handed children (8-12 years) from the Cape Coloured community in Cape Town, South Africa, using Proximity Judgment and Exact Addition (EA) tasks. The findings were consistent with the hypothesis that, as in adults, the anterior horizontal intraparietal sulcus (HIPS) plays a major role in the representation and manipulation of quantity in children. The posterior medial frontal cortex, believed to be involved in performance monitoring in more complex arithmetic manipulations in adults, was extensively activated even for relatively simple symbolic number processing in the children. Other areas activated to a greater degree in the children included the left precentral sulcus, which may mediate number knowledge and, for EA, the head of the caudate nucleus, which is part of a fronto-subcortical circuit involved in the behavioral execution of sequences. Two regions that have been linked to number processing in adults - the angular gyrus and posterior superior parietal lobule - were not activated in the children. The data are consistent with the inference that although the functional specialization of the anterior HIPS may increase as symbolic number processing becomes increasingly automatic, this region and other elements of the parietofrontal network identified in adults are already reliably and robustly activated by middle childhood.

Unconscious numerical priming despite interocular suppression

Whether high-level properties of stimuli rendered invisible by interocular competition can influence perception and behavior remains controversial. We studied whether suppressed and invisible symbolic and nonsymbolic numerical stimuli can elicit priming. First, we established that participants were objectively unable to discriminate numerical prime stimuli when interocular suppression rendered them invisible. Next, we asked participants to enumerate a visible target set of items after being exposed to a suppressed, invisible (nonsymbolic or symbolic) prime set. Both symbolic and nonsymbolic unconsciously perceived numerical primes induced robust priming effects that were specific to the numerical distance between the target and prime. Comparison with a no-prime condition revealed that primes larger than targets interfered with target enumeration and primes the same as or smaller than targets facilitated target enumeration. Taken together, our findings provide clear evidence for high-level processing of stimuli rendered invisible through interocular suppression.

Numerical representation in the parietal lobes: abstract or not abstract?

The study of neuronal specialisation in different cognitive and perceptual domains is important for our understanding of the human brain, its typical and atypical development, and the evolutionary precursors of cognition. Central to this understanding is the issue of numerical representation, and the question of whether numbers are represented in an abstract fashion. Here we discuss and challenge the claim that numerical representation is abstract. We discuss the principles of cortical organisation with special reference to number and also discuss methodological and theoretical limitations that apply to numerical cognition and also to the field of cognitive neuroscience in general. We argue that numerical representation is primarily non-abstract and is supported by different neuronal populations residing in the parietal cortex.

Common and segregated neural pathways for the processing of symbolic and nonsymbolic numerical magnitude: an fMRI study

Numbers are everywhere in modern life. Looking out a window, one might see both symbolic numbers, like the numerals on a thermometer, and nonsymbolic quantities, such as the number of chickadees at a bird feeder. Although differences between symbolic and nonsymbolic numbers appear very salient, most research on numerical cognition has focused on similarities rather than differences between numerical stimulus formats. Thus, little is known about differences in the processing of symbolic and nonsymbolic numerical magnitudes. A recent computational model proposed that symbolic and nonsymbolic quantities undergo distinct encoding processes which then converge on a common neural representation of numerical magnitude (Verguts, T., Fias, W., 2004. Representation of number in animals and humans: a neural model. J. Cogn. Neurosci. 16 (9), 1493-1504.). Moreover, this model predicted that discrete brain regions underlie these encoding processes. Using functional magnetic resonance imaging, the present study tested the predictions of this model by examining the functional neuroanatomy of symbolic and nonsymbolic number processing. Nineteen adults compared the relative numerical magnitude of symbolic and nonsymbolic stimuli. An initial conjunction analysis revealed the right inferior parietal lobule to be significantly active in both symbolic and nonsymbolic numerical comparison. A contrast of the activation associated with symbolic and nonsymbolic stimuli revealed that both the left angular and superior temporal gyri were more activated for symbolic compared to nonsymbolic numerical magnitude judgments. The reverse comparison (nonsymbolic>symbolic) revealed several regions including the right posterior superior parietal lobe. These results reveal both format-general and format-specific processing of numerical stimuli in the brain. The potential roles of these regions in symbolic and nonsymbolic numerical processing are discussed.