Notation-dependent and -independent representations of numbers in the parietal lobes

Repetition suppression in aging: A near-infrared spectroscopy study on the size-congruity effect

Age-related changes in the hemodynamic response regarding inhibition capacity and repetition suppression were examined using a modified version of the numerical Stroop task. Young (20-38 yrs; M = 28 yrs; N = 18), middle-aged (47-59 yrs; M = 52 yrs; N = 17), and older participants (60-78 yrs; M = 69 yrs; N = 19) solved a physical and numerical version of the size-congruity task, in which trials from the same experimental condition were presented in triplets. Response times revealed a strong Stroop effect in both tasks (faster reaction times during neutral than during incongruent trials) and increased with participants' age. Reaction times decreased with item repetition. In line with previous studies, the hemodynamic response (relative concentration changes in oxygenated and deoxygenated hemoglobin) assessed with near-infrared spectroscopy was comparable across incongruent and neutral trials. Strong repetition suppression of the oxygenated hemoglobin response was observed in frontal brain regions as well as in the left parietal region in all age groups. In middle and right parietal regions, repetition suppression decreased with age and was absent among older participants. These results indicate a reduced adaptation of the hemodynamic response in middle and right parietal regions of older individuals' brains in response to repeated interference control.

Primitive Concepts of Number and the Developing Human Brain

Counting is an evolutionarily recent cultural invention of the human species. In order for humans to have conceived of counting in the first place, certain representational and logical abilities must have already been in place. The focus of this review is the origins and nature of those fundamental mechanisms that promoted the emergence of the human number concept. Five claims are presented that support an evolutionary view of numerical development: 1) number is an abstract concept with an innate basis in humans, 2) maturational processes constrain the development of humans' numerical representations between infancy and adulthood, 3) there is evolutionary continuity in the neural processes of numerical cognition in primates, 4) primitive logical abilities support verbal counting development in humans, and 5) primitive neural processes provide the foundation for symbolic numerical development in the human brain. We support these claims by examining current evidence from animal cognition, child development, and human brain function. The data show that at the basis of human numerical concepts are primitive perceptual and logical mechanisms that have evolutionary homologs in other primates and form the basis of numerical development in the human brain. In the final section of the review, we discuss some hypotheses for what makes human numerical reasoning unique by drawing on evidence from human and non-human primate neuroimaging research.

Transcranial Direct Current Stimulation over the Parietal Cortex Improves Approximate Numerical Averaging

The parietal cortex has been implicated in a variety of numerosity and numerical cognition tasks and was proposed to encompass dedicated neural populations that are tuned for analogue magnitudes as well as for symbolic numerals. Nonetheless, it remains unknown whether the parietal cortex plays a role in approximate numerical averaging (rapid, yet coarse computation of numbers' mean)—a process that is fundamental to preference formation and decision-making. To causally investigate the role of the parietal cortex in numerical averaging, we have conducted a transcranial direct current stimulation (tDCS) study, in which participants were presented with rapid sequences of numbers and asked to convey their intuitive estimation of each sequence's average. During the task, the participants underwent anodal (excitatory) tDCS (or sham), applied either on a parietal or a frontal region. We found that, although participants exhibit above-chance accuracy in estimating the average of numerical sequences, they did so with higher precision under parietal stimulation. In a second experiment, we have replicated this finding and confirmed that the effect is number-specific rather than domain-general or attentional. We present a neurocomputational model postulating population-coding underlying rapid numerical averaging to account for our findings. According to this model, stimulation of the parietal cortex elevates neural activity in number-tuned dedicated detectors, leading to increase in the system's signal-to-noise level and thus resulting in more precise estimations.

Counteracting Implicit Conflicts by Electrical Inhibition of the Prefrontal Cortex

Cognitive conflicts and distractions by task-irrelevant information often counteract effective and goal-directed behaviors. In some cases, conflicting information can even emerge implicitly, without an overt distractor, by the automatic activation of mental representations. For instance, during number processing, magnitude information automatically elicits spatial associations resembling a mental number line. This spatial–numerical association of response codes (SNARC) effect can modulate cognitive-behavioral performance but is also highly flexible and context-dependent, which points toward a critical involvement of working memory functions. Transcranial direct current stimulation to the PFC, in turn, has been effective in modulating working memory-related cognitive performance. In a series of experiments, we here demonstrate that decreasing activity of the left PFC by cathodal transcranial direct current stimulation consistently and specifically eliminates implicit cognitive conflicts based on the SNARC effect, but explicit conflicts based on visuospatial distraction remain unaffected. This dissociation is polarity-specific and appears unrelated to functional magnitude processing as classified by regular numerical distance effects. These data demonstrate a causal involvement of the left PFC in implicit cognitive conflicts based on the automatic activation of spatial–numerical processing. Corroborating the critical interaction of brain stimulation and neurocognitive functions, our findings suggest that distraction from goal-directed behavior by automatic activation of implicit, task-irrelevant information can be blocked by the inhibition of prefrontal activity.

Effects of Non-Symbolic Approximate Number Practice on Symbolic Numerical Abilities in Pakistani Children

Current theories of numerical cognition posit that uniquely human symbolic number abilities connect to an early developing cognitive system for representing approximate numerical magnitudes, the approximate number system (ANS). In support of this proposal, recent laboratory-based training experiments with U.S. children show enhanced performance on symbolic addition after brief practice comparing or adding arrays of dots without counting: tasks that engage the ANS. Here we explore the nature and generality of this effect through two brief training experiments. In Experiment 1, elementary school children in Pakistan practiced either a non-symbolic numerical addition task or a line-length addition task with no numerical content, and then were tested on symbolic addition. After training, children in the numerical training group completed the symbolic addition test faster than children in the line length training group, suggesting a causal role of brief, non-symbolic numerical training on exact, symbolic addition. These findings replicate and extend the core findings of a recent U.S. laboratory-based study to non-Western children tested in a school setting, attesting to the robustness and generalizability of the observed training effects. Experiment 2 tested whether ANS training would also enhance the consistency of performance on a symbolic number line task. Over several analyses of the data there was some evidence that approximate number training enhanced symbolic number line placements relative to control conditions. Together, the findings suggest that engagement of the ANS through brief training procedures enhances children's immediate attention to number and engagement with symbolic number tasks.

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.

Neuronal foundations of human numerical representations

The human species has developed complex mathematical skills which likely emerge from a combination of multiple foundational abilities. One of them seems to be a preverbal capacity to extract and manipulate the numerosity of sets of objects which is shared with other species and in humans is thought to be integrated with symbolic knowledge to result in a more abstract representation of numerical concepts. For what concerns the functional neuroanatomy of this capacity, neuropsychology and functional imaging have localized key substrates of numerical processing in parietal and frontal cortex. However, traditional fMRI mapping relying on a simple subtraction approach to compare numerical and nonnumerical conditions is limited to tackle with sufficient precision and detail the issue of the underlying code for number, a question which more easily lends itself to investigation by methods with higher spatial resolution, such as neurophysiology. In recent years, progress has been made through the introduction of approaches sensitive to within-category discrimination in combination with fMRI (adaptation and multivariate pattern recognition), and the present review summarizes what these have revealed so far about the neural coding of individual numbers in the human brain, the format of these representations and parallels between human and monkey neurophysiology findings.

Probing the Neural Correlates of Number Processing

The cognitive and neural mechanisms that enable humans to encode and manipulate numerical information have been subject to an increasing number of experimental studies over the past 25 years or so. Here, I highlight recent findings about how numerical information is neurally coded, focusing on the theoretical implications derived from the most influential theoretical framework in numerical cognition—the Triple Code Model. At the core of this model is the assumption that bilateral parietal cortex hosts an approximate number system that codes for the cardinal value of perceived numerals. I will review studies that ask whether or not the numerical coding within this system is invariant to varying input notation, format, or modality, and whether or not the observed parietal activity is number-specific over and above the parietal involvement in response-related processes. Extant computational models of numerosity (the number of objects in a set) perception are summarized and related to empirical data from human neuroimaging and monkey neurophysiology.

Symbolic, Nonsymbolic and Conceptual: An Across-Notation Study on the Space Mapping of Numerals

Previous studies suggested that there are interconnections between two numeral modalities of symbolic notation and nonsymbolic notation (array of dots), differences and similarities of the processing, and representation of the two modalities have both been found in previous research. However, whether there are differences between the spatial representation and numeral-space mapping of the two numeral modalities of symbolic notation and nonsymbolic notation is still uninvestigated. The present study aims to examine whether there are differences between the spatial representation and numeral-space mapping of the two numeral modalities of symbolic notation and nonsymbolic notation; especially how zero, as both a symbolic magnitude numeral and a nonsymbolic conceptual numeral, mapping onto space; and if the mapping happens automatically at an early stage of the numeral information processing. Results of the two experiments demonstrate that the low-level processing of symbolic numerals including zero and nonsymbolic numerals except zero can mapping onto space, whereas the low-level processing of nonsymbolic zero as a semantic conceptual numeral cannot mapping onto space, which indicating the specialty of zero in the numeral domain. The present study indicates that the processing of non-semantic numerals can mapping onto space, whereas semantic conceptual numerals cannot mapping onto space.

Bidirectional Modulation of Numerical Magnitude

Numerical cognition is critical for modern life; however, the precise neural mechanisms underpinning numerical magnitude allocation in humans remain obscure. Based upon previous reports demonstrating the close behavioral and neuro-anatomical relationship between number allocation and spatial attention, we hypothesized that these systems would be subject to similar control mechanisms, namely dynamic interhemispheric competition. We employed a physiological paradigm, combining visual and vestibular stimulation, to induce interhemispheric conflict and subsequent unihemispheric inhibition, as confirmed by transcranial direct current stimulation (tDCS). This allowed us to demonstrate the first systematic bidirectional modulation of numerical magnitude toward either higher or lower numbers, independently of either eye movements or spatial attention mediated biases. We incorporated both our findings and those from the most widely accepted theoretical framework for numerical cognition to present a novel unifying computational model that describes how numerical magnitude allocation is subject to dynamic interhemispheric competition. That is, numerical allocation is continually updated in a contextual manner based upon relative magnitude, with the right hemisphere responsible for smaller magnitudes and the left hemisphere for larger magnitudes.

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.

Evidence for distinct magnitude systems for symbolic and non-symbolic number

Cognitive models of magnitude representation are mostly based on the results of studies that use a magnitude comparison task. These studies show similar distance or ratio effects in symbolic (Arabic numerals) and non-symbolic (dot arrays) variants of the comparison task, suggesting a common abstract magnitude representation system for processing both symbolic and non-symbolic numerosities. Recently, however, it has been questioned whether the comparison task really indexes a magnitude representation. Alternatively, it has been hypothesized that there might be different representations of magnitude: an exact representation for symbolic magnitudes and an approximate representation for non-symbolic numerosities. To address the question whether distinct magnitude systems exist, we used an audio-visual matching paradigm in two experiments to explore the relationship between symbolic and non-symbolic magnitude processing. In Experiment 1, participants had to match visually and auditory presented numerical stimuli in different formats (digits, number words, dot arrays, tone sequences). In Experiment 2, they were instructed only to match the stimuli after processing the magnitude first. The data of our experiments show different results for non-symbolic and symbolic number and are difficult to reconcile with the existence of one abstract magnitude representation. Rather, they suggest the existence of two different systems for processing magnitude, i.e., an exact symbolic system next to an approximate non-symbolic system.

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.

Neural foundations and functional specificity of number representations

Number is a complex category, as with the word "number" we may refer to different entities. First, it is a perceptual property that characterizes any set of individual items, namely its cardinality. The ability to extract the (approximate) cardinality of sets is almost universal in the animal domain and present in humans since birth. In primates, posterior parietal cortex seems to be a crucial site for this ability, even if the degree of selectivity of numerical representations in parietal cortex reported to date appears much lower compared to that of other semantic categories in the ventral stream. Number can also be intended as a mathematical object, which we humans use to count, measure, and order: a (verbal or visual) symbol that stands for the cardinality of a set, the intensity of a continuous quantity or the position of an item on a list. Evidence points to a convergence towards parietal cortex for the semantic coding of numerical symbols and to the bilateral occipitotemporal cortex for the shape coding of Arabic digits and other number symbols.

Is 9 louder than 1? Audiovisual cross-modal interactions between number magnitude and judged sound loudness

The cross-modal impact of number magnitude (i.e. Arabic digits) on perceived sound loudness was examined. Participants compared a target sound's intensity level against a previously heard reference sound (which they judged as quieter or louder). Paired with each target sound was a task irrelevant Arabic digit that varied in magnitude, being either small (1, 2, 3) or large (7, 8, 9). The degree to which the sound and the digit were synchronized was manipulated, with the digit and sound occurring simultaneously in Experiment 1, and the digit preceding the sound in Experiment 2. Firstly, when target sounds and digits occurred simultaneously, sounds paired with large digits were categorized as loud more frequently than sounds paired with small digits. Secondly, when the events were separated, number magnitude ceased to bias sound intensity judgments. In Experiment 3, the events were still separated, however the participants held the number in short-term memory. In this instance the bias returned.

Numerical magnitude processing in abacus-trained children with superior mathematical ability: an EEG study

Distance effect has been regarded as the best established marker of basic numerical magnitude processes and is related to individual mathematical abilities. A larger behavioral distance effect is suggested to be concomitant with lower mathematical achievement in children. However, the relationship between distance effect and superior mathematical abilities is unclear. One could get superior mathematical abilities by acquiring the skill of abacus-based mental calculation (AMC), which can be used to solve calculation problems with exceptional speed and high accuracy. In the current study, we explore the relationship between distance effect and superior mathematical abilities by examining whether and how the AMC training modifies numerical magnitude processing. Thus, mathematical competencies were tested in 18 abacus-trained children (who accepted the AMC training) and 18 non-trained children. Electroencephalography (EEG) waveforms were recorded when these children executed numerical comparison tasks in both Arabic digit and dot array forms. We found that: (a) the abacus-trained group had superior mathematical abilities than their peers; (b) distance effects were found both in behavioral results and on EEG waveforms; (c) the distance effect size of the average amplitude on the late negative-going component was different between groups in the digit task, with a larger effect size for abacus-trained children; (d) both the behavioral and EEG distance effects were modulated by the notation. These results revealed that the neural substrates of magnitude processing were modified by AMC training, and suggested that the mechanism of the representation of numerical magnitude for children with superior mathematical abilities was different from their peers. In addition, the results provide evidence for a view of non-abstract numerical representation.

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.

Link between cognitive neuroscience and education: the case of clinical assessment of developmental dyscalculia

In recent years, cognitive neuroscience research has identified several biological and cognitive features of number processing deficits that may now make it possible to diagnose mental or educational impairments in arithmetic, even earlier and more precisely than is possible using traditional assessment tools. We provide two sets of recommendations for improving cognitive assessment tools, using the important case of mathematics as an example. (1) neurocognitive tests would benefit substantially from incorporating assessments (based on findings from cognitive neuroscience) that entail systematic manipulation of fundamental aspects of number processing. Tests that focus on evaluating networks of core neurocognitive deficits have considerable potential to lead to more precise diagnosis and to provide the basis for designing specific intervention programs tailored to the deficits exhibited by the individual child. (2) implicit knowledge, derived from inspection of variables that are irrelevant to the task at hand, can also provide a useful assessment tool. Implicit knowledge is powerful and plays an important role in human development, especially in cases of psychiatric or neurological deficiencies (such as math learning disabilities or math anxiety).

Common magnitude representation of fractions and decimals is task dependent

Although several studies have compared the representation of fractions and decimals, no study has investigated whether fractions and decimals, as two types of rational numbers, share a common representation of magnitude. The current study aimed to answer the question of whether fractions and decimals share a common representation of magnitude and whether the answer is influenced by task paradigms. We included two different number pairs, which were presented sequentially: fraction-decimal mixed pairs and decimal-fraction mixed pairs in all four experiments. Results showed that when the mixed pairs were very close numerically with the distance 0.1 or 0.3, there was a significant distance effect in the comparison task but not in the matching task. However, when the mixed pairs were further apart numerically with the distance 0.3 or 1.3, the distance effect appeared in the matching task regardless of the specific stimuli. We conclude that magnitudes of fractions and decimals can be represented in a common manner, but how they are represented is dependent on the given task. Fractions and decimals could be translated into a common representation of magnitude in the numerical comparison task. In the numerical matching task, fractions and decimals also shared a common representation. However, both of them were represented coarsely, leading to a weak distance effect. Specifically, fractions and decimals produced a significant distance effect only when the numerical distance was larger.

Musicians and music making as a model for the study of brain plasticity

Playing a musical instrument is an intense, multisensory, and motor experience that usually commences at an early age and requires the acquisition and maintenance of a range of sensory and motor skills over the course of a musician's lifetime. Thus, musicians offer an excellent human model for studying behavioral-cognitive as well as brain effects of acquiring, practicing, and maintaining these specialized skills. Research has shown that repeatedly practicing the association of motor actions with specific sound and visual patterns (musical notation), while receiving continuous multisensory feedback will strengthen connections between auditory and motor regions (e.g., arcuate fasciculus) as well as multimodal integration regions. Plasticity in this network may explain some of the sensorimotor and cognitive enhancements that have been associated with music training. Furthermore, the plasticity of this system as a result of long term and intense interventions suggest the potential for music making activities (e.g., forms of singing) as an intervention for neurological and developmental disorders to learn and relearn associations between auditory and motor functions such as vocal motor functions.

Implicit interactions between number and space in digit-color synesthesia

In digit-color synesthesia, a variant of grapheme-color synesthesia, digits trigger an additional color percept. Recent work on number processing in synesthesia suggests that colors can implicitly elicit numerical representations in digit-color synesthetes implying that synesthesia is bidirectional. Furthermore, morphometric investigations revealed structural differences in the parietal cortex of grapheme-color synesthetes, i.e., in the brain region where interactions between number and space occur in non-synesthetic subjects. Based upon these previous findings, we here examined whether implicitly evoked numerical representations interact with spatial representations in synesthesia in such a way that even a non-numerical, visuo-spatial task (here: line bisection) is modulated, i.e., whether synesthetes exhibit a systematic bisection bias for colored lines. Thirteen digit-color synesthetes were asked to bisect two sets of lines which were colored in their individual synesthetic colors associated with a small or a large digit, respectively. For all colored line stimuli combined, digit-color synesthetes showed--like control subjects (n = 13, matched for age, gender, IQ and handedness)--a pseudo-neglect when bisecting colored lines. Measuring the color-induced change of the bisection bias (i.e., comparing the biases when bisecting lines colored according to a small number vs those lines corresponding to a large number) revealed that only digit-color synesthetes were significantly influenced by line color. The results provide further evidence for the bidirectional nature of synesthesia and support the concept of a mental number line. In addition, they extend previous reports on bidirectionality in synesthesia by showing that even non-numerical, visuo-spatial performance can be modulated by implicit bidirectional processes.

Development of brain systems for nonsymbolic numerosity and the relationship to formal math academic achievement

A central question in cognitive and educational neuroscience is whether brain operations supporting nonlinguistic intuitive number sense (numerosity) predict individual acquisition and academic achievement for symbolic or "formal" math knowledge. Here, we conducted a developmental functional magnetic resonance imaging (MRI) study of nonsymbolic numerosity task performance in 44 participants including 14 school age children (6-12 years old), 14 adolescents (13-17 years old), and 16 adults and compared a brain activity measure of numerosity precision to scores from the Woodcock-Johnson III Broad Math index of math academic achievement. Accuracy and reaction time from the numerosity task did not reliably predict formal math achievement. We found a significant positive developmental trend for improved numerosity precision in the parietal cortex and intraparietal sulcus specifically. Controlling for age and overall cognitive ability, we found a reliable positive relationship between individual math achievement scores and parietal lobe activity only in children. In addition, children showed robust positive relationships between math achievement and numerosity precision within ventral stream processing areas bilaterally. The pattern of results suggests a dynamic developmental trajectory for visual discrimination strategies that predict the acquisition of formal math knowledge. In adults, the efficiency of visual discrimination marked by numerosity acuity in ventral occipital-temporal cortex and hippocampus differentiated individuals with better or worse formal math achievement, respectively. Overall, these results suggest that two different brain systems for nonsymbolic numerosity acuity may contribute to individual differences in math achievement and that the contribution of these systems differs across development.

Continuity and change in children's longitudinal neural responses to numbers

Human children possess the ability to approximate numerical quantity nonverbally from a young age. Over the course of early childhood, children develop increasingly precise representations of numerical values, including a symbolic number system that allows them to conceive of numerical information as Arabic numerals or number words. Functional brain imaging studies of adults report that activity in bilateral regions of the intraparietal sulcus (IPS) represents a key neural correlate of numerical cognition. Developmental neuroimaging studies indicate that the right IPS develops its number-related neural response profile more rapidly than the left IPS during early childhood. One prediction that can be derived from previous findings is that there is longitudinal continuity in the number-related neural responses of the right IPS over development while the development of the left IPS depends on the acquisition of numerical skills. We tested this hypothesis using fMRI in a longitudinal design with children ages 4 to 9. We found that neural responses in the right IPS are correlated over a 1-2-year period in young children whereas left IPS responses change systematically as a function of children's numerical discrimination acuity. The data are consistent with the hypothesis that functional properties of the right IPS in numerical processing are stable over early childhood whereas the functions of the left IPS are dynamically modulated by the development of numerical skills.

How number-space relationships are assessed before formal schooling: A taxonomy proposal

The last years of research on numerical development have provided evidence that spatial-numerical associations (SNA) can be formed independent of formal school training. However, most of these studies used various experimental paradigms that referred to slightly different aspects of number and space processing. This poses a question of whether all SNAs described in the developmental literature can be interpreted as a unitary construct, or whether they are rather examples of different, but related phenomena. Our review aims to provide a starting point for a systematic classification of SNA measures used from infancy to late preschool years, and their underlying representations. We propose to distinguish among four basic SNA categories: (i) cross-dimensional magnitude processing, (ii) associations between spatial and numerical intervals, (iii) associations between cardinalities and spatial directions, (iv) associations between ordinalities and spatial directions. Such systematization allows for identifying similarities and differences between processes and representations that underlie the described measures, and also for assessing the adequacy of using different SNA tasks at different developmental stages.

The Neural Mechanism Underlying Ordinal Numerosity Processing

Changes in the sensory properties of numerosity stimuli have a direct effect on the outcomes of nonsymbolic number tasks. This suggests a prominent role of sensory properties in numerosity processing. However, the current consensus holds that numerosity is processed independent of its sensory properties. To investigate the role of sensory cues in ordinal number processes, we manipulated both dimensions orthogonally. Participants passively viewed the stimuli while their brain activity was measured using EEG. The results revealed an interaction between numerosity and its sensory properties in the absence of main effects. Different neural responses were present for trials where numerosity and sensory cues changed in the same direction compared with trials where they changed in opposite directions. These results show that the sensory cues are expected to change in concert with numerosity and support the notion that the visual cues are taken into account when judging numerosity.

Number Representation: A Question of Look? The Distance Effect in Comparison of English and Turkish Number Words

Cohen Kadosh found that Hebrew number words and Arabic digits elicited different distance effects. The numerical distance effect (i.e., faster reaction times and smaller error rates for numbers with larger distance to a standard number, e.g., 5) was smaller for number words than for digits. The author suggested that the result indicated a notation-dependent (i.e., nonabstract) numerical representation. Alternatively, however, it is possible that the distance effect was decreased by perceptual features of the Hebrew number words used as stimuli. To test this hypothesis, we replicated the study of Cohen Kadosh, but with English and Turkish number words compared to Arabic digits. The perceptual features of Turkish number words lead to the hypothesis that the opposite effect should be present: an increased distance effect for number words compared to Arabic digits. On the other hand, English number words show features similar to those of Hebrew number words, thus leading to the expectation of a decreased distance effect for them. Our results confirmed our hypotheses. We conclude that the distance effect can be influenced also by perceptual features in addition to the impact of numerical representations.

Functional clustering of the human inferior parietal lobule by whole-brain connectivity mapping of resting-state functional magnetic resonance imaging signals

The human inferior parietal lobule (IPL) comprised the lateral bank of the intraparietal sulcus, angular gyrus, and supramarginal gyrus, defined on the basis of anatomical landmarks and cytoarchitectural organization of neurons. However, it is not clear as to whether the three areas represent functional subregions within the IPL. For instance, imaging studies frequently identified clusters of activities that cut across areal boundaries. Here, we used resting-state functional magnetic resonance imaging (fMRI) data to examine how individual voxels within the IPL are best clustered according to their connectivity to the whole brain. The results identified a best estimate of seven clusters that are hierarchically arranged as the anterior, middle, and posterior subregions. The anterior, middle, and posterior IPL are each significantly connected to the somatomotor areas, superior/middle/inferior frontal gyri, and regions of the default mode network. This functional segregation is supported by recent cytoarchitechtonics and tractography studies. IPL showed hemispheric differences in connectivity that accord with a predominantly left parietal role in tool use and language processing and a right parietal role in spatial attention and mathematical cognition. The functional clusters may also provide a more parsimonious and perhaps even accurate account of regional activations of the IPL during a variety of cognitive challenges, as reported in earlier fMRI studies.

Common and dissociated mechanisms for estimating large and small dot arrays: value-specific fMRI adaptation

An fMRI pair-adaptation paradigm was used to identify the brain regions linked to the apprehension of small and large numbers of items. Participants classified stimuli on the basis of their numerosities (fewer or more than five dots). We manipulated the type of repetition within pairs of dot arrays. Overall processing of pairs with small as opposed to large quantities was associated with a decreased BOLD response in the midline structures and inferior parietal cortex. The opposite pattern was observed in middle cingulate cortex. Pairs in which the same numerosity category was repeated, were associated with a decreased signal in the left prefrontal and the left inferior parietal cortices, compared with when numerosities changed. Repetitions of exact numerosities irrespective of sample size were associated with decreased responses in bi-lateral prefrontal, sensory-motor regions, posterior occipital and left intraparietal sulcus (IPS). More importantly, we found value-specific adaptation specific to repeated small quantity in the left lateral occipito-temporal cortex, irrespective of whether the exact same stimulus pattern repeated. Our results indicate that a large network of regions (including the IPS) support visual quantity processing independent of the number of items present; however assimilation of small quantities is associated with additional support from regions within the left occipito-temporal cortex. We propose that processing of small quantities is aided by a subitizing-specific network. This network may account for the increased processing efficiency often reported for numerosities in the subitizing range.

Brain mechanisms underlying the impact of attachment-related stress on social cognition

Mentalizing, in particular the successful attribution of complex mental states to others, is crucial for navigating social interactions. This ability is highly influenced by external factors within one's daily life, such as stress. We investigated the impact of stress on the brain basis of mentalization in adults. Using a novel modification of the Reading the Mind in the Eyes Test (RMET-R) we compared the differential effects of two personalized stress induction procedures: a general stress induction (GSI) and an attachment-related stress induction (ASI). Participants performed the RMET-R at baseline and after each of the two inductions. Baseline results replicated and extended previous findings regarding the neural correlates of the RMET-R. Additionally, we identified brain regions associated with making complex age judgments from the same stimuli. Results after stress exposure showed that the ASI condition resulted in reduced mentalization-related activation in the left posterior superior temporal sulcus (STS), left inferior frontal gyrus and left temporoparietal junction (TPJ). Moreover, the left middle frontal gyrus and left anterior insula showed greater functional connectivity to the left posterior STS after the ASI. Our findings indicate that attachment-related stress has a unique effect on the neural correlates of mentalization.

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.

Task- and experience-dependent cortical selectivity to features informative for categorization

In this study, we bridge the gap between monkey electrophysiological recordings that showed selective responses to informative features and human fMRI data that demonstrated increased and selective responses to trained objects. Human participants trained with computer-generated fish stimuli. For each participant, two features of the fish were informative for category membership and two features were uninformative. After training, participants showed higher perceptual sensitivity to the informative dimensions. An fMRI adaptation paradigm revealed that during categorization the right inferior frontal gyrus and occipitotemporal cortex were selectively responsive to the informative features. These selective cortical responses were experience dependent; they were not present for the entire trained object, but specific for those features that were informative for categorization. Responses in the inferior frontal gyrus showed category selectivity. Moreover, selectivity to the informative features correlated with performance on the categorization task during scanning. This all suggests that the frontal cortex is involved in actively categorizing objects and that it uses informative features to do so while ignoring those features that do not contribute category information. Occipitotemporal cortex also showed selectivity to the informative features during the categorization task. Interestingly, this area showed a positive correlation of performance during training and selectivity to the informative features and a negative correlation with selectivity to the uninformative features. This indicates that training enhanced sensitivity to trained items and decreased sensitivity to uninformative features. The absence of sensitivity for informative features during a color change detection task indicates that there is a strong component of task-related processing of these features.

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.

Bilateral bi-cephalic tDCS with two active electrodes of the same polarity modulates bilateral cognitive processes differentially [corrected]

Transcranial direct current stimulation (tDCS) is an innovative method to explore the causal structure-function relationship of brain areas. We investigated the specificity of bilateral bi-cephalic tDCS with two active electrodes of the same polarity (e.g., cathodal on both hemispheres) applied to intraparietal cortices bilaterally using a combined between- and within-task approach. Regarding between-task specificity, we observed that bilateral bi-cephalic tDCS affected a numerical (mental addition) but not a control task (colour word Stroop), indicating a specific influence of tDCS on numerical but not on domain general cognitive processes associated with the bilateral IPS. In particular, the numerical effect of distractor distance was more pronounced under cathodal than under anodal stimulation. Moreover, with respect to within-task specificity we only found the numerical distractor distance effect in mental addition to be modulated by direct current stimulation, whereas the effect of target identity was not affected. This implies a differential influence of bilateral bi-cephalic tDCS on the recruitment of different processing components within the same task (number magnitude processing vs. recognition of familiarity). In sum, this first successful application of bilateral bi-cephalic tDCS with two active electrodes of the same polarity in numerical cognition research corroborates the specific proposition of the Triple Code Model that number magnitude information is represented bilaterally in the intraparietal cortices.

Design and evaluation of the computer-based training program Calcularis for enhancing numerical cognition

This article presents the design and a first pilot evaluation of the computer-based training program Calcularis for children with developmental dyscalculia (DD) or difficulties in learning mathematics. The program has been designed according to insights on the typical and atypical development of mathematical abilities. The learning process is supported through multimodal cues, which encode different properties of numbers. To offer optimal learning conditions, a user model completes the program and allows flexible adaptation to a child's individual learning and knowledge profile. Thirty-two children with difficulties in learning mathematics completed the 6–12-weeks computer training. The children played the game for 20 min per day for 5 days a week. The training effects were evaluated using neuropsychological tests. Generally, children benefited significantly from the training regarding number representation and arithmetic operations. Furthermore, children liked to play with the program and reported that the training improved their mathematical abilities.

Cross-format physical similarity effects and their implications for the numerical cognition architecture

The sound |faɪv| is visually depicted as a written number word "five" and as an Arabic digit "5." Here, we present four experiments--two quantity same/different experiments and two magnitude comparison experiments--that assess whether auditory number words (|faɪv|), written number words ("five"), and Arabic digits ("5") directly activate one another and/or their associated quantity. The quantity same/different experiments reveal that the auditory number words, written number words, and Arabic digits directly activate one another without activating their associated quantity. That is, there are cross-format physical similarity effects but no numerical distance effects. The cross-format magnitude comparison experiments reveal significant effects of both physical similarity and numerical distance. We discuss these results in relation to the architecture of numerical cognition.

Left parietal TMS disturbs priming between symbolic and non-symbolic number representations

An amodal number representation activated by all types of numerical input, irrespective of the input notation, has often been proposed to be located in the left or right intraparietal sulcus (IPS). Two cross-notational priming experiments were carried out to test the existence of a notation-independent magnitude representation in the left or right parietal lobes. In Experiment 1, stimuli were Arabic digits and number words. Results revealed no significant effect of repetitive transcranial magnetic stimulation (rTMS) over left or right IPS during prime presentation. In contrast, in Experiment 2, digits and dot patterns were intermixed and here the priming distance effect (PDE) was reduced in the right TMS condition and absent for stimulation over left IPS. These findings suggest: (1) that TMS over left but not right IPS disrupts processes that are crucial for priming when symbolic and non-symbolic stimuli are intermixed, and (2) that disruption of the left IPS on its own is not sufficient to disrupt cross-notational priming when purely symbolic number notations are used. Our results point towards a crucial role of the left hemisphere for the mapping between small symbolic and non-symbolic numerosities.

Effects of number magnitude and notation at 7T: separating the neural response to small and large, symbolic and nonsymbolic number

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

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.

Semantic and Perceptual Processing of Number Symbols: Evidence from a Cross-linguistic fMRI Adaptation Study

The ability to process the numerical magnitude of sets of items has been characterized in many animal species. Neuroimaging data have associated this ability to represent nonsymbolic numerical magnitudes (e.g., arrays of dots) with activity in the bilateral parietal lobes. Yet the quantitative abilities of humans are not limited to processing the numerical magnitude of nonsymbolic sets. Humans have used this quantitative sense as the foundation for symbolic systems for the representation of numerical magnitude. Although numerical symbol use is widespread in human cultures, the brain regions involved in processing of numerical symbols are just beginning to be understood. Here, we investigated the brain regions underlying the semantic and perceptual processing of numerical symbols. Specifically, we used an fMRI adaptation paradigm to examine the neural response to Hindu-Arabic numerals and Chinese numerical ideographs in a group of Chinese readers who could read both symbol types and a control group who could read only the numerals. Across groups, the Hindu-Arabic numerals exhibited ratio-dependent modulation in the left IPS. In contrast, numerical ideographs were associated with activation in the right IPS, exclusively in the Chinese readers. Furthermore, processing of the visual similarity of both digits and ideographs was associated with activation of the left fusiform gyrus. Using culture as an independent variable, we provide clear evidence for differences in the brain regions associated with the semantic and perceptual processing of numerical symbols. Additionally, we reveal a striking difference in the laterality of parietal activation between the semantic processing of the two symbols types.

Why overlearned sequences are special: distinct neural networks for ordinal sequences

Several observations suggest that overlearned ordinal categories (e.g., letters, numbers, weekdays, months) are processed differently than non-ordinal categories in the brain. In synesthesia, for example, anomalous perceptual experiences are most often triggered by members of ordinal categories (Rich et al., 2005; Eagleman, 2009). In semantic dementia (SD), the processing of ordinal stimuli appears to be preserved relative to non-ordinal ones (Cappelletti et al., 2001). Moreover, ordinal stimuli often map onto unconscious spatial representations, as observed in the SNARC effect (Dehaene et al., 1993; Fias, 1996). At present, little is known about the neural representation of ordinal categories. Using functional neuroimaging, we show that words in ordinal categories are processed in a fronto-temporo-parietal network biased toward the right hemisphere. This differs from words in non-ordinal categories (such as names of furniture, animals, cars, and fruit), which show an expected bias toward the left hemisphere. Further, we find that increased predictability of stimulus order correlates with smaller regions of BOLD activation, a phenomenon we term prediction suppression. Our results provide new insights into the processing of ordinal stimuli, and suggest a new anatomical framework for understanding the patterns seen in synesthesia, unconscious spatial representation, and SD.

Is order the defining feature of magnitude representation? An ERP study on learning numerical magnitude and spatial order of artificial symbols

Using an artificial-number learning paradigm and the ERP technique, the present study investigated neural mechanisms involved in the learning of magnitude and spatial order. 54 college students were divided into 2 groups matched in age, gender, and school major. One group was asked to learn the associations between magnitude (dot patterns) and the meaningless Gibson symbols, and the other group learned the associations between spatial order (horizontal positions on the screen) and the same set of symbols. Results revealed differentiated neural mechanisms underlying the learning processes of symbolic magnitude and spatial order. Compared to magnitude learning, spatial-order learning showed a later and reversed distance effect. Furthermore, an analysis of the order-priming effect showed that order was not inherent to the learning of magnitude. Results of this study showed a dissociation between magnitude and order, which supports the numerosity code hypothesis of mental representations of magnitude.

Dissociated neural correlates of quantity processing of quantifiers, numbers, and numerosities

Quantities can be represented using either mathematical language (i.e., numbers) or natural language (i.e., quantifiers). Previous studies have shown that numerical processing elicits greater activation in the brain regions around the intraparietal sulcus (IPS) relative to other semantic processes. However, little research has been conducted to investigate whether the IPS is also critical for the semantic processing of quantifiers in natural language. In this study, 20 adults were scanned with functional magnetic resonance imaging while they performed semantic distance judgment involving six types of materials (i.e., frequency adverbs, quantity pronouns and nouns, animal names, Arabic digits, number words, and dot arrays). Conjunction analyses of brain activation showed that numbers and dot arrays elicited greater activation in the right IPS than did words (i.e., animal names) or quantifiers (i.e., frequency adverbs and quantity pronouns and nouns). Quantifiers elicited more activation in left middle temporal gyrus and inferior frontal gyrus than did numbers and dot arrays. No differences were found between quantifiers and animal names. These findings suggest that, although quantity processing for numbers and dot arrays typically relies on the right IPS region, quantity processing for quantifiers typically relies on brain regions for general semantic processing. Thus, the IPS does not appear to be the only brain region for quantity processing.