Numerical magnitude in the human parietal lobe; tests of representational generality and domain specificity

Behavioral evidence suggests that human adults have a single system for representing the numerical magnitude of both symbolic numbers (e.g., Arabic digits) and nonsymbolic number stimuli (e.g., dot arrays). Brain imaging studies have implicated a specific parietal region in symbolic number processing, leading to the influential hypothesis that this region is the locus of a dedicated, domain-specific number system. Here we evaluated a prediction of this hypothesis, that this region should be activated not only by symbolic but also nonsymbolic number processing. Using nonsymbolic stimuli, we tested for higher parietal activations for number than for nonnumber comparison tasks (experiment 1), fMRI adaptation for numerosity repetition (experiment 2), and greater fMRI increases with increasing task difficulty for number than nonnumber tasks (experiment 3). None of these predictions were supported by the data, posing a serious challenge to the hypothesis that a single, domain-specific parietal region underlies both symbolic and nonsymbolic number representation.     

Arithmetic Memory Is Modality Specific

In regards to numerical cognition and working memory, it is an open question as to whether numbers are stored into and retrieved from a central abstract representation or from separate notation-specific representations. This study seeks to help answer this by utilizing the numeral modality effect (NME) in three experiments to explore how numbers are processed by the human brain. The participants were presented with numbers (1–9) as either Arabic digits or written number words (Arabic digits and dot matrices in Experiment 2) at the first (S1) and second (S2) stimuli. The participant’s task was to add the first two stimuli together and verify whether the answer (S3), presented simultaneously with S2, was correct. We hypothesized that if reaction time (RT) at S2/S3 depends on the modality of S1 then numbers are retrieved from modality specific memory stores. Indeed, RT depended on the modality of S1 whenever S2 was an Arabic digit which argues against the concept of numbers being stored and retrieved from a central, abstract representation.     

Mental representation of number in different numerical forms

How are numerical operations implemented within the human brain? It has been suggested that there are at least three different codes for representing number: a verbal code that is used to manipulate number words and perform mental numerical operations (e.g., multiplication), a visual code that is used to decode frequently used visual number forms (e.g., Arabic digits), and an abstract analog code that may be used to represent numerical quantities. Furthermore, each of these codes is associated with a different neural substrate. We extend these studies using dense-sensor event-related EEG recording techniques to investigate the temporal pattern of notation-specific effects observed in a parity judgement (odd versus even) task in which single numbers were presented in one of four different numerical notations. Contrasts between different notations demonstrated clear modulations in the visual evoked potentials (VEP) recorded. We observed increased amplitudes for the P1 and N1 components of the VEP that were specific to Arabic numerals and to dot configurations but differed for random and recognizable (die-face) dot configurations. These results demonstrate clear, notation-specific differences in the time course of numerical information processing and provide electrophysiological support for the triple-code model of numerical representation.     

The Mediating Relation between Symbolic and Nonsymbolic Foundations of Math Competence

This study investigated the relation between symbolic and nonsymbolic magnitude processing abilities with 2 standardized measures of math competence (WRAT Arithmetic and KeyMath Numeration) in 150 3^rd- grade children (mean age 9.01 years). Participants compared sets of dots and pairs of Arabic digits with numerosities 1–9 for relative numerical magnitude. In line with previous studies, performance on both symbolic and nonsymbolic magnitude processing was related to math ability. Performance metrics combining reaction and accuracy, as well as weber fractions, were entered into mediation models with standardized math test scores. Results showed that symbolic magnitude processing ability fully mediates the relation between nonsymbolic magnitude processing and math ability, regardless of the performance metric or standardized test.     

A Two-Minute Paper-and-Pencil Test of Symbolic and Nonsymbolic Numerical Magnitude Processing Explains Variability in Primary School Children's Arithmetic Competence

Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children’s ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children’s performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children’s ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1–3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics.     

cDNA cloning and gene expression analysis of human myo-inositol 1-phosphate synthase

myo-Inositol 1-phosphate synthase (EC 5.5.1.4) (IPS) is a key enzyme in myo-inositol biosynthesis pathway. This study describes the molecular cloning of the full length human myo-inositol 1-phosphate synthase (hIPS) cDNA, tissue distribution of its mRNA and characterizes its gene expression in cultured HepG2 cells. Human testis, ovary, heart, placenta, and pancreas express relatively high level of hIPS mRNA, while blood leukocyte, thymus, skeletal muscle, and colon express low or marginal amount of the mRNA. In the presence of glucose, hIPS mRNA level increases 2- to 4-fold in HepG2 cells. hIPS mRNA is also up-regulated 2- to 3-fold by 2.5 microM lovastain. This up-regulation is prevented by mevalonic acid, farnesol, and geranylgeraniol, suggesting a G-protein mediated signal transduction mechanism in the regulation of hIPS gene expression. hIPS mRNA expression is 50% suppressed by 10mM lithium ion in these cells. Neither 5mM myo-inositol nor the three hormones: estrogen, thyroid hormone, and insulin altered hIPS mRNA expression in these cells.     

The Processing of Symbolic and Nonsymbolic Ratios in School-Age Children

This study tested the processing of ratios of natural numbers in school-age children. Nine- and eleven-year-olds were presented collections made up of orange and grey dots (i.e., nonsymbolic format) and fractions (i.e., symbolic format). They were asked to estimate ratios between the number of orange dots and the total number of dots and fractions by producing an equivalent ratio of surface areas (filling up a virtual glass). First, we tested whether symbolic notation of ratios affects their processing by directly comparing performance on fractions with that on dot sets. Second, we investigated whether children’s estimates of nonsymbolic ratios of natural numbers relied at least in part on ratios of surface areas by contrasting a condition in which the ratio of surface areas occupied by dots covaried with the ratio of natural numbers and a condition in which this ratio of surface areas was kept constant across ratios of natural numbers. The results showed that symbolic notation did not really have a negative impact on performance among 9-year-olds, while it led to more accurate estimates in 11-year-olds. Furthermore, in dot conditions, children’s estimates increased consistently with ratios between the number of orange dots and the total number of dots even when the ratio of surface areas was kept constant but were less accurate in that condition than when the ratio of surface areas covaried with the ratio of natural numbers. In summary, these results indicate that mental magnitude representation is more accurate when it is activated from symbolic ratios in children as young as 11 years old and that school-age children rely at least in part on ratios of surface areas to process nonsymbolic ratios of natural numbers when given the opportunity to do so.     

An ERP Study of the Processing of Common and Decimal Fractions: How Different They Are

This study explored event-related potential (ERP) correlates of common fractions (1/5) and decimal fractions (0.2). Thirteen subjects performed a numerical magnitude matching task under two conditions. In the common fraction condition, a nonsymbolic fraction was asked to be judged whether its magnitude matched the magnitude of a common fraction; in the decimal fraction condition, a nonsymbolic fraction was asked to be matched with a decimal fraction. Behavioral results showed significant main effects of condition and numerical distance, but no significant interaction of condition and numerical distance. Electrophysiological data showed that when nonsymbolic fractions were compared to common fractions, they displayed larger N1 and P3 amplitudes than when they were compared to decimal fractions. This finding suggested that the visual identification for nonsymbolic fractions was different under the two conditions, which was not due to perceptual differences but to task demands. For symbolic fractions, the condition effect was observed in the N1 and P3 components, revealing stimulus-specific visual identification processing. The effect of numerical distance as an index of numerical magnitude representation was observed in the P2, N3 and P3 components under the two conditions. However, the topography of the distance effect was different under the two conditions, suggesting stimulus specific semantic processing of common fractions and decimal fractions.     

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.     

Letter Position Coding Across Modalities: The Case of Braille Readers

Background

The question of how the brain encodes letter position in written words has attracted increasing attention in recent years. A number of models have recently been proposed to accommodate the fact that transposed-letter stimuli like jugde or caniso are perceptually very close to their base words.

Methodology

Here we examined how letter position coding is attained in the tactile modality via Braille reading. The idea is that Braille word recognition may provide more serial processing than the visual modality, and this may produce differences in the input coding schemes employed to encode letters in written words. To that end, we conducted a lexical decision experiment with adult Braille readers in which the pseudowords were created by transposing/replacing two letters.

Principal Findings

We found a word-frequency effect for words. In addition, unlike parallel experiments in the visual modality, we failed to find any clear signs of transposed-letter confusability effects. This dissociation highlights the differences between modalities.

Conclusions

The present data argue against models of letter position coding that assume that transposed-letter effects (in the visual modality) occur at a relatively late, abstract locus.     

The number domain- can we count on parietal cortex?

Does the primate brain contain a dedicated and localized neural circuitry for processing generic numerical information? The human parietal cortex, particularly the intraparietal sulcus (IPS), has long been implicated in processing symbolic (verbal) number information. If the IPS is indeed the site of generic numerical processing, however, its neurons should also encode nonsymbolic numerosity information. Two recent studies by Shuman and Kanwisher and by Piazza et al. published in this issue of Neuron tested this assumption...with quite different results.     

Relative numerical middle in rhesus monkeys

Animals show vast numerical competence in tasks that require both ordinal and cardinal numerical representations, but few studies have addressed whether animals can identify the numerical middle in a sequence. Two rhesus monkeys (Macaca mulatta) learned to select the middle dot in a horizontal sequence of three dots on a touchscreen. When subsequently presented with longer sequences composed of 5, 7 or 9 items, monkeys transferred the middle rule. Accuracy decreased as the length of the sequence increased. In a second test, we presented monkeys with asymmetrical sequences composed of nine items, where the numerical and spatial middle were distinct and both monkeys selected the numerical middle over the spatial middle. Our results demonstrate that rhesus macaques can extract an abstract numerical rule to bisect a discrete set of items.     

Children’s Mapping between Non-Symbolic and Symbolic Numerical Magnitudes and Its Association with Timed and Untimed Tests of Mathematics Achievement

The ability to map between non-symbolic numerical magnitudes and Arabic numerals has been put forward as a key factor in children’s mathematical development. This mapping ability has been mainly examined indirectly by looking at children’s performance on a symbolic magnitude comparison task. The present study investigated mapping in a more direct way by using a task in which children had to choose which of two choice quantities (Arabic digits or dot arrays) matched the target quantity (dot array or Arabic digit), thereby focusing on small quantities ranging from 1 to 9. We aimed to determine the development of mapping over time and its relation to mathematics achievement. Participants were 36 first graders (M = 6 years 8 months) and 46 third graders (M = 8 years 8 months) who all completed mapping tasks, symbolic and non-symbolic magnitude comparison tasks and standardized timed and untimed tests of mathematics achievement. Findings revealed that children are able to map between non-symbolic and symbolic representations and that this mapping ability develops over time. Moreover, we found that children’s mapping ability is related to timed and untimed measures of mathematics achievement, over and above the variance accounted for by their numerical magnitude comparison skills.     

Semantic integration by pattern priming: experiment and cortical network model

Neural network models describe semantic priming effects by way of mechanisms of activation of neurons coding for words that rely strongly on synaptic efficacies between pairs of neurons. Biologically inspired Hebbian learning defines efficacy values as a function of the activity of pre- and post-synaptic neurons only. It generates only pair associations between words in the semantic network. However, the statistical analysis of large text databases points to the frequent occurrence not only of pairs of words (e.g., “the way”) but also of patterns of more than two words (e.g., “by the way”). The learning of these frequent patterns of words is not reducible to associations between pairs of words but must take into account the higher level of coding of three-word patterns. The processing and learning of pattern of words challenges classical Hebbian learning algorithms used in biologically inspired models of priming. The aim of the present study was to test the effects of patterns on the semantic processing of words and to investigate how an inter-synaptic learning algorithm succeeds at reproducing the experimental data. The experiment manipulates the frequency of occurrence of patterns of three words in a multiple-paradigm protocol. Results show for the first time that target words benefit more priming when embedded in a pattern with the two primes than when only associated with each prime in pairs. A biologically inspired inter-synaptic learning algorithm is tested that potentiates synapses as a function of the activation of more than two pre- and post-synaptic neurons. Simulations show that the network can learn patterns of three words to reproduce the experimental results.     

Identification of repertoires of surface antigens on leukemias using an antibody microarray

We have previously described a microarray of cluster of differentiation (CD) antibodies that enables concurrent determination of more than 60 CD antigens on leukocytes. This procedure does not require protein purification or labeling, or a secondary detection system. Whole cells are captured by a microarray of 10 nL antibody dots immobilized on a nitrocellulose film on a microscope slide. Distinct patterns of cell binding are observed for different leukemias or lymphomas. These haematological malignancies arise from precursor cells of T- or B-lymphocytic, or myeloid lineages of hematopoiesis. The dot patterns obtained from patients are distinct from those of peripheral blood leukocytes from normal subjects. This microarray technology has recently undergone a number of refinements. The microarray now contains more CD antibodies, and a scanner for imaging dot patterns and software for data analysis provide an extensive immunophenotype sufficient for diagnosis of common leukemias. The technology is being evaluated for diagnosis of leukemias with parallel use of conventional diagnostic criteria.     

Testing Whether and When Abstract Symmetric Patterns Produce Affective Responses

Symmetry has a central role in visual art, it is often linked to beauty, and observers can detect it efficiently in the lab. We studied what kind of fast and automatic responses are generated by visual presentation of symmetrical patterns. Specifically, we tested whether a brief presentation of novel symmetrical patterns engenders positive affect using a priming paradigm. The abstract patterns were used as primes in a pattern-word interference task. To ensure that familiarity was not a factor, no pattern and no word was ever repeated within each experiment. The task was to classify words that were selected to have either positive or negative valence. We tested irregular patterns, patterns containing vertical and horizontal reflectional symmetry, and patterns containing a 90 deg rotation. In a series of 7 experiments we found that the effect of affective congruence was present for both types of regularity but only when observers had to classify the regularity of the pattern after responding to the word. The findings show that processing abstract symmetrical shapes or random pattern can engender positive or negative affect as long as the regularity of the pattern is a feature that observers have to attend to and classify.     

Towards theoretical formalisms

This article deals with the relationship between vocabulary (total number of distinct oligomers or “words”) and text-length (total number of oligomers or “words”) for a coding DNA sequence (CDS). For natural human languages, Heaps established a mathematical formula known as Heaps’ law, which relates vocabulary to text-length. Our analysis shows that Heaps’ law fails to model this relationship for CDSs. Here we develop a mathematical model to establish the relationship between the number of type of words (vocabulary) and the number of words sampled (text-length) for CDSs, when non-overlapping nucleotide strings with the same length are treated as words. We use tangent-hyperbolic function, which captures the saturation property of vocabulary. Based on the parameters of the model, we formulate a mathematical equation, known as “equation of word organization”, whose parameters essentially indicate that nucleotide organization of coding sequences are different from one another. We also compare the word organization of CDSs with the random word distribution and conclude that a CDS is neither similar to a natural human language nor to a random one. Moreover, these sequences have their unique nucleotide organization and it is completely structured for specific biological functioning.     

Retinotopic mapping of categorical and coordinate spatial relation processing in early visual cortex

Spatial relations are commonly divided in two global classes. Categorical relations concern abstract relations which define areas of spatial equivalence, whereas coordinate relations are metric and concern exact distances. Categorical and coordinate relation processing are thought to rely on at least partially separate neurocognitive mechanisms, as reflected by differential lateralization patterns, in particular in the parietal cortex. In this study we address this textbook principle from a new angle. We studied retinotopic activation in early visual cortex, as a reflection of attentional distribution, in a spatial working memory task with either a categorical or a coordinate instruction. Participants were asked to memorize a dot position, with regard to a central cross, and to indicate whether a subsequent dot position matched the first dot position, either categorically (opposite quadrant of the cross) or coordinately (same distance to the centre of the cross). BOLD responses across the retinotopic maps of V1, V2, and V3 indicate that the spatial distribution of cortical activity was different for categorical and coordinate instructions throughout the retention interval; a more local focus was found during categorical processing, whereas focus was more global for coordinate processing. This effect was strongest for V3, approached significance in V2 and was absent in V1. Furthermore, during stimulus presentation the two instructions led to different levels of activation in V3 during stimulus encoding; a stronger increase in activity was found for categorical processing. Together this is the first demonstration that instructions for specific types of spatial relations may yield distinct attentional patterns which are already reflected in activity early in the visual cortex.     

Tilt aftereffects generated by symmetrical dot patterns with two or four axes of symmetry

This paper follows from studies by Joung, van der Zwan and Latimer (2000) in which symmetrical dot patterns with one axis of symmetry were used to produce tilt aftereffects (TAEs). The present paper investigates TAE functions produced by symmetrical dot patterns with multiple axes of symmetry. In Experiments 1 and 2, TAE functions produced by dot patterns with two axes of symmetry were compared with TAE functions produced by line stimuli arranged in the same orientation and location as the axes of symmetry in the dot patterns. Similar functions were found. In Experiments 3 and 4, functions produced by dot patterns with four axes of symmetry were compared with functions produced by line stimuli arranged in the same orientation and location as the four axes of symmetry. Again, similar functions were found. These experiments demonstrate that line stimuli and dot stimuli produce similar TAE functions. The implications of these results are discussed.