Learning complex arithmetic—an fMRI study

Flexible transfer of knowledge in mental arithmetic--an fMRI study

Recent imaging studies could show that fact acquisition in arithmetic is associated with decreasing activation in several frontal and parietal areas, and relatively increasing activation within the angular gyrus, indicating a switch from direct calculation to retrieval of a learned fact from memory. So far, however, little is known about the transfer of learned facts between arithmetic operations. The aim of the present fMRI study was to investigate whether and how newly acquired arithmetic knowledge might transfer from trained multiplication problems to related division problems. On the day before scanning, ten complex multiplication problems were trained. Within the scanner, trained multiplication problems were compared with untrained multiplication problems, and division problems related to multiplication (transfer condition) were compared with unrelated division problems (no-transfer condition). Replicating earlier results, untrained multiplication problems activated several frontal and parietal brain areas more strongly than trained multiplication problems, while trained multiplication problems showed relatively stronger activation in the left angular gyrus than untrained multiplication problems. Concerning division, an ROI analysis indicated that activation in the left angular gyrus was relatively stronger for the transfer condition than for the no-transfer condition. We also observed distinct inter-individual differences with regard to transfer that modulated activation within the left angular gyrus. Activation within the left angular gyrus was generally higher for participants who showed a transfer effect for division problems. In conclusion, the present study yielded some evidence that successful transfer of knowledge between arithmetic operations is accompanied by modifications of brain activation patterns. The left angular gyrus seems not only to be involved in the retrieval of stored arithmetic facts, but also in the transfer between arithmetic operations.

Arithmetic and brain connectivity: mental calculation in adolescents with periventricular lesions

The ability for mental calculation represents a fundamental prerequisite for development of intelligence, which is predictive for educational and professional success in life. Many individuals with calculation difficulties are survivors of premature birth. The brain mechanisms of these deficits are, however, largely unknown. In this work, we clarify whether and, if so, how calculation abilities in adolescents who were born premature are related to the extent and topography of periventricular lesions that affect brain connectivity. Performance on a set of mental calculation tasks is lower in adolescents with periventricular leukomalacia (PVL) than in former preterms and term-born peers without signs of brain abnormalities on a magnetic resonance imaging scan. No difference in the calculation ability was found between term-born and preterm adolescents without PVL. Calculation abilities in PVL patients were unrelated to volumetric extent and topography of lesions in both brain hemispheres. Whereas previous work clearly reveals the link between the extent and topography of lesions and severity of impairments in visual cognition ranging from body motion processing to visual navigation and social cognition, no such association occurs for mental calculation. We assume that the lack of relationship between calculation abilities and the extent and topography of periventricular lesions point to topographically restricted neural substrate that serves as the keystone for mental calculation. The findings suggest that periventricular brain damage does not substantially affect the connectivity of this region with other brain structures engaged in the mental calculation network.

Using brain imaging to extract the structure of complex events at the rational time band

A functional magnetic resonance imaging (fMRI) study was performed in which participants performed a complex series of mental calculations that spanned about 2 min. An Adaptive Control of Thought--Rational (ACT-R) model [Anderson, J. R. How can the human mind occur in the physical universe? New York: Oxford University Press, 2007] was developed that successfully fit the distribution of latencies. This model generated predictions for the fMRI signal in six brain regions that have been associated with modules in the ACT-R theory. The model's predictions were confirmed for a fusiform region that reflects the visual module, for a prefrontal region that reflects the retrieval module, and for an anterior cingulate region that reflects the goal module. In addition, the only significant deviations to the motor region that reflects the manual module were anticipatory hand movements. In contrast, the predictions were relatively poor for a parietal region that reflects an imaginal module and for a caudate region that reflects the procedural module. Possible explanations of these poor fits are discussed. In addition, exploratory analyses were performed to find regions that might correspond to the predictions of the modules.

Development of neural networks for exact and approximate calculation: a FMRI study

Neuroimaging findings in adults suggest exact and approximate number processing relying on distinct neural circuits. In the present study we are investigating whether this cortical specialization is already established in 9- and 12-year-old children. Using fMRI, brain activation was measured in 10 third- and 10 sixth-grade school children and 20 adults during trials of symbolic approximate (AP) and exact (EX) calculation, as well as non-symbolic magnitude comparison (MC) of objects. Children activated similar networks like adults, denoting an availability and a similar spatial extent of specified networks as early as third grade. However, brain areas related to number processing become further specialized with schooling. Children showed weaker activation in the intraparietal sulcus during all three tasks, in the left inferior frontal gyrus during EX and in occipital areas during MC. In contrast, activation in the anterior cingulate gyrus, a region associated with attentional effort and working memory load, was enhanced in children. Moreover, children revealed reduced or absent deactivation of regions involved in the so-called default network during symbolic calculation, suggesting a rather general developmental effect. No difference in brain activation patterns between AP and EX was found. Behavioral results indicated major differences between children and adults in AP and EX, but not in MC. Reaction time and accuracy rate were not correlated to brain activation in regions showing developmental changes suggesting rather effects of development than performance differences between children and adults. In conclusion, increasing expertise with age may lead to more automated processing of mental arithmetic, which is reflected by improved performance and by increased brain activation in regions related to number processing and decreased activation in supporting areas.

The Neuroscience of Stigma and Stereotype Threat

This article reviews social neuroscience research on the experience of stigma from the target's perspective. More specifically, we discuss several research programs that employ electroencephalography, event-related potentials, or functional magnetic resonance imaging methods to examine neural correlates of stereotype and social identity threat. We present neuroimaging studies that show brain activation related to the experience of being stereotyped and ERP studies that shed light on the cognitive processes underlying social identity processes. Among these are two projects from our own lab. The first project reveals the important role of the neurocognitive conflict-detection system in stereotype threat effects, especially as it pertains to stereotype threat `spillover'. The second project examines the role of automatic ingroup evaluations as a neural mediator between social identity threats and compensatory ingroup bias. We conclude with a discussion of the benefits, limitations, and unique contributions of social neuroscience to our understanding of stigma and social identity threat.

Effects of development and enculturation on number representation in the brain

A striking way in which humans differ from non-human primates is in their ability to represent numerical quantity using abstract symbols and to use these 'mental tools' to perform skills such as exact calculations. How do functional brain circuits for the symbolic representation of numerical magnitude emerge? Do neural representations of numerical magnitude change as a function of development and the learning of mental arithmetic? Current theories suggest that cultural number symbols acquire their meaning by being mapped onto non-symbolic representations of numerical magnitude. This Review provides an evaluation of this contention and proposes hypotheses to guide investigations into the neural mechanisms that constrain the acquisition of cultural representations of numerical magnitude.

How verbal and spatial manipulation networks contribute to calculation: an fMRI study

The manipulation of numbers required during calculation is known to rely on working memory (WM) resources. Here, we investigated the respective contributions of verbal and/or spatial WM manipulation brain networks during the addition of four numbers performed by adults, using functional magnetic resonance imaging (fMRI). Both manipulation and maintenance tasks were proposed with syllables, locations, or two-digit numbers. As compared to their maintenance, numbers manipulation (addition) elicited increased activation within a widespread cortical network including inferior temporal, parietal, and prefrontal regions. Our results demonstrate that mastery of arithmetic calculation requires the cooperation of three WM manipulation systems: an executive manipulation system conjointly recruited by the three manipulation tasks, including the anterior cingulate cortex (ACC), the orbital part of the inferior frontal gyrus, and the caudate nuclei; a left-lateralized, language-related, inferior fronto-temporal system elicited by numbers and syllables manipulation tasks required for retrieval, selection, and association of symbolic information; and a right superior and posterior fronto-parietal system elicited by numbers and locations manipulation tasks for spatial WM and attentional processes. Our results provide new information that the anterior intraparietal sulcus (IPS) is involved in tasks requiring a magnitude processing with symbolic (numbers) and nonsymbolic (locations) stimuli. Furthermore, the specificity of arithmetic processing is mediated by a left-hemispheric specialization of the anterior and posterior parts of the IPS as compared to a spatial task involving magnitude processing with nonsymbolic material.

The Negative Consequences of Threat

This study used functional magnetic resonance imaging to identify the neural structures associated with women's underperformance on math tasks. Although women in a control condition recruited neural networks that are associated with mathematical learning (i.e., angular gyrus, left parietal and prefrontal cortex), women who were reminded of gender stereotypes about math ability did not recruit these regions, and instead revealed heightened activation in a neural region associated with social and emotional processing (ventral anterior cingulate cortex).

Fast reproducible identification and large-scale databasing of individual functional cognitive networks

BACKGROUND

Although cognitive processes such as reading and calculation are associated with reproducible cerebral networks, inter-individual variability is considerable. Understanding the origins of this variability will require the elaboration of large multimodal databases compiling behavioral, anatomical, genetic and functional neuroimaging data over hundreds of subjects. With this goal in mind, we designed a simple and fast acquisition procedure based on a 5-minute functional magnetic resonance imaging (fMRI) sequence that can be run as easily and as systematically as an anatomical scan, and is therefore used in every subject undergoing fMRI in our laboratory. This protocol captures the cerebral bases of auditory and visual perception, motor actions, reading, language comprehension and mental calculation at an individual level.

RESULTS

81 subjects were successfully scanned. Before describing inter-individual variability, we demonstrated in the present study the reliability of individual functional data obtained with this short protocol. Considering the anatomical variability, we then needed to correctly describe individual functional networks in a voxel-free space. We applied then non-voxel based methods that automatically extract main features of individual patterns of activation: group analyses performed on these individual data not only converge to those reported with a more conventional voxel-based random effect analysis, but also keep information concerning variance in location and degrees of activation across subjects.

CONCLUSION

This collection of individual fMRI data will help to describe the cerebral inter-subject variability of the correlates of some language, calculation and sensorimotor tasks. In association with demographic, anatomical, behavioral and genetic data, this protocol will serve as the cornerstone to establish a hybrid database of hundreds of subjects suitable to study the range and causes of variation in the cerebral bases of numerous mental processes.

Relationship between regional hemodynamic activity and simultaneously recorded EEG-theta associated with mental arithmetic-induced workload

Theta increases with workload and is associated with numerous processes including working memory, problem solving, encoding, or self monitoring. These processes, in turn, involve numerous structures of the brain. However, the relationship between regional brain activity and the occurrence of theta remains unclear. In the present study, simultaneous EEG-fMRI recordings were used to investigate the functional topography of theta. EEG-theta was enhanced by mental arithmetic-induced workload. For the EEG-constrained fMRI analysis, theta-reference time-series were extracted from the EEG, reflecting the strength of theta occurrence during the time course of the experiment. Theta occurrence was mainly associated with activation of the insular cortex, hippocampus, superior temporal areas, cingulate cortex, superior parietal, and frontal areas. Though observation of temporal and insular activation is in accord with the theory that theta specifically reflects encoding processes, the involvement of several other brain regions implies that surface-recorded theta represents comprehensive functional brain states rather than specific processes in the brain. The results provide further evidence for the concept that emergent theta band oscillations represent dynamic functional binding of widely distributed cortical assemblies, essential for cognitive processing. This binding process may form the source of surface-recorded EEG theta.

Frontal and parietal ERPs associated with duration discriminations with or without task interference

The main objective of this study was to examine fronto-parietal networks underlying visual duration discriminations. Two types of interference tasks were used to augment cognitive load: line orientation associated with the right hemisphere and multiplication with the left. Both subtasks deteriorated duration discriminations, more severely for line orientation. Relative to the condition without interference, the dual task paradigm decreased amplitudes of the contingent negative variation (CNV) wave, predominant at frontal sites, and the P300 wave, predominant at parietal sites. Inversely, amplitudes of a later appearing positive component (LPC) and its parietal counterpart of opposite polarity (LNC) increased with spatial or numeric task interference. These results are concordant with the view that fronto-parietal networks underlying duration discriminations act in a concerted fashion, with the LPC/LNC waves acting as a warning signal to mitigate errors during high cognitive load.

Imaging early practice effects in arithmetic

A better understanding of learning processes in arithmetic in healthy adults can guide research into learning disabilities such as dyscalculia. The goal of the present functional magnetic resonance imaging study was to investigate the ongoing process of learning itself. No training was provided prior to the scanning session. Training consisted in a higher frequency of repetition for one set of complex multiplication problems (repeated) and a lower frequency for the other set (novel). Repeated and novel problems were presented randomly in an event-related design. We observed activation decreases due to training in fronto-parietal areas and the caudate nucleus, and activation increases in temporo-parietal regions such as the left angular gyrus. Training effects became significant after approximately eight repetitions of a problem and remained stable over the course of the experiment. The change in brain activation patterns observed was similar to the results of previous neuroimaging studies investigating training effects in arithmetic after a week of extensive training. The paradigm employed seems to be a suitably sensitive tool to investigate and compare learning processes on group level for different populations. Furthermore, on a more general level, the early and robust changes in brain activation in healthy adults observed here indicate that repeating stimuli can profoundly and quickly affect fMRI results.

When brightness counts: the neuronal correlate of numerical-luminance interference

Previous studies showed that the processing of numerical information and spatial information such as physical size causes a mutual interference. The neuronal correlate of such interference was suggested to be in the parietal lobe. However, a previous study showed that such interference does not occur between numerical information and nonspatial dimensions such as luminance level (Pinel P, Piazza M, Le Bihan D, Dehaene S. 2004. Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron. 41:983-993). Here it is shown that numerical value and luminance level do cause a behavioral interference and that this interference modulates the activity in the parietal lobe. The current results support the idea that the parietal lobe might be equipped with neuronal substrates for magnitude processing even for nonspatial dimensions.

Cerebral activation in patients with somatoform pain disorder exposed to pain and stress: An fMRI study

Patients with somatoform pain disorders are supposed to suffer from an early acquired defect in stress regulation. In order to look for common alterations of the pain- and stress-responsive cortical areas, we prospectively recorded cerebral activations induced by pin-prick pain, by cognitive stress and emotional stress using functional magnetic resonance imaging (fMRI) in a group of 17 patients and an age-matched control group. In addition, the hippocampal volumes of both groups were measured. Patients showed increased activations of the known pain-processing areas (thalamus, basal ganglia, operculo-insular cortex), but also of some prefrontal, temporal and parietal regions during first pain exposure and of temporal and parietal areas during cognitive stress, but reduced activations during emotional stress. In contrast to these functional differences, hippocampal volume was not significantly reduced in patients. Although the superior temporal gyrus was the only common area of an "overactivation" in patients in the pain and stress condition, findings of our study support the current concept of mechanisms involved in somatoform pain disorders: central processing of pain and of cognitive stress is increased in patients possibly due to exaggerated memory and/or anticipation of pain exposure and to a disturbance of stress-regulating systems which has to be worked out on a cortical level in more detail. Our finding of a reduced responsiveness to emotional stress is surprising, but not contradictive to these results because some sort of neglect or coping mechanisms may have developed over time as a response to early adversities.

Dissociated brain organization for single-digit addition and multiplication

This study compared the patterns of brain activation elicited by single-digit addition and multiplication problems. 20 Chinese undergraduates were asked to verify whether arithmetic equations were true or false during functional magnetic resonance imaging. Results showed that both addition and multiplication were supported by a broad neural system that involved regions within SMA, precentral gyrus, intraparietal sulcus, occipital gyri, superior temporal gyrus, and middle frontal gyrus, as well as some subcortical structures. Nevertheless, addition problems elicited more activation in the intraparietal sulcus and middle occipital gyri at the right hemisphere, and superior occipital gyri at both hemispheres, whereas multiplication had more activation in precentral gyrus, supplementary motor areas, and posterior and anterior superior temporal gyrus at the left hemisphere. This pattern of dissociated activation supports our hypothesis that addition has greater reliance on visuospatial processing and multiplication on verbal processing.

Automatic behaviour: efficient not mindless

Automaticity is a core construct underpinning theoretical accounts of human performance and cognition. In spite of this, its current conceptualisation is plagued by circularity - automaticity is typically defined in terms of the very behaviour it seeks to explain - and a lack of internal consistency-defining features of automaticity do not reliably co-occur. Furthermore, invoking automaticity tends to be post hoc as it is used to explain violations of dominant theories of attention. Prevailing models of automaticity explain automatic processing as merely faster processing than controlled processing. We present an alternative conceptualisation of automaticity as efficient, elegant and economical but not fast. This is supported by functional imaging studies, which reveal a pattern of reduced global activation as well as a shift in activation from cortical to subcortical areas once automaticity has been achieved. Were automaticity to be faster processing, functional imaging would indicate greater activation when an automatic task is performed. We propose possible circuitry of automaticity incorporating the direct pathways of the basal ganglia.

Effects of age and mild cognitive impairment on direct and indirect access to arithmetic knowledge

The present study aimed at investigating age-related changes and mild cognitive impairment (MCI) related effects in simple arithmetic. To pursue this goal, MCI patients, healthy old adults and young adults performed three computerised tasks. The production (e.g., 3 x 4=?) and the verification task (3 x 4 12?) evaluated direct access to multiplication knowledge, the number-matching task (3 x 4 34?, 'do 3 x 4 and 34 have the same digits?') tested indirect access. In verification and number-matching, interference from related distractors (e.g., 3 x 4 followed by 16) relative to unrelated distractors (17) reflects access to stored fact representations as well as efficiency of inhibition processes. Results indicated that, compared to young adults, MCI and healthy old adults were slower in responding across tasks. In production and verification, analyses of individual latency regression slopes and intercepts suggested that these age effects were related to differences at peripheral processing stages (e.g., encoding) rather than at the central (arithmetic retrieval) stage. Differences between MCI and healthy elderly emerged only in the number-matching task. While in verification effects were comparable between groups, in number-matching MCI patients were more susceptible to interference from irrelevant information than healthy old participants. Overall, the present findings indicate that aging has a general effect on peripheral processing speed, but not on arithmetic memory retrieval. Parietal cortico-subcortical circuits mediating arithmetic fact retrieval (Dehaene, S., & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83-120; Dehaene, S., & Cohen, L. (1997). Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33, 219-250) thus seem to be preserved in normal aging and MCI. In contrast, MCI patients show enhanced interference in number-matching. This task-specific lack of inhibition may point to dysfunctional frontal cortico-subcortical networks in MCI.

Age-related changes in the activation of the intraparietal sulcus during nonsymbolic magnitude processing: an event-related functional magnetic resonance imaging study

Numerical magnitude processing is an essential everyday skill. Functional brain imaging studies with human adults have repeatedly revealed that bilateral regions of the intraparietal sulcus are correlated with various numerical and mathematical skills. Surprisingly little, however, is known about the development of these brain representations. In the present study, we used functional neuroimaging to compare the neural correlates of nonsymbolic magnitude judgments between children and adults. Although behavioral performance was similar across groups, in comparison to the group of children the adult participants exhibited greater effects of numerical distance on the left intraparietal sulcus. Our findings are the first to reveal that even the most basic aspects of numerical cognition are subject to age-related changes in functional neuroanatomy. We propose that developmental impairments of number may be associated with atypical specialization of cortical regions underlying magnitude processing.

Neural correlates of arithmetic and language comprehension: a common substrate?

There is debate as to the relationship between mathematical ability and language. Some research has suggested that common processes underlie arithmetic and grammar while other research has suggested that these are distinct processes. The current study aimed to address this issue in a large group of 68 left hemisphere stroke patients who were all tested on analogous arithmetic and language comprehension measures. The behavioral data revealed a significant correlation between performance on the comprehension and arithmetic measures, although a subset of patients showed a dissociation in performance on the two tasks. To determine the brain regions critical for performance on each measure, patients' lesions were analyzed using Voxel-based Lesion Symptom Mapping. Arithmetic was associated with a small number of foci, with the most significant region located in the left inferior parietal lobule (Brodmann areas 39 and 40). Comprehension was associated with a larger number of brain regions, most extensively in the left middle and superior temporal gyri. There was also overlap between the arithmetic and comprehension maps in a number of regions, such as the inferior frontal gyrus. Our findings suggest that arithmetic and language comprehension are mediated by partially overlapping brain networks. These findings are discussed in light of previous work on the neural basis of arithmetic ability and its relationship to language.

Acalculia from a right hemisphere lesion dealing with "where" in multiplication procedures

The present study describes in detail, for the first time, a case of failure with multiplication procedures in a right hemisphere damaged patient (PN). A careful, step-by-step, error analysis made possible to show that an important portion of PN's errors could be better explained as spatial in nature and specifically related to the demands of a multi-digit multiplication. These errors can be distinguished from other types of errors, including those, expected after a right hemisphere lesion, determined by a generic inability to deal with spatial material, or from other deficits, like neglect, affecting cognitive capacities across the board. The best explanation for PN's problems is that he might have difficulties in relying on a visuo-spatial store containing a layout representation specific to multiplication. As a consequence, while knowing what, when and how to carry out the various steps, PN does not know where. What he may thus lack is a spatial schema of multiplication. Such schema is thought to help normal calculators in overcoming working memory demands of complex calculation by representing the information of where exactly each sub-step should be placed.

Second to fourth digit ratio and numerical competence in children

The ratio between the 2nd and 4th fingers (2D:4D)-a potential proxy for prenatal testosterone (T) exposure-shows a sex difference, with males usually having lower mean values; the latter potentially indicates higher prenatal T exposure. We studied relations between 2D:4D and competencies in the domains of counting, number knowledge, and visual-number representation in 73 children aged 6-11 years. Significant negative correlations between numerical performance in all of these areas and right and left hand 2D:4D ratios were found for boys but not girls. To the extent that 2D:4D ratios reflects prenatal exposure to T, the implications are (i) high prenatal T may be associated with better performance on some basic numerical measures for boys, and (ii) prenatal exposure to T may affect boys and girls differently with respect to some numerical competencies.

Neural substrates for forward and backward recitation of numbers and the alphabet: a close examination of the role of intraparietal sulcus and perisylvian areas

Despite numerous studies on the neural basis of numerical processing, few studies have examined the neural substrates of one of the most basic numerical processing-number sequence recitation. The present study used fMRI to investigate neural substrates of number sequence recitation, focusing on the intraparietal sulcus (IPS) and perisylvian areas. This study used a 2 (number versus alphabet) x 2 (forward versus backward recitation) design. 12 Chinese undergraduates were asked to recite overtly but gently numerical and alphabetical sequences forward and backward. Results showed that, for both numerical and alphabetic sequences, the left IPS was activated when performing backward recitation, but not when performing forward recitation. In terms of perisylvian areas, all four tasks elicited activation in bilateral superior temporal gyrus and inferior frontal gyrus, but forward recitation elicited greater activation in the left posterior superior temporal gyrus than did backward recitation, whereas backward recitation elicited greater activation in the left inferior frontal gyrus than did forward recitation. These results suggest that forward recitation of numbers and the alphabet is typically based on verbal processing of numbers implemented in the perisylvian area, whereas backward recitation would likely require additional neural resources in the IPS.

Principles underlying the design of "The Number Race", an adaptive computer game for remediation of dyscalculia

Background

Adaptive game software has been successful in remediation of dyslexia. Here we describe the cognitive and algorithmic principles underlying the development of similar software for dyscalculia. Our software is based on current understanding of the cerebral representation of number and the hypotheses that dyscalculia is due to a "core deficit" in number sense or in the link between number sense and symbolic number representations.

Methods

"The Number Race" software trains children on an entertaining numerical comparison task, by presenting problems adapted to the performance level of the individual child. We report full mathematical specifications of the algorithm used, which relies on an internal model of the child's knowledge in a multidimensional "learning space" consisting of three difficulty dimensions: numerical distance, response deadline, and conceptual complexity (from non-symbolic numerosity processing to increasingly complex symbolic operations).

Results

The performance of the software was evaluated both by mathematical simulations and by five weeks of use by nine children with mathematical learning difficulties. The results indicate that the software adapts well to varying levels of initial knowledge and learning speeds. Feedback from children, parents and teachers was positive. A companion article [1] describes the evolution of number sense and arithmetic scores before and after training.

Conclusion

The software, open-source and freely available online, is designed for learning disabled children aged 5–8, and may also be useful for general instruction of normal preschool children. The learning algorithm reported is highly general, and may be applied in other domains.

How specifically do we learn? Imaging the learning of multiplication and subtraction

The present functional magnetic resonance imaging (fMRI) study investigates modifications of brain activation patterns related to the training of two different arithmetic operations, multiplication and subtraction. Healthy young adults were trained in five sessions to answer multiplication and subtraction problems. In the following fMRI session, trained and new untrained problems closely matched for difficulty were presented in blocked order. Contrasts between untrained and trained operations showed stronger activation of inferior frontal and parietal regions, especially along the banks of the intraparietal sulcus. The reverse contrasts, trained minus untrained operations, yielded significantly higher activation in the left angular gyrus for multiplication but no significantly activated area for subtraction. This suggests that training leads to a reduction of general purpose processes, such as working memory and executive control in both operations, indicated by the decrease of activation in inferior frontal areas. For multiplication, however, the increase of activation in the left angular gyrus indicates a switching of cognitive processes. Trained subtraction therefore seems to lead to faster and more efficient strategies, while trained multiplication showed a shift from quantity-based processing (supported by the areas along the intraparietal sulci) to more automatic retrieval (supported by the left angular gyrus). The same training method caused changes in brain activation patterns that depended on the given operation. The effects of learning on the brain therefore seem not only to depend on the method of learning but also on its content.

Isolated numerical skills in posterior cortical atrophy--an fMRI study

Posterior cortical atrophy (PCA) is characterized by bilateral parieto-occipito-temporal atrophy and hypometabolism. Neuropsychological impairments include complex visual disturbances, alexia, agraphia, finger agnosia, right-left disorientation and dyscalculia. A recent case study reported severe numerical deficits with some selectively preserved numerical skills in a patient affected by PCA [Delazer, M., Karner, E., Zamarian, L., Donnemiller, E., & Benke, T. (2006). Number processing in posterior cortical atrophy--a neuropsycholgical case study. Neuropsychologia]. In a functional magnetic resonance imaging (fMRI) study brain activation patterns related to these selectively preserved numerical skills were analyzed. Recitation of multiplication tables and counting forward were contrasted to word recitation in a block design. Contrasts between experimental conditions and control condition yielded significant activation of inferior and medial temporal structures. Since numerical processing is generally associated with parietal activation, it was hypothesized that preserved brain structures would compensate for the functional deficits.

The BOLD response during Stroop task-like inhibition paradigms: Effects of task difficulty and task-relevant modality

Previous studies of the Stroop task propose two key mediators: the prefrontal and cingulate cortices but hints exist of functional specialization within these regions. This study aimed to examine the effect of task modality upon the prefrontal and cingulate response by examining the response to colour, number, and shape Stroop tasks whilst BOLD fMRI images were acquired on a Siemens 3T MRI scanner. Behavioural analyses indicated facilitation and interference effects and a noticeable effect of task difficulty. Some modular effects of modality were observed in the prefrontal cortex that survived exclusion of task difficulty related activations. No effect of task-relevant information was observed in the anterior cingulate. Future comparisons of the mediation of selective attention need to consider the effects of task context and task difficulty.

Effect of Language Switching on Arithmetic: A Bilingual fMRI Study

The role of language in performing numerical computations has been a topic of special interest in cognition. The “Triple Code Model” proposes the existence of a language-dependent verbal code involved in retrieving arithmetic facts related to addition and multiplication, and a language-independent analog magnitude code subserving tasks such as number comparison and estimation. Neuroimaging studies have shown dissociation between dependence of arithmetic computations involving exact and approximate processing on language-related circuits. However, a direct manipulation of language using different arithmetic tasks is necessary to assess the role of language in forming arithmetic representations and in solving problems in different languages. In the present study, 20 English-Chinese bilinguals were trained in two unfamiliar arithmetic tasks in one language and scanned using fMRI on the same problems in both languages (English and Chinese). For the exact “base-7 addition” task, language switching effects were found in the left inferior frontal gyrus (LIFG) and left inferior parietal lobule extending to the angular gyrus. In the approximate “percentage estimation” task, language switching effects were found predominantly in the bilateral posterior intraparietal sulcus and LIFG, slightly dorsal to the LIFG activation seen for the base-7 addition task. These results considerably strengthen the notion that exact processing relies on verbal and language-related networks, whereas approximate processing engages parietal circuits typically involved in magnitude-related processing.

On the functional role of human parietal cortex in number processing: How gender mediates the impact of a ‘virtual lesion’ induced by rTMS

Areas around the horizontal part of the intraparietal sulcus (hIPS) have repeatedly been reported to participate in processing numerical magnitude. Using transcranial magnetic stimulation (TMS), we investigated the functional role of the hIPS by examining two effects from the domain of numerical cognition: in magnitude comparison tasks response latencies are inversely related to the numerical distance between two numbers. This distance effect indexes access to the mental number representation. In magnitude comparison tasks responses are faster when decade and unit comparison would lead to the same decision (e.g. 42_57, 4 < 5 and 2 < 7) than when they would not (e.g. 47_62, 4 < 6 but 7 > 2). This compatibility effect reflects unit-decade integration processes. Differential susceptibility of (fe)male participants to TMS was examined. We applied repetitive TMS (rTMS; 1Hz for 10 min) over the left hIPS in 12 participants (6 female). No stimulation and vertex stimulation served as control conditions. The effect of rTMS was mediated by gender: in male participants, the distance effect decreased after TMS over hIPS. For female participants distance and compatibility effect both increased. This modulation of the compatibility effect was limited in duration to no more than 4 min. The hIPS seems to be functionally involved both in number magnitude processing and in integrating unit-decade magnitude information of two-digit numbers. Relative hemispheric specialization of the hIPS with respect to two-digit magnitude comparison is discussed.

Hemispheric lateralization of number comparison

In order to clarify the respective contribution of the right and left posterior parietal cortex (PPC) to number comparison, transcranial magnetic stimulation (TMS) was used to disrupt PPC processing in subjects instructed to determine whether a digit was smaller or larger than 5. Single pulse TMS was applied over the PPC, either unilaterally or bilaterally, 150, 200, or 250 ms after digit presentation. Sham TMS was used as a control condition to take into account the unspecific effects of TMS on reaction time (RT). The main finding of the present study is a significant increase in RTs when comparing digits close to 5 following a disruption either of the left PPC alone or of both PPC simultaneously. The comparison of digits far from 5 was unaltered by disrupting only one PPC but RTs were found increased after bilateral PPC stimulation. These disruptive effects were observed irrespective of the TMS delay. We concluded that coding precise numerical values requires the integrity of the left PPC, as suggested by the deficit in discriminating close digits consequent to its disruption. In contrast, approximate comparisons can be processed either by the left or right PPC, since simultaneous bilateral TMS was needed to alter the comparison of digits far from 5.

Neuro-functional differences associated with arithmetic processing in Turner syndrome

Turner syndrome (TS) is a neurogenetic disorder characterized by the absence of one X chromosome in a phenotypic female. Individuals with TS are at risk for impairments in mathematics. We investigated the neural mechanisms underlying arithmetic processing in TS. Fifteen subjects with TS and 15 age-matched typically developing controls were scanned using functional MRI while they performed easy (two-operand) and difficult (three-operand) versions of an arithmetic processing task. Both groups activated fronto-parietal regions involved in arithmetic processing during the math tasks. Compared with controls, the TS group recruited additional neural resources in frontal and parietal regions during the easier, two-operand math task. During the more difficult three-operand task, individuals with TS demonstrated significantly less activation in frontal, parietal and subcortical regions than controls. However, the TS group's performance on both math tasks was comparable to controls. Individuals with TS demonstrate activation differences in fronto-parietal areas during arithmetic tasks compared with controls. They must recruit additional brain regions during a relatively easy task and demonstrate a potentially inefficient response to increased task difficulty compared with controls.

Language and arithmetic – a study using the intracarotid amobarbital procedure

The intracarotid amobarbital procedure is used as a standard procedure in presurgical evaluation to assess hemispheric lateralization of language and memory, but has not been applied to investigate numerical processing. Patients with medically intractable epilepsy (n=20) were consecutively recruited during a presurgical evaluation programme. All 14 patients with left-lateralized language showed better arithmetic performance with the left hemisphere (intracarotid amobarbital procedure right), while five out of six patients with bilateral or right-hemispheric language representation showed better performance with the right hemisphere (intracarotid amobarbital procedure left). Furthermore, in patients with left-lateralized language, an interaction between intracarotid amobarbital procedure and type of arithmetic operation was found. The study suggests a close association between language lateralization and hemispheric specialization for arithmetic processing.

Number processing in posterior cortical atrophy--a neuropsycholgical case study

Posterior cortical atrophy (PCA) is an uncommon syndrome of dementia with early onset, characterised by disorders of higher visual function, variable symptoms of Balint's syndrome, visual agnosia, alexia, agraphia, finger agnosia, right-left disorientation and dyscalculia [Benson D. F., Davis R. J., & Snyder B. D. (1988). Posterior cortical atrophy. Archives of Neurology, 45, 789-793]. In a single case study specific numerical deficits were observed which may be predicted by parietal neurodegeneration (more pronounced on the right side; verified by SPECT). Besides impairments in all tasks involving visuo-spatial abilities (e.g., dot counting, analog number scale task), deficits appeared in tasks requiring access to an internal representation of numbers such as mental number bisection, approximation, estimation and semantic facts. In number comparison an increased distance effect was found. In simple arithmetic, a striking dissociation between operations was found-multiplication and addition facts being preserved at a superficial level, subtraction and division being severely impaired. The study confirms the close relation between spatial and numerical processing and highlights the modular organisation of the semantic system (number semantics impaired). Moreover, the study adds evidence about the clinical manifestation of the particular degenerative syndrome.

Learning by strategies and learning by drill—evidence from an fMRI study

The present fMRI study investigates, first, whether learning new arithmetic operations is reflected by changing cerebral activation patterns, and second, whether different learning methods lead to differential modifications of brain activation. In a controlled design, subjects were trained over a week on two new complex arithmetic operations, one operation trained by the application of back-up strategies, i.e., a sequence of arithmetic operations, the other by drill, i.e., by learning the association between the operands and the result. In the following fMRI session, new untrained items, items trained by strategy and items trained by drill, were assessed using an event-related design. Untrained items as compared to trained showed large bilateral parietal activations, with the focus of activation along the right intraparietal sulcus. Further foci of activation were found in both inferior frontal gyri. The reverse contrast, trained vs. untrained, showed a more focused activation pattern with activation in both angular gyri. As suggested by the specific activation patterns, newly acquired expertise was implemented in previously existing networks of arithmetic processing and memory. Comparisons between drill and strategy conditions suggest that successful retrieval was associated with different brain activation patterns reflecting the underlying learning methods. While the drill condition more strongly activated medial parietal regions extending to the left angular gyrus, the strategy condition was associated to the activation of the precuneus which may be accounted for by visual imagery in memory retrieval.

Agrammatic but numerate

A central question in cognitive neuroscience concerns the extent to which language enables other higher cognitive functions. In the case of mathematics, the resources of the language faculty, both lexical and syntactic, have been claimed to be important for exact calculation, and some functional brain imaging studies have shown that calculation is associated with activation of a network of left-hemisphere language regions, such as the angular gyrus and the banks of the intraparietal sulcus. We investigate the integrity of mathematical calculations in three men with large left-hemisphere perisylvian lesions. Despite severe grammatical impairment and some difficulty in processing phonological and orthographic number words, all basic computational procedures were intact across patients. All three patients solved mathematical problems involving recursiveness and structure-dependent operations (for example, in generating solutions to bracket equations). To our knowledge, these results demonstrate for the first time the remarkable independence of mathematical calculations from language grammar in the mature cognitive system.

Automatized clustering and functional geometry of human parietofrontal networks for language, space, and number

Human functional MRI studies frequently reveal the joint activation of parietal and of lateral and mesial frontal areas during various cognitive tasks. To analyze the geometrical organization of those networks, we used an automatized clustering algorithm that parcels out sets of areas based on their similar profile of task-related activations or deactivations. This algorithm allowed us to reanalyze published fMRI data (Simon, O., Mangin, J.F., Cohen, L., Le Bihan, D., Dehaene, S., 2002. Topographical layout of hand, eye, calculation, and language-related areas in the human parietal lobe. Neuron 33, 475-487) and to reproduce the previously observed geometrical organization of activations for saccades, attention, grasping, pointing, calculation, and language processing in the parietal lobe. Further, we show that this organization extends to lateral and mesial prefrontal regions. Relative to the parietal lobe, the prefrontal functional geometry is characterized by a partially symmetrical anteroposterior ordering of activations, a decreased representation of effector-specific tasks, and a greater emphasis on higher cognitive functions of attention, higher-order spatial representation, calculation, and language. Anatomically, our results in humans are closely homologous to the known connectivity of parietal and frontal regions in the macaque monkey.

Neural networks involved in mathematical thinking: evidence from linear and non-linear analysis of electroencephalographic activity

Using linear and non-linear methods, electroencephalographic (EEG) signals were measured at various brain regions to provide information regarding patterns of local and coordinated activity during performance of three arithmetic tasks (number comparison, single-digit multiplication, and two-digit multiplication) and two control tasks that did not require arithmetic operations. It was hypothesized that these measures would reveal the engagement of local and increasingly complex cortical networks as a function of task specificity and complexity. Results indicated regionally increased neuronal signalling as a function of task complexity at frontal, temporal and parietal brain regions, although more robust task-related changes in EEG-indices of activation were derived over the left hemisphere. Both linear and non-linear indices of synchronization among EEG signals recorded from over different brain regions were consistent with the notion of more "local" processing for the number comparison task. Conversely, multiplication tasks were associated with a widespread pattern of distant signal synchronizations, which could potentially indicate increased demands for neural networks cooperation during performance of tasks that involve a greater number of cognitive operations.

Arithmetic and the brain

Recent studies in human neuroimaging, primate neurophysiology, and developmental neuropsychology indicate that the human ability for arithmetic has a tangible cerebral substrate. The human intraparietal sulcus is systematically activated in all number tasks and could host a central amodal representation of quantity. Areas of the precentral and inferior prefrontal cortex also activate when subjects engage in mental calculation. A monkey analogue of these parieto-frontal regions has recently been identified, and a neuronal population code for number has been characterized. Finally, pathologies of this system, leading to acalculia in adults or to developmental dyscalculia in children, are beginning to be understood, thus paving the way for brain-oriented intervention studies.