Algebra in a man with severe aphasia

The calculating brain

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills require a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of nonsymbolic numerical quantity and then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. Although it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

Language is primarily a tool for communication rather than thought

Language is a defining characteristic of our species, but the function, or functions, that it serves has been debated for centuries. Here we bring recent evidence from neuroscience and allied disciplines to argue that in modern humans, language is a tool for communication, contrary to a prominent view that we use language for thinking. We begin by introducing the brain network that supports linguistic ability in humans. We then review evidence for a double dissociation between language and thought, and discuss several properties of language that suggest that it is optimized for communication. We conclude that although the emergence of language has unquestionably transformed human culture, language does not appear to be a prerequisite for complex thought, including symbolic thought. Instead, language is a powerful tool for the transmission of cultural knowledge; it plausibly co-evolved with our thinking and reasoning capacities, and only reflects, rather than gives rise to, the signature sophistication of human cognition.

High-definition transcranial direct current stimulation (HD-tDCS) of the left middle temporal gyrus (LMTG) improves mathematical reasoning

The role of the visuospatial network in mathematical processing has been established, but the role of the semantic neural network in mathematical processing is still poorly understood. The current study used high-definition transcranial direct current stimulation (HD-tDCS) to examine whether the semantic network supports mathematical processing. Using a single-blind, randomized, sham-controlled experimental design, 48 participants were randomly assigned to receive either anodal or sham HD-tDCS on the left middle temporal gyrus (LMTG), a core region of the semantic network. A number series completion task was used to measure mathematical reasoning and an arithmetical computation task was used as a control condition. Both tasks were administered before and after the 20 min HD-tDCS. The results showed that anodal HD-tDCS on the LMTG enhanced performance on the number series completion task, but not on the arithmetical computation task. Trial-level analysis further showed greater improvement at the more difficult problems of the number series completion task. These results demonstrate that the semantic network plays an important role in mathematical processing.

The involvement of the semantic neural network in rule identification of mathematical processing

The role of the visuospatial network in mathematical processing has been established, but the involvement of the semantic network in mathematical processing is still poorly understood. The current study utilized a number series completion paradigm with the event-related potential (ERP) technique to examine whether the semantic network supports mathematical processing and to find the corresponding spatiotemporal neural marker. In total, 32 right-handed undergraduate students were recruited and asked to complete the number series completion as well as the arithmetical computation task in which numbers were presented in sequence. The event-related potential and multi-voxel pattern analysis showed that the rule identification process involves more semantic processing when compared with the arithmetical computation processes, and it elicited higher amplitudes for the late negative component (LNC) in left frontal and temporal lobes. These results demonstrated that the semantic network supports the rule identification in mathematical processing, with the LNC acting as the neural marker.

Entropy, complexity, and maturity in children’s neural responses to naturalistic video lessons

Temporal characteristics of neural signals are often overlooked in traditional fMRI developmental studies but are critical to studying brain functions in ecologically valid settings. In the present study, we explore the temporal properties of children's neural responses during naturalistic mathematics and grammar tasks. To do so, we introduce a novel measure in developmental fMRI: neural entropy, which indicates temporal complexity of BOLD signals. We show that temporal patterns of neural activity have lower complexity and greater variability in children than in adults in the association cortex but not in the sensory-motor cortex. We also show that neural entropy is associated with both child-adult similarity in functional connectivity and neural synchrony, and that neural entropy increases with the size of functionally connected networks in the association cortex. In addition, neural entropy increases with functional maturity (i.e., child-adult neural synchrony) in content-specific regions. These exploratory findings suggest the hypothesis that neural entropy indexes the increasing breadth and diversity of neural processes available to children for analyzing mathematical information over development.

Common Neural Functions during Children's Learning from Naturalistic and Controlled Mathematics Paradigms

Two major goals of human neuroscience are to understand how the brain functions in the real world and to measure neural processes under conditions that are ecologically valid. A critical step toward these goals is understanding how brain activity during naturalistic tasks that mimic the real world relates to brain activity in more traditional laboratory tasks. In this study, we used intersubject correlations to locate reliable stimulus-driven cerebral processes among children and adults in a naturalistic video lesson and a laboratory forced-choice task that shared the same arithmetic concept. We show that relative to a control condition with grammatical content, naturalistic and laboratory arithmetic tasks evoked overlapping activation within brain regions previously associated with math semantics. The regions of specific functional overlap between the naturalistic mathematics lesson and laboratory mathematics task included bilateral intraparietal cortex, which confirms that this region processes mathematical content independently of differences in task mode. These findings suggest that regions of the intraparietal cortex process mathematical content when children are learning about mathematics in a naturalistic setting.

Cortical collateralization induced by language and arithmetic in non-right-handers

The functional overlap of language and arithmetic is debatable. Although some studies have reported independent representations of arithmetic and language in the brain, other studies have reported shared activity of the two cognitive domains in the inferior frontal gyrus. Although most previous studies have evaluated right-handed individuals, variability of hemispheric dominance in non-right-handed individuals should provide important information on the functional collateralization of these two cognitive domains. The present study evaluated the cortical lateralization patterns of the two cognitive domains using functional magnetic resonance imaging in 30 non-right-handed participants who performed language and arithmetic tasks. We found that language and arithmetic tasks demonstrated shared activity in the bilateral inferior frontal gyrus (IFG). Furthermore, the lateralization patterns of language and arithmetic tasks were correlated with each other. Most participants with language dominance in the left hemisphere also exhibited dominance of arithmetic tasks in the left hemisphere; similarly, most participants with language dominance in the right hemisphere exhibited dominance of arithmetic tasks in the right hemisphere. Among all the brain regions, the precentral gyrus, which is located slightly posterior to the IFG, exhibited the highest correlation coefficient between laterality indices of language and arithmetic tasks. These results suggest a shared functional property between language and arithmetic in the brain.

Neural Evidence of Cross-domain Structural Interaction between Language and Arithmetic

The presence of a shared neural system for the syntactic processing in language and arithmetic is controversial. Recent behavioral studies reported a cross-domain structural priming between language and arithmetic. Using functional magnetic resonance imaging, we examined whether the neural activation reflects the structural interaction between language and arithmetic. We prepared sentences and arithmetic expressions (A-expressions) with same and different syntactic structures and presented structurally congruent/incongruent pairs consecutively. By directly comparing activations in the congruent and incongruent conditions, we observed significant repetition suppression effect in the regions including the bilateral inferior frontal gyrus, i.e., neural activation with an A-expression decreased after a sentence with the same syntactic structure (and vice versa). The results strongly support the idea that arithmetic and language share the neural basis for processing syntactic structures.

Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain's semantic networks

Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that 'the sine function is periodical' or that 'London buses are red', activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Origins of the brain networks for advanced mathematics in expert mathematicians

The origins of human abilities for mathematics are debated: Some theories suggest that they are founded upon evolutionarily ancient brain circuits for number and space and others that they are grounded in language competence. To evaluate what brain systems underlie higher mathematics, we scanned professional mathematicians and mathematically naive subjects of equal academic standing as they evaluated the truth of advanced mathematical and nonmathematical statements. In professional mathematicians only, mathematical statements, whether in algebra, analysis, topology or geometry, activated a reproducible set of bilateral frontal, Intraparietal, and ventrolateral temporal regions. Crucially, these activations spared areas related to language and to general-knowledge semantics. Rather, mathematical judgments were related to an amplification of brain activity at sites that are activated by numbers and formulas in nonmathematicians, with a corresponding reduction in nearby face responses. The evidence suggests that high-level mathematical expertise and basic number sense share common roots in a nonlinguistic brain circuit.

Neural mechanisms underlying the computation of hierarchical tree structures in mathematics

Whether mathematical and linguistic processes share the same neural mechanisms has been a matter of controversy. By examining various sentence structures, we recently demonstrated that activations in the left inferior frontal gyrus (L. IFG) and left supramarginal gyrus (L. SMG) were modulated by the Degree of Merger (DoM), a measure for the complexity of tree structures. In the present study, we hypothesize that the DoM is also critical in mathematical calculations, and clarify whether the DoM in the hierarchical tree structures modulates activations in these regions. We tested an arithmetic task that involved linear and quadratic sequences with recursive computation. Using functional magnetic resonance imaging, we found significant activation in the L. IFG, L. SMG, bilateral intraparietal sulcus (IPS), and precuneus selectively among the tested conditions. We also confirmed that activations in the L. IFG and L. SMG were free from memory-related factors, and that activations in the bilateral IPS and precuneus were independent from other possible factors. Moreover, by fitting parametric models of eight factors, we found that the model of DoM in the hierarchical tree structures was the best to explain the modulation of activations in these five regions. Using dynamic causal modeling, we showed that the model with a modulatory effect for the connection from the L. IPS to the L. IFG, and with driving inputs into the L. IFG, was highly probable. The intrinsic, i.e., task-independent, connection from the L. IFG to the L. IPS, as well as that from the L. IPS to the R. IPS, would provide a feedforward signal, together with negative feedback connections. We indicate that mathematics and language share the network of the L. IFG and L. IPS/SMG for the computation of hierarchical tree structures, and that mathematics recruits the additional network of the L. IPS and R. IPS.

A functional dissociation between language and multiple-demand systems revealed in patterns of BOLD signal fluctuations

What is the relationship between language and other high-level cognitive functions? Neuroimaging studies have begun to illuminate this question, revealing that some brain regions are quite selectively engaged during language processing, whereas other "multiple-demand" (MD) regions are broadly engaged by diverse cognitive tasks. Nonetheless, the functional dissociation between the language and MD systems remains controversial. Here, we tackle this question with a synergistic combination of functional MRI methods: we first define candidate language-specific and MD regions in each subject individually (using functional localizers) and then measure blood oxygen level-dependent signal fluctuations in these regions during two naturalistic conditions ("rest" and story-comprehension). In both conditions, signal fluctuations strongly correlate among language regions as well as among MD regions, but correlations across systems are weak or negative. Moreover, data-driven clustering analyses based on these inter-region correlations consistently recover two clusters corresponding to the language and MD systems. Thus although each system forms an internally integrated whole, the two systems dissociate sharply from each other. This independent recruitment of the language and MD systems during cognitive processing is consistent with the hypothesis that these two systems support distinct cognitive functions.

The role of number words: the phonological length effect in multidigit addition

We explored the role of phonological representations of number words in exact calculation. The reaction times and accuracy of responses in multidigit addition problems were compared across three groups of participants (young healthy, older healthy, and 3 patients with severe aphasia) and two types of addition problems: phonologically long in English (containing the bisyllabic number word "seven") and short in English (monosyllabic number words-e.g., "six"). Older healthy participants were significantly faster and more accurate in calculation than younger healthy participants. The older participants showed no evidence of a phonological length effect. However this effect was apparent in the younger adults, with longer reaction times on phonologically long problems. Furthermore, there was an association between the presence of a phonological length effect and the overall speed of response, suggesting that less proficient calculators were more reliant on phonological mediation of performance. The aphasic participants retained the ability to complete multidigit additions and were as accurate as the younger healthy group, although the response times of two of the 3 patients were slow. The aphasic participants varied with regard to the presence of a phonological length effect. Two participants showed no evidence of phonological mediation, while 1 displayed a phonological length effect. The results suggest that language resources are not mandatory for exact addition, although they may be used to scaffold math performance in less competent calculators. Evidence of phonological mediation of performance in aphasic participants may provide insight into the integrity or otherwise of inner speech in severe aphasia.

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.