Behavioral and prefrontal representation of spatial proportions in the monkey

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.

Non-symbolic estimation of big and small ratios with accurate and noisy feedback

The ratio of two magnitudes can take one of two values depending on the order they are operated on: a 'big' ratio of the larger to smaller magnitude, or a 'small' ratio of the smaller to larger. Although big and small ratio scales have different metric properties and carry divergent predictions for perceptual comparison tasks, no psychophysical studies have directly compared them. Two experiments are reported in which subjects implicitly learned to compare pairs of brightnesses and line lengths by non-symbolic feedback based on the scaled big ratio, small ratio or difference of the magnitudes presented. Results of Experiment 1 showed all three operations were learned quickly and estimated with a high degree of accuracy that did not significantly differ across groups or between intensive and extensive modalities, though regressions on individual data suggested an overall predisposition towards differences. Experiment 2 tested whether subjects learned to estimate the operation trained or to associate stimulus pairs with correct responses. For each operation, Gaussian noise was added to the feedback that was constant for repetitions of each pair. For all subjects, coefficients for the added noise component were negative when entered in a regression model alongside the trained differences or ratios, and were statistically significant in 80% of individual cases. Thus, subjects learned to estimate the comparative operations and effectively ignored or suppressed the added noise. These results suggest the perceptual system is highly flexible in its capacity for non-symbolic computation, which may reflect a deeper connection between perceptual structure and mathematics.

Decoding of spatial proportions using somatosensory feedback in sighted and visually impaired children

BACKGROUND AND PURPOSE

Humans can naturally operate with ratios of continuous magnitudes (proportions). We asked if sighted children (S) and visually impaired children (VI) can discriminate proportions via somatosensory feedback.

PROCEDURES

Children formed a proportion by tracing a pair of straight lines with their finger, and compared this proportion with a second proportion resulting from the tracing of another pair of lines.

MAIN FINDINGS

Performance was 68% in S, thus significantly lower (p < 0.001) compared to VI (75%). Tracing velocity (p < 0.01) and trial-to-trial variability of tracing velocity (p < 0.05) was higher in S compared to VI.

CONCLUSIONS

Operating with proportions solely from somatosensory feedback is possible, thus tracing lines might support learning in mathematics education. Kinematic variables point to the reason for the difference between S and VI, in that higher trial-to-trial variability in velocity in S leads to biased estimation of absolute line lengths.

Comparison of visual quantities in untrained neural networks

The ability to compare quantities of visual objects with two distinct measures, proportion and difference, is observed even in newborn animals. However, how this function originates in the brain, even before visual experience, remains unknown. Here, we propose a model in which neuronal tuning for quantity comparisons can arise spontaneously in completely untrained neural circuits. Using a biologically inspired model neural network, we find that single units selective to proportions and differences between visual quantities emerge in randomly initialized feedforward wirings and that they enable the network to perform quantity comparison tasks. Notably, we find that two distinct tunings to proportion and difference originate from a random summation of monotonic, nonlinear neural activities and that a slight difference in the nonlinear response function determines the type of measure. Our results suggest that visual quantity comparisons are primitive types of functions that can emerge spontaneously before learning in young brains.

Crows flexibly apply statistical inferences based on previous experience

Statistical inference, the ability to use limited information to draw conclusions about the likelihood of an event, is critical for decision-making during uncertainty. The ability to make statistical inferences was thought to be a uniquely human skill requiring verbal instruction and mathematical reasoning.1 However, basic inferences have been demonstrated in both preliterate and pre-numerate individuals,2,3,4,5,6,7 as well as non-human primates.8 More recently, the ability to make statistical inferences has been extended to members outside of the primate lineage in birds.9,10 True statistical inference requires subjects use relative rather than absolute frequency of previously experienced events. Here, we show that crows can relate memorized reward probabilities to infer reward-maximizing decisions. Two crows were trained to associate multiple reward probabilities ranging from 10% to 90% to arbitrary stimuli. When later faced with the choice between various stimulus combinations, crows retrieved the reward probabilities associated with individual stimuli from memory and used them to gain maximum reward. The crows showed behavioral distance and size effects when judging reward values, indicating that the crows represented probabilities as abstract magnitudes. When controlling for absolute reward frequency, crows still made reward-maximizing choices, which is the signature of true statistical inference. Our study provides compelling evidence of decision-making by relative reward frequency in a statistical inference task.

Categorical representation of abstract spatial magnitudes in the executive telencephalon of crows

The ability to group abstract continuous magnitudes into meaningful categories is cognitively demanding but key to intelligent behavior. To explore its neuronal mechanisms, we trained carrion crows to categorize lines of variable lengths into arbitrary "short" and "long" categories. Single-neuron activity in the nidopallium caudolaterale (NCL) of behaving crows reflected the learned length categories of visual stimuli. The length categories could be reliably decoded from neuronal population activity to predict the crows' conceptual decisions. NCL activity changed with learning when a crow was retrained with the same stimuli assigned to more categories with new boundaries ("short", "medium," and "long"). Categorical neuronal representations emerged dynamically so that sensory length information at the beginning of the trial was transformed into behaviorally relevant categorical representations shortly before the crows' decision making. Our data show malleable categorization capabilities for abstract spatial magnitudes mediated by the flexible networks of the crow NCL.

The developmental relationship between nonsymbolic and symbolic fraction abilities

A fundamental research question in quantitative cognition concerns the developmental relationship between nonsymbolic and symbolic quantitative abilities. This study examined this developmental relationship in abilities to process nonsymbolic and symbolic fractions. There were 99 6th graders (M_age = 11.86 years), 101 10th graders (M_age = 15.71 years), and 102 undergraduate and graduate students (M_age = 21.97 years) participating in this study, and their nonsymbolic and symbolic fraction abilities were measured with nonsymbolic and symbolic fraction comparison tasks, respectively. Nonsymbolic and symbolic fraction abilities were significantly correlated in all age groups even after controlling for the ability to process nonsymbolic absolute quantity and general cognitive abilities, including working memory and inhibitory control. Moreover, the strength of nonsymbolic-symbolic correlations was higher in 6th graders than in 10th graders and adults. These findings suggest a weakened association between nonsymbolic and symbolic fraction abilities during development, and this developmental pattern may be related with participants' increasing proficiency in symbolic fractions.

Circling around number: People can accurately extract numeric values from circle area ratios

It has long been known that people have the ability to estimate numerical quantities without counting. A standard account is that people develop a sense of the size of symbolic numbers by learning to map symbolic numbers (e.g., 6) to their corresponding numerosities (e.g. :::) and concomitant approximate magnitude system (ANS) representations. However, we here demonstrate that adults are capable of extracting fractional numerical quantities from non-symbolic visual ratios (i.e., labeling a ratio of two circle areas with the appropriate symbolic fraction). Not only were adult participants able to perform this task, but they were remarkably accurate: linear regressions on median estimates yielded slopes near 1, and accounted for 97% of the variability. Participants also performed at least as well on line-estimation and ratio-estimation tasks using non-numeric circular stimuli as they did in earlier experiments using non-symbolic numerosities, which are frequently considered to be numeric stimuli. We discuss results as consistent with accounts suggesting that non-symbolic ratios have the potential to act as a reliable and stable ground for symbolic number, even when composed of non-numeric stimuli.

Approach direction and accuracy, but not response times, show spatial-numerical association in chicks

Chicks trained to identify a target item in a sagittally-oriented series of identical items show a higher accuracy for the target on the left, rather than that on the right, at test when the series was rotated by 90°. Such bias seems to be due to a right hemispheric dominance in visuospatial tasks. Up to now, the bias was highlighted by looking at accuracy, the measure mostly used in non-human studies to detect spatial numerical association, SNA. In the present study, processing by each hemisphere was assessed by scoring three variables: accuracy, response times and direction of approach. Domestic chicks were tested under monocular vision conditions, as in the avian brain input to each eye is mostly processed by the contralateral hemisphere. Four-day-old chicks learnt to peck at the 4th element in a sagittal series of 10 identical elements. At test, when facing a series oriented fronto-parallel, birds confined their responses to the visible hemifield, with high accuracy for the 4th element. The first element in the series was also highly selected, suggesting an anchoring strategy to start the proto-counting at one end of the series. In the left monocular condition, chicks approached the series starting from the left, and in the right monocular condition, they started from the right. Both hemispheres appear to exploit the same strategy, scanning the series from the most lateral element in the clear hemifield. Remarkably, there was no effect in the response times: equal latency was scored for correct or incorrect and for left vs. right responses. Overall, these data indicate that the measures implying a direction of choice, accuracy and direction of approach, and not velocity, i.e., response times, can highlight SNA in this paradigm. We discuss the relevance of the selected measures to unveil SNA.

More than the sum of its parts: Exploring the development of ratio magnitude versus simple magnitude perception

Humans perceptually extract quantity information from our environments, be it from simple stimuli in isolation, or from relational magnitudes formed by taking ratios of pairs of simple stimuli. Some have proposed that these two types of magnitude are processed by a common system, whereas others have proposed separate systems. To test these competing possibilities, the present study examined the developmental trajectories of simple and relational magnitude discrimination and relations among these abilities for preschoolers (n = 42), 2nd-graders (n = 31), 5th-graders (n = 29), and adults (n = 32). Participants completed simple magnitude and ratio discrimination tasks in four different nonsymbolic formats, using dots, lines, circles, and irregular blobs. All age cohorts accurately discriminated both simple and ratio magnitudes. Discriminability differed by format such that performance was highest with line and lowest with dot stimuli. Moreover, developmental trajectories calculated for each format were similar across simple and ratio discriminations. Although some characteristics were similar for both types of discrimination, ratio acuity in a given format was more closely related with ratio acuities in alternate formats than to within-format simple magnitude acuity. Results demonstrate that ratio magnitude processing shares several similarities to simple magnitude processing, but is also substantially different.

First and Second Graders Successfully Reason About Ratios With Both Dot Arrays and Arabic Numerals

Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio reasoning in both formats and improved with age. Children who engaged in the nonsymbolic task first performed better on the symbolic task compared to children tested in the reverse order, suggesting that nonsymbolic ratio reasoning may function as a scaffold for symbolic ratio reasoning. Nonsymbolic ratio reasoning mediated the relation between children's numerosity comparison performance and symbolic mathematics performance in the domain of probabilities, but numerosity comparison performance explained significant unique variance in general numeration skills.

Symbolic fractions elicit an analog magnitude representation in school-age children

A fundamental question about fractions is whether they are grounded in an abstract nonsymbolic magnitude code similar to that postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving nonsymbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second- and fifth-grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, nonsymbolic ratios, and mixed symbolic-nonsymbolic pairs. This paradigm allowed us to test three key questions: (a) whether children show an analog magnitude code for rational numbers, (b) whether that code is compatible with mental representations of symbolic fractions, and (c) how formal education with fractions affects the symbolic-nonsymbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth graders' reaction times and error rates showed classic distance and notation effects. Nonsymbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions but had a smaller advantage for nonsymbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance and that fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a nonsymbolic ratio magnitude code and that symbolic fractions can be translated into that magnitude code.

One-to-one correspondence without language

A logical rule important in counting and representing exact number is one-to-one correspondence, the understanding that two sets are equal if each item in one set corresponds to exactly one item in the second set. The role of this rule in children's development of counting remains unclear, possibly due to individual differences in the development of language. We report that non-human primates, which do not have language, have at least a partial understanding of this principle. Baboons were given a quantity discrimination task where two caches were baited with different quantities of food. When the quantities were baited in a manner that highlighted the one-to-one relation between those quantities, baboons performed significantly better than when one-to-one correspondence cues were not provided. The implication is that one-to-one correspondence, which requires intuitions about equality and is a possible building block of counting, has a pre-linguistic origin.

The Relational SNARC: Spatial Representation of Nonsymbolic Ratios

Recent research in numerical cognition has begun to systematically detail the ability of humans and nonhuman animals to perceive the magnitudes of nonsymbolic ratios. These relationally defined analogs to rational numbers offer new potential insights into the nature of human numerical processing. However, research into their similarities with and connections to symbolic numbers remains in its infancy. The current research aims to further explore these similarities by investigating whether the magnitudes of nonsymbolic ratios are associated with space just as symbolic numbers are. In two experiments, we found that responses were faster on the left for smaller nonsymbolic ratio magnitudes and faster on the right for larger nonsymbolic ratio magnitudes. These results further elucidate the nature of nonsymbolic ratio processing, extending the literature of spatial-numerical associations to nonsymbolic relative magnitudes. We discuss potential implications of these findings for theories of human magnitude processing in general and how this general processing relates to numerical processing.

Task Constraints Affect Mapping From Approximate Number System Estimates to Symbolic Numbers

The Approximate Number System (ANS) allows individuals to assess nonsymbolic numerical magnitudes (e.g., the number of apples on a tree) without counting. Several prominent theories posit that human understanding of symbolic numbers is based – at least in part – on mapping number symbols (e.g., 14) to their ANS-processed nonsymbolic analogs. Number-line estimation – where participants place numerical values on a bounded number-line – has become a key task used in research on this mapping. However, some research suggests that such number-line estimation tasks are actually proportion judgment tasks, as number-line estimation requires people to estimate the magnitude of the to-be-placed value, relative to set upper and lower endpoints, and thus do not so directly reflect magnitude representations. Here, we extend this work, assessing performance on nonsymbolic tasks that should more directly interface with the ANS. We compared adults’ (n = 31) performance when placing nonsymbolic numerosities (dot arrays) on number-lines to their performance with the same stimuli on two other tasks: Free estimation tasks where participants simply estimate the cardinality of dot arrays, and ratio estimation tasks where participants estimate the ratio instantiated by a pair of arrays. We found that performance on these tasks was quite different, with number-line and ratio estimation tasks failing to the show classic psychophysical error patterns of scalar variability seen in the free estimation task. We conclude the constraints of tasks using stimuli that access the ANS lead to considerably different mapping performance and that these differences must be accounted for when evaluating theories of numerical cognition. Additionally, participants showed typical underestimation patterns in the free estimation task, but were quite accurate on the ratio task. We discuss potential implications of these findings for theories regarding the mapping between ANS magnitudes and symbolic numbers.

Natural Alternatives to Natural Number: The Case of Ratio

The overwhelming majority of efforts to cultivate early mathematical thinking rely primarily on counting and associated natural number concepts. Unfortunately, natural numbers and discretized thinking do not align well with a large swath of the mathematical concepts we wish for children to learn. This misalignment presents an important impediment to teaching and learning. We suggest that one way to circumvent these pitfalls is to leverage students' non-numerical experiences that can provide intuitive access to foundational mathematical concepts. Specifically, we advocate for explicitly leveraging a) students' perceptually based intuitions about quantity and b) students' reasoning about change and variation, and we address the affordances offered by this approach. We argue that it can support ways of thinking that may at times align better with to-be-learned mathematical ideas, and thus may serve as a productive alternative for particular mathematical concepts when compared to number. We illustrate this argument using the domain of ratio, and we do so from the distinct disciplinary lenses we employ respectively as a cognitive psychologist and as a mathematics education researcher. Finally, we discuss the potential for productive synthesis given the substantial differences in our preferred methods and general epistemologies.

Spatially Tuned Neurons in Corvid Nidopallium Caudolaterale Signal Target Position During Visual Search

The avian pallial endbrain area nidopallium caudolaterale (NCL) shows important similarities to mammalian prefrontal cortex in connectivity, dopamine neurochemistry, and function. Neuronal processing in NCL has been studied with respect to sensory, cognitive, and reward information, but little is known about its role in more direct control of motor behavior. We investigated NCL activity during the choice period of a delayed match-to-sample task, as 2 trained crows searched and selected a previously remembered visual target among an array of 4 pictures. The crows exhibited behavioral response patterns consistent with serial visual search. Many single NCL neurons were spatially tuned to specific target positions during visual search and directed motor behavior. Moreover, single NCL neurons dynamically changed their tuning properties to represent different behaviorally relevant task variables across the trial. In consecutive task periods, single neurons responded to visual stimuli, stored stimulus information in working memory, guided goal-directed behavior depending on the remembered target picture, and encoded trial outcomes. This flexible encoding of all task-relevant aspects in the executive control of goal-directed behavior represents a striking convergence to neuronal encoding in primate prefrontal cortex. These data highlight key properties of associative endbrain areas underlying flexible cognitive behavior in corvids and primates.

From continuous magnitudes to symbolic numbers: The centrality of ratio

Leibovich et al.'s theory neither accounts for the deep connections between whole numbers and other classes of number nor provides a potential mechanism for mapping continuous magnitudes to symbolic numbers. We argue that focusing on non-symbolic ratio processing abilities can furnish a more expansive account of numerical cognition that remedies these shortcomings.

Experimental Evidence From Newborn Chicks Enriches Our Knowledge on Human Spatial-Numerical Associations

Núñez and Fias raised concerns on whether our results demonstrate a linear number-space mapping. Patro and Nuerk urge caution on the use of animal models to understand the origin (cultural vs. biological) of the orientation of spatial-numerical association. Here, we discuss why both objections are unfounded.

Numerical Proportion Representation: A Neurocomputational Account

Proportion representation is an emerging subdomain in numerical cognition. However, its nature and its correlation with simple number representation remain elusive, especially at the theoretical level. To fill this gap, we propose a gain-field model of proportion representation to shed light on the neural and computational basis of proportion representation. The model is based on two well-supported neuroscientific findings. The first, gain modulation, is a general mechanism for information integration in the brain; the second relevant finding is how simple quantity is neurally represented. Based on these principles, the model accounts for recent relevant proportion representation data at both behavioral and neural levels. The model further addresses two key computational problems for the cognitive processing of proportions: invariance and generalization. Finally, the model provides pointers for future empirical testing.

Spontaneous, modality-general abstraction of a ratio scale

The existence of a generalized magnitude system in the human mind and brain has been studied extensively but remains elusive because it has not been clearly defined. Here we show that one possibility is the representation of relative magnitudes via ratio calculations: ratios are a naturally dimensionless or abstract quantity that could qualify as a common currency for magnitudes measured on vastly different psychophysical scales and in different sensory modalities like size, number, duration, and loudness. In a series of demonstrations based on comparisons of item sequences, we demonstrate that subjects spontaneously use knowledge of inter-item ratios within and across sensory modalities and across magnitude domains to rate sequences as more or less similar on a sliding scale. Moreover, they rate ratio-preserved sequences as more similar to each other than sequences in which only ordinal relations are preserved, indicating that subjects are aware of differences in levels of relative-magnitude information preservation. The ubiquity of this ability across many different magnitude pairs, even those sharing no sensory information, suggests a highly general code that could qualify as a candidate for a generalized magnitude representation.

Comparative cognition of number and space: the case of geometry and of the mental number line

Evidence is discussed about the use of geometric information for spatial orientation and the association between space and numbers in non-human animals. A variety of vertebrate species can reorient using simple Euclidian geometry of the environmental surface layout, i.e. in accord with metric and sense (right/left) relationships among extended surfaces. There seems to be a primacy of geometric over non-geometric information in spatial reorientation and, possibly, innate encoding of the sense of direction. The hippocampal formation plays a key role in geometry-based reorientation in mammals, birds, amphibians and fish. Although some invertebrate species show similar behaviours, it is unclear whether the underlying mechanisms are the same as in vertebrates. As to the links between space and number representations, a disposition to associate numerical magnitudes onto a left-to-right-oriented mental number line appears to exist independently of socio-cultural factors, and can be observed in animals with very little numerical experience, such as newborn chicks and human infants. Such evidence supports a nativistic foundation of number-space association. Some speculation about the possible underlying mechanisms is provided together with consideration on the difficulties inherent to any comparison among species of different taxonomic groups.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Fractions We Cannot Ignore: The Nonsymbolic Ratio Congruity Effect

Although many researchers theorize that primitive numerosity processing abilities may lay the foundation for whole number concepts, other classes of numbers, like fractions, are sometimes assumed to be inaccessible to primitive architectures. This research presents evidence that the automatic processing of nonsymbolic magnitudes affects processing of symbolic fractions. Participants completed modified Stroop tasks in which they selected the larger of two symbolic fractions while the ratios of the fonts in which the fractions were printed and the overall sizes of the compared fractions were manipulated as irrelevant dimensions. Participants were slower and less accurate when nonsymbolic dimensions of printed fractions were incongruent with the symbolic comparison decision. Results indicated a robust basic sensitivity to nonsymbolic ratios that exceeds prior conceptions about the accessibility of fraction values. Results also indicated a congruity effect for overall fraction size, contrary to findings of prior research. These findings have implications for extending theory about the nature of human number sense and mathematical cognition more generally.

Ratio abstraction over discrete magnitudes by newly hatched domestic chicks (Gallus gallus)

A large body of literature shows that non-human animals master a variety of numerical tasks, but studies involving proportional discrimination are sparse and primarily done with mature animals. Here we trained 4-day-old domestic chicks (Gallus gallus) to respond to stimuli depicting multiple examples of the proportion 4:1 when compared with the proportion 2:1. Stimuli were composed of green and red dot arrays; for the rewarded 4:1 proportion, 4 green dots for every red dot (e.g. ratios: 32:8, 12:3, and 44:11). The birds continued to discriminate when presented with new ratios at test (such as 20:5), characterized by new numbers of dots and new spatial configurations (Experiment 1). This indicates that chicks can extract the common proportional value shared by different ratios and apply it to new ones. In Experiment 2, chicks identified a specific proportion (2:1) from either a smaller (4:1) or a larger one (1:1), demonstrating an ability to represent the specific, and not relative, value of a particular proportion. Again, at test, chicks selectively responded to the previously reinforced proportion from new ratios. These findings provide strong evidence for very young animals’ ability to extract, identify, and productively use proportion information across a range of different amounts.

Mapping number to space in the two hemispheres of the avian brain

Pre-verbal infants and non-human animals associate small numbers with the left space and large numbers with the right space. Birds and primates, trained to identify a given position in a sagittal series of identical positions, whenever required to respond on a left/right oriented series, referred the given position starting from the left end. Here, we extended this evidence by selectively investigating the role of either cerebral hemisphere, using the temporary monocular occlusion technique. In birds, lacking the corpus callosum, visual input is fed mainly to the contralateral hemisphere. We trained 4-day-old chicks to identify the 4th element in a sagittal series of 10 identical elements. At test, the series was identical but left/right oriented. Test was conducted in right monocular, left monocular or binocular condition of vision. Right monocular chicks pecked at the 4th right element; left monocular and binocular chicks pecked at the 4th left element. Data on monocular chicks demonstrate that both hemispheres deal with an ordinal (sequential) task. Data on binocular chicks indicate that the left bias is linked to a right hemisphere dominance, that allocates the attention toward the left hemispace. This constitutes a first step towards understanding the neural basis of number space mapping.

The neuronal code for number

Humans and non-human primates share an elemental quantification system that resides in a dedicated neural network in the parietal and frontal lobes. In this cortical network, 'number neurons' encode the number of elements in a set, its cardinality or numerosity, irrespective of stimulus appearance across sensory motor systems, and from both spatial and temporal presentation arrays. After numbers have been extracted from sensory input, they need to be processed to support goal-directed behaviour. Studying number neurons provides insights into how information is maintained in working memory and transformed in tasks that require rule-based decisions. Beyond an understanding of how cardinal numbers are encoded, number processing provides a window into the neuronal mechanisms of high-level brain functions.

Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance

What basic capacities lay the foundation for advanced numerical cognition? Are there basic nonsymbolic abilities that support the understanding of advanced numerical concepts, such as fractions? To date, most theories have posited that previously identified core numerical systems, such as the approximate number system (ANS), are ill-suited for learning fraction concepts. However, recent research in developmental psychology and neuroscience has revealed a ratio-processing system (RPS) that is sensitive to magnitudes of nonsymbolic ratios and may be ideally suited for supporting fraction concepts. We provide evidence for this hypothesis by showing that individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university entrance exam in algebra. We suggest that the nonsymbolic RPS may support symbolic fraction understanding much as the ANS supports whole-number concepts. Thus, even abstract mathematical concepts, such as fractions, may be grounded not only in higher-order logic and language, but also in basic nonsymbolic processing abilities.

Comparison of discrete ratios by rhesus macaques (Macaca mulatta)

Perceiving and comparing ratios are crucial skills for humans. Little is known about whether other animals can compare ratios. We trained two rhesus macaques (Macaca mulatta) to choose arrays that contained the greater ratio of positive to negative stimuli, regardless of the absolute number of stimuli in each of the two choice arrays. Subjects learned this task, and their performance generalized to novel ratios. Moreover, performance was modulated by the ratio between ratios; subjects responded more quickly and accurately when the ratio between ratios was higher. Control conditions ruled out the possibility that subjects were relying on surface area, although the ratio between ratios of surface area did seem to influence their choices. Our results demonstrate that rhesus monkeys can compare discrete ratios, demonstrating not only proportional reasoning ability but also the ability to reason about relations between relations.

Audio-vocal interaction in single neurons of the monkey ventrolateral prefrontal cortex

Complex audio-vocal integration systems depend on a strong interconnection between the auditory and the vocal motor system. To gain cognitive control over audio-vocal interaction during vocal motor control, the PFC needs to be involved. Neurons in the ventrolateral PFC (VLPFC) have been shown to separately encode the sensory perceptions and motor production of vocalizations. It is unknown, however, whether single neurons in the PFC reflect audio-vocal interactions. We therefore recorded single-unit activity in the VLPFC of rhesus monkeys (Macaca mulatta) while they produced vocalizations on command or passively listened to monkey calls. We found that 12% of randomly selected neurons in VLPFC modulated their discharge rate in response to acoustic stimulation with species-specific calls. Almost three-fourths of these auditory neurons showed an additional modulation of their discharge rates either before and/or during the monkeys' motor production of vocalization. Based on these audio-vocal interactions, the VLPFC might be well positioned to combine higher order auditory processing with cognitive control of the vocal motor output. Such audio-vocal integration processes in the VLPFC might constitute a precursor for the evolution of complex learned audio-vocal integration systems, ultimately giving rise to human speech.

Neurons selective to the number of visual items in the corvid songbird endbrain

It is unknown whether anatomical specializations in the endbrains of different vertebrates determine the neuronal code to represent numerical quantity. Therefore, we recorded single-neuron activity from the endbrain of crows trained to judge the number of items in displays. Many neurons were tuned for numerosities irrespective of the physical appearance of the items, and their activity correlated with performance outcome. Comparison of both behavioral and neuronal representations of numerosity revealed that the data are best described by a logarithmically compressed scaling of numerical information, as postulated by the Weber-Fechner law. The behavioral and neuronal numerosity representations in the crow reflect surprisingly well those found in the primate association cortex. This finding suggests that distantly related vertebrates with independently developed endbrains adopted similar neuronal solutions to process quantity.

Eye movements reflect and shape strategies in fraction comparison

The comparison of fractions is a difficult task that can often be facilitated by separately comparing components (numerators and denominators) of the fractions—that is, by applying so-called component-based strategies. The usefulness of such strategies depends on the type of fraction pair to be compared. We investigated the temporal organization and the flexibility of strategy deployment in fraction comparison by evaluating sequences of eye movements in 20 young adults. We found that component-based strategies could account for the response times and the overall number of fixations observed for the different fraction pairs. The analysis of eye movement sequences showed that the initial eye movements in a trial were characterized by stereotypical scanning patterns indicative of an exploratory phase that served to establish the kind of fraction pair presented. Eye movements that followed this phase adapted to the particular type of fraction pair and indicated the deployment of specific comparison strategies. These results demonstrate that participants employ eye movements systematically to support strategy use in fraction comparison. Participants showed a remarkable flexibility to adapt to the most efficient strategy on a trial-by-trial basis. Our results confirm the value of eye movement measurements in the exploration of strategic adaptation in complex tasks.

Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes

This study presents evidence that humans have intuitive, perceptually based access to the abstract fraction magnitudes instantiated by nonsymbolic ratio stimuli. Moreover, it shows these perceptually accessed magnitudes can be easily compared with symbolically represented fractions. In cross-format comparisons, participants picked the larger of two ratios. Ratios were presented either symbolically as fractions or nonsymbolically as paired dot arrays or as paired circles. Response patterns were consistent with participants comparing specific analog fractional magnitudes independently of the particular formats in which they were presented. These results pose a challenge to accounts that argue human cognitive architecture is ill-suited for processing fractions. Instead, it seems that humans can process nonsymbolic ratio magnitudes via perceptual routes and without recourse to conscious symbolic algorithms, analogous to the processing of whole number magnitudes. These findings have important implications for theories regarding the nature of human number sense - they imply that fractions may in some sense be natural numbers, too.

The use of proportion by young domestic chicks (Gallus gallus)

We investigated whether 4-day-old domestic chicks can discriminate proportions. Chicks were trained to respond, via food reinforcement, to one of the two stimuli, each characterized by different proportions of red and green areas (¼ vs. ¾). In Experiment 1, chicks approached the proportion associated with food, even if at test the spatial dispositions of the two areas were novel. In Experiment 2, chicks responded on the basis of proportion even when the testing stimuli were of enlarged dimensions, creating a conflict between the absolute positive area experienced during training and the relative proportion of the two areas. However, chicks could have responded on the basis of the overall colour (red or green) of the figures rather than proportion per se. To control for this objection, in Experiment 3, we used new pairs of testing stimuli, each depicting a different number of small squares on a white background (i.e. 1 green and 3 red vs. 3 green and 1 red or 5 green and 15 red vs. 5 red and 15 green). Chicks were again able to respond to the correct proportion, showing they discriminated on the basis of proportion of continuous quantities and not on the basis of the prevalent colour or on the absolute amount of it. Data indicate that chicks can track continuous quantities through various manipulations, suggesting that proportions are information that can be processed by very young animals.

The development of the mental representations of the magnitude of fractions

We investigated the development of the mental representation of the magnitude of fractions during the initial stages of fraction learning in grade 5, 6 and 7 children as well as in adults. We examined the activation of global fraction magnitude in a numerical comparison task and a matching task. There were global distance effects in the comparison task, but not in the matching task. This suggests that the activation of the global magnitude representation of fractions is not automatic in all tasks involving magnitude judgments. The slope of the global distance effect increased during early fraction learning and declined by adulthood, demonstrating that the development of the fraction global distance effect differs from that of the integer distance effect.

The mental representations of fractions: adults' same-different judgments

Two experiments examined whether the processing of the magnitude of fractions is global or componential. Previously, some authors concluded that adults process the numerators and denominators of fractions separately and do not access the global magnitude of fractions. Conversely, others reported evidence suggesting that the global magnitude of fractions is accessed. We hypothesized that in a fraction matching task, participants automatically extract the magnitude of the components but that the activation of the global magnitude of the whole fraction is only optional or strategic. Participants carried out same/different judgment tasks. Two different tasks were used: a physical matching task and a numerical matching task. Pairs of fractions were presented either simultaneously or sequentially. Results showed that participants only accessed the representation of the global magnitude of fractions in the numerical matching task. The mode of stimulus presentation did not affect the processing of fractions. The present study allows a deeper understanding of the conditions in which the magnitude of fractions is mentally represented by using matching tasks and two different modes of presentation.

Comparison of abstract decision encoding in the monkey prefrontal cortex, the presupplementary, and cingulate motor areas

Deciding between alternatives is a critical element of flexible behavior. Perceptual decisions have been studied extensively in an action-based framework. Recently, we have shown that abstract perceptual decisions are encoded in prefrontal cortex (PFC) neurons ( Merten and Nieder 2012 ). However, the role of other frontal cortex areas remained elusive. Here, we trained monkeys to perform a rule-based visual detection task that disentangled abstract perceptual decisions from motor preparation. We recorded the single-neuron activity in the presupplementary (preSMA) and the rostral part of the cingulate motor area (CMAr) and compared it to the results previously found in the PFC. Neurons in both areas traditionally identified with motor planning process the abstract decision independently of any motor preparatory activity by similar mechanisms as the PFC. A larger proportion of decision neurons and a higher strength of decision encoding was found in the preSMA than in the PFC. Neurons in both areas reliably predicted the monkeys' decisions. The fraction of CMAr decision neurons and their strength of the decision encoding were comparable to the PFC. Our findings highlight the role of both preSMA and CMAr in abstract cognitive processing and emphasize that both frontal areas encode decisions prior to the preparation of a motor output.

Non-symbolic halving in an Amazonian indigene group

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS.

Coding of abstract quantity by 'number neurons' of the primate brain

Humans share with nonhuman animals a quantification system for representing the number of items as nonverbal mental magnitudes. Over the past decade, the anatomical substrates and neuronal mechanisms of this quantification system have been unraveled down to the level of single neurons. Work with behaviorally trained nonhuman primates identified a parieto-frontal cortical network with individual neurons selectively tuned to the number of items. Such 'number neurons' can track items across space, time, and modality to encode numerosity in a most abstract, supramodal way. The physiological properties of these neurons can explain fundamental psychophysical phenomena during numerosity judgments. Functionally overlapping groups of parietal neurons represent not only numerable-discrete quantity (numerosity), but also innumerable-continuous quantity (extent) and relations between quantities (proportions), supporting the idea of a generalized magnitude system in the brain. These studies establish putative homologies between the monkey and human brain and demonstrate the suitability of nonhuman primates as model system to explore the neurobiological roots of the brain's nonverbal quantification system, which may constitute the evolutionary foundation of all further, more elaborate numerical skills in humans.

Numerical rule coding in the prefrontal, premotor, and posterior parietal cortices of macaques

Switching flexibly between behavioral goals is a hallmark of executive control and requires integration of external and internal information. We recorded single-neuron correlates of different numerical representations (sensory-, working memory-, and rule-related activity) in the dorsal premotor area (PMd), the cingulate motor areas (CMA), and the ventral intraparietal sulcus (VIP) and compared them to previous recordings in the lateral prefrontal cortex (PFC). Two monkeys were trained to encode and memorize numerosities and flexibly switch between two abstract quantitative rules based on rule cues. Almost 20% of randomly selected PFC and PMd neurons significantly represented the numerical rule in a behaviorally relevant manner, approximately twice as many as in the CMA and VIP. Rule selectivity was significantly better for PMd neurons than for PFC cells. Seemingly at the expense of rule selectivity, however, sensory- and memory-related numerosity activity was greatly diminished compared with previous delayed match-to-numerosity studies. These findings suggest the involvement of the frontal premotor areas in strategic planning such as rule following. Moreover, the results emphasize that the coding capacities of neurons in association cortical areas are far more dynamic depending on task demands than previously thought.

Supramodal numerosity selectivity of neurons in primate prefrontal and posterior parietal cortices

Numerosity, the number of elements in a set, is a most abstract quantitative category. As such, it is independent of the sensory modality of its elements, i.e., supramodal. Because neuronal numerosity selectivity had never been compared directly across different sensory modalities, it remained elusive if and where single neurons encode numerosity irrespective of the items' modality. Here, monkeys were trained to discriminate both the number of auditory sounds and visual items within the same session. While the monkeys performed this task, the activity of neurons was recorded in the lateral prefrontal cortex and ventral intraparietal sulcus, structures critically involved in numerical cognition. Groups of neurons in both areas encoded either the number of auditory pulses, visual items, or both. The finding of neurons responding to numerosity irrespective of the sensory modality supports the idea of a nonverbal, supramodal neuronal code of numerical quantity in the primate brain.

Relating magnitudes: the brain's code for proportions

Whereas much is known about how we categorize and reason based on absolute quantity, data exploring ratios of quantities, as in proportions and fractions, are comparatively sparse. Until recently, it remained elusive whether these two representations of number are connected, how proportions are implemented by neurons and how language shapes this code. New data derived with complementary methods and from different model systems now shed light on the mechanisms of magnitude ratio representations. A coding scheme for proportions has emerged that is remarkably reminiscent of the representation of absolute number. These novel findings suggest a sense for ratios that grants the brain automatic access to proportions independently of language and the format of presentation.