Representation of number in the brain

Zero-shot counting with a dual-stream neural network model

To understand a visual scene, observers need to both recognize objects and encode relational structure. For example, a scene comprising three apples requires the observer to encode concepts of "apple" and "three." In the primate brain, these functions rely on dual (ventral and dorsal) processing streams. Object recognition in primates has been successfully modeled with deep neural networks, but how scene structure (including numerosity) is encoded remains poorly understood. Here, we built a deep learning model, based on the dual-stream architecture of the primate brain, which is able to count items "zero-shot"-even if the objects themselves are unfamiliar. Our dual-stream network forms spatial response fields and lognormal number codes that resemble those observed in the macaque posterior parietal cortex. The dual-stream network also makes successful predictions about human counting behavior. Our results provide evidence for an enactive theory of the role of the posterior parietal cortex in visual scene understanding.

Single-neuron representation of nonsymbolic and symbolic number zero in the human medial temporal lobe

The number zero holds a special status among numbers, indispensable for developing a comprehensive number theory.1,2,3,4 Despite its importance in mathematics, the neuronal foundation of zero in the human brain is unknown. We conducted single-neuron recordings in neurosurgical patients5,6,7 while they made judgments involving nonsymbolic number representations (dot numerosity), including the empty set, and symbolic numbers (Arabic numerals), including numeral zero. Neurons showed responsiveness to either the empty set or numeral zero, but not both. Neuronal activity to zero in both nonsymbolic and symbolic formats exhibited a numerical distance effect, indicating that zero representations are integrated together with countable numerosities and positive integers at the low end of the number line.8,9 A boundary in neuronal coding existed between the nonsymbolic empty set and small numerosities, correlating with the relative difficulty in discriminating numerosity zero behaviorally. Conversely, no such boundary was found for symbolic zero activity, suggesting that symbolic representations integrate zero with other numerals along the number line, reconciling its outlier role. The status of zero as a special nonsymbolic numerical quantity is reflected in the activity of neurons in the human brain, which seems to serve as a scaffold for more advanced representations of zero as a symbolic number.

Spatial attention in mental arithmetic: A literature review and meta-analysis

We review the evidence for the conceptual association between arithmetic and space and quantify the effect size in meta-analyses. We focus on three effects: (a) the operational momentum effect (OME), which has been defined as participants' tendency to overestimate results of addition problems and underestimate results of subtraction problems; (b) the arithmetic cueing effect, in which arithmetic problems serve as spatial cues in target detection or temporal order judgment tasks; and (c) the associations between arithmetic and space observed with eye- and hand-tracking studies. The OME was consistently found in paradigms that provided the participants with numerical response alternatives. The OME shows a large effect size, driven by an underestimation during subtraction while addition was unbiased. In contrast, paradigms in which participants indicated their estimate by transcoding their final estimate to a spatial reference frame revealed no consistent OME. Arithmetic cueing studies show a reliable small to medium effect size, driven by a rightward bias for addition. Finally, eye- and hand-tracking studies point to replicable associations between arithmetic and eye or hand movements. To account for the complexity of the observed pattern, we introduce the Adaptive Pathways in Mental Arithmetic (APiMA) framework. The model accommodates central notions of numerical and arithmetic processing and helps identifying which pathway a given paradigm operates on. It proposes that the divergence between OME and arithmetic cueing studies comes from the predominant use of non-symbolic versus symbolic stimuli, respectively. Overall, our review and findings clearly support an association between arithmetic and spatial processing.

Short-term number sense training recapitulates long-term neurodevelopmental changes from childhood to adolescence

Number sense is fundamental to the development of numerical problem-solving skills. In early childhood, children establish associations between non-symbolic (e.g., a set of dots) and symbolic (e.g., Arabic numerals) representations of quantity. The developmental estrangement theory proposes that the relationship between non-symbolic and symbolic representations of quantity evolves with age, with increased dissociation across development. Consistent with this theory, recent research suggests that cross-format neural representational similarity (NRS) between non-symbolic and symbolic quantities is correlated with arithmetic fluency in children but not in adolescents. However, it is not known if short-term training (STT) can induce similar changes as long-term development. In this study, children aged 7-10 years underwent a theoretically motivated 4-week number sense training. Using multivariate neural pattern analysis, we investigated whether short-term learning could modify the relation between cross-format NRS and arithmetic skills. Our results revealed a significant correlation between cross-format NRS and arithmetic fluency in distributed brain regions, including the parietal and prefrontal cortices, prior to training. However, this association was no longer observed after training, and multivariate predictive models confirmed these findings. Our findings provide evidence that intensive STT during early childhood can promote behavioral improvements and neural plasticity that resemble and recapitulate long-term neurodevelopmental changes that occur from childhood to adolescence. More generally, our study contributes to our understanding of the malleability of number sense and highlights the potential for targeted interventions to shape neurodevelopmental trajectories in early childhood. RESEARCH HIGHLIGHTS: We tested the hypothesis that short-term number sense training induces the dissociation of symbolic numbers from non-symbolic representations of quantity in children. We leveraged a theoretically motivated intervention and multivariate pattern analysis to determine training-induced neurocognitive changes in the relation between number sense and arithmetic problem-solving skills. Neural representational similarity between non-symbolic and symbolic quantity representations was correlated with arithmetic skills before training but not after training. Short-term training recapitulates long-term neurodevelopmental changes associated with numerical problem-solving from childhood to adolescence.

Cracking the neural code for word recognition in convolutional neural networks

Learning to read places a strong challenge on the visual system. Years of expertise lead to a remarkable capacity to separate similar letters and encode their relative positions, thus distinguishing words such as FORM and FROM, invariantly over a large range of positions, sizes and fonts. How neural circuits achieve invariant word recognition remains unknown. Here, we address this issue by recycling deep neural network models initially trained for image recognition. We retrain them to recognize written words and then analyze how reading-specialized units emerge and operate across the successive layers. With literacy, a small subset of units becomes specialized for word recognition in the learned script, similar to the visual word form area (VWFA) in the human brain. We show that these units are sensitive to specific letter identities and their ordinal position from the left or the right of a word. The transition from retinotopic to ordinal position coding is achieved by a hierarchy of "space bigram" unit that detect the position of a letter relative to a blank space and that pool across low- and high-frequency-sensitive units from early layers of the network. The proposed scheme provides a plausible neural code for written words in the VWFA, and leads to predictions for reading behavior, error patterns, and the neurophysiology of reading.

Symbolic and non-symbolic representations of numerical zero in the human brain

Representing the quantity zero as a symbolic concept is considered a unique achievement of abstract human thought.1,2 To conceptualize zero, one must abstract away from the (absence of) sensory evidence to construct a representation of numerical absence: creating "something" out of "nothing."2,3,4 Previous investigations of the neural representation of natural numbers reveal distinct numerosity-selective neural populations that overlap in their tuning curves with adjacent numerosities.5,6 Importantly, a component of this neural code is thought to be invariant across non-symbolic and symbolic numerical formats.7,8,9,10,11 Although behavioral evidence indicates that zero occupies a place at the beginning of this mental number line,12,13,14 in humans zero is also associated with unique behavioral and developmental profiles compared to natural numbers,4,15,16,17 suggestive of a distinct neural basis for zero. We characterized the neural representation of zero in the human brain by employing two qualitatively different numerical tasks18,19 in concert with magnetoencephalography (MEG) recordings. We assay both neural representations of non-symbolic numerosities (dot patterns), including zero (empty sets), and symbolic numerals, including symbolic zero. Our results reveal that neural representations of zero are situated along a graded neural number line shared with other natural numbers. Notably, symbolic representations of zero generalized to predict non-symbolic empty sets. We go on to localize abstract representations of numerical zero to posterior association cortex, extending the purview of parietal cortex in human numerical cognition to encompass representations of zero.10,20.

(Dis)similarities between non-symbolic and symbolic number representations: Insights from vector space models

Empirical evidence in support of a shared system for non-symbolic and symbolic number processing has been inconclusive. The current study aims to address this question in a novel way, specifically by testing whether the efficient coding principle based on co-occurrence of number symbols in natural language holds for both non-symbolic and symbolic number processing. The efficient coding principle postulates that perception is optimized when stimuli frequently co-occur in a natural environment. We hypothesized that both numerical ratios and co-occurrence frequencies of symbolic numbers would significantly influence participants' performance on a non-symbolic and symbolic number comparison task. To test this hypothesis, we employed latent semantic analysis on a TASA corpus to quantify number co-occurrence in natural language and calculate language similarity estimates. We engaged 73 native English speakers (mean age = 19.36, standard deviation = 1.83) with normal or corrected vision and no learning disorders in a number comparison task involving non-symbolic (dot arrays) and symbolic stimuli (Arabic numerals and English number words). Results showed that numerical ratios significantly predicted participants' performances across all number formats (ps < 0.001). Language similarity estimates derived from everyday language also predicted performance on the non-symbolic task and the symbolic task involving number words (ps < 0.007). Our results highlight the complex nature of numerical processing, pointing to the co-occurrence of number symbols in natural language as an auxiliary factor in understanding the shared characteristics between non-symbolic and symbolic number representations. Given that our study focused on a limited number range (5 to 16) and a specific task type, future studies should explore a wider range of tasks and numbers to further test the role of the efficient coding principle in number processing.

Not all core knowledge systems are created equal, and they are subject to revision in both children and adults

Core knowledge systems play an important role in theories of cognitive development. However, recent studies suggest that fundamental principles of the object and agent systems can be revised by adults and preschoolers, when given small amounts of counterevidence. We argue that not all core knowledge systems are created equal, and they may be subject to revision throughout development.

Temporal-spatial deciphering mental subtraction in the human brain

Mental subtraction, involving numerical processing and operation, requires a complex interplay among several brain regions. Diverse studies have utilized scalp electroencephalograph, electrocorticogram, or functional magnetic resonance imaging to resolve the structure pattern and functional activity during subtraction operation. However, a high resolution of the spatial-temporal understanding of the neural mechanisms involved in mental subtraction is unavailable. Thus, this study obtained intracranial stereoelectroencephalography recordings from 20 patients with pharmacologically resistant epilepsy. Specifically, two sample-delayed mismatch paradigms of numeric comparison and subtracting results comparison were used to help reveal the time frame of mental subtraction. The brain sub-regions were chronologically screened using the stereoelectroencephalography recording for mental subtraction. The results indicated that the anterior cortex, containing the frontal, insular, and parahippocampous, worked for preparing for mental subtraction; moreover, the posterior cortex, such as parietal, occipital, limbic, and temporal regions, cooperated during subtraction. Especially, the gamma band activities in core regions within the parietal-cingulate-temporal cortices mediated the critical mental subtraction. Overall, this research is the first to describe the spatiotemporal activities underlying mental subtraction in the human brain. It provides a comprehensive insight into the cognitive control activity underlying mental arithmetic.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-023-09937-z.

Form perception is a cognitive correlate of the relation between subitizing ability and math performance

"Subitizing" defines a phenomenon whereby approximately four items can be quickly and accurately processed. Studies have shown the close association between subitizing and math performance, however, the mechanism for the association remains unclear. The present study was conducted to investigate whether form perception assessed on a serial figure matching task is a potential non-numerical mechanism between subitizing ability and math performance. Three-hundred and seventy-three Chinese primary school students completed four kinds of dot comparison tasks, serial figure matching task, math performance tasks (including three arithmetic computation tasks and math word problem task), and other cognitive tasks as their general cognitive abilities were observed as covariates. A series of hierarchical regression analyses showed that after controlling for age, gender, nonverbal matrix reasoning, and visual tracking, subitizing comparison (subitizing vs. subitizing, subitizing vs. estimation) still contributed to simple addition or simple subtraction but not to complex subtraction ability or math word problem. After taking form perception as an additional control variable, the predictive power of different dot comparison conditions disappeared. A path model also showed that form perception fully mediates the relation between numerosity comparison (within and beyond the subitizing range) and arithmetic performance. These findings support the claim that form perception is a non-numerical cognitive correlate of the relation between subitizing ability and math performance (especially arithmetic computation).

Probing Our Built-in Calculator: A Systematic Narrative Review of Noninvasive Brain Stimulation Studies on Arithmetic Operation-Related Brain Areas

This systematic review presented a comprehensive survey of studies that applied transcranial magnetic stimulation and transcranial electrical stimulation to parietal and nonparietal areas to examine the neural basis of symbolic arithmetic processing. All findings were compiled with regard to the three assumptions of the triple-code model (TCM) of number processing. Thirty-seven eligible manuscripts were identified for review (33 with healthy participants and 4 with patients). Their results are broadly consistent with the first assumption of the TCM that intraparietal sulcus both hold a magnitude code and engage in operations requiring numerical manipulations such as subtraction. However, largely heterogeneous results conflicted with the second assumption of the TCM that the left angular gyrus subserves arithmetic fact retrieval, such as the retrieval of rote-learned multiplication results. Support is also limited for the third assumption of the TCM, namely, that the posterior superior parietal lobule engages in spatial operations on the mental number line. Furthermore, results from the stimulation of brain areas outside of those postulated by the TCM show that the bilateral supramarginal gyrus is involved in online calculation and retrieval, the left temporal cortex in retrieval, and the bilateral dorsolateral prefrontal cortex and cerebellum in online calculation of cognitively demanding arithmetic problems. The overall results indicate that multiple cortical areas subserve arithmetic skills.

Electrophysiological correlates of symbolic numerical order processing

Determining if a sequence of numbers is ordered or not is one of the fundamental aspects of numerical processing linked to concurrent and future arithmetic skills. While some studies have explored the neural underpinnings of order processing using functional magnetic resonance imaging, our understanding of electrophysiological correlates is comparatively limited. To address this gap, we used a three-item symbolic numerical order verification task (with Arabic numerals from 1 to 9) to study event-related potentials (ERPs) in 73 adult participants in an exploratory approach. We presented three-item sequences and manipulated their order (ordered vs. unordered) as well as their inter-item numerical distance (one vs. two). Participants had to determine if a presented sequence was ordered or not. They also completed a speeded arithmetic fluency test, which measured their arithmetic skills. Our results revealed a significant mean amplitude difference in the grand average ERP waveform between ordered and unordered sequences in a time window of 500-750 ms at left anterior-frontal, left parietal, and central electrodes. We also identified distance-related amplitude differences for both ordered and unordered sequences. While unordered sequences showed an effect in the time window of 500-750 ms at electrode clusters around anterior-frontal and right-frontal regions, ordered sequences differed in an earlier time window (190-275 ms) in frontal and right parieto-occipital regions. Only the mean amplitude difference between ordered and unordered sequences showed an association with arithmetic fluency at the left anterior-frontal electrode. While the earlier time window for ordered sequences is consistent with a more automated and efficient processing of ordered sequential items, distance-related differences in unordered sequences occur later in time.

A number sense as an emergent property of the manipulating brain

The ability to understand and manipulate numbers and quantities emerges during childhood, but the mechanism through which humans acquire and develop this ability is still poorly understood. We explore this question through a model, assuming that the learner is able to pick up and place small objects from, and to, locations of its choosing, and will spontaneously engage in such undirected manipulation. We further assume that the learner's visual system will monitor the changing arrangements of objects in the scene and will learn to predict the effects of each action by comparing perception with a supervisory signal from the motor system. We model perception using standard deep networks for feature extraction and classification. Our main finding is that, from learning the task of action prediction, an unexpected image representation emerges exhibiting regularities that foreshadow the perception and representation of numbers and quantity. These include distinct categories for zero and the first few natural numbers, a strict ordering of the numbers, and a one-dimensional signal that correlates with numerical quantity. As a result, our model acquires the ability to estimate numerosity, i.e. the number of objects in the scene, as well as subitization, i.e. the ability to recognize at a glance the exact number of objects in small scenes. Remarkably, subitization and numerosity estimation extrapolate to scenes containing many objects, far beyond the three objects used during training. We conclude that important aspects of a facility with numbers and quantities may be learned with supervision from a simple pre-training task. Our observations suggest that cross-modal learning is a powerful learning mechanism that may be harnessed in artificial intelligence.

Perceptual addition of continuous magnitudes in an 'artificial algebra'

Although there is substantial evidence for an innate 'number sense' that scaffolds learning about mathematics, whether the underlying representations are based on discrete or continuous perceptual magnitudes has been controversial. Yet the nature of the computations supported by these representations has been neglected in this debate. While basic computation of discrete non-symbolic quantities has been reliably demonstrated in adults, infants, and non-humans, far less consideration has been given to the capacity for computation of continuous perceptual magnitudes. Here we used a novel experimental task to ask if humans can learn to add non-symbolic, continuous magnitudes in accord with the properties of an algebraic group, by feedback and without explicit instruction. Three pairs of experiments tested perceptual addition under the group properties of commutativity (Experiments 1a-b), identity and inverses (Experiments 2a-b) and associativity (Experiments 3a-b), with both line length and brightness modalities. Transfer designs were used in which participants responded on trials with feedback based on sums of magnitudes and later were tested with novel stimulus configurations. In all experiments, correlations of average responses with magnitude sums were high on trials with feedback. Responding on transfer trials was accurate and provided strong support for addition under all of the group axioms with line length, and for all except associativity with brightness. Our results confirm that adult human subjects can implicitly add continuous quantities in a manner consistent with symbolic addition over the integers, and that an 'artificial algebra' task can be used to study implicit computation.

Neurobiology of numerical learning

Numerical abilities are complex cognitive skills essential for dealing with requirements of the modern world. Although the brain structures and functions underlying numerical cognition in different species have long been appreciated, genetic and molecular techniques have more recently expanded the knowledge about the mechanisms underlying numerical learning. In this review, we discuss the status of the research related to the neurobiological bases of numerical abilities. We consider how genetic factors have been associated with mathematical capacities and how these link to the current knowledge of brain regions underlying these capacities in human and non-human animals. We further discuss the extent to which significant variations in the levels of specific neurotransmitters may be used as potential markers of individual performance and learning difficulties and take into consideration the therapeutic potential of brain stimulation methods to modulate learning and improve interventional outcomes. The implications of this research for formulating a more comprehensive view of the neural basis of mathematical learning are discussed.

How learning influences non-symbolic numerical processing: effects of feedback in the dot comparison task

It has long been debated how humans estimate the numerosity of sets of elements and what role continuous visual properties play in this process. The dot comparison task, in which the more numerous of two dot arrays must be selected, is a dominant method to investigate this phenomenon. It has been shown that the visual properties of the two dot patterns strongly influence the comparison. This influence can be systematically investigated by manipulating visual properties congruently and incongruently with numerosity. However, it remains unclear how learning and prior experience affect the influence of the visual properties. To address this question, we introduced feedback into the classical dot comparison task: during the learning phase, participants in the experimental group received feedback after each trial indicating whether their answer was correct whereas participants in the control group did not. After the learning phase, neither group received feedback. The convex hull of the dot patterns and the average dot diameter were manipulated congruently and incongruently with numerosity. Our results show that feedback had no effect on overall performance. However, when manipulated separately, dot diameter no longer affected performance in the experimental group after the learning phase, but it did in the control group. Moreover, this effect remained visible even when diameter and convex hull were manipulated simultaneously. This pattern of results is consistent with the notion of sensory integration which proposes that weights are assigned to different visual cues and that numerical judgments depend on an additive combination of these weights. We also found a correlation between performance on an arithmetic task and performance on trials in which dot size was manipulated incongruently with numerosity. However, there were no correlations between an inhibition task and performance in the dot comparison task. Taken together, the current results suggest that learning with feedback may affect some visual properties but not others. Future studies should further investigate a wider range of visual properties to examine which of them can be influenced by learning and under what conditions learning occurs.

Human brain representations of internally generated outcomes of approximate calculation revealed by ultra-high-field brain imaging

Much of human culture's advanced technology owes its existence to the ability to mentally manipulate quantities. Neuroscience has described the brain regions overall recruited by numerical tasks and the neuronal codes representing individual quantities during perceptual tasks. Nevertheless, it remains unknown how quantity representations are combined or transformed during mental computations and how specific quantities are coded in the brain when generated as the result of internal computations rather than evoked by a stimulus. Here, we imaged the brains of adult human subjects at 7 Tesla during an approximate calculation task designed to disentangle in- and outputs of the computation from the operation itself. While physically presented sample numerosities were distinguished in activity patterns along the dorsal visual pathway and within frontal and occipito-temporal regions, a representation of the internally generated result was most prominently detected in higher order regions such as angular gyrus and lateral prefrontal cortex. Behavioral precision in the task was related to cross-decoding performance between sample and result representations in medial IPS regions. This suggests the transformation of sample into result may be carried out within dorsal stream sensory-motor integration regions, and resulting outputs maintained for task purposes in higher-level regions in a format possibly detached from sensory-evoked inputs.

Emergence of number sense through the integration of multimodal information: developmental learning insights from neural network models

Introduction

Associating multimodal information is essential for human cognitive abilities including mathematical skills. Multimodal learning has also attracted attention in the field of machine learning, and it has been suggested that the acquisition of better latent representation plays an important role in enhancing task performance. This study aimed to explore the impact of multimodal learning on representation, and to understand the relationship between multimodal representation and the development of mathematical skills.

Methods

We employed a multimodal deep neural network as the computational model for multimodal associations in the brain. We compared the representations of numerical information, that is, handwritten digits and images containing a variable number of geometric figures learned through single- and multimodal methods. Next, we evaluated whether these representations were beneficial for downstream arithmetic tasks.

Results

Multimodal training produced better latent representation in terms of clustering quality, which is consistent with previous findings on multimodal learning in deep neural networks. Moreover, the representations learned using multimodal information exhibited superior performance in arithmetic tasks.

Discussion

Our novel findings experimentally demonstrate that changes in acquired latent representations through multimodal association learning are directly related to cognitive functions, including mathematical skills. This supports the possibility that multimodal learning using deep neural network models may offer novel insights into higher cognitive functions.

The parietal cortex has a causal role in ambiguity computations in humans

Humans often face the challenge of making decisions between ambiguous options. The level of ambiguity in decision-making has been linked to activity in the parietal cortex, but its exact computational role remains elusive. To test the hypothesis that the parietal cortex plays a causal role in computing ambiguous probabilities, we conducted consecutive fMRI and TMS-EEG studies. We found that participants assigned unknown probabilities to objective probabilities, elevating the uncertainty of their decisions. Parietal cortex activity correlated with the objective degree of ambiguity and with a process that underestimates the uncertainty during decision-making. Conversely, the midcingulate cortex (MCC) encodes prediction errors and increases its connectivity with the parietal cortex during outcome processing. Disruption of the parietal activity increased the uncertainty evaluation of the options, decreasing cingulate cortex oscillations during outcome evaluation and lateral frontal oscillations related to value ambiguous probability. These results provide evidence for a causal role of the parietal cortex in computing uncertainty during ambiguous decisions made by humans.

Efficient, continual, and generalized learning in the brain - neural mechanism of Mental Schema 2.0

There has been tremendous progress in artificial neural networks (ANNs) over the past decade; however, the gap between ANNs and the biological brain as a learning device remains large. With the goal of closing this gap, this paper reviews learning mechanisms in the brain by focusing on three important issues in ANN research: efficiency, continuity, and generalization. We first discuss the method by which the brain utilizes a variety of self-organizing mechanisms to maximize learning efficiency, with a focus on the role of spontaneous activity of the brain in shaping synaptic connections to facilitate spatiotemporal learning and numerical processing. Then, we examined the neuronal mechanisms that enable lifelong continual learning, with a focus on memory replay during sleep and its implementation in brain-inspired ANNs. Finally, we explored the method by which the brain generalizes learned knowledge in new situations, particularly from the mathematical generalization perspective of topology. Besides a systematic comparison in learning mechanisms between the brain and ANNs, we propose "Mental Schema 2.0," a new computational property underlying the brain's unique learning ability that can be implemented in ANNs.

The role of outgroup homogeneity and the neurodynamics of the frontal cortex during beauty comparisons

The distance effect states that the closer two compared magnitudes (e.g., two numbers, physical attractiveness in two faces), the more difficult the comparison, and the greater the activity of the frontoparietal control network. However, it is unclear whether this network is also recruited to the same extent when we perform ingroup and outgroup beauty comparisons and whether the activation of these networks is tracked by interindividual variation in the perceptions we hold about an outgroup. We recorded brain activity with fMRI, where participants compared the beauty of two women ostensibly either from their ingroup or from an outgroup. Low-distance conditions produced longer response times than the high-distance conditions, and this was found in both the ingroup and outgroup conditions. However, our neuroimaging analyses revealed that the left IFG/anterior insula showed the classic distance effect only during ingroup processing but not during outgroup processing. Notably, interaction-specific activity within the left IFG/anterior insula was related to perceptions of outgroup homogeneity assessed via a questionnaire. This set of findings reveals the dynamic role of the prefrontal cortex and its interplay with perceptions of outgroup homogeneity in shaping ingroup and outgroup decision-making.

Cortico-spinal modularity in the parieto-frontal system: A new perspective on action control

Classical neurophysiology suggests that the motor cortex (MI) has a unique role in action control. In contrast, this review presents evidence for multiple parieto-frontal spinal command modules that can bypass MI. Five observations support this modular perspective: (i) the statistics of cortical connectivity demonstrate functionally-related clusters of cortical areas, defining functional modules in the premotor, cingulate, and parietal cortices; (ii) different corticospinal pathways originate from the above areas, each with a distinct range of conduction velocities; (iii) the activation time of each module varies depending on task, and different modules can be activated simultaneously; (iv) a modular architecture with direct motor output is faster and less metabolically expensive than an architecture that relies on MI, given the slow connections between MI and other cortical areas; (v) lesions of the areas composing parieto-frontal modules have different effects from lesions of MI. Here we provide examples of six cortico-spinal modules and functions they subserve: module 1) arm reaching, tool use and object construction; module 2) spatial navigation and locomotion; module 3) grasping and observation of hand and mouth actions; module 4) action initiation, motor sequences, time encoding; module 5) conditional motor association and learning, action plan switching and action inhibition; module 6) planning defensive actions. These modules can serve as a library of tools to be recombined when faced with novel tasks, and MI might serve as a recombinatory hub. In conclusion, the availability of locally-stored information and multiple outflow paths supports the physiological plausibility of the proposed modular perspective.

Conceptual foundations of early numeracy: Evidence from infant brain data

Understanding the conceptual resources that children bring to mathematics learning is crucial for developing effective instruction and interventions. Despite the considerable number of studies examining the neural underpinnings of number representations in adults and the growing number of reports in children, very few studies have examined the neural correlates of the link between foundational resources related to numerical information and symbolic number representations in infants. There is currently an active debate about which foundational resources are critical for symbolic mathematics. Is early numerical discrimination best explained by a holistic and generalized sense of magnitude rather than a number sense? Does early number sense provide the conceptual basis for mapping numerical symbols to their meaning? Are foundational number systems marginal while children learn to count and perform symbolic arithmetic, and only later children map non symbolic representations of numerical magnitudes onto symbols? After describing the mainstream theories of numerical cognition and the sources of controversy, we review recent studies of the neural bases of human infants' numerical performance with the aim of clarifying the link between early conceptual resources and symbolic number systems as children's mathematical minds develop.

Dissecting the chain of information processing and its interplay with neurochemicals and fluid intelligence across development

Previous research has highlighted the role of glutamate and gamma-aminobutyric acid (GABA) in perceptual, cognitive, and motor tasks. However, the exact involvement of these neurochemical mechanisms in the chain of information processing, and across human development, is unclear. In a cross-sectional longitudinal design, we used a computational approach to dissociate cognitive, decision, and visuomotor processing in 293 individuals spanning early childhood to adulthood. We found that glutamate and GABA within the intraparietal sulcus (IPS) explained unique variance in visuomotor processing, with higher glutamate predicting poorer visuomotor processing in younger participants but better visuomotor processing in mature participants, while GABA showed the opposite pattern. These findings, which were neurochemically, neuroanatomically and functionally specific, were replicated ~21 mo later and were generalized in two further different behavioral tasks. Using resting functional MRI, we revealed that the relationship between IPS neurochemicals and visuomotor processing is mediated by functional connectivity in the visuomotor network. We then extended our findings to high-level cognitive behavior by predicting fluid intelligence performance. We present evidence that fluid intelligence performance is explained by IPS GABA and glutamate and is mediated by visuomotor processing. However, this evidence was obtained using an uncorrected alpha and needs to be replicated in future studies. These results provide an integrative biological and psychological mechanistic explanation that links cognitive processes and neurotransmitters across human development and establishes their potential involvement in intelligent behavior.

Changes in the superior longitudinal fasciculus and anterior thalamic radiation in the left brain are associated with developmental dyscalculia

Developmental dyscalculia is a neurodevelopmental disorder specific to arithmetic learning even with normal intelligence and age-appropriate education. Difficulties often persist from childhood through adulthood lowering the individual's quality of life. However, the neural correlates of developmental dyscalculia are poorly understood. This study aimed to identify brain structural connectivity alterations in developmental dyscalculia. All participants were recruited from a large scale, non-referred population sample in a longitudinal design. We studied 10 children with developmental dyscalculia (11.3 ± 0.7 years) and 16 typically developing peers (11.2 ± 0.6 years) using diffusion-weighted magnetic resonance imaging. We assessed white matter microstructure with tract-based spatial statistics in regions-of-interest tracts that had previously been related to math ability in children. Then we used global probabilistic tractography for the first time to measure and compare tract length between developmental dyscalculia and typically developing groups. The high angular resolution diffusion-weighted magnetic resonance imaging and crossing-fiber probabilistic tractography allowed us to evaluate the length of the pathways compared to previous studies. The major findings of our study were reduced white matter coherence and shorter tract length of the left superior longitudinal/arcuate fasciculus and left anterior thalamic radiation in the developmental dyscalculia group. Furthermore, the lower white matter coherence and shorter pathways tended to be associated with the lower math performance. These results from the regional analyses indicate that learning, memory and language-related pathways in the left hemisphere might be related to developmental dyscalculia in children.

Differential involvement of the anterior and posterior hippocampus, parahippocampus, and retrosplenial cortex in making precise judgments of spatial distance and object size for remotely acquired memories of environments and objects

The hippocampus is known to support processing of precise spatial information in recently learned environments. It is less clear, but crucial for theories of systems consolidation, to know whether it also supports processing of precise spatial information in familiar environments learned long ago and whether such precision extends to objects and numbers. In this fMRI study, we asked participants to make progressively more refined spatial distance judgments among well-known Toronto landmarks (whether landmark A is closer to landmark B or C) to examine hippocampal involvement. We also tested whether the hippocampus was similarly engaged in estimating magnitude regarding sizes of familiar animals and numbers. We found that the hippocampus was only engaged in spatial judgment. Activation was greater and lasted longer in the posterior than anterior hippocampus, which instead showed greater modulation as discrimination between spatial distances became more fine grained. These findings suggest that the anterior and posterior hippocampus have different functions which are influenced differently by estimation of differential distance. Similarly, parahippocampal-place-area and retrosplenial cortex were involved only in the spatial condition. By contrast, activation of the intraparietal sulcus was modulated by precision in all conditions. Therefore, our study supports the idea that the hippocampus and related structures are implicated in retrieving and operating even on remote spatial memories whenever precision is required, as posted by some theories of systems consolidation.

Characterizing exact arithmetic abilities before formal schooling

Children appear to have some arithmetic abilities before formal instruction in school, but the extent of these abilities as well as the mechanisms underlying them are poorly understood. Over two studies, an initial exploratory study of preschool children in the U.S. (N = 207; Age = 2.89-4.30 years) and a pre-registered replication of preschool children in Italy (N = 130; Age = 3-6.33 years), we documented some basic behavioral signatures of exact arithmetic using a non-symbolic subtraction task. Furthermore, we investigated the underlying mechanisms by analyzing the relationship between individual differences in exact subtraction and assessments of other numerical and non-numerical abilities. Across both studies, children performed above chance on the exact non-symbolic arithmetic task, generally showing better performance on problems involving smaller quantities compared to those involving larger quantities. Furthermore, individual differences in non-verbal approximate numerical abilities and exact cardinal number knowledge were related to different aspects of subtraction performance. Specifically, non-verbal approximate numerical abilities were related to subtraction performance in older but not younger children. Across both studies we found evidence that cardinal number knowledge was related to performance on subtraction problems where the answer was zero (i.e., subtractive negation problems). Moreover, subtractive negation problems were only solved above chance by children who had a basic understanding of cardinality. Together these finding suggest that core non-verbal numerical abilities, as well as emerging knowledge of symbolic numbers provide a basis for some, albeit limited, exact arithmetic abilities before formal schooling.

The transitive inference task to study the neuronal correlates of memory-driven decision making: A monkey neurophysiology perspective

A vast amount of literature agrees that rank-ordered information as A>B>C>D>E>F is mentally represented in spatially organized schemas after learning. This organization significantly influences the process of decision-making, using the acquired premises, i.e. deciding if B is higher than D is equivalent to comparing their position in this space. The implementation of non-verbal versions of the transitive inference task has provided the basis for ascertaining that different animal species explore a mental space when deciding among hierarchically organized memories. In the present work, we reviewed several studies of transitive inference that highlighted this ability in animals and, consequently, the animal models developed to study the underlying cognitive processes and the main neural structures supporting this ability. Further, we present the literature investigating which are the underlying neuronal mechanisms. Then we discuss how non-human primates represent an excellent model for future studies, providing ideal resources for better understanding the neuronal correlates of decision-making through transitive inference tasks.

Individual risk attitudes arise from noise in neurocognitive magnitude representations

Humans are generally risk averse, preferring smaller certain over larger uncertain outcomes. Economic theories usually explain this by assuming concave utility functions. Here, we provide evidence that risk aversion can also arise from relative underestimation of larger monetary payoffs, a perceptual bias rooted in the noisy logarithmic coding of numerical magnitudes. We confirmed this with psychophysics and functional magnetic resonance imaging, by measuring behavioural and neural acuity of magnitude representations during a magnitude perception task and relating these measures to risk attitudes during separate risky financial decisions. Computational modelling indicated that participants use similar mental magnitude representations in both tasks, with correlated precision across perceptual and risky choices. Participants with more precise magnitude representations in parietal cortex showed less variable behaviour and less risk aversion. Our results highlight that at least some individual characteristics of economic behaviour can reflect capacity limitations in perceptual processing rather than processes that assign subjective values to monetary outcomes.

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.

Contrasting symbolic and non-symbolic numerical representations in a joint classification task

Both symbolic (digits) and non-symbolic (dots) numerals are spatially coded, with relatively small numbers being responded faster with a left key and large numbers being responded faster with a right key (spatial-numerical association of response codes [SNARC]). The idea of format independent SNARC seems to support the existence of a common system for symbolic and non-symbolic numerical representations, although evidence in the field is still mixed. The aim of the present study is to investigate whether symbolic and non-symbolic numerals interact in the SNARC effect when both information is simultaneously displayed. To do so, participants were presented with dice-like patterns, with digits being used instead of dots. In two separate magnitude classification tasks, participants had to respond either to the number of digits presented on the screen or to their numerical size. In the non-symbolic task, they had to judge whether the digits on the screen were more or less than three, irrespective of the numerical value of the digits. In the symbolic task, participants had to judge whether the digits on the screen were numerically smaller or larger than three, irrespective of the number of digits being present. The results show a consistent SNARC effect in the symbolic task and no effect in the non-symbolic one. Furthermore, congruency between symbolic and non-symbolic numerals did not modulate the response patterns, thus supporting the idea of independent representations and questioning some propositions of current theoretical accounts.

Electrophysiological Comparison of Cumulative Area and Non-Symbolic Number Judgments

Despite the importance of representing different magnitudes (i.e., number and cumulative area) for action planning and formal mathematics, there is much debate about the nature of these representations, particularly the extent to which magnitudes interact in the mind and brain. Early interaction views suggest that there are shared perceptual processes that form overlapping magnitude representations. However, late interaction views hold that representations of different magnitudes remain distinct, interacting only when preparing a motor response. The present study sheds light on this debate by examining the temporal onset of ratio and congruity effects as participants made ordinal judgments about number and cumulative area. Event-related potentials (ERPs) were recorded to identify whether the onset of such effects aligned with early versus late views. Ratio effects for both magnitudes were observed starting in the P100. Moreover, a congruity effect emerged within the P100. That interactions were observed early in processing, at the same time that initial ratio effects occurred, suggests that number and cumulative area processes interacted when magnitude representations were being formed, prior to preparing a decision response. Our findings are consistent with an early interaction view of magnitude processing, in which number and cumulative area may rely on shared perceptual mechanisms.

Dissecting the neurocomputational bases of patch-switching

The survival and well-being of humans require solving the patch-switching problem: we must decide when to stop collecting rewards in a current patch and travel somewhere else where gains may be higher. Previous studies suggested that frontal regions are underpinned by several processes in the context of foraging decisions such as tracking task difficulty, and/or the value of exploring the environment. To dissociate between these processes, participants completed an fMRI patch-switching learning task inspired by behavioral ecology. By analyzing >11,000 trials collected across 21 participants, we found that the activation in the cingulate cortex was closely related to several patch-switching-related variables including the decision to leave the current patch, the encounter of a new patch, the harvest value, and the relative forage value. Learning-induced changes in the patch-switching threshold were tracked by activity within frontoparietal regions including the superior frontal gyrus and angular gyrus. Our findings suggest that frontoparietal regions shape patch-switching learning apart from encoding classical non-learning foraging processes. These findings provide a novel neurobiological understanding of how learning emerges neurocomputationally shaping patch-switching behavior with implications in real-life choices such as job selection and pave the way for future studies to probe the causal role of these neurobiological mechanisms.

Précis of What Babies Know

Where does human knowledge begin? Research on human infants, children, adults, and nonhuman animals, using diverse methods from the cognitive, brain, and computational sciences, provides evidence for six early emerging, domain-specific systems of core knowledge. These automatic, unconscious systems are situated between perceptual systems and systems of explicit concepts and beliefs. They emerge early in infancy, guide children's learning, and function throughout life.

Implicit Processing of Numerical Order: Evidence from a Continuous Interocular Flash Suppression Study

Processing the ordered relationships between sequential items is a key element in many cognitive abilities that are important for survival. Specifically, order may play a crucial role in numerical processing. Here, we assessed the existence of a cognitive system designed to implicitly evaluate numerical order, by combining continuous flash suppression with a priming method in a numerical enumeration task. In two experiments and diverse statistical analysis, targets that required numerical enumeration were preceded by an invisibly ordered or non-ordered numerical prime sequence. The results of both experiments showed that enumeration for targets that appeared after an ordered prime was significantly faster, while the ratio of the prime sequences produced no significant effect. The findings suggest that numerical order is processed implicitly and affects a basic cognitive ability: enumeration of quantities.

A-B-3-Associations and dissociations of reading and arithmetic: Is domain-specific prediction outdated?

Reading and arithmetic are core domains of academic achievement with marked impact on career opportunities and socioeconomic status. While associations between reading and arithmetic are well established, evidence on underlying mechanisms is inconclusive. The main goal of this study was to reevaluate the domain-specificity of established predictors and to enhance our understanding of the (co-)development of reading and arithmetic. In a sample of 885 German-speaking children, standard domain-specific predictors of reading and arithmetic were assessed before and/or at the onset of formal schooling. Reading and arithmetic skills were measured at the beginning and end of second grade. Latent variables were extracted for all relevant constructs: Grapheme-phoneme processing (phonological awareness, letter identification), RAN (RAN-objects, RAN-digits), number system knowledge (number identification, successor knowledge), and magnitude processing (non-symbolic and symbolic magnitude comparison), as well as the criterion measures reading and arithmetic. Four structural equation models tested distinct research questions. Grapheme-phoneme processing was a specific predictor of reading, and magnitude processing explained variance specific to arithmetic. RAN explained variance in both domains, and it explained variance in reading even after controlling for arithmetic. RAN and number system knowledge further explained variance in skills shared between reading and arithmetic. Reading and arithmetic entail domain-specific cognitive components, and they both require tight networks of visual, verbal, and semantic information, as reflected by RAN. This perspective provides a useful background to explain associations and dissociations between reading and arithmetic performance.

Longitudinal relationship between number line estimation and other mathematical abilities in Chinese preschool children

Many cross-sectional studies have shown that number line estimation is associated with other mathematical skills, but there has been limited longitudinal research. To systematically examine such associations longitudinally at the earliest stage of mathematical learning, the current study tested 40 Chinese preschoolers (mean age = 4.97 years, SD = 0.18) and followed them up 8 months later. For both waves of data collection, children were administered six tasks: number line estimation, dot counting, comparison of two dot arrays, comparison of triple dot arrays, symbolic number comparison, and simple addition. Results of two-wave cross-lagged panel analysis showed that (a) dot counting and non-symbolic numerical comparison at Time 1 had significant longitudinal associations with number line estimation at Time 2, (b) number line estimation had bidirectional associations with symbolic number comparison, and (c) number line estimation at Time 1 had a marginally significant longitudinal association with simple addition at Time 2. These results extend the small but accumulating literature on the longitudinal relations between number line estimation and other mathematical skills and specify the important role of number line estimation in the early development of mathematical skills.

Sequence Order in the Range 1 to 19 by Chimpanzees on a Touchscreen Task: Processing Two-Digit Arabic Numerals

The sequence of Arabic numerals from 1 to 19 was taught to six chimpanzees, three pairs of mother and child. Each chimpanzee participant sat facing a touchscreen on which the numerals appeared in random positions within an imaginary 5-by-8 matrix. They had to touch the numerals in ascending order. Baseline training involved touching the adjacent numerals from 1 to X or from the numeral X to 19. Systematic tests revealed the following results: (1) The range 1 to 9 was easier than 1 to 19. (2) Adjacent numerals were easier than nonadjacent ones. (3) The "masking" (memory task) caused deterioration of performance. All these factors depended on the number of numerals simultaneously presented on the screen. A chimpanzee named Pal mastered the skill of ordering two-digit numerals with 100% accuracy. Human participants were tested in the same experiment with the same procedure. Both species showed relative difficulty in handling two-digit numerals. Global-local information processing is known to be different between humans and other primates. The assessment of chimpanzee performance and comparison with humans were discussed in terms of the possible difference in the global-local dual information processing of two-digit numerals.

Numerosity as a visual property: Evidence from two highly evolutionary distant species

Most animals, from humans to invertebrates, possess an ability to estimate numbers. This evolutionary advantage facilitates animals' choice of environments with more food sources, more conspecifics to increase mating success, and/or reduced predation risk among others. However, how the brain processes numerical information remains largely unknown. There are currently two lines of research interested in how numerosity of visual objects is perceived and analyzed in the brain. The first argues that numerosity is an advanced cognitive ability processed in high-order brain areas, while the second proposes that "numbers" are attributes of the visual scene and thus numerosity is processed in the visual sensory system. Recent evidence points to a sensory involvement in estimating magnitudes. In this Perspective, we highlight this evidence in two highly evolutionary distant species: humans and flies. We also discuss the advantages of studying numerical processing in fruit flies in order to dissect the neural circuits involved in and required for numerical processing. Based on experimental manipulation and the fly connectome, we propose a plausible neural network for number sense in invertebrates.

Representations within the Intraparietal Sulcus Distinguish Numerical Tasks and Formats

How does our brain understand the number five when it is written as an Arabic numeral, and when presented as five fingers held up? Four facets have been implicated in adult numerical processing: semantic, visual, manual, and phonological/verbal. Here, we ask how the brain represents each, using a combination of tasks and stimuli. We collected fMRI data from adult participants while they completed our novel "four number code" paradigm. In this paradigm, participants viewed one of two stimulus types to tap into the visual and manual number codes, respectively. Concurrently, they completed one of two tasks to tap into the semantic and phonological/verbal number codes, respectively. Classification analyses revealed that neural codes representing distinctions between the number comparison and phonological tasks were generalizable across format (e.g., Arabic numerals to hands) within intraparietal sulcus (IPS), angular gyrus, and precentral gyrus. Neural codes representing distinctions between formats were generalizable across tasks within visual areas such as fusiform gyrus and calcarine sulcus, as well as within IPS. Our results identify the neural facets of numerical processing within a single paradigm and suggest that IPS is sensitive to distinctions between semantic and phonological/verbal, as well as visual and manual, facets of number representations.

Cortical representations of numbers and nonsymbolic quantities expand and segregate in children from 5 to 8 years of age

Number symbols, such as Arabic numerals, are cultural inventions that have transformed human mathematical skills. Although their acquisition is at the core of early elementary education in children, it remains unknown how the neural representations of numerals emerge during that period. It is also unclear whether these relate to an ontogenetically earlier sense of approximate quantity. Here, we used multivariate fMRI adaptation coupled with within- and between-format machine learning to probe the cortical representations of Arabic numerals and approximate nonsymbolic quantity in 89 children either at the beginning (age 5) or four years into formal education (age 8). Although the cortical representations of both numerals and nonsymbolic quantities expanded from age 5 to age 8, these representations also segregated with learning and development. Specifically, a format-independent neural representation of quantity was found in the right parietal cortex, but only for 5-year-olds. These results are consistent with the so-called symbolic estrangement hypothesis, which argues that the relation between symbolic and nonsymbolic quantity weakens with exposure to formal mathematics in children.

Assessing evidence accumulation and rule learning in humans with an online game

Evidence accumulation, an essential component of perception and decision making, is frequently studied with psychophysical tasks involving noisy or ambiguous stimuli. In these tasks, participants typically receive verbal or written instructions that describe the strategy that should be used to guide decisions. Although convenient and effective, explicit instructions can influence learning and decision making strategies and can limit comparisons with animal models, in which behaviors are reinforced through feedback. Here, we developed an online video game and nonverbal training pipeline, inspired by pulse-based tasks for rodents, as an alternative to traditional psychophysical tasks used to study evidence accumulation. Using this game, we collected behavioral data from hundreds of participants trained with an explicit description of the decision rule or with experiential feedback. Participants trained with feedback alone learned the game rules rapidly and used strategies and displayed biases similar to those who received explicit instructions. Finally, by leveraging data across hundreds of participants, we show that perceptual judgments were well described by an accumulation process in which noise scaled nonlinearly with evidence, consistent with previous animal studies but inconsistent with diffusion models widely used to describe perceptual decisions in humans. These results challenge the conventional description of the accumulation process and suggest that online games provide a valuable platform to examine perceptual decision making and learning in humans. In addition, the feedback-based training pipeline developed for this game may be useful for evaluating perceptual decision making in human populations with difficulty following verbal instructions.NEW & NOTEWORTHY Perceptual uncertainty sets critical constraints on our ability to accumulate evidence and make decisions; however, its sources remain unclear. We developed a video game, and feedback-based training pipeline, to study uncertainty during decision making. Leveraging choices from hundreds of subjects, we demonstrate that human choices are inconsistent with popular diffusion models of human decision making and instead are best fit by models in which perceptual uncertainty scales nonlinearly with the strength of sensory evidence.

EEG-representational geometries and psychometric distortions in approximate numerical judgment

When judging the average value of sample stimuli (e.g., numbers) people tend to either over- or underweight extreme sample values, depending on task context. In a context of overweighting, recent work has shown that extreme sample values were overly represented also in neural signals, in terms of an anti-compressed geometry of number samples in multivariate electroencephalography (EEG) patterns. Here, we asked whether neural representational geometries may also reflect a relative underweighting of extreme values (i.e., compression) which has been observed behaviorally in a great variety of tasks. We used a simple experimental manipulation (instructions to average a single-stream or to compare dual-streams of samples) to induce compression or anti-compression in behavior when participants judged rapid number sequences. Model-based representational similarity analysis (RSA) replicated the previous finding of neural anti-compression in the dual-stream task, but failed to provide evidence for neural compression in the single-stream task, despite the evidence for compression in behavior. Instead, the results indicated enhanced neural processing of extreme values in either task, regardless of whether extremes were over- or underweighted in subsequent behavioral choice. We further observed more general differences in the neural representation of the sample information between the two tasks. Together, our results indicate a mismatch between sample-level EEG geometries and behavior, which raises new questions about the origin of common psychometric distortions, such as diminishing sensitivity for larger values.

The influence of visual illusion perception on numerosity estimation could be evolutionarily conserved: exploring the numerical Delboeuf illusion in humans (Homo sapiens) and fish (Poecilia reticulata)

Discriminating between different quantities is an essential ability in daily life that has been demonstrated in a variety of non-human vertebrates. Nonetheless, what drives the estimation of numerosity is not fully understood, as numerosity intrinsically covaries with several other physical characteristics. There is wide debate as to whether the numerical and spatial abilities of vertebrates are processed by a single magnitude system or two different cognitive systems. Adopting a novel approach, we aimed to investigate this issue by assessing the interaction between area size and numerosity, which has never been conceptualized with consideration for subjective experience in non-human animals. We examined whether the same perceptual biases underlying one of the best-known size illusions, the Delboeuf illusion, can be also identified in numerical estimation tasks. We instructed or trained human participants and guppies, small teleost fish, to select a target numerosity (larger or smaller) of squares between two sets that actually differed in their numerosity. Subjects were also presented with illusory trials in which the same numerosity was presented in two different contexts, against a large and a small background, resembling the Delboeuf illusion. In these trials, both humans and fish demonstrated numerical biases in agreement with the perception of the classical version of the Delboeuf illusion, with the array perceived as larger appearing more numerous. Thus, our results support the hypothesis of a single magnitude system, as perceptual biases that influence spatial decisions seem to affect numerosity judgements in the same way.

More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties (e.g., equidistance, ordinality) can easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children (3–5-years old) estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural (left-to-right) directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space.

Assessing the ecological validity of numerosity-selective neuronal populations with real-world natural scenes

Animals and humans are able to quickly and effortlessly estimate the number of items in a set: their numerosity. Numerosity perception is thought to be critical to behavior, from feeding to escaping predators to human mathematical cognition. Virtually, all scientific studies on numerosity mechanisms use well controlled but artificial stimuli to isolate the numerosity dimension from other physical quantities. Here, we probed the ecological validity of these artificial stimuli and evaluate whether an important component in numerosity processing, the numerosity-selective neural populations, also respond to numerosity of items in real-world natural scenes. Using 7T MRI and natural images from a wide range of categories, we provide evidence that the numerosity-tuned neuronal populations show numerosity-selective responses when viewing images from a real-world natural scene. Our findings strengthen the role of numerosity-selective neurons in numerosity perception and provide an important link to their function in numerosity perception in real-world settings.

Spatial and chromatic properties of numerosity estimation in isolation and context

Numerosity estimation around the subitizing range is facilitated by a shape-template matching process and shape-coding mechanisms are selective to visual features such as colour and luminance contrast polarity. Objects in natural scenes are often embedded within other objects or textured surfaces. Numerosity estimation is improved when objects are grouped into small clusters of the same colour, a phenomenon termed groupitizing, which is thought to leverage on the subitizing system. Here we investigate whether numerosity mechanisms around the subitizing range are selective to colour, luminance contrast polarity and orientation, and how spatial organisation of context and target elements modulates target numerosity estimation. Stimuli consisted of a small number (3-to-6) of target elements presented either in isolation or embedded within context elements. To examine selectivity to colour, luminance polarity and orientation, we compared target-only conditions in which all elements were either the same or different along one of these feature dimensions. We found comparable performance in the same and different feature conditions, revealing that subitizing mechanism do not depend on ‘on-off’ luminance-polarity, colour or orientation channel interactions. We also measured the effect of varying spatial organisation of (i) context, by arranging the elements either in a grid, mirror-symmetric, translation-symmetric or random; (ii) target, by placing the elements either mirror-symmetric, on the vertices of simple shapes or random. Our results indicate higher accuracy and lower RTs in the grid compared to all other context types, with mirror symmetric, translation and random arrangements having comparable effects on target numerosity. We also found improved performance with shape-target followed by symmetric and random target arrangements in the absence and presence of context. These findings indicate that numerosity mechanisms around the subitizing range are not selective to colour, luminance polarity and orientation, and that symmetric, translation and random contexts organisations inhibit target-numerosity encoding stronger than regular/grid context.

Associations among socioeconomic status and preschool-aged children's, number skills, and spatial skills: The role of executive function

Extensive literature has documented socioeconomic status (SES) disparities in young children's standardized math achievement, which primarily reflect differences in basic number and arithmetic skills. In addition, growing evidence indicates that direct assessments of executive function (EF) both predict standardized math achievement and mediate SES differences in standardized math tests. However, early spatial skills and children's approximate number system (ANS) acuity, critical components of later math competence, have been largely absent in this past research. The current study examined SES associations with multiple direct assessments of early ANS, cardinality, and spatial skills, as well as standardized math achievement, in a socioeconomically diverse sample of 4-year-old children (N = 149). Structural equation modeling revealed SES effect sizes of .21 for geometric sensitivity skills, .23 for ANS acuity, .39 for cardinality skills, and .28 for standardized math achievement. Furthermore, relations between SES and children's spatial skills, ANS acuity, cardinality, and standardized math skills were mediated by a composite measure of children's EF skills. Implications of pervasive SES disparities across multiple domains of early math development, as well as the mitigating role of EF, are discussed.

A shared numerical magnitude representation evidenced by the distance effect in frequency-tagging EEG

Humans can effortlessly abstract numerical information from various codes and contexts. However, whether the access to the underlying magnitude information relies on common or distinct brain representations remains highly debated. Here, we recorded electrophysiological responses to periodic variation of numerosity (every five items) occurring in rapid streams of numbers presented at 6 Hz in randomly varying codes—Arabic digits, number words, canonical dot patterns and finger configurations. Results demonstrated that numerical information was abstracted and generalized over the different representation codes by revealing clear discrimination responses (at 1.2 Hz) of the deviant numerosity from the base numerosity, recorded over parieto-occipital electrodes. Crucially, and supporting the claim that discrimination responses reflected magnitude processing, the presentation of a deviant numerosity distant from the base (e.g., base “2” and deviant “8”) elicited larger right-hemispheric responses than the presentation of a close deviant numerosity (e.g., base “2” and deviant “3”). This finding nicely represents the neural signature of the distance effect, an interpretation further reinforced by the clear correlation with individuals’ behavioral performance in an independent numerical comparison task. Our results therefore provide for the first time unambiguously a reliable and specific neural marker of a magnitude representation that is shared among several numerical codes.

The dopamine circuit as a reward-taxis navigation system

Studying the brain circuits that control behavior is challenging, since in addition to their structural complexity there are continuous feedback interactions between actions and sensed inputs from the environment. It is therefore important to identify mathematical principles that can be used to develop testable hypotheses. In this study, we use ideas and concepts from systems biology to study the dopamine system, which controls learning, motivation, and movement. Using data from neuronal recordings in behavioral experiments, we developed a mathematical model for dopamine responses and the effect of dopamine on movement. We show that the dopamine system shares core functional analogies with bacterial chemotaxis. Just as chemotaxis robustly climbs chemical attractant gradients, the dopamine circuit performs ‘reward-taxis’ where the attractant is the expected value of reward. The reward-taxis mechanism provides a simple explanation for scale-invariant dopaminergic responses and for matching in free operant settings, and makes testable quantitative predictions. We propose that reward-taxis is a simple and robust navigation strategy that complements other, more goal-directed navigation mechanisms.