The Evolutionary History of Brains for Numbers

Recruitment of magnitude representations to understand graded words

Language understanding and mathematics understanding are two fundamental forms of human thinking. Prior research has largely focused on the question of how language shapes mathematical thinking. The current study considers the converse question. Specifically, it investigates whether the magnitude representations that are thought to anchor understanding of number are also recruited to understand the meanings of graded words. These are words that come in scales (e.g., Anger) whose members can be ordered by the degree to which they possess the defining property (e.g., calm, annoyed, angry, furious). Experiment 1 uses the comparison paradigm to find evidence that the distance, ratio, and boundary effects that are taken as evidence of the recruitment of magnitude representations extend from numbers to words. Experiment 2 uses a similarity rating paradigm and multi-dimensional scaling to find converging evidence for these effects in graded word understanding. Experiment 3 evaluates an alternative hypothesis - that these effects for graded words simply reflect the statistical structure of the linguistic environment - by using machine learning models of distributional word semantics: LSA, word2vec, GloVe, counterfitted word vectors, BERT, RoBERTa, and GPT-2. These models fail to show the full pattern of effects observed of humans in Experiment 2, suggesting that more is needed than mere statistics. This research paves the way for further investigations of the role of magnitude representations in sentence and text comprehension, and of the question of whether language understanding and number understanding draw on shared or independent magnitude representations. It also informs the role of machine learning models in cognitive psychology research.

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.

Disparate processing of numerosity and associated continuous magnitudes in rats

The studies of number sense in different species are severely hampered by the inevitable entanglement of non-numerical attributes inherent in nonsymbolic stimuli representing numerosity, resulting in contrasting theories of numerosity processing. Here, we developed an algorithm and associated analytical methods to generate stimuli that not only minimized the impact of non-numerical magnitudes in numerosity perception but also allowed their quantification. We trained number-naïve rats with these stimuli as sound pulses representing two or three numbers and demonstrated that their numerical discrimination ability mainly relied on numerosity. Also, studying the learning process revealed that rats used numerosity before using magnitudes for choices. This numerical processing could be impaired specifically by silencing the posterior parietal cortex. Furthermore, modeling this capacity by neural networks shed light on the separation of numerosity and magnitudes extraction. Our study helps dissect the relationship between magnitude and numerosity processing, and the above different findings together affirm the independent existence of innate number and magnitudes sense in rats.

Convergent Circuit Computation for Categorization in the Brains of Primates and Songbirds

Categorization is crucial for behavioral flexibility because it enables animals to group stimuli into meaningful classes that can easily be generalized to new circumstances. A most abstract quantitative category is set size, the number of elements in a set. This review explores how categorical number representations are realized by the operations of excitatory and inhibitory neurons in associative telencephalic microcircuits in primates and songbirds. Despite the independent evolution of the primate prefrontal cortex and the avian nidopallium caudolaterale, the neuronal computations of these associative pallial circuits show surprising correspondence. Comparing cellular functions in distantly related taxa can inform about the evolutionary principles of circuit computations for cognition in distinctly but convergently realized brain structures.

Association neurons in the crow telencephalon link visual signs to numerical values

Many animals can associate signs with numerical values and use these signs in a goal-directed way during task performance. However, the neuronal basis of this semantic association has only rarely been investigated, and so far only in primates. How mechanisms of number associations are implemented in the distinctly evolved brains of other animal taxa such as birds is currently unknown. Here, we explored this semantic number-sign mapping by recording single-neuron activity in the crows' nidopallium caudolaterale (NCL), a brain structure critically involved in avian numerical cognition. Crows were trained to associate visual shapes with varying numbers of items in a number production task. The responses of many NCL neurons during stimulus presentation reflected the numerical values associated with visual shapes in a behaviorally relevant way. Consistent with the crow's better behavioral performance with signs, neuronal representations of numerical values extracted from shapes were more selective compared to those from dot arrays. The existence of number association neurons in crows points to a phylogenetic preadaptation of the brains of cognitively advanced vertebrates to link visual shapes with numerical meaning.

Number sense at 12 months predicts 4-year-olds' maths skills

Preverbal infants spontaneously represent the number of objects in collections. Is this 'sense of number' (also referred to as Approximate Number System, ANS) part of the cognitive foundations of mathematical skills? Multiple studies reported a correlation between the ANS and mathematical achievement in children. However, some have suggested that such correlation might be mediated by general-purpose inhibitory skills. We addressed the question using a longitudinal approach: we tested the ANS of 60 12 months old infants and, when they were 4 years old (final N = 40), their symbolic math achievement as well as general intelligence and inhibitory skills. Results showed that the ANS at 12 months is a specific predictor of later maths skills independent from general intelligence or inhibitory skills. The correlation between ANS and maths persists when both abilities are measured at four years. These results confirm that the ANS has an early, specific and longstanding relation with mathematical abilities in childhood. RESEARCH HIGHLIGHTS: In the literature there is a lively debate about the correlation between the ANS and maths skills. We longitudinally tested a sample of 60 preverbal infants at 12 months and rested them at 4 years (final sample of 40 infants). The ANS tested at 12 months predicted later symbolic mathematical skills at 4 years, even when controlling for inhibition, general intelligence and perceptual skills. The ANS tested at 4 years remained linked with symbolic maths skills, confirming this early and longstanding relation in childhood.

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.

Visual number sense for real-world scenes shared by deep neural networks and humans

Recently, visual number sense has been identified from deep neural networks (DNNs). However, whether DNNs have the same capacity for real-world scenes, rather than the simple geometric figures that are often tested, is unclear. In this study, we explore the number perception of scenes using AlexNet and find that numerosity can be represented by the pattern of group activation of the category layer units. The global activation of these units increases with the number of objects in the scene, and the variations in their activation decrease accordingly. By decoding the numerosity from this pattern, we reveal that the embedding coefficient of a scene determines the likelihood of potential objects to contribute to numerical perception. This was demonstrated by the more optimized performance for pictures with relatively high embedding coefficients in both DNNs and humans. This study for the first time shows that a distinct feature in visual environments, revealed by DNNs, can modulate human perception, supported by a group-coding mechanism.

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.

Neural representations of the perception of handwritten digits and visual objects from a convolutional neural network compared to humans

We investigated neural representations for visual perception of 10 handwritten digits and six visual objects from a convolutional neural network (CNN) and humans using functional magnetic resonance imaging (fMRI). Once our CNN model was fine-tuned using a pre-trained VGG16 model to recognize the visual stimuli from the digit and object categories, representational similarity analysis (RSA) was conducted using neural activations from fMRI and feature representations from the CNN model across all 16 classes. The encoded neural representation of the CNN model exhibited the hierarchical topography mapping of the human visual system. The feature representations in the lower convolutional (Conv) layers showed greater similarity with the neural representations in the early visual areas and parietal cortices, including the posterior cingulate cortex. The feature representations in the higher Conv layers were encoded in the higher-order visual areas, including the ventral/medial/dorsal stream and middle temporal complex. The neural representations in the classification layers were observed mainly in the ventral stream visual cortex (including the inferior temporal cortex), superior parietal cortex, and prefrontal cortex. There was a surprising similarity between the neural representations from the CNN model and the neural representations for human visual perception in the context of the perception of digits versus objects, particularly in the primary visual and associated areas. This study also illustrates the uniqueness of human visual perception. Unlike the CNN model, the neural representation of digits and objects for humans is more widely distributed across the whole brain, including the frontal and temporal areas.

Visual categories and concepts in the avian brain

Birds are excellent model organisms to study perceptual categorization and concept formation. The renewed focus on avian neuroscience has sparked an explosion of new data in the field. At the same time, our understanding of sensory and particularly visual structures in the avian brain has shifted fundamentally. These recent discoveries have revealed how categorization is mediated in the avian brain and has generated a theoretical framework that goes beyond the realm of birds. We review the contribution of avian categorization research-at the methodical, behavioral, and neurobiological levels. To this end, we first introduce avian categorization from a behavioral perspective and the common elements model of categorization. Second, we describe the functional and structural organization of the avian visual system, followed by an overview of recent anatomical discoveries and the new perspective on the avian 'visual cortex'. Third, we focus on the neurocomputational basis of perceptual categorization in the bird's visual system. Fourth, an overview of the avian prefrontal cortex and the prefrontal contribution to perceptual categorization is provided. The fifth section outlines how asymmetries of the visual system contribute to categorization. Finally, we present a mechanistic view of the neural principles of avian visual categorization and its putative extension to concept learning.

The current state of carnivore cognition

The field of animal cognition has advanced rapidly in the last 25 years. Through careful and creative studies of animals in captivity and in the wild, we have gained critical insights into the evolution of intelligence, the cognitive capacities of a diverse array of taxa, and the importance of ecological and social environments, as well as individual variation, in the expression of cognitive abilities. The field of animal cognition, however, is still being influenced by some historical tendencies. For example, primates and birds are still the majority of study species in the field of animal cognition. Studies of diverse taxa improve the generalizability of our results, are critical for testing evolutionary hypotheses, and open new paths for understanding cognition in species with vastly different morphologies. In this paper, we review the current state of knowledge of cognition in mammalian carnivores. We discuss the advantages of studying cognition in Carnivorans and the immense progress that has been made across many cognitive domains in both lab and field studies of carnivores. We also discuss the current constraints that are associated with studying carnivores. Finally, we explore new directions for future research in studies of carnivore cognition.

Number selective sensorimotor neurons in the crow translate perceived numerosity into number of actions

Translating a perceived number into a matching number of self-generated actions is a hallmark of numerical reasoning in humans and animals alike. To explore this sensorimotor transformation, we trained crows to judge numerical values in displays and to flexibly plan and perform a matching number of pecks. We report number selective sensorimotor neurons in the crow telencephalon that signaled the impending number of self-generated actions. Neuronal population activity during the sensorimotor transformation period predicted whether the crows mistakenly planned fewer or more pecks than instructed. During sensorimotor transformation, both a static neuronal code characterized by persistently number-selective neurons and a dynamic code originating from neurons carrying rapidly changing numerical information emerged. The findings indicate there are distinct functions of abstract neuronal codes supporting the sensorimotor number system.

An insect brain organizes numbers on a left-to-right mental number line

The ability to judge numbers exists in various vertebrate species but also in honey bees, thus raising the question of the phylogenetic origins of numerosity systems. Here, we studied if bees, like humans, organize numbers spatially from left to right according to their magnitude. As the cultural vs. biological origins of this mental number line (MNL) are a subject of debate, our study provides an important perspective for this discussion. We show that bees order numbers from left to right according to their magnitude and that the location of a number on that line varies with the reference number previously trained. Thus, the MNL is a biological numeric representation that is common to the nervous system with distant evolutionary origins.

The “mental number line” (MNL) is a form of spatial numeric representation that associates small and large numbers with the left and right spaces, respectively. This spatio-numeric organization can be found in adult humans and has been related to cultural factors such as writing and reading habits. Yet, both human newborns and birds order numbers consistently with an MNL, thus raising the question of whether culture is a main explanation for MNL. Here, we explored the numeric sense of honey bees and show that after being trained to associate numbers with a sucrose reward, they order numbers not previously experienced from left to right according to their magnitude. Importantly, the location of a number on that scale varies with the reference number previously trained and does not depend on low-level cues present on numeric stimuli. We provide a series of neural explanations for this effect based on the extensive knowledge accumulated on the neural underpinnings of visual processing in honey bees and conclude that the MNL is a form of numeric representation that is evolutionarily conserved across nervous systems endowed with a sense of number, irrespective of their neural complexity.

A neural theory for counting memories

Keeping track of the number of times different stimuli have been experienced is a critical computation for behavior. Here, we propose a theoretical two-layer neural circuit that stores counts of stimulus occurrence frequencies. This circuit implements a data structure, called a count sketch, that is commonly used in computer science to maintain item frequencies in streaming data. Our first model implements a count sketch using Hebbian synapses and outputs stimulus-specific frequencies. Our second model uses anti-Hebbian plasticity and only tracks frequencies within four count categories ("1-2-3-many"), which trades-off the number of categories that need to be distinguished with the potential ethological value of those categories. We show how both models can robustly track stimulus occurrence frequencies, thus expanding the traditional novelty-familiarity memory axis from binary to discrete with more than two possible values. Finally, we show that an implementation of the "1-2-3-many" count sketch exists in the insect mushroom body.

Underestimation in temporal numerosity judgments computationally explained by population coding model

The ability to judge numerosity is essential to an animal’s survival. Nevertheless, the number of signals presented in a sequence is often underestimated. We attempted to elucidate the mechanism for the underestimation by means of computational modeling based on population coding. In the model, the population of neurons which were selective to the logarithmic number of signals responded to sequential signals and the population activity was integrated by a temporal window. The total number of signals was decoded by a weighted average of the integrated activity. The model predicted well the general trends in the human data while the prediction was not fully sufficient for the novel aging effect wherein underestimation was significantly greater for the elderly than for the young in specific stimulus conditions. Barring the aging effect, we can conclude that humans judge the number of signals in sequence by temporally integrating the neural representations of numerosity.

Number neurons in the nidopallium of young domestic chicks

Numerosity, that is, the number of items in a set, is a significant aspect in the perception of the environment. Behavioral and in silico experiments suggest that number sense belongs to a core knowledge system and can be present already at birth. However, neurons sensitive to the number of visual items have been so far described only in the brain of adult animals. Therefore, it remained unknown to what extent their selectivity would depend on visual learning and experience. We found number neurons in the caudal nidopallium (a higher associative area functionally similar to the mammalian prefrontal cortex) of very young, numerically naïve domestic chicks. This result suggests that numerosity perception is possibly an inborn feature of the vertebrate brain.

Numerical cognition is ubiquitous in the animal kingdom. Domestic chicks are a widely used developmental model for studying numerical cognition. Soon after hatching, chicks can perform sophisticated numerical tasks. Nevertheless, the neural basis of their numerical abilities has remained unknown. Here, we describe number neurons in the caudal nidopallium (functionally equivalent to the mammalian prefrontal cortex) of young domestic chicks. Number neurons that we found in young chicks showed remarkable similarities to those in the prefrontal cortex and caudal nidopallium of adult animals. Thus, our results suggest that numerosity perception based on number neurons might be an inborn feature of the vertebrate brain.

Quantity as a Fish Views It: Behavior and Neurobiology

An ability to estimate quantities, such as the number of conspecifics or the size of a predator, has been reported in vertebrates. Fish, in particular zebrafish, may be instrumental in advancing the understanding of magnitude cognition. We review here the behavioral studies that have described the ecological relevance of quantity estimation in fish and the current status of the research aimed at investigating the neurobiological bases of these abilities. By combining behavioral methods with molecular genetics and calcium imaging, the involvement of the retina and the optic tectum has been documented for the estimation of continuous quantities in the larval and adult zebrafish brain, and the contributions of the thalamus and the dorsal-central pallium for discrete magnitude estimation in the adult zebrafish brain. Evidence for basic circuitry can now be complemented and extended to research that make use of transgenic lines to deepen our understanding of quantity cognition at genetic and molecular levels.

Archerfish number discrimination

Debates have arisen as to whether non-human animals actually can learn abstract non-symbolic numerousness or whether they always rely on some continuous physical aspect of the stimuli, covarying with number. Here, we investigated archerfish (Toxotes jaculatrix) non-symbolic numerical discrimination with accurate control for covarying continuous physical stimulus attributes. Archerfish were trained to select one of two groups of black dots (Exp. 1: 3 vs 6 elements; Exp. 2: 2 vs 3 elements); these were controlled for several combinations of physical variables (elements' size, overall area, overall perimeter, density, and sparsity), ensuring that only numerical information was available. Generalization tests with novel numerical comparisons (2 vs 3, 5 vs 8, and 6 vs 9 in Exp. 1; 3 vs 4, 3 vs 6 in Exp. 2) revealed choice for the largest or smallest numerical group according to the relative number that was rewarded at training. None of the continuous physical variables, including spatial frequency, were affecting archerfish performance. Results provide evidence that archerfish spontaneously use abstract relative numerical information for both small and large numbers when only numerical cues are available.

Dogs (canis familiaris) underestimate the quantity of connected items: first demonstration of susceptibility to the connectedness illusion in non-human animals

In humans, numerical estimation is affected by perceptual biases, such as those originating from the spatial arrangement of elements. Different animal species can also make relative quantity judgements. This includes dogs, who have been proposed as a good model for comparative neuroscience. However, dogs do not show the same perceptual biases observed in humans. Thus, the exact perceptual/cognitive mechanisms underlying quantity estimations in dogs and their degree of similarity with humans are still a matter of debate. Here we explored whether dogs are susceptible to the connectedness illusion, an illusion based on the tendency to underestimate the quantity of interconnected items. Dogs were first trained to choose the larger of two food arrays. Then, they were presented with two arrays containing the same quantity of food, of which one had items interconnected by lines. Dogs significantly selected the array with unconnected items, suggesting that, like in humans, connectedness determines underestimation biases, possibly disrupting the perceptual system's ability to segment the display into discrete objects. The similarity in dogs' and humans' susceptibility to the connectedness, but not to other numerical illusions, suggests that different mechanisms are involved in the estimation of quantity of stimuli with different characteristics.

The role of neural tuning in quantity perception

Perception of quantities, such as numerosity, timing, and size, is essential for behavior and cognition. Accumulating evidence demonstrates neurons processing quantities are tuned, that is, have a preferred quantity amount, not only for numerosity, but also other quantity dimensions and sensory modalities. We argue that quantity-tuned neurons are fundamental to understanding quantity perception. We illustrate how the properties of quantity-tuned neurons can underlie a range of perceptual phenomena. Furthermore, quantity-tuned neurons are organized in distinct but overlapping topographic maps. We suggest that this overlap in tuning provides the neural basis for perceptual interactions between different quantities, without the need for a common neural representational code.

The Sense of Number in Fish, with Particular Reference to Its Neurobiological Bases

Simple Summary

The ability to deal with quantity, both discrete (numerosities) and continuous (spatial or temporal extent) developed from an evolutionarily conserved system for approximating numerical magnitude. Non-symbolic number cognition based on an approximate sense of magnitude has been documented in a variety of vertebrate species, including fish. Fish, in particular zebrafish, are widely used as models for the investigation of the genetics and molecular mechanisms of behavior, and thus may be instrumental to development of a neurobiology of number cognition. We review here the behavioural studies that have permitted to identify numerical abilities in fish, and the current status of the research related to the neurobiological bases of these abilities with special reference to zebrafish. Combining behavioural tasks with molecular genetics, molecular biology and confocal microscopy, a role of the retina and optic tectum in the encoding of continuous magnitude in larval zebrafish has been reported, while the thalamus and the dorso-central subdivision of pallium in the encoding of discrete magnitude (number) has been documented in adult zebrafish. Research in fish, in particular zebrafish, may reveal instrumental for identifying and characterizing the molecular signature of neurons involved in quantity discrimination processes of all vertebrates, including humans.

Abstract

It is widely acknowledged that vertebrates can discriminate non-symbolic numerosity using an evolutionarily conserved system dubbed Approximate Number System (ANS). Two main approaches have been used to assess behaviourally numerosity in fish: spontaneous choice tests and operant training procedures. In the first, animals spontaneously choose between sets of biologically-relevant stimuli (e.g., conspecifics, food) differing in quantities (smaller or larger). In the second, animals are trained to associate a numerosity with a reward. Although the ability of fish to discriminate numerosity has been widely documented with these methods, the molecular bases of quantities estimation and ANS are largely unknown. Recently, we combined behavioral tasks with molecular biology assays (e.g c-fos and egr1 and other early genes expression) showing that the thalamus and the caudal region of dorso-central part of the telencephalon seem to be activated upon change in numerousness in visual stimuli. In contrast, the retina and the optic tectum mainly responded to changes in continuous magnitude such as stimulus size. We here provide a review and synthesis of these findings.

Consciousness without cortex

Sensory consciousness - the awareness and ability to report subjective experiences - is a property of biological nervous systems that has evolved out of unconscious processing over hundreds of millions of years. From which brain structures and based on which mechanisms can conscious experience emerge? Based on the body of work in human and nonhuman primates, the emergence of consciousness is intimately associated with the workings of the mammalian cerebral cortex with its specific cell types and layered structure. However, recent neurophysiological recordings demonstrate a neuronal correlate of consciousness in the pallial endbrain of crows. These telencephalic integration centers in birds originate embryonically from other pallial territories, lack a layered architecture characteristic for the cerebral cortex, and exhibit independently evolved pallial cell types. This argues that the mammalian cerebral cortex is not a prerequisite for consciousness to emerge in all vertebrates. Rather, it seems that the anatomical and physiological principles of the telencephalic pallium offer this structure as a brain substrate for consciousness to evolve independently across vertebrate phylogeny.