The neuronal code for number

The effect of visual parameters on neural activation during nonsymbolic number comparison and its relation to math competency

Nonsymbolic numerical comparison task performance (whereby a participant judges which of two groups of objects is numerically larger) is thought to index the efficiency of neural systems supporting numerical magnitude perception, and performance on such tasks has been related to individual differences in math competency. However, a growing body of research suggests task performance is heavily influenced by visual parameters of the stimuli (e.g. surface area and dot size of object sets) such that the correlation with math is driven by performance on trials in which number is incongruent with visual cues. Almost nothing is currently known about whether the neural correlates of nonsymbolic magnitude comparison are also affected by visual congruency. To investigate this issue, we used functional magnetic resonance imaging (fMRI) to analyze neural activity during a nonsymbolic comparison task as a function of visual congruency in a sample of typically developing high school students (n = 36). Further, we investigated the relation to math competency as measured by the preliminary scholastic aptitude test (PSAT) in 10th grade. Our results indicate that neural activity was modulated by the ratio of the dot sets being compared in brain regions previously shown to exhibit an effect of ratio (i.e. left anterior cingulate, left precentral gyrus, left intraparietal sulcus, and right superior parietal lobe) when calculated from the average of congruent and incongruent trials, as it is in most studies, and that the effect of ratio within those regions did not differ as a function of congruency condition. However, there were significant differences in other regions in overall task-related activation, as opposed to the neural ratio effect, when congruent and incongruent conditions were contrasted at the whole-brain level. Math competency negatively correlated with ratio-dependent neural response in the left insula across congruency conditions and showed distinct correlations when split across conditions. There was a positive correlation between math competency in the right supramarginal gyrus during congruent trials and a negative correlation in the left angular gyrus during incongruent trials. Together, these findings support the idea that performance on the nonsymbolic comparison task relates to math competency and ratio-dependent neural activity does not differ by congruency condition. With regards to math competency, congruent and incongruent trials showed distinct relations between math competency and individual differences in ratio-dependent neural activity.

Zebrafish prefer larger to smaller shoals: analysis of quantity estimation in a genetically tractable model organism

Numerical abilities have been demonstrated in a variety of non-human vertebrates. However, underlying biological mechanisms have been difficult to study due to a paucity of experimental tools. Powerful genetic and neurobiological tools already exist for the zebrafish, but numerical abilities remain scarcely explored with this species. Here, we investigate the choice made by single experimental zebrafish between numerically different shoals of conspecifics presented concurrently on opposite sides of the experimental tank. We examined this choice using the AB strain and pet store zebrafish. We found zebrafish of both populations to generally prefer the numerically larger shoal to the smaller one. This preference was significant for contrasted ratios above or equalling 2:1 (i.e. 4 vs. 0, 4 vs. 1, 8 vs. 2, 6 vs. 2 and 6 vs. 3). Interestingly, zebrafish showed no significant preference when each of the two contrasted shoals had at least 4 members, e.g. in a contrast 8 versus 4. These results confirm that zebrafish possess the ability to distinguish larger numbers of items from smaller number of items, in a shoaling context, with a potential limit above 4. Our findings confirm the utility of the zebrafish for the exploration of both the behavioural and the biological mechanisms underlying numerical abilities in vertebrates.

Numerosity processing in early visual cortex

While parietal cortex is thought to be critical for representing numerical magnitudes, we recently reported an event-related potential (ERP) study demonstrating selective neural sensitivity to numerosity over midline occipital sites very early in the time course, suggesting the involvement of early visual cortex in numerosity processing. However, which specific brain area underlies such early activation is not known. Here, we tested whether numerosity-sensitive neural signatures arise specifically from the initial stages of visual cortex, aiming to localize the generator of these signals by taking advantage of the distinctive folding pattern of early occipital cortices around the calcarine sulcus, which predicts an inversion of polarity of ERPs arising from these areas when stimuli are presented in the upper versus lower visual field. Dot arrays, including 8-32dots constructed systematically across various numerical and non-numerical visual attributes, were presented randomly in either the upper or lower visual hemifields. Our results show that neural responses at about 90ms post-stimulus were robustly sensitive to numerosity. Moreover, the peculiar pattern of polarity inversion of numerosity-sensitive activity at this stage suggested its generation primarily in V2 and V3. In contrast, numerosity-sensitive ERP activity at occipito-parietal channels later in the time course (210-230ms) did not show polarity inversion, indicating a subsequent processing stage in the dorsal stream. Overall, these results demonstrate that numerosity processing begins in one of the earliest stages of the cortical visual stream.

Learning Recruits Neurons Representing Previously Established Associations in the Corvid Endbrain

Crows quickly learn arbitrary associations. As a neuronal correlate of this behavior, single neurons in the corvid endbrain area nidopallium caudolaterale (NCL) change their response properties during association learning. In crows performing a delayed association task that required them to map both familiar and novel sample pictures to the same two choice pictures, NCL neurons established a common, prospective code for associations. Here, we report that neuronal tuning changes during learning were not distributed equally in the recorded population of NCL neurons. Instead, such learning-related changes relied almost exclusively on neurons which were already encoding familiar associations. Only in such neurons did behavioral improvements during learning of novel associations coincide with increasing selectivity over the learning process. The size and direction of selectivity for familiar and newly learned associations were highly correlated. These increases in selectivity for novel associations occurred only late in the delay period. Moreover, NCL neurons discriminated correct from erroneous trial outcome based on feedback signals at the end of the trial, particularly in newly learned associations. Our results indicate that task-relevant changes during association learning are not distributed within the population of corvid NCL neurons but rather are restricted to a specific group of association-selective neurons. Such association neurons in the multimodal cognitive integration area NCL likely play an important role during highly flexible behavior in corvids.

Magnitude Codes for Cross-Modal Working Memory in the Primate Frontal Association Cortex

Quantitative features of stimuli may be ordered along a magnitude continuum, or line. Magnitude refers to parameters of different types of stimulus properties. For instance, the frequency of a sound relates to sensory and continuous stimulus properties, whereas the number of items in a set is an abstract and discrete property. In addition, within a stimulus property, magnitudes need to be processed not only in one modality, but across multiple modalities. In the sensory domain, for example, magnitude applies to both to the frequency of auditory sounds and tactile vibrations. Similarly, both the number of visual items and acoustic events constitute numerical quantity, or numerosity. To support goal-directed behavior and executive functions across time, magnitudes need to be held in working memory, the ability to briefly retain and manipulate information in mind. How different types of magnitudes across multiple modalities are represented in working memory by single neurons has only recently been explored in primates. These studies show that neurons in the frontal lobe can encode the same magnitude type across sensory modalities. However, while multimodal sensory magnitude in relative comparison tasks is represented by monotonically increasing or decreasing response functions ("summation code"), multimodal numerical quantity in absolute matching tasks is encoded by neurons tuned to preferred numerosities ("labeled-line code"). These findings indicate that most likely there is not a single type of cross-modal working-memory code for magnitudes, but rather a flexible code that depends on the stimulus dimension as well as on the task requirements.

Computational Architecture of the Parieto-Frontal Network Underlying Cognitive-Motor Control in Monkeys

The statistical structure of intrinsic parietal and parieto-frontal connectivity in monkeys was studied through hierarchical cluster analysis. Based on their inputs, parietal and frontal areas were grouped into different clusters, including a variable number of areas that in most instances occupied contiguous architectonic fields. Connectivity tended to be stronger locally: that is, within areas of the same cluster. Distant frontal and parietal areas were targeted through connections that in most instances were reciprocal and often of different strength. These connections linked parietal and frontal clusters formed by areas sharing basic functional properties. This led to five different medio-laterally oriented pillar domains spanning the entire extent of the parieto-frontal system, in the posterior parietal, anterior parietal, cingulate, frontal, and prefrontal cortex. Different information processing streams could be identified thanks to inter-domain connectivity. These streams encode fast hand reaching and its control, complex visuomotor action spaces, hand grasping, action/intention recognition, oculomotor intention and visual attention, behavioral goals and strategies, and reward and decision value outcome. Most of these streams converge on the cingulate domain, the main hub of the system. All of them are embedded within a larger eye–hand coordination network, from which they can be selectively set in motion by task demands.

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

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

An emergentist perspective on the origin of number sense

The finding that human infants and many other animal species are sensitive to numerical quantity has been widely interpreted as evidence for evolved, biologically determined numerical capacities across unrelated species, thereby supporting a 'nativist' stance on the origin of number sense. Here, we tackle this issue within the 'emergentist' perspective provided by artificial neural network models, and we build on computer simulations to discuss two different approaches to think about the innateness of number sense. The first, illustrated by artificial life simulations, shows that numerical abilities can be supported by domain-specific representations emerging from evolutionary pressure. The second assumes that numerical representations need not be genetically pre-determined but can emerge from the interplay between innate architectural constraints and domain-general learning mechanisms, instantiated in deep learning simulations. We show that deep neural networks endowed with basic visuospatial processing exhibit a remarkable performance in numerosity discrimination before any experience-dependent learning, whereas unsupervised sensory experience with visual sets leads to subsequent improvement of number acuity and reduces the influence of continuous visual cues. The emergent neuronal code for numbers in the model includes both numerosity-sensitive (summation coding) and numerosity-selective response profiles, closely mirroring those found in monkey intraparietal neurons. We conclude that a form of innatism based on architectural and learning biases is a fruitful approach to understanding the origin and development of number sense.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Primitive Concepts of Number and the Developing Human Brain

Counting is an evolutionarily recent cultural invention of the human species. In order for humans to have conceived of counting in the first place, certain representational and logical abilities must have already been in place. The focus of this review is the origins and nature of those fundamental mechanisms that promoted the emergence of the human number concept. Five claims are presented that support an evolutionary view of numerical development: 1) number is an abstract concept with an innate basis in humans, 2) maturational processes constrain the development of humans' numerical representations between infancy and adulthood, 3) there is evolutionary continuity in the neural processes of numerical cognition in primates, 4) primitive logical abilities support verbal counting development in humans, and 5) primitive neural processes provide the foundation for symbolic numerical development in the human brain. We support these claims by examining current evidence from animal cognition, child development, and human brain function. The data show that at the basis of human numerical concepts are primitive perceptual and logical mechanisms that have evolutionary homologs in other primates and form the basis of numerical development in the human brain. In the final section of the review, we discuss some hypotheses for what makes human numerical reasoning unique by drawing on evidence from human and non-human primate neuroimaging research.

The Distributed Nature of Working Memory

Studies in humans and non-human primates have provided evidence for storage of working memory contents in multiple regions ranging from sensory to parietal and prefrontal cortex. We discuss potential explanations for these distributed representations: (i) features in sensory regions versus prefrontal cortex differ in the level of abstractness and generalizability; and (ii) features in prefrontal cortex reflect representations that are transformed for guidance of upcoming behavioral actions. We propose that the propensity to produce persistent activity is a general feature of cortical networks. Future studies may have to shift focus from asking where working memory can be observed in the brain to how a range of specialized brain areas together transform sensory information into a delayed behavioral response.

Effects of adaptation on numerosity decoding in the human brain

Psychophysical studies have shown that numerosity is a sensory attribute susceptible to adaptation. Neuroimaging studies have reported that, at least for relatively low numbers, numerosity can be accurately discriminated in the intra-parietal sulcus. Here we developed a novel rapid adaptation paradigm where adapting and test stimuli are separated by pauses sufficient to dissociate their BOLD activity. We used multivariate pattern recognition to classify brain activity evoked by non-symbolic numbers over a wide range (20-80), both before and after psychophysical adaptation to the highest numerosity. Adaptation caused underestimation of all lower numerosities, and decreased slightly the average BOLD responses in V1 and IPS. Using support vector machine, we showed that the BOLD response of IPS, but not in V1, classified numerosity well, both when tested before and after adaptation. However, there was no transfer from training pre-adaptation responses to testing post-adaptation, and vice versa, indicating that adaptation changes the neuronal representation of the numerosity. Interestingly, decoding was more accurate after adaptation, and the amount of improvement correlated with the amount of perceptual underestimation of numerosity across subjects. These results suggest that numerosity adaptation acts directly on IPS, rather than indirectly via other low-level stimulus parameters analysis, and that adaptation improves the capacity to discriminate numerosity.

Counteracting Implicit Conflicts by Electrical Inhibition of the Prefrontal Cortex

Cognitive conflicts and distractions by task-irrelevant information often counteract effective and goal-directed behaviors. In some cases, conflicting information can even emerge implicitly, without an overt distractor, by the automatic activation of mental representations. For instance, during number processing, magnitude information automatically elicits spatial associations resembling a mental number line. This spatial–numerical association of response codes (SNARC) effect can modulate cognitive-behavioral performance but is also highly flexible and context-dependent, which points toward a critical involvement of working memory functions. Transcranial direct current stimulation to the PFC, in turn, has been effective in modulating working memory-related cognitive performance. In a series of experiments, we here demonstrate that decreasing activity of the left PFC by cathodal transcranial direct current stimulation consistently and specifically eliminates implicit cognitive conflicts based on the SNARC effect, but explicit conflicts based on visuospatial distraction remain unaffected. This dissociation is polarity-specific and appears unrelated to functional magnitude processing as classified by regular numerical distance effects. These data demonstrate a causal involvement of the left PFC in implicit cognitive conflicts based on the automatic activation of spatial–numerical processing. Corroborating the critical interaction of brain stimulation and neurocognitive functions, our findings suggest that distraction from goal-directed behavior by automatic activation of implicit, task-irrelevant information can be blocked by the inhibition of prefrontal activity.

Cell-type-specific modulation of targets and distractors by dopamine D1 receptors in primate prefrontal cortex

The prefrontal cortex (PFC) is crucial for maintaining relevant information in working memory and resisting interference. PFC neurons are strongly regulated by dopamine, but it is unknown whether dopamine receptors are involved in protecting target memories from distracting stimuli. We investigated the prefrontal circuit dynamics and dopaminergic modulation of targets and distractors in monkeys trained to ignore interfering stimuli in a delayed-match-to-numerosity task. We found that dopamine D1 receptors (D1Rs) modulate the recovery of task-relevant information following a distracting stimulus. The direction of modulation is cell-type-specific: in putative pyramidal neurons, D1R inhibition enhances and D1R stimulation attenuates coding of the target stimulus after the interference, while the opposite pattern is observed in putative interneurons. Our results suggest that dopaminergic neuromodulation of PFC circuits regulates mental representations of behaviourally relevant stimuli that compete with task-irrelevant input and could play a central role for cognitive functioning in health and disease.

The Symbol Grounding Problem Revisited: A Thorough Evaluation of the ANS Mapping Account and the Proposal of an Alternative Account Based on Symbol-Symbol Associations

Recently, a lot of studies in the domain of numerical cognition have been published demonstrating a robust association between numerical symbol processing and individual differences in mathematics achievement. Because numerical symbols are so important for mathematics achievement, many researchers want to provide an answer on the ‘symbol grounding problem,’ i.e., how does a symbol acquires its numerical meaning? The most popular account, the approximate number system (ANS) mapping account, assumes that a symbol acquires its numerical meaning by being mapped on a non-verbal and ANS. Here, we critically evaluate four arguments that are supposed to support this account, i.e., (1) there is an evolutionary system for approximate number processing, (2) non-symbolic and symbolic number processing show the same behavioral effects, (3) non-symbolic and symbolic numbers activate the same brain regions which are also involved in more advanced calculation and (4) non-symbolic comparison is related to the performance on symbolic mathematics achievement tasks. Based on this evaluation, we conclude that all of these arguments and consequently also the mapping account are questionable. Next we explored less popular alternative, where small numerical symbols are initially mapped on a precise representation and then, in combination with increasing knowledge of the counting list result in an independent and exact symbolic system based on order relations between symbols. We evaluate this account by reviewing evidence on order judgment tasks following the same four arguments. Although further research is necessary, the available evidence so far suggests that this symbol–symbol association account should be considered as a worthy alternative of how symbols acquire their meaning.

A shared numerical representation for action and perception

Humans and other species have perceptual mechanisms dedicated to estimating approximate quantity: a sense of number. Here we show a clear interaction between self-produced actions and the perceived numerosity of subsequent visual stimuli. A short period of rapid finger-tapping (without sensory feedback) caused subjects to underestimate the number of visual stimuli presented near the tapping region; and a period of slow tapping caused overestimation. The distortions occurred both for stimuli presented sequentially (series of flashes) and simultaneously (clouds of dots); both for magnitude estimation and forced-choice comparison. The adaptation was spatially selective, primarily in external, real-world coordinates. Our results sit well with studies reporting links between perception and action, showing that vision and action share mechanisms that encode numbers: a generalized number sense, which estimates the number of self-generated as well as external events.