Can responses to basic non-numerical visual features explain neural numerosity responses?

Cortical quantity representations of visual numerosity and timing overlap increasingly into superior cortices but remain distinct

Many sensory brain areas are organized as topographic maps where neural response preferences change gradually across the cortical surface. Within association cortices, 7-Tesla fMRI and neural model-based analyses have also revealed many topographic maps for quantities like numerosity and event timing, often in similar locations. Numerical and temporal quantity estimations also show behavioral similarities and even interactions. For example, the duration of high-numerosity displays is perceived as longer than that of low-numerosity displays. Such interactions are often ascribed to a generalized magnitude system with shared neural responses across quantities. Anterior quantity responses are more closely linked to behavior. Here, we investigate whether common quantity representations hierarchically emerge by asking whether numerosity and timing maps become increasingly closely related in their overlap, response preferences, and topography. While the earliest quantity maps do not overlap, more superior maps overlap increasingly. In these overlapping areas, some intraparietal maps have consistently correlated numerosity and timing preferences, and some maps have consistent angles between the topographic progressions of numerosity and timing preferences. However, neither of these relationships increases hierarchically like the amount of overlap does. Therefore, responses to different quantities are initially derived separately, then progressively brought together, without generally becoming a common representation. Bringing together distinct responses to different quantities may underlie behavioral interactions and allow shared access to comparison and action planning systems.

The Interaction between Congruency and Numerical Ratio Effects in the Nonsymbolic Comparison Test

The nonsymbolic comparison task is used to investigate the precision of the Approximate Number Sense, the ability to process discrete numerosity without counting and symbols. There is an ongoing debate regarding the extent to which the ANS is influenced by the processing of non-numerical visual cues. To address this question, we assessed the congruency effect in a nonsymbolic comparison task, examining its variability across different stimulus presentation formats and numerical proportions. Additionally, we examined the variability of the numerical ratio effect with the format and congruency. Utilizing generalized linear mixed-effects models with a sample of 290 students (89% female, mean age 19.33 years), we estimated the congruency effect and numerical ratio effect for separated and intermixed formats of stimulus presentation, and for small and large numerical proportions. The findings indicated that the congruency effect increased in large numerical proportion conditions, but this pattern was observed only in the separated format. In the intermixed format, the congruency effect was insignificant for both types of numerical proportion. Notably, the numerical ratio effect varied for congruent and incongruent trials in different formats. The results may suggest that the processing of visual non-numerical parameters may be crucial when numerosity processing becomes noisier, specifically when numerical proportion becomes larger. The implications of these findings for refining the ANS theory are discussed.

Phenomenology, Quantity, and Numerosity

There are many situations in which we interact with collections of objects, from a crowd of people to a bowl of blackberries. There is an experience of the quantity of these items, although not a precise number, and we have this impression quickly and effortlessly. It can be described as an expressive property of the whole. In the literature, the study of this sense of numerosity has a long history, which is reviewed here with examples. I argue that numerosity is a direct perceptual experience, and that all experiences of numerosity, not only estimations, are affected by perceptual organisation.

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.

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.

Intracranial recordings show evidence of numerosity tuning in human parietal cortex

Numerosity is the set size of a group of items. Numerosity perception is a trait shared across numerous species. Numerosity-selective neural populations are thought to underlie numerosity perception. These neurons have been identified primarily using electrical recordings in animal models and blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) in humans. Here we use electrical intracranial recordings to investigate numerosity tuning in humans, focusing on high-frequency transient activations. These recordings combine a high spatial and temporal resolution and can bridge the gap between animal models and human recordings. In line with previous studies, we find numerosity-tuned responses at parietal sites in two out of three participants. Neuronal populations at these locations did not respond to other visual stimuli, i.e. faces, houses, and letters, in contrast to several occipital sites. Our findings further corroborate the specificity of numerosity tuning of in parietal cortex, and further link fMRI results and electrophysiological recordings.

Attention drives human numerosity-selective responses

Numerosity, the set size of a group of items, helps guide behavior and decisions. Previous studies have shown that neural populations respond selectively to numerosities. How numerosity is extracted from the visual scene is a longstanding debate, often contrasting low-level visual with high-level cognitive processes. Here, we investigate how attention influences numerosity-selective responses. The stimuli consisted of black and white dots within the same display. Participants' attention was focused on either black or white dots, while we systematically changed the numerosity of black, white, and total dots. Using 7 T fMRI, we show that the numerosity-tuned neural populations respond only when attention is focused on their preferred numerosity, irrespective of the unattended or total numerosities. Without attention, responses to preferred numerosity are suppressed. Unlike traditional effects of attention in the visual cortex, where attention enhances already existing responses, these results suggest that attention is required to drive numerosity-selective responses.

Auditory timing-tuned neural responses in the human auditory cortices

Perception of sub-second auditory event timing supports multisensory integration, and speech and music perception and production. Neural populations tuned for the timing (duration and rate) of visual events were recently described in several human extrastriate visual areas. Here we ask whether the brain also contains neural populations tuned for auditory event timing, and whether these are shared with visual timing. Using 7T fMRI, we measured responses to white noise bursts of changing duration and rate. We analyzed these responses using neural response models describing different parametric relationships between event timing and neural response amplitude. This revealed auditory timing-tuned responses in the primary auditory cortex, and auditory association areas of the belt, parabelt and premotor cortex. While these areas also showed tonotopic tuning for auditory pitch, pitch and timing preferences were not consistently correlated. Auditory timing-tuned response functions differed between these areas, though without clear hierarchical integration of responses. The similarity of auditory and visual timing tuned responses, together with the lack of overlap between the areas showing these responses for each modality, suggests modality-specific responses to event timing are computed similarly but from different sensory inputs, and then transformed differently to suit the needs of each modality.

Numerosity tuning in human association cortices and local image contrast representations in early visual cortex

Human early visual cortex response amplitudes monotonically increase with numerosity (object number), regardless of object size and spacing. However, numerosity is typically considered a high-level visual or cognitive feature, while early visual responses follow image contrast in the spatial frequency domain. We find that, at fixed contrast, aggregate Fourier power (at all orientations and spatial frequencies) follows numerosity closely but nonlinearly with little effect of object size, spacing or shape. This would allow straightforward numerosity estimation from spatial frequency domain image representations. Using 7T fMRI, we show monotonic responses originate in primary visual cortex (V1) at the stimulus's retinotopic location. Responses here and in neural network models follow aggregate Fourier power more closely than numerosity. Truly numerosity tuned responses emerge after lateral occipital cortex and are independent of retinotopic location. We propose numerosity's straightforward perception and neural responses may result from the pervasive spatial frequency analyses of early visual processing.

Groupitizing modifies neural coding of numerosity

Numerical estimation of arrays of objects is faster and more accurate when items can be clustered into groups, a phenomenon termed “groupitizing.” Grouping can facilitate segregation into subitizable “chunks,” each easily estimated, then summed. The current study investigates whether spatial grouping of arrays drives specific neural responses during numerical estimation, reflecting strategies such as exact calculation and fact retrieval. Fourteen adults were scanned with fMRI while estimating either the numerosity or shape of arrays of items, either randomly distributed or spatially grouped. Numerosity estimation of both classes of stimuli elicited common activation of a right lateralized frontoparietal network. Grouped stimuli additionally recruited regions in the left hemisphere and bilaterally in the angular gyrus. Multivariate pattern analysis showed that classifiers trained with the pattern of neural activations read out from parietal regions, but not from the primary visual areas, can decode different numerosities both within and across spatial arrangements. The behavioral numerical acuity correlated with the decoding performance of the parietal but not with occipital regions. Overall, this experiment suggests that the estimation of grouped stimuli relies on the approximate number system for numerosity estimation, but additionally recruits regions involved in calculation.

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.

Decoding location-specific and location-invariant stages of numerosity processing in subitizing

Extracting the number of objects in perceived scenes is a fundamental cognitive ability. Number processing is proposed to rely on two consecutive stages: an early object location map that captures individuated objects in a location‐specific way and a subsequent location‐invariant representation that captures numerosity at an abstract level. However, it is unclear whether this framework applies to small numerosities that can be individuated at once (“subitized”). Here, we reanalyzed data from two electroencephalography (EEG) experiments using multivariate pattern decoding to identify location‐specific and location‐invariant stages of numerosity processing in the subitizing range. In these experiments, one to three targets were presented in the left or right hemifield, which allowed for decoding target numerosity within each hemifield separately (location specific) or across hemifields (location invariant). Experiment 1 indicated the presence of a location‐specific stage (180–200 ms after stimulus), followed by a location‐invariant stage (300 ms after stimulus). A time‐by‐channel searchlight analysis revealed that the early location‐specific stage is most evident at occipital channels, whereas the late location‐invariant stage is most evident at parietal channels. Experiment 2 showed that both location‐specific and location‐invariant components are engaged only during tasks that explicitly require numerosity processing, ruling out automatic, and passive recording of numerosity. These results suggest that numerosity coding in subitizing is strongly grounded on an attention‐based, location‐specific stage. This stage overlaps with the subsequent activation of a location‐invariant stage, where a full representation of numerosity is finalized. Taken together, our findings provide clear evidence for a temporal and spatial segregation of location‐specific and location‐invariant numerosity coding of small object numerosities.

Topographic numerosity maps cover subitizing and estimation ranges

Numerosity, the set size of a group of items, helps guide behaviour and decisions. Non-symbolic numerosities are represented by the approximate number system. However, distinct behavioural performance suggests that small numerosities, i.e. subitizing range, are implemented differently in the brain than larger numerosities. Prior work has shown that neural populations selectively responding (i.e. hemodynamic responses) to small numerosities are organized into a network of topographical maps. Here, we investigate how neural populations respond to large numerosities, well into the ANS. Using 7 T fMRI and biologically-inspired analyses, we found a network of neural populations tuned to both small and large numerosities organized within the same topographic maps. These results demonstrate a continuum of numerosity preferences that progressively cover both the subitizing range and beyond within the same numerosity map, suggesting a single neural mechanism. We hypothesize that differences in map properties, such as cortical magnification and tuning width, underlie known differences in behaviour.

Typical Crossmodal Numerosity Perception in Preterm Newborns

Premature birth is associated with a high risk of damage in the parietal cortex, a key area for numerical and non-numerical magnitude perception and mathematical reasoning. Children born preterm have higher rates of learning difficulties for school mathematics. In this study, we investigated how preterm newborns (born at 28-34 weeks of gestation age) and full-term newborns respond to visual numerosity after habituation to auditory stimuli of different numerosities. The results show that the two groups have a similar preferential looking response to visual numerosity, both preferring the incongruent set after crossmodal habituation. These results suggest that the numerosity system is resistant to prematurity.

Adaptation to visual numerosity changes neural numerosity selectivity

Perceiving numerosity, i.e. the set size of a group of items, is an evolutionarily preserved ability found in humans and animals. A useful method to infer the neural underpinnings of a given perceptual property is sensory adaptation. Like other primary perceptual attributes, numerosity is susceptible to adaptation. Recently, we have shown numerosity-selective neural populations with a topographic organization in the human brain. Here, we investigated whether numerosity adaptation can affect the numerosity selectivity of these populations using ultra-high field (7 Tesla) functional magnetic resonance imaging (fMRI). Participants viewed stimuli of changing numerosity (1 to 7 dots), which allowed the mapping of numerosity selectivity. We interleaved a low or high numerosity adapter stimulus with these mapping stimuli, repeatedly presenting 1 or 20 dots respectively to adapt the numerosity-selective neural populations. We analyzed the responses using custom-build population receptive field neural models of numerosity encoding and compared estimated numerosity preferences between adaptation conditions. We replicated our previous studies where we found several topographic maps of numerosity-selective responses. We found that overall, numerosity adaptation altered the preferred numerosities within the numerosity maps, resulting in predominantly attractive biases towards the numerosity of the adapter. The differential biases could be explained by the difference between the unadapted preferred numerosity and the numerosity of the adapter, with attractive biases being observed with higher difference. The results could link perceptual numerosity adaptation effects to changes in neural numerosity selectivity.

Topographic maps representing haptic numerosity reveals distinct sensory representations in supramodal networks

Dedicated maps for cognitive quantities such as timing, size and numerosity support the view that topography is a general principle of brain organization. To date, however, all of these maps were driven by the visual system. Here, we ask whether there are supramodal topographic maps representing cognitive dimensions irrespective of the stimulated sensory modality. We measured haptically and visually driven numerosity-selective neural responses using model-based analyses and ultra-high field (7T) fMRI. We found topographically organized neural populations tuned to haptic numerosity. The responses to visual or haptic numerosity shared a similar cortical network. However, the maps of the two modalities only partially overlap. Thus, although both visual and haptic numerosities are processed in a similar supramodal functional network, the underlying neural populations may be related, but distinct. Therefore, we hypothesize that overlap between modality-specific maps facilitates cross-modal interactions and supramodal representation of cognitive quantities.

Topographically organized tuned responses to haptic numerosity were found in the human brain. The responses to visual or haptic numerosity shared a similar large-scale cortical network, yet the maps of the two modalities only partially overlapped, suggesting distinct underlying neural populations.

Fast saccadic eye-movements in humans suggest that numerosity perception is automatic and direct

Fast saccades are rapid automatic oculomotor responses to salient and ecologically important visual stimuli such as animals and faces. Discriminating the number of friends, foe, or prey may also have an evolutionary advantage. In this study, participants were asked to saccade rapidly towards the more numerous of two arrays. Participants could discriminate numerosities with high accuracy and great speed, as fast as 190 ms. Intermediate numerosities were more likely to elicit fast saccades than very low or very high numerosities. Reaction-times for vocal responses (collected in a separate experiment) were slower, did not depend on numerical range, and correlated only with the slow not the fast saccades, pointing to different systems. The short saccadic reaction-times we observe are surprising given that discrimination using numerosity estimation is thought to require a relatively complex neural circuit, with several relays of information through the parietal and prefrontal cortex. Our results suggest that fast numerosity-driven saccades may be generated on a single feed-forward pass of information recruiting a primitive system that cuts through the cortical hierarchy and rapidly transforms the numerosity information into a saccade command.

A Network of Topographic Maps in Human Association Cortex Hierarchically Transforms Visual Timing-Selective Responses

Accurately timing sub-second sensory events is crucial when perceiving our dynamic world. This ability allows complex human behaviors that require timing-dependent multisensory integration and action planning. Such behaviors include perception and performance of speech, music, driving, and many sports. How are responses to sensory event timing processed for multisensory integration and action planning? We measured responses to viewing systematically changing visual event timing using ultra-high-field fMRI. We analyzed these responses with neural population response models selective for event duration and frequency, following behavioral, computational, and macaque action planning results and comparisons to alternative models. We found systematic local changes in timing preferences (recently described in supplementary motor area) in an extensive network of topographic timing maps, mirroring sensory cortices and other quantity processing networks. These timing maps were partially left lateralized and widely spread, from occipital visual areas through parietal multisensory areas to frontal action planning areas. Responses to event duration and frequency were closely linked. As in sensory cortical maps, response precision varied systematically with timing preferences, and timing selectivity systematically varied between maps. Progressing from posterior to anterior maps, responses to multiple events were increasingly integrated, response selectivity narrowed, and responses focused increasingly on the middle of the presented timing range. These timing maps largely overlap with numerosity and visual field map networks. In both visual timing map and visual field map networks, selective responses and topographic map organization may facilitate hierarchical transformations by allowing neural populations to interact over minimal distances.

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Many brain areas show neural responses to specific ranges of visual event timing

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Timing preferences change gradually in these areas, forming topographic timing maps

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Neural response properties hierarchically transform from visual to premotor areas

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Timing, numerosity, and visual field map networks are distinct but largely overlap

Challenging the neurobiological link between number sense and symbolic numerical abilities

A significant body of research links individual differences in symbolic numerical abilities, such as arithmetic, to number sense, the neurobiological system used to approximate and manipulate quantities without language or symbols. However, recent findings from cognitive neuroscience challenge this influential theory. Our current review presents an overview of evidence for the number sense account of symbolic numerical abilities and then reviews recent studies that challenge this account, organized around the following four assertions. (1) There is no number sense as traditionally conceived. (2) Neural substrates of number sense are more widely distributed than common consensus asserts, complicating the neurobiological evidence linking number sense to numerical abilities. (3) The most common measures of number sense are confounded by other cognitive demands, which drive key correlations. (4) Number sense and symbolic number systems (Arabic digits, number words, and so on) rely on distinct neural mechanisms and follow independent developmental trajectories. The review follows each assertion with comments on future directions that may bring resolution to these issues.

Attentional amplification of neural codes for number independent of other quantities along the dorsal visual stream

Humans and other animals base important decisions on estimates of number, and intraparietal cortex is thought to provide a crucial substrate of this ability. However, it remains debated whether an independent neuronal processing mechanism underlies this ‘number sense’, or whether number is instead judged indirectly on the basis of other quantitative features. We performed high-resolution 7 Tesla fMRI while adult human volunteers attended either to the numerosity or an orthogonal dimension (average item size) of visual dot arrays. Along the dorsal visual stream, numerosity explained a significant amount of variance in activation patterns, above and beyond non-numerical dimensions. Its representation was selectively amplified and progressively enhanced across the hierarchy when task relevant. Our results reveal a sensory extraction mechanism yielding information on numerosity separable from other dimensions already at early visual stages and suggest that later regions along the dorsal stream are most important for explicit manipulation of numerical quantity.

Numbers and the ability to count and calculate are an essential part of human culture. They are part of everyday life, featuring in calendars, computers or the weekly shop, but also in some of humanity’s biggest achievements: without them the pyramids or space travel would not exist. A precursor of sophisticated mathematical skill could reside in a simpler mental ability: the capacity to assess numerical quantities at a glance. This ‘number sense’ appears in humans in early childhood and it is also present in other animals, but it is still poorly understood.

Brain imaging techniques have identified the parts of the brain that are active when perceiving numbers or making calculations. As techniques have advanced, it has become possible to resolve fine differences in brain activity that occur when people switch their attention between different visual tasks. But how exactly does the human brain process visual information to make sense of numbers? One theory suggests that humans use visual cues, such as the size of a group of objects or how densely packed objects are, to estimate numbers. On the other hand, it is also possible that humans can sense number directly, without reference to other properties of the group being observed.

Castaldi et al. presented twenty adult volunteers with groups of dots and asked them to focus either on the number of dots or on the size of the dots during a brain scan. This approach allowed the separation of brain signals specific to number from signals corresponding to other visual cues, such as size or density of the group. The experiment revealed that brain activity changed depending on the number of dots displayed. The signal related to number became stronger when people focused on the number of dots, while signals related to other properties of the group remained unchanged. Moreover, brain signals for number were observed at the very early stages of visual processing, in the parts of the brain that receive input from the eyes first.

These results suggest that the human visual system perceives number directly, and not by processing information about the size or density of a group of objects. This finding provides insights into how human brains encode numbers, which could be important to understand disorders where number sense can be impaired leading to difficulties learning math and operating with numbers.

A MEG Study on the Processing of Time and Quantity: Parietal Overlap but Functional Divergence

A common magnitude system for the processing of time and numerosity, supported by areas in the posterior parietal cortex, has been proposed by some authors. The present study aims to investigate possible intersections between the neural processing of non-numerical (time) and numerical magnitudes in the posterior parietal lobe. Using Magnetoencephalography for the comparison of brain source activations during the processing of duration and numerosity contrasts, we demonstrate parietal overlap as well as dissociations between these two dimensions. Within the parietal cortex, the main areas of overlap were bilateral precuneus, bilateral intraparietal sulci, and right supramarginal gyrus. Interestingly, however, these regions did not equivalently correlated with the behavior for the two dimensions: left and right precuneus together with the right supramarginal gyrus accounted functionally for durational judgments, whereas numerosity judgments were accounted by the activation pattern in the right intraparietal sulcus. Present results, indeed, demonstrate an overlap between the neural substrates for processing duration and quantity. However, the functional relevance of parietal overlapping areas for each dimension is not the same. In fact, our data indicates that the same parietal sites rule differently non-numerical and numerical dimensions, as parts of broader networks.

Number, time, and space are not singularly represented: Evidence against a common magnitude system beyond early childhood

Our ability to represent temporal, spatial, and numerical information is critical for understanding the world around us. Given the prominence of quantitative representations in the natural world, numerous cognitive, neurobiological, and developmental models have been proposed as a means of describing how we track quantity. One prominent theory posits that time, space, and number are represented by a common magnitude system, or a common neural locus (i.e., Bonn & Cantlon in Cognitive Neuropsychology, 29(1/2), 149-173, 2012; Cantlon, Platt, & Brannon in Trends in Cognitive Sciences, 13(2), 83-91, 2009; Meck & Church in Animal Behavior Processes, 9(3), 320, 1983; Walsh in Trends in Cognitive Sciences, 7(11), 483-488, 2003). Despite numerous similarities in representations of time, space, and number, an increasing body of literature reveals striking dissociations in how each quantity is processed, particularly later in development. These findings have led many researchers to consider the possibility that separate systems may be responsible for processing each quantity. This review will analyze evidence in favor of a common magnitude system, particularly in infancy, which will be tempered by counter evidence, the majority of which comes from experiments with children and adult participants. After reviewing the current data, we argue that although the common magnitude system may account for quantity representations in infancy, the data do not provide support for this system throughout the life span. We also identify future directions for the field and discuss the likelihood of the developmental divergence model of quantity representation, like that of Newcombe (Ecological Psychology, 2, 147-157, 2014), as a more plausible account of quantity development.

Asymmetrical interference between number and item size perception provides evidence for a domain specific impairment in dyscalculia

Dyscalculia, a specific learning disability that impacts arithmetical skills, has previously been associated to a deficit in the precision of the system that estimates the approximate number of objects in visual scenes (the so called 'number sense' system). However, because in tasks involving numerosity comparisons dyscalculics' judgements appears disproportionally affected by continuous quantitative dimensions (such as the size of the items), an alternative view linked dyscalculia to a domain-general difficulty in inhibiting task-irrelevant responses. To arbitrate between these views, we evaluated the degree of reciprocal interference between numerical and non-numerical quantitative dimensions in adult dyscalculics and matched controls. We used a novel stimulus set orthogonally varying in mean item size and numerosity, putting particular attention into matching both features' perceptual discriminability. Participants compared those stimuli based on each of the two dimensions. While control subjects showed no significant size interference when judging numerosity, dyscalculics' numerosity judgments were strongly biased by the unattended size dimension. Importantly however, both groups showed the same degree of interference from the unattended dimension when judging mean size. Moreover, only the ability to discard the irrelevant size information when comparing numerosity (but not the reverse) significantly predicted calculation ability across subjects. Overall, our results show that numerosity discrimination is less prone to interference than discrimination of another quantitative feature (mean item size) when the perceptual discriminability of these features is matched, as here in control subjects. By quantifying, for the first time, dyscalculic subjects' degree of interference on another orthogonal dimension of the same stimuli, we are able to exclude a domain-general inhibition deficit as explanation for their poor / biased numerical judgement. We suggest that enhanced reliance on non-numerical cues during numerosity discrimination can represent a strategy to cope with a less precise number sense.

Psychophysical evidence for the number sense

It is now clear that most animals, including humans, possess an ability to rapidly estimate number. Some have questioned whether this ability arises from dedicated numerosity mechanisms, or is derived indirectly from judgements of density or other attributes. We describe a series of psychophysical experiments, largely using adaptation techniques, which demonstrate clearly the existence of a number sense in humans. The number sense is truly general, extending over space, time and sensory modality, and is closely linked with action. We further show that when multiple cues are present, numerosity emerges as the natural dimension for discrimination. However, when element density increases past a certain level, the elements become too crowded to parse, and the scene is perceived as a texture rather than array of elements. The two different regimes are psychophysically discriminable in that they follow distinct psychophysical laws, and show different dependencies on eccentricity, luminance levels and effects of perceptual grouping. The distinction is important, as the ability to discriminate numerosity, but not texture, correlates with formal maths skills.

This article is part of the discussion meeting issue ‘The origins of numerical abilities’.

Numerical encoding in early visual cortex

The ability to estimate numerosity in a visual array arose early in evolution, develops early in human development, and is correlated with mathematical ability. Previous work with visually presented arrays indicates that the intraparietal sulcus (IPS) represents number. However, it is not clear if the number signal originates in IPS or is propagated from earlier visual areas. Previous work from our group has demonstrated a rapidly instantiated representation of number in low-level regions of visual cortex using the high temporal resolution of event-related electro-encephalography (EEG). Here, we use a rapid event-related functional magnetic resonance imaging (fMRI) paradigm and find convergent evidence for a number signal in low-level visual cortex (areas V1, V2, and V3). Employing a stringent set of stimulus controls, we demonstrate that this signal cannot be explained by the total extent of the array, the density of the items in the array, the aggregate visual area of the items, the size of individual items, the proportion of the array covered by items, nor the overall scale of the array and items. Our findings thus provide strong support for the hypothesis that number is rapidly and directly encoded early in the visual processing stream.

The role of sensorimotor processes in social group contagion

Although it is well known that action observation triggers an imitative response, not much is known about how these responses develop as a function of group size. Research on social contagion suggests that imitative tendencies initially increase but then stabilize as groups become larger. However, these findings have mainly been explained in terms of interpretative processes. Across seven experiments (N = 322), the current study investigated the contribution of sensorimotor processes to social group contagion by looking at the relation between group size and automatic imitation in a task that involved minimal interpretation. The results of Experiments 1-2 revealed that automatic imitation increased with group size according to an asymptotic curve on congruent trials but a linear curve on incongruent trials. The results of Experiments 3-7 showed that the asymptote on congruent trials disappeared when no control was needed, namely in the absence of incongruent trials. This suggests that the asymptote in the relation between group size and automatic imitation can be explained in terms of strategic control mechanisms that aim to prevent unintended imitative responses. The findings of the current study are in close correspondence with previous research in the social domain and as such support the hypothesis that sensorimotor processes contribute to the relation between group size and social contagion.

Effect of perceived length on numerosity estimation: Evidence from the Müller-Lyer illusion

Previous studies showed that the magnitude information conveyed by sensory cues, such as length or surface, influences the ability to compare the numerosity of sets of objects. However, the perceptual nature of this representation and how it interacts with the processes involved in numerical judgements remain unclear. This study aims to address these issues by studying the interference of length on numerosity under different perceptual and response conditions. The first experiment shows that the influence of length does not depend on the actual length but on subjective values reflecting the way length is perceived in a given visual context. The Müller-Lyer illusion was used to manipulate the perceived length of two dot arrays independently of their actual length. When the length of two dot arrays was equal but perceived as different due to the illusion, participants erroneously reported differences in the number of dots contained in each array, evidencing a similar effect of Müller-Lyer illusion on length and numerosity comparison. This finding was replicated in a second experiment where participants had to give a verbal estimate of the number of dots contained in a given array, thereby eliminating the choice between a small or large response. Compared with a neutral condition, estimations were systematically larger than the actual number of dots as the illusory length increased. These results demonstrate that the illusory-induced experience of length influences numerosity estimation over and beyond objective cues and that this influence is not a response selection bias.

Data describing the accuracy of non-numerical visual features in predicting fMRI responses to numerosity

Here we took several stimulus configurations that have the same numerosity progression but vary considerably in their non-numerical visual features. We collected responses to these stimuli using ultra-high-field (7T) fMRI in a posterior parietal area that responds to changes in these stimuli. We first quantify the relationships between numerosity and several non-numerical visual features in each stimulus configuration. We then use population receptive field (pRF) modeling to quantify how well responses to each of these visual features predicts the observed responses to each stimulus configuration, and observed responses to all stimulus configurations together. We compare the predictive accuracy of responses to numerosity and to non-numerical visual features in explaining the observed responses. This provides the details of the analysis outcomes summarized in an accompanying article (10.1016/j.neuroimage.2017.02.012, NIMG-16-1350).