Number is more than meets the eye: Unveiling segmentation mechanisms in numerosity perception with visual illusions

Animals and humans possess an adaptive ability to rapidly estimate approximate numerosity, yet the visual mechanisms underlying this process remain poorly understood. Evidence suggests that approximate numerosity relies on segmented perceptual units modulated by grouping cues, with perceived numerosity decreasing when objects are connected by irrelevant lines, independent of low-level features. However, most studies have focused on physical objects. Illusory contours (ICs) are powerful tools for exploring visual segmentation mechanisms, as "illusory" objects exhibit perceptual biases (e.g., tilt aftereffect) similar to real objects, suggesting shared processing mechanisms. To investigate whether approximate numerosity perception of ICs is influenced by connectedness, we conducted a psychophysical forced-choice task. Participants compared Ehrenstein-like ICs ensembles of varying numerosities interspersed with four task-irrelevant lines. We manipulated the number of connected pairs (0, 2, or 4) by aligning lines with the ICs-triggering gaps, while controlling low-level features across conditions. Our results revealed a monotonic underestimation of numerosity as connections increased, with constant precision reflecting Weber-like encoding. Reaction times proportionally increased with connectedness, suggesting an underlying recurrent neural mechanism. These findings demonstrate that ICs ensembles are subject to the same connectedness effect as real objects, supporting a shared visual mechanism for numerosity extraction. This work highlights the parallels between real and illusory object processing and provides insights into segmentation mechanisms relevant to models of artificial intelligence and visual perception.

Innate numerosity perception and its roots in early vision

In the current issue of Cerebral Cortex, Lorenzi et al. (2025) discuss evidence for an innate sense of object number (numerosity) in the brains of many species, without the need for visual experience. This commentary discusses how numerosity processing can be understood as an innate property of vision, derived from the spatial frequency-based representation of images in the visual systems of many animals.

The Mechanisms and Neural Signature of Time-averaged Numerosity Perception

The animal brain is endowed with an innate sense of number allowing to intuitively perceive the approximate quantity of items in a scene, or "numerosity." This ability is not limited to items distributed in space, but also to events unfolding in time and to the average numerosity of dynamic scenes. How the brain computes and represents the average numerosity over time, however, remains unclear. Here, we investigate the mechanisms and EEG signature of the perception of average numerosity over time. To do so, we used stimuli composed of a variable number (3-12) of briefly presented dot arrays (50 msec each) and asked participants to judge the average numerosity of the sequence. We first show that the weight of different portions of the stimuli in determining the judgment depends on how many arrays are included in the sequence itself: the longer the sequence, the lower the weight of the latest arrays. Second, we show systematic adaptation effects across stimuli in consecutive trials. Importantly, the EEG results highlight two processing stages whereby the amplitude of occipital ERPs reflects the adaptation effect (∼300 msec after stimulus onset) and the accuracy and precision of average numerosity judgments (∼450-700 msec). These two stages are consistent with processes involved with the representation of perceived average numerosity and with perceptual decision-making, respectively. Overall, our findings provide new evidence showing how the visual system computes the average numerosity of dynamic visual stimuli, and support the existence of a dedicated, relatively low-level perceptual mechanism mediating this process.

Feature-selective adaptation of numerosity perception

Perceptual adaptation has been widely used to infer the existence of numerosity detectors, enabling animals to quickly estimate the number of objects in a scene. Here, we investigated, in humans, whether numerosity adaptation is influenced by stimulus feature changes as previous research suggested that adaptation is reduced when the colour of adapting and test stimuli did not match. We tested whether such adaptation reduction is due to unspecific novelty effects or changes of stimuli identity. Numerosity adaptation was measured for stimuli matched or unmatched for low-level (colour, luminance, shape and motion) or high-level (letters' identity and face emotions) features. Robust numerosity adaptation occurred in all conditions, but it was reduced when adapting and test stimuli differed for colour, luminance and shape. However, no reduction was observed between moving and still stimuli, a readable change that did not affect the item's identity. Similarly, changes in letters' spatial rotations or face features did not affect adaptation magnitude. Overall, changes in stimulus identity defined by low-level features, rather than novelty in general, determined the strength of the adaptation effects, provided these changes were readily noticeable. These findings suggest that numerosity mechanisms operate on categorized items in addition to the total quantity of the set.

The neural link between stimulus duration and spatial location in the human visual hierarchy

Integrating spatial and temporal information is essential for our sensory experience. While psychophysical evidence suggests spatial dependencies in duration perception, few studies have directly tested the neural link between temporal and spatial processing. Using ultra-high-field functional MRI and neuronal-based modeling, we investigated how and where the processing and the representation of a visual stimulus duration is linked to that of its spatial location. Our results show a transition in duration coding: from monotonic and spatially-dependent in early visual cortex to unimodal and spatially-invariant in frontal cortex. Along the dorsal visual stream, particularly in the intraparietal sulcus (IPS), neuronal populations show common selective responses to both spatial and temporal information. In the IPS, spatial and temporal topographic organizations are also linked, although duration maps are smaller, less clustered, and more variable across participants. These findings help identify the mechanisms underlying human perception of visual duration and characterize the functional link between time and space processing, highlighting the importance of their interactions in shaping brain responses.

The divisive normalization model of visual number sense: model predictions and experimental confirmation

Our intuitive sense of number allows rapid estimation for the number of objects (numerosity) in a scene. How does the continuous nature of neural information processing create a discrete representation of number? A neurocomputational model with divisive normalization explains this process and existing data; however, a successful model should not only explain existing data but also generate novel predictions. Here, we experimentally test novel predictions of this model to evaluate its merit for explaining mechanisms of numerosity perception. We did so by consideration of the coherence illusion: the underestimation of number for arrays containing heterogeneous compared to homogeneous items. First, we established the existence of the coherence illusion for homogeneity manipulations of both area and orientation of items in an array. Second, despite the behavioral similarity, the divisive normalization model predicted that these two illusions should reflect activity in different stages of visual processing. Finally, visual evoked potentials from an electroencephalography experiment confirmed these predictions, showing that area and orientation coherence modulate brain responses at distinct latencies and topographies. These results demonstrate the utility of the divisive normalization model for explaining numerosity perception, according to which numerosity perception is a byproduct of canonical neurocomputations that exist throughout the visual pathway.

The interplay between spatial and non-spatial grouping cues over approximate number perception

Humans and animals share the cognitive ability to quickly extract approximate number information from sets. Main psychophysical models suggest that visual approximate numerosity relies on segmented units, which can be affected by Gestalt rules. Indeed, arrays containing spatial grouping cues, such as connectedness, closure, and even symmetry, are underestimated compared to ungrouped arrays with equal low-level features. Recent evidence suggests that non-spatial cues, such as color-similarity, also trigger numerosity underestimation. However, in natural vision, several grouping cues may coexist in the scene. Notably, conjunction of grouping cues (color and closure) reduces perceived numerosity following an additive rule. To test whether the conjunction-effect holds for other Gestalt cues, we investigated the effect of connectedness and symmetry over numerosity perception both in isolation and, critically, in conjunction with luminance similarity. Participants performed a comparison-task between a reference and a test stimulus varying in numerosity. In Experiment 1, test stimuli contained two isolated groupings (connectedness or luminance), a conjunction (connectedness and luminance), and a neutral condition (no groupings). Results show that point of subjective equality was higher in both isolated grouping conditions compared to the neutral condition. Furthermore, in the conjunction condition, the biases from isolated grouping cues added linearly, resulting in a numerosity underestimation equal to the sum of the isolated biases. In Experiment 2 we found that conjunction of symmetry and luminance followed the same additive rule. These findings strongly suggest that both spatial and non-spatial isolated cues affect numerosity perception. Crucially, we show that their conjunction effect extends to symmetry and connectedness.

Emergence of behavioral phenomena and adaptation effects in human numerosity decoder using recurrent neural networks

Humans possess an innate ability to visually perceive numerosities, which refers to the cardinality of a set. Numerous studies indicate that the lateral intraparietal cortex (LIP) and other intraparietal sulcus (IPS) regions (region) of the brain contain the neurological substrates responsible for number processing. Existing computational models of number perception often focus on a limited range of numbers and fail to account for important behavioral characteristics like adaptation effects, despite simulating fundamental aspects such as size and distance effects. To address these limitations, our study develops (introduces) a novel computational model of number perception utilizing a network of neurons with self-excitatory and mutual inhibitory properties. Our approach assumes that the mean activation of the network at steady state can encode numerosity by exhibiting a monotonically increasing relationship with the input variable set size. By optimizing the total number of inhibition strengths required, we achieve coverage of the full range of numbers through three distinct intervals: 1 to 4, 5 to 17, and 21 to 50. Remarkably, this division aligns closely with the breakpoints in numerosity perception identified in behavioral studies. Furthermore, our study develops a method for decoding the mean activation into a continuous scale of numbers spanning from 1 to 50. Additionally, we propose a mechanism for dynamically selecting the inhibition strength based on current inputs, enabling the network to operate effectively across an extended (entire) range of numerosities. Our model not only sheds new light on the generation of diverse behavioral phenomena in the brain but also elucidates how continuous visual attributes and adaptation effects influence perceived numerosity.

Fundamental units of numerosity estimation

Humans can approximately enumerate a large number of objects at a single glance. While several mechanisms have been proposed to account for this ability, the fundamental units over which they operate remain unclear. Previous studies have argued that estimation mechanisms act only on topologically distinct units or on units formed by spatial grouping cues such as proximity and connectivity, but not on units grouped by similarity. Over four experiments, we tested this claim by systematically assessing and demonstrating that similarity grouping leads to underestimation, just as spatial grouping does. Ungrouped objects with the same low-level properties as grouped objects did not cause underestimation. Further, the underestimation caused by spatial and similarity grouping was additive, suggesting that these grouping processes operate independently. These findings argue against the proposal that estimation mechanisms operate solely on topological units. Instead, we conclude that estimation processes act on representations constructed after Gestalt grouping principles, whether similarity based or spatial, have organised incoming visual input.

The symmetry-induced numerosity illusion depends on visual attention

Symmetry is an important and strong cue we rely on to organize the visual world. Although it is at the basis of objects segmentation in a visual scene, it can sometimes bias our perception. When asked to discriminate numerical quantities between symmetric and asymmetric arrays, individuals tend to underestimate the number of items in the symmetric stimuli. The reason for this underestimation is currently unknown. In this study we investigated whether the symmetry-induced numerosity underestimation depends on perceptual grouping mechanisms by depriving attentional resources. Twenty-six adults judged the numerosity of dot arrays arranged symmetrically or randomly, while ignoring a visual distractor (single task) or while simultaneously judging its color and orientation (dual-task). Diverting attention to the concurrent color-orientation conjunction task halved the symmetry-induced numerosity underestimation. Taken together these results showed that the bias in numerosity perception of symmetric arrays depends-at least partially-on attentional resources and suggested that it might originate from the recruitment of attentional dependent incremental grouping mechanisms.

Near-optimal integration of the magnitude information of time and numerosity

Magnitude information is often correlated in the external world, providing complementary information about the environment. As if to reflect this relationship, the perceptions of different magnitudes (e.g. time and numerosity) are known to influence one another. Recent studies suggest that such magnitude interaction is similar to cue integration, such as multisensory integration. Here, we tested whether human observers could integrate the magnitudes of two quantities with distinct physical units (i.e. time and numerosity) as abstract magnitude information. The participants compared the magnitudes of two visual stimuli based on time, numerosity, or both. Consistent with the predictions of the maximum-likelihood estimation model, the participants integrated time and numerosity in a near-optimal manner; the weight of each dimension was proportional to their relative reliability, and the integrated estimate was more reliable than either the time or numerosity estimate. Furthermore, the integration approached a statistical optimum as the temporal discrepancy of the acquisition of each piece of information became smaller. These results suggest that magnitude interaction arises through a similar computational mechanism to cue integration. They are also consistent with the idea that different magnitudes are processed by a generalized magnitude system.

Spontaneous supra-modal encoding of number in the infant brain

The core knowledge hypothesis postulates that infants automatically analyze their environment along abstract dimensions, including numbers. According to this view, approximate numbers should be encoded quickly, pre-attentively, and in a supra-modal manner by the infant brain. Here, we directly tested this idea by submitting the neural responses of sleeping 3-month-old infants, measured with high-density electroencephalography (EEG), to decoders designed to disentangle numerical and non-numerical information. The results show the emergence, in approximately 400 ms, of a decodable number representation, independent of physical parameters, that separates auditory sequences of 4 vs. 12 tones and generalizes to visual arrays of 4 vs. 12 objects. Thus, the infant brain contains a number code that transcends sensory modality, sequential or simultaneous presentation, and arousal state.

Symmetry as a grouping cue for numerosity perception

To estimate the number of objects in an image, each element needs to be segregated as a single unit. Several principles guide the process of element identification, one of the strongest being symmetry. In the current study, we investigated how symmetry affects the ability to rapidly estimate the number of objects (numerosity). Participants judged the numerosity of asymmetric or symmetric arrays of various numerosities. The results show that the numerosity of symmetrical arrays was significantly underestimated at low numerosities, but the effect was greatly reduced at higher numerosities. Adding an additional axis of symmetry (double symmetry) further reduced perceived numerosity. The magnitude of the symmetry-driven underestimation was inversely correlated with autistic personality traits, consistent with previous work associating autistic traits with perceptual grouping. Overall, these results support the idea that perceived numerosity relies on object segmentation and grouping cues, with symmetry playing a key role.

Incentive motivation improves numerosity discrimination in children and adolescents

We recently showed that incentive motivation improves the precision of the Approximate Number System (ANS) in young adults. To shed light on the development of incentive motivation, the present study investigated whether this effect and its underlying mechanisms may also be observed in younger samples. Specifically, seven-year-old children (n = 23; 12 girls) and 14-year-old adolescents (n = 30; 15 girls) performed a dot comparison task with monetary reward incentives. Both age groups showed higher accuracy in a reward compared to a neutral condition and, similarly, higher processing efficiency as revealed by the drift rate parameter of the EZ-diffusion model. Furthermore, in line with the Incentive Salience Hypothesis, phasic pupil dilations—indicating the activation of the brain’s salience network—were greater in incentivized trials in both age groups. Together these finding suggest that incentive modulation improves numerosity discrimination in children and adolescents by enhancing the perceptual saliency of numerosity information. However, the observed reward anticipation effects were less pronounced in children relative to adolescents. Furthermore, unlike previous findings regarding young adults, the decision thresholds of children and adolescents were not raised by the monetary reward, which may indicate a more protracted development of incentive regulation of response caution than perceptual evidence accumulation.

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.

Deprivation of Auditory Experience Influences Numerosity Discrimination, but Not Numerosity Estimation

Number sense is the ability to estimate the number of items, and it is common to many species. Despite the numerous studies dedicated to unveiling how numerosity is processed in the human brain, to date, it is not clear whether the representation of numerosity is supported by a single general mechanism or by multiple mechanisms. Since it is known that deafness entails a selective impairment in the processing of temporal information, we assessed the approximate numerical abilities of deaf individuals to disentangle these two hypotheses. We used a numerosity discrimination task (2AFC) and an estimation task, in both cases using sequential (temporal) or simultaneous (spatial) stimuli. The results showed a selective impairment of the deaf participants compared with the controls (hearing) in the temporal numerosity discrimination task, while no difference was found to discriminate spatial numerosity. Interestingly, the deaf and hearing participants did not differ in spatial or temporal numerosity estimation. Overall, our results suggest that the deficit in temporal processing induced by deafness also impacts perception in other domains such as numerosity, where sensory information is conveyed in a temporal format, which further suggests the existence of separate mechanisms subserving the processing of temporal and spatial numerosity.

Nonsymbolic numerosity in sets with illusory-contours exploits a context-sensitive, but contrast-insensitive, visual boundary formation process

The visual mechanisms underlying approximate numerical representation are still intensely debated because numerosity information is often confounded with continuous sensory cues (e.g., texture density, area, convex hull). However, numerosity is underestimated when a few items are connected by illusory contours (ICs) lines without changing other physical cues, suggesting in turn that numerosity processing may rely on discrete visual input. Yet, in these previous works, ICs were generated by black-on-gray inducers producing an illusory brightness enhancement, which could represent a further continuous sensory confound. To rule out this possibility, we tested participants in a numerical discrimination task in which we manipulated the alignment of 0, 2, or 4 pairs of open/closed inducers and their contrast polarity. In Experiment 1, aligned open inducers had only one polarity (all black or all white) generating ICs lines brighter or darker than the gray background. In Experiment 2, open inducers had always opposite contrast polarity (one black and one white inducer) generating ICs without strong brightness enhancement. In Experiment 3, reverse-contrast inducers were aligned but closed with a line preventing ICs completion. Results showed that underestimation triggered by ICs lines was independent of inducer contrast polarity in both Experiment 1 and Experiment 2, whereas no underestimation was found in Experiment 3. Taken together, these results suggest that mere brightness enhancement is not the primary cause of the numerosity underestimation induced by ICs lines. Rather, a boundary formation mechanism insensitive to contrast polarity may drive the effect, providing further support to the idea that numerosity processing exploits discrete inputs.

Numbers, numerosities, and new directions

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system (ANS) represents numbers or numerosities, and why the ANS represents rational (but not irrational) numbers.

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 dynamics of grouping-induced biases in apparent numerosity revealed by a continuous tracking technique

Connecting pairs of items causes robust underestimation of the numerosity of an ensemble, presumably by invoking grouping mechanisms. Here we asked whether this underestimation in numerosity judgments could be revealed and further explored by continuous tracking, a newly developed technique that allows for fast and efficient data acquisition and monitors the dynamics of the responses. Participants continuously reproduced the perceived numerosity of a cloud of dots by moving a cursor along a number line, while the number of dots and the proportion connected by lines varied over time following two independent random walks. The technique was robust and efficient, and correlated well with results obtained with a standard psychophysics task. Connecting objects with lines caused an underestimation of approximately 15% during tracking, agreeing with previous studies. The response to the lines was slower than the response to the physical numerosity, with a delay of approximately 150 ms, suggesting that this extra time is necessary for processing the grouping effect.

Spontaneous representation of numerosity zero in a deep neural network for visual object recognition

Conceiving "nothing" as a numerical value zero is considered a sophisticated numerical capability that humans share with cognitively advanced animals. We demonstrate that representation of zero spontaneously emerges in a deep learning neural network without any number training. As a signature of numerical quantity representation, and similar to real neurons from animals, numerosity zero network units show maximum activity to empty sets and a gradual decrease in activity with increasing countable numerosities. This indicates that the network spontaneously ordered numerosity zero as the smallest numerical value along the number line. Removal of empty-set network units caused specific deficits in the network's judgment of numerosity zero, thus reflecting these units' functional relevance. These findings suggest that processing visual information is sufficient for a visual number sense that includes zero to emerge and explains why cognitively advanced animals with whom we share a nonverbal number system exhibit rudiments of numerosity zero.

Context-Dependent Modulation of Early Visual Cortical Responses to Numerical and Nonnumerical Magnitudes

Whether and how the brain encodes discrete numerical magnitude differently from continuous nonnumerical magnitude is hotly debated. In a previous set of studies, we orthogonally varied numerical (numerosity) and nonnumerical (size and spacing) dimensions of dot arrays and demonstrated a strong modulation of early visual evoked potentials (VEPs) by numerosity and not by nonnumerical dimensions. Although very little is known about the brain's response to systematic changes in continuous dimensions of a dot array, some authors intuit that the visual processing stream must be more sensitive to continuous magnitude information than to numerosity. To address this possibility, we measured VEPs of participants viewing dot arrays that changed exclusively in one nonnumerical magnitude dimension at a time (size or spacing) while holding numerosity constant and compared this to a condition where numerosity was changed while holding size and spacing constant. We found reliable but small neural sensitivity to exclusive changes in size and spacing; however, exclusively changing numerosity elicited a much more robust modulation of the VEPs. Together with previous work, these findings suggest that sensitivity to magnitude dimensions in early visual cortex is context dependent: The brain is moderately sensitive to changes in size and spacing when numerosity is held constant, but sensitivity to these continuous variables diminishes to a negligible level when numerosity is allowed to vary at the same time. Neurophysiological explanations for the encoding and context dependency of numerical and nonnumerical magnitudes are proposed within the framework of neuronal normalization.

The pupil responds spontaneously to perceived numerosity

Although luminance is the main determinant of pupil size, the amplitude of the pupillary light response is also modulated by stimulus appearance and attention. Here we ask whether perceived numerosity modulates the pupillary light response. Participants passively observed arrays of black or white dots of matched physical luminance but different physical or illusory numerosity. In half the patterns, pairs of dots were connected by lines to create dumbbell-like shapes, inducing an illusory underestimation of perceived numerosity; in the other half, connectors were either displaced or removed. Constriction to white arrays and dilation to black were stronger for patterns with higher perceived numerosity, either physical or illusory, with the strength of the pupillary light response scaling with the perceived numerosity of the arrays. Our results show that even without an explicit task, numerosity modulates a simple automatic reflex, suggesting that numerosity is a spontaneously encoded visual feature.

The specious interaction of time and numerosity perception

Magnitude information is essential to create a representation of the external environment and successfully interact with it. Duration and numerosity, for example, can shape our predictions and bias each other (i.e. the greater the number of people queuing, the longer we expect to wait). While these biases suggest the existence of a generalized magnitude system, asymmetric effects (i.e. numerosity affecting duration but not vice versa) challenged this idea. Here, we propose that such asymmetric integration depends on the stimuli used and the neural processing dynamics they entail. Across multiple behavioural experiments employing different stimulus presentation displays (static versus dynamic) and experimental manipulations known to bias numerosity and duration perceptions (i.e. connectedness and multisensory integration), we show that the integration between numerosity and time can be symmetrical if the stimuli entail a similar neural time-course and numerosity unfolds over time. Overall, these findings support the idea of a generalized magnitude system, but also highlight the role of early sensory processing in magnitude representation and integration.

Evidence for an A-Modal Number Sense: Numerosity Adaptation Generalizes Across Visual, Auditory, and Tactile Stimuli

Humans and other species share a perceptual mechanism dedicated to the representation of approximate quantities that allows to rapidly and reliably estimate the numerosity of a set of objects: an Approximate Number System (ANS). Numerosity perception shows a characteristic shared by all primary visual features: it is susceptible to adaptation. As a consequence of prolonged exposure to a large/small quantity (“adaptor”), the apparent numerosity of a subsequent (“test”) stimulus is distorted yielding a robust under- or over-estimation, respectively. Even if numerosity adaptation has been reported across several sensory modalities (vision, audition, and touch), suggesting the idea of a central and a-modal numerosity processing system, evidence for cross-modal effects are limited to vision and audition, two modalities that are known to preferentially encode sensory stimuli in an external coordinate system. Here we test whether numerosity adaptation for visual and auditory stimuli also distorts the perceived numerosity of tactile stimuli (and vice-versa) despite touch being a modality primarily coded in an internal (body-centered) reference frame. We measured numerosity discrimination of stimuli presented sequentially after adaptation to series of either few (around 2 Hz; low adaptation) or numerous (around 8 Hz; high adaptation) impulses for all possible combinations of visual, auditory, or tactile adapting and test stimuli. In all cases, adapting to few impulses yielded a significant overestimation of the test numerosity with the opposite occurring as a consequence of adaptation to numerous stimuli. The overall magnitude of adaptation was robust (around 30%) and rather similar for all sensory modality combinations. Overall, these findings support the idea of a truly generalized and a-modal mechanism for numerosity representation aimed to process numerical information independently from the sensory modality of the incoming signals.

Serial dependence in time and numerosity perception is dimension-specific

The perception of a visual event (e.g., a flock of birds) at the present moment can be biased by a previous perceptual experience (e.g., the perception of an earlier flock). Serial dependence is a perceptual bias whereby a current stimulus appears more similar to a previous one than it actually is. Whereas serial dependence emerges within several visual stimulus dimensions, whether it could simultaneously operate across different dimensions of the same stimulus (e.g., the numerosity and the duration of a visual pattern) remains unclear. Here we address this question by assessing the presence of serial dependence across duration and numerosity, two stimulus dimensions that are often associated and can bias each other. Participants performed either a duration or a numerosity discrimination task, in which they compared a constant reference with a variable test stimulus, varying along the task-relevant dimension (either duration or numerosity). Serial dependence was induced by a task-irrelevant inducer, that is, a stimulus presented before the reference and always varying in both duration and numerosity. The results show systematic serial dependencies only within the task-relevant stimulus dimension, that is, stimulus numerosity affects numerosity perception only, and duration affects duration perception only. Additionally, at least in the numerosity condition, the task-irrelevant dimension of the inducer (duration) had an opposite, repulsive effect. These findings thus show that attractive serial dependence operates in a highly specific fashion and does not transfer across different stimulus dimensions. Instead, the repulsive influence, possibly reflecting perceptual adaptation, can transfer from one dimension to another.

Numerosity representation in a deep convolutional neural network

Enumerating objects in the environment (i.e., “number sense”) is crucial for survival in many animal species, and foundational for the construction of more abstract and complex mathematical knowledge in humans. Perhaps surprisingly, deep convolutional neural networks (DCNNs) spontaneously emerge a similar number sense even without any explicit training for numerosity estimation. However, little is known about how the number sense emerges, and the extent to which it is comparable with human number sense. Here, we examined whether the numerosity underestimation effect, a phenomenon indicating that numerosity perception acts upon the perceptual number rather than the physical number, can be observed in DCNNs. In a typical DCNN, AlexNet, we found that number-selective units at late layers operated on the perceptual number, like humans do. More importantly, this perceptual number sense did not emerge abruptly, rather developed progressively along the hierarchy in the DCNN, shifting from the physical number sense at early layers to perceptual number sense at late layers. Our finding hence provides important implications for the neural implementation of number sense in the human brain and advocates future research to determine whether the representation of numerosity also develops gradually along the human visual stream from physical number to perceptual number.

The Number Sense Represents (Rational) Numbers

On a now orthodox view, humans and many other animals possess a "number sense," or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique-the arguments from congruency, confounds, and imprecision-and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as "numerosities" or "quanticals," as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

The ratio effect in visual numerosity comparisons is preserved despite spatial frequency equalisation

How non-symbolic numerosity is visually extracted remains a matter of intense debate. Most evidence suggests that numerosity is directly extracted on individual objects following Weber's law, at least for a moderate numerical range. Alternative accounts propose that, whatever the range, numerosity is indirectly derived from summary texture-statistics of the raw image such as spatial frequency (SF). Here, to disentangle these accounts, we tested whether the well-known behavioural signature of numerosity encoding (ratio effect) is preserved despite the equalisation of the SF content. In Experiment 1, participants had to select the numerically larger of two briefly presented moderate-range numerical sets (i.e., 8-18 dots) carefully matched for SF; the ratio between numerosities was manipulated by levels of increasing difficulty (e.g., 0.66, 0.75, 0.8). In Experiment 2, participants performed the same task, but they were presented with both the original and SF equalised stimuli. In both experiments, the results clearly showed a ratio-dependence of the performance: numerosity discrimination became harder and slower as the ratio between numerosities increased. Moreover, this effect was found to be independent of the stimulus type, although the overall performance was better with the original rather than the SF equalised stimuli (Experiment 2). Taken together, these findings indicate that the power spectrum per se cannot explain the main behavioural signature of Weber-like encoding of numerosities (the ratio effect), at least over the tested numerical range, partially challenging alternative indirect accounts of numerosity processing.

Similarly oriented objects appear more numerous

Several non-numerical factors influence the numerical estimation of visual arrays, including the spacing of items and whether they are arranged randomly or symmetrically. Here we report a novel numerosity illusion we term the coherence illusion. When items in an array have a coherent orientation (all pointing in the same direction) they seem to be more numerous than when items are oriented randomly. Participants show parametric effects of orientation coherence in three distinct numerical judgment tasks. These findings are not predicted by any current model of numerical estimation. We discuss array entropy as a possible framework for explaining both the coherence illusion and the previously reported regular-random illusion.

Disentangling feedforward versus feedback processing in numerosity representation

Numerosity is a fundamental aspect of the external environment, needed to guide our behavior in an effective manner. Previous studies show that numerosity processing involves at least two temporal stages (~100 and ~150 msec after stimulus onset) in early visual cortex. One possibility is that the two stages reflect an initial feedforward processing followed by feedback signals from higher-order cortical areas that underlie segmentation of visual inputs into perceptual units that define numerosity. Alternatively, multiple stages of feedforward processing might progressively refine the input leading to the segmented representation. Here, we distinguish these two hypotheses by exploiting the connectedness illusion (i.e., the systematic underestimation of pairwise-connected dots), backward masking (to suppress feedback signals), and serial dependence (i.e., a perceptual bias making a stimulus appear to be more similar to its preceding one). Our results show that a connected dot array biases the numerosity representation of the subsequent dot array based on its illusory perception, irrespective of whether it is visible or suppressed by masking. These findings demonstrate that feedback processing is not strictly necessary for the perceptual segmentation that gives rise to perceived numerosity, and instead suggest that different stages of feedforward activity presumably carrying low and high spatial frequency information are sufficient to create a numerosity representation in early visual areas.

Tactile numerosity is coded in external space

Humans, and several non-human species, possess the ability to make approximate but reliable estimates of the number of objects around them. Alike other perceptual features, numerosity perception is susceptible to adaptation: exposure to a high number of items causes underestimation of the numerosity of a subsequent set of items, and vice versa. Several studies have investigated adaptation in the auditory and visual modality, whereby stimuli are preferentially encoded in an external coordinate system. As tactile stimuli are primarily coded in an internal (body-centered) reference frame, here we ask whether tactile numerosity adaptation operates based on internal or external spatial coordinates as it occurs in vision or audition. Twenty participants performed an adaptation task with their right hand located either in the right (uncrossed) or left (crossed) hemispace, in order for the two hands to occupy either two completely different positions, or the same position in space, respectively. Tactile adaptor and test stimuli were passively delivered either to the same (adapted) or different (non-adapted) hands. Our results show a clear signature of tactile numerosity adaptation aftereffects with a pattern of over- and under-estimation according to the adaptation rate (low and high, respectively). In the uncrossed position, we observed stronger adaptation effects when adaptor and test stimuli were delivered to the "adapted" hand. However, when both hands were aligned in the same spatial position (crossed condition), the magnitude of adaptation was similar irrespective of which hand received adaptor and test stimuli. These results demonstrate that numerosity information is automatically coded in external coordinates even in the tactile modality, suggesting that such a spatial reference frame is an intrinsic property of numerosity processing irrespective of the sensory modality.

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.

Shape facilitates number: brain potentials and microstates reveal the interplay between shape and numerosity in human vision

Recognition of simple shapes and numerosity estimation for small quantities are often studied independently of each other, but we know that these processes are both rapid and accurate, suggesting that they may be mediated by common neural mechanisms. Here we address this issue by examining how spatial configuration, shape complexity, and luminance polarity of elements affect numerosity estimation. We directly compared the Event Related Potential (ERP) time-course for numerosity estimation under shape and random configurations and found a larger N2 component for shape over lateral-occipital electrodes (250–400 ms), which also increased with higher numbers. We identified a Left Mid Frontal (LMF; 400–650 ms) component over left-lateralised medial frontal sites that specifically separated low and high numbers of elements, irrespective of their spatial configuration. Different luminance-polarities increased N2 amplitude only, suggesting that shape but not numerosity is selective to polarity. Functional microstates confined numerosity to a strict topographic distribution occurring within the LMF time-window, while a microstate responding only to shape-configuration was evidenced earlier, in the N2 time-window. We conclude that shape-coding precedes numerosity estimation, which can be improved when the number of elements and shape vertices are matched. Thus, numerosity estimation around the subitizing range is facilitated by a shape-template matching process.

Unconscious Number Discrimination in the Human Visual System

How do humans compute approximate number? According to one influential theory, approximate number representations arise in the intraparietal sulcus and are amodal, meaning that they arise independent of any sensory modality. Alternatively, approximate number may be computed initially within sensory systems. Here we tested for sensitivity to approximate number in the visual system using steady state visual evoked potentials. We recorded electroencephalography from humans while they viewed dotclouds presented at 30 Hz, which alternated in numerosity (ranging from 10 to 20 dots) at 15 Hz. At this rate, each dotcloud backward masked the previous dotcloud, disrupting top-down feedback to visual cortex and preventing conscious awareness of the dotclouds' numerosities. Spectral amplitude at 15 Hz measured over the occipital lobe (Oz) correlated positively with the numerical ratio of the stimuli, even when nonnumerical stimulus attributes were controlled, indicating that subjects' visual systems were differentiating dotclouds on the basis of their numerical ratios. Crucially, subjects were unable to discriminate the numerosities of the dotclouds consciously, indicating the backward masking of the stimuli disrupted reentrant feedback to visual cortex. Approximate number appears to be computed within the visual system, independently of higher-order areas, such as the intraparietal sulcus.

Distinct strategies for estimating the temporal average of numerical and perceptual information

Humans can estimate global trends in dynamic information presented either as perceptual features or as symbolic codes such as numbers. Previous studies on temporal statistics estimation have shown that observers judge the temporal average of visual attributes according to information from the last few frames of the presentation sequence (in what is referred to as the recency effect). Here, we investigated how humans estimate the temporal average of number vs. orientation using identical stimuli for the two tasks. In Experiment 1, a randomly-selected single-digit number was serially presented at orientations randomly varying over time. In Experiment 2, a texture comprising a random number of Gabor elements was shown at orientations randomly varying over time. In both experiments, observers judged the temporal averages of the numerical values and orientations in separate blocks. Results showed that observers judging the temporal average of orientation relied upon information from later frames as predicted by a typical model of perceptual decision making. By contrast, for the judgement of numerical values, we found that the impacts of each temporal frame were constant or varied little across temporal frames regardless of whether the numerical information was given as digits or by the number of texture elements. The results are interpreted as evidence that distinct computational strategies may be involved in estimating the temporal averages of perceptual features and numerical information.

Anisotropy of perceived numerosity: Evidence for a horizontal-vertical numerosity illusion

Many studies have investigated whether numerical and spatial abilities share similar cognitive systems. A novel approach to this issue consists of investigating whether the same perceptual biases underlying size illusions can be identified in numerical estimation tasks. In this study, we required adult participants to estimate the number of white dots in arrays made of white and black dots displayed in such a way as to generate horizontal-vertical illusions with inverted T and L configurations. In agreement with previous literature, we found that participants tended to underestimate the target numbers. However, in the presence of the illusory patterns, participants were less inclined to underestimate the number of vertically aligned white dots. This reflects the perceptual biases underlying horizontal-vertical illusions. In addition, we identified an enhanced illusory effect when participants observed vertically aligned white dots in the T shape compared to the L shape, a result that resembles the length bisection bias reported in the spatial domain. Overall, we found the first evidence that numerical estimation differs as a function of the vertical or horizontal displacement of the stimuli. In addition, the involvement of the same perceptual biases observed in spatial tasks supports the idea that spatial and numerical abilities share similar cognitive processes.

Neural Dynamics of Serial Dependence in Numerosity Perception

Serial dependence—an attractive perceptual bias whereby a current stimulus is perceived to be similar to previously seen ones—is thought to represent the process that facilitates the stability and continuity of visual perception. Recent results demonstrate a neural signature of serial dependence in numerosity perception, emerging very early in the time course during perceptual processing. However, whether such a perceptual signature is retained after the initial processing remains unknown. Here, we address this question by investigating the neural dynamics of serial dependence using a recently developed technique that allowed a reactivation of hidden memory states. Participants performed a numerosity discrimination task during EEG recording, with task-relevant dot array stimuli preceded by a task-irrelevant stimulus inducing serial dependence. Importantly, the neural network storing the representation of the numerosity stimulus was perturbed (or pinged) so that the hidden states of that representation can be explicitly quantified. The results first show that a neural signature of serial dependence emerges early in the brain signals, starting soon after stimulus onset. Critical to the central question, the pings at a later latency could successfully reactivate the biased representation of the initial stimulus carrying the signature of serial dependence. These results provide one of the first pieces of empirical evidence that the biased neural representation of a stimulus initially induced by serial dependence is preserved throughout a relatively long period.

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.

Serial dependence in numerosity perception

Our conscious experience of the external world is remarkably stable and seamless, despite the intrinsically discontinuous and noisy nature of sensory information. Serial dependencies in visual perception-reflecting attractive biases making a current stimulus to appear more similar to previous ones-have been recently hypothesized to be involved in perceptual continuity. However, while these effects have been observed across a variety of visual features and at the neural level, several aspects of serial dependence and how it generalizes across visual dimensions is still unknown. Here we explore the behavioral signature of serial dependence in numerosity perception by assessing how the perceived numerosity of dot-array stimuli is biased by a task-irrelevant "inducer" stimulus presented before task-relevant stimuli. First, although prior work suggests that numerosity perception starts in the subcortex, the current study rules out a possible involvement of subcortical processing in serial dependence, confirming that the effect likely starts in the visual cortex. Second, we show that the effect is coarsely spatially localized to the position of the inducer stimulus. Third, we demonstrate that the effect is present even with a stimulus presentation procedure minimizing the involvement of post-perceptual processes, but only when participants actively pay attention to the inducer stimulus. Overall, these results provide a comprehensive characterization of serial dependencies in numerosity perception, demonstrating that attractive biases occur by means of spatially localized attentional modulations of early sensory activity.