Mechanisms for perception of numerosity or texture-density are governed by crowding-like effects

Hierarchical representations of relative numerical magnitudes in the human frontoparietal cortex

The ability to estimate numerical magnitude is essential for decision-making and is thought to underlie arithmetic skills. In humans, neural populations in the frontoparietal regions are tuned to represent numerosity. However, it remains unclear whether their response properties are fixed to a specific numerosity (i.e., absolute code) or dynamically scaled according to the range of numerosities relevant to the context (i.e., relative code). Here, using functional magnetic resonance imaging combined with multivariate pattern analysis, we uncover evidence that representations of relative numerosity coding emerge gradually as visual information processing advances in the frontoparietal regions. In contrast, the early sensory areas predominantly exhibit absolute coding. These findings indicate a hierarchical organization of relative numerosity representations that adapt their response properties according to the context. Our results highlight the existence of a context-dependent optimization mechanism in numerosity representation, enabling the efficient processing of infinite magnitude information with finite neural resources.

Estimating the distribution of numerosity and non-numerical visual magnitudes in natural scenes using computer vision

Humans share with many animal species the ability to perceive and approximately represent the number of objects in visual scenes. This ability improves throughout childhood, suggesting that learning and development play a key role in shaping our number sense. This hypothesis is further supported by computational investigations based on deep learning, which have shown that numerosity perception can spontaneously emerge in neural networks that learn the statistical structure of images with a varying number of items. However, neural network models are usually trained using synthetic datasets that might not faithfully reflect the statistical structure of natural environments, and there is also growing interest in using more ecological visual stimuli to investigate numerosity perception in humans. In this work, we exploit recent advances in computer vision algorithms to design and implement an original pipeline that can be used to estimate the distribution of numerosity and non-numerical magnitudes in large-scale datasets containing thousands of real images depicting objects in daily life situations. We show that in natural visual scenes the frequency of appearance of different numerosities follows a power law distribution. Moreover, we show that the correlational structure for numerosity and continuous magnitudes is stable across datasets and scene types (homogeneous vs. heterogeneous object sets). We suggest that considering such "ecological" pattern of covariance is important to understand the influence of non-numerical visual cues on numerosity judgements.

Adaptation biases the parallel perception of subitized numerosities

Numerosity adaptation is a phenomenon in which prolonged exposure to a stimulus of greater numerosity makes subsequent stimuli appear less numerous, and vice versa. It has been confined to moderated numerosities outside the subitizing range (> 4). This study investigated whether the estimation of small numerosities (1-4), which is performed rapidly and accurately due to the mechanism of subitizing, is susceptible to adaptation. After adapting to a 50-dot stimulus, participants were presented with stimuli consisting of 1-5 color sets. In some trials, participants were informed of the target color-set before the presentation of stimuli, while in others, they were instructed afterwards. When estimating 1-4 dots in the single-color set or superset (the total dots), no adaptation effect was observed. The coefficient of variation (CV) was below 0.05, indicating the effective function of subitizing. However, when enumerating subsets in parallel, adaptation biased the estimation. The CV in estimating subitized numerosities was comparable to and correlated with that of estimating moderate numerosities (5-12), suggesting that subitizing was superseded by numerosity estimation. Greater effects arise when the targets were probed afterwards, with elevated CV. The prior adaptor may be more weighted to optimize detection of number deviations, especially under higher perceptual uncertainty.

Visual properties and perceived restorativeness in green offices: a photographic evaluation of office environments with various degrees of greening

Office environments play a critical role in employee wellbeing and productivity. While the benefits of incorporating nature into workspaces have been recognized, the specific visual characteristics that contribute to restorativeness remain unclear. This study investigates how visual characteristics of office environments, specifically the presence of greenery and color complexity, are associated with perceived restorativeness. In Study 1, we developed a scale based on Attention Restoration Theory to measure the restorative characteristics of office environments, consisting of three subscales: Being Away, Fascination, and Extent. In Study 2, we used this scale to examine the correlation between the restorative characteristics of offices and the visual properties of office photographs. The results showed that the square root of the percentage of green area, the color fractal dimension, and the brightness fractal predicted perceived restorativeness. Notably, the color fractal dimension often showed a stronger effect than the amount of greenery per se. These findings suggest that both the presence of greenery and the overall complexity of color transitions in office spaces contribute to their restorative potential. Our study provides insights for designing more restorative office environments, emphasizing the importance of not only increasing greenery but also mimicking natural color patterns. The predictive model developed provides a practical tool for estimating the restorative potential of office designs. Although there are limitations such as the use of photographic assessments and the inability to fully explain the Extent component of restorativeness, this study contributes to our understanding of how to create more psychologically supportive work environments.

The radial-tangential anisotropy of numerosity perception

Humans can estimate the number of visually presented items without counting. In most studies on numerosity perception, items are uniformly distributed across displays, with identical distributions in central and eccentric parts. However, the neural and perceptual representation of the human visual field differs between the fovea and the periphery. For example, in peripheral vision, there are strong asymmetries with regard to perceptual interferences between visual items. In particular, items arranged radially usually interfere more strongly with each other than items arranged tangentially (the radial-tangential anisotropy). This has been shown for crowding (the deleterious effect of clutter on target identification) and redundancy masking (the reduction of the number of perceived items in repeating patterns). In the present study, we tested how the radial-tangential anisotropy of peripheral vision impacts numerosity perception. In four experiments, we presented displays with varying numbers of discs that were predominantly arranged radially or tangentially, forming strong and weak interference conditions, respectively. Participants were asked to report the number of discs. We found that radial displays were reported as less numerous than tangential displays for all radial and tangential manipulations: weak (Experiment 1), strong (Experiment 2), and when using displays with mixed contrast polarity discs (Experiments 3 and 4). We propose that numerosity perception exhibits a significant radial-tangential anisotropy, resulting from local spatial interactions between items.

Relative numerosity is constructed from size and density information: Evidence from adaptation

To dissociate aftereffects of size and density in the perception of relative numerosity, large or small adapter sizes were crossed with high or low adapter densities. A total of 48 participants were included in this preregistered design. To adapt the same retinotopic region as the large adapters, the small adapters were flashed in a sequence so as to "paint" the adapting density across the large region. Perceived numerosities and sizes in the adapted region were then compared to those in an unadapted region in separate blocks of trials, so that changes in density could be inferred. These density changes were found to be bidirectional and roughly symmetric, whereas the aftereffects of size and number were not symmetric. A simple account of these findings is that local adaptations to retinotopic density as well as global adaptations to size combine in producing numerosity aftereffects measured by assessing perceived relative number. Accounts based on number adaptation are contraindicated, in particular, by the result of adapting to a large, sparse adapter and testing with a stimulus with a double the density but half number of dots.

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.

Attentional Strategies and the Transition From Subitizing to Estimation in Numerosity Perception

The common view of the transition between subitizing and numerosity estimation regimes is that there is a hard bound on the subitizing range, and beyond this range, people estimate. However, this view does not adequately address the behavioral signatures of enumeration under conditions of attentional load or in the immediate post-subitizing range. The possibility that there might exist a numerosity range where both processes of subitizing and estimation operate in conjunction has so far been ignored. Here, we investigate this new proposal, that people strategically combine the processes of subitizing and estimation to maximize accuracy and precision, given time or attentional constraints. We present a process-level account of how subitizing and estimation can be combined through strategic deployment of attention to maximize the precision of perceived numerosity given time constraints. We then describe a computational model of this account and apply it in two experimental simulations to demonstrate how it can explain key findings in prior enumeration research. While recent modeling work has argued that the behavioral signatures of enumeration can best be explained through a single numerosity system with a single form of representation, we argue that our model demonstrates how the traditional two-systems view of numerical representation accounts for behavioral data through coordination with a unified attentional mechanism, rather than a unified representation.

Autistic individuals show less grouping-induced bias in numerosity judgments

Introduction

When items are connected together, they tend to be perceived as an integrated whole rather than as individual dots, causing a strong underestimation of the numerosity of the ensemble. Previous evidence on grouping-induced biases of numerosity has shown a dependency on autistic-like personality traits in neurotypical adults, with a weaker tendency for grouping into meaningful segmented objects in individuals with strong autistic traits. Here we asked whether this result would generalize to the autistic population.

Methods

Twenty-two adults with a diagnosis of Autism Spectrum Disorder (ASD) and 22 matched neurotypical controls judged the numerosity of clouds of dot-pairs connected by thin lines.

Results

Results showed no significant group difference in discrimination precision, suggesting that both groups were equally capable performing the task. However, while connecting pairs of dots at moderate numerosities caused large changes in apparent numerosity in the neurotypical controls, particularly those with low autistic-like traits, it had little effect in the group of autistic participants, suggesting significant differences in numerosity estimation between autistic and neurotypical perception. Consistent with earlier studies, the magnitude of the effect covaried strongly with AQ-defined autistic traits in the neurotypical range, reinforcing the idea that autistic traits predict the strength of grouping.

Discussion

These results provide strong support for the theories of autistic perception that highlight dissimilarities in global vs. local processing, and open the door to study grouping mechanisms indirectly, by asking participants to report on the apparent numerosity rather than on the grouping organization per se.

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.

Different impacts of long-term abacus training on symbolic and non-symbolic numerical magnitude processing in children

Abacus-based mental calculation (AMC) has been shown to be effective in promoting math ability in children. Given that AMC relies on a visuospatial strategy to perform rapid and precise arithmetic, previous studies mostly focused on the promotion of AMC training on arithmetic ability and mathematical visual-spatial ability, as well as its transfer of advanced cognitive ability. However, little attention has been given to its impact on basic numerical comparison ability. Here, we aim to examine whether and how long-term AMC training impacts symbolic and non-symbolic numerical comparisons. The distance effect (DE) was utilized as a marker, indicating that the comparison between two numbers becomes faster as their numerical distance enlarges. In the current study, forty-one children matched for age and sex were recruited at primary school entry and randomly assigned to the AMC group and the control group. After three years of training, the event-related potential (ERP) recording technique was used to explore the temporal dynamics of number comparison, of which tasks were given in symbolic (Arabic number) or non-symbolic (dot array) format. In the symbolic task, the children in the AMC group showed a smaller DE than those in the control group. Two ERP components, N1 and P2p, located in parietal areas (PO7, PO8) were selected as neural markers of numerical processing. Both groups showed DE in the P2p component in both tasks, but only the children in the AMC group showed DE in the N1 component in the non-symbolic task. In addition, the DE size calculated from reaction times and ERP amplitudes was correlated with higher cognitive capacities, such as coding ability. Taken together, the present results provide evidence that long-term AMC training may be beneficial for numerical processing in children, which may be associated with neurocognitive indices of parietal brain regions.

Quantum spin models for numerosity perception

Humans share with animals, both vertebrates and invertebrates, the capacity to sense the number of items in their environment already at birth. The pervasiveness of this skill across the animal kingdom suggests that it should emerge in very simple populations of neurons. Current modelling literature, however, has struggled to provide a simple architecture carrying out this task, with most proposals suggesting the emergence of number sense in multi-layered complex neural networks, and typically requiring supervised learning; while simple accumulator models fail to predict Weber's Law, a common trait of human and animal numerosity processing. We present a simple quantum spin model with all-to-all connectivity, where numerosity is encoded in the spectrum after stimulation with a number of transient signals occurring in a random or orderly temporal sequence. We use a paradigmatic simulational approach borrowed from the theory and methods of open quantum systems out of equilibrium, as a possible way to describe information processing in neural systems. Our method is able to capture many of the perceptual characteristics of numerosity in such systems. The frequency components of the magnetization spectra at harmonics of the system's tunneling frequency increase with the number of stimuli presented. The amplitude decoding of each spectrum, performed with an ideal-observer model, reveals that the system follows Weber's law. This contrasts with the well-known failure to reproduce Weber's law with linear system or accumulators models.

What the Solitaire illusion tells us about perception of numerosity

In four experiments we investigated the Solitaire illusion. In this illusion, most observers see as more numerous a set of dots that forms a single central group, compared to dots on the outside of that group. We confirmed and extended the effect to configurations with much higher numerosity than the original and of various colours. Contrary to prediction, separating the two groups, so that they are presented side by side, reduced but did not abolish or reverse the illusion. In this illusion, therefore, neither total size of the region (area), not average distance of the elements has the expected effect. In Experiments 3 and 4 we eliminated the regularity of the pattern, by sampling 50% (Exp 3) or only a 10% (Exp 4) of the elements. These produces quasi-random configurations. For these configurations the bias for the inner groups was still present, and it was only eliminated when the groups were shown as separate. However, the effect never reversed (no bias for the outer group, despite its larger area). We conclude that the Solitaire illusion is evidence of a strong bias in favour of centrally located elements, a bias that can overcome other factors.

Anisotropic representations of visual space modulate visual numerosity estimation

Humans can estimate the number of visually displayed items without counting. This capacity of numerosity perception has often been attributed to a dedicated system to estimate numerosity, or alternatively to the exploitation of various stimulus features, such as density, convex hull, the size of items, and occupancy area. The distribution of the presented items is usually not varied with eccentricity in the visual field. However, our visual fields are highly asymmetric. To date, it is unclear how inhomogeneities of the visual field impact numerosity perception. Besides eccentricity, a pronounced asymmetry is the radial-tangential anisotropy. For example, in crowding, radially placed flankers interfere more strongly with target perception than tangentially placed flankers. Similarly, in redundancy masking, the number of perceived items in repeating patterns is reduced when the items are arranged radially but not when they are arranged tangentially. Here, we investigated whether numerosity perception is subject to the radial-tangential anisotropy of spatial vision to shed light on the underlying topology of numerosity perception. In Experiment 1, observers were presented with varying numbers of discs, predominantly arranged radially or tangentially, and asked to report their perceived number. In Experiment 2, observers were presented with the same displays as in Experiment 1, and were asked to encircle items that were perceived as a group. We found that numerosity estimation depended on the arrangement of discs, suggesting a radial-tangential anisotropy of numerosity perception. Grouping among discs did not seem to explain our results. We suggest that the topology of spatial vision modulates numerosity estimation and that asymmetries of visual space should be taken into account when investigating numerosity estimation.

Number comparison under the Ebbinghaus illusion

A series of studies show interest in how visual attributes affect the estimate of object numbers in a scene. In comparison tasks, it is suggested that larger patches are perceived as more numerous. However, the inequality of density, which changes inversely with the area when numerosity remains constant, may mediate the influence of area on numerosity perception. This study aims to explore the role of area and density in the judgment of numerosity. The Ebbinghaus illusion paradigm was adopted to induce differences in the perceived, rather than the physical, area of the two patches to be compared. Participants were asked to compare the area, density, and the number of the two patches in three tasks. To this end, no PSE (point of subjective equality) bias was found in number comparison with randomly distributed dots, although a significant difference was revealed in the perceived area of the two patches. No PSE bias was found in the density comparison, either. For a comparison, density and number tasks were also conducted with regularly distributed dots. No PSE bias was found in density comparison. By contrast, significant PSE bias showed up in number comparison, and larger patches appeared to be more numerous than smaller patches. The density mechanism was proposed as the basis for number comparison with regular patterns. The individual Weber fractions for regular patterns were not correlated with those for random patterns in the number task, but they were correlated with those for both patterns in the density task. To summarize, numerosity is directly sensed, and numerosity perception is not affected by area inequality induced by the Ebbinghaus illusion. In contrast, density and area are combined to infer numerosity when the approximate numerosity mechanism is disrupted by dot distribution.

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.

Untypical Contrast Normalization Explains the “Weak Outnumber Strong” Numerosity Illusion

Less salient, lower contrast disks appear to be more numerous than more salient, higher contrast disks when intermingled in equal numbers into the same display (Lei and Reeves, 2018), but they are equal in perceived numerosity when segregated into different displays. Comparative judgements indicate that the apparent numerosity of the lower contrast disks is unaffected by being intermingled with high contrast disks, whereas the high contrast disks are reduced in numerosity by being intermingled with the low contrast ones (Lei and Reeves, 2018). Here, we report that this illusion also occurs for absolute judgements of the numerosities of displays of from 20 to 80 disks. A model based on luminance-difference contrast normalization (LDCN) explains the illusory loss of high-contrast (salient) items along with veridical perception of the low-contrast ones. The model correctly predicts that perceived numerosity is linearly related to the square-root of the number of disks, with the extent of the illusion depending on an attentionally-weighted function of contrast and assimilation.

How much time does it take to discriminate two sets by their numbers of elements?

The ability to evaluate the number of elements in a set-numerosity-without symbolic representation is a form of primitive perceptual intelligence. A simple binomial model was proposed to explain how observers discriminate the numerical proportion between two sets of elements distinct in color or orientation (Raidvee et al., 2017, Attention Perception & Psychophysics, 79[1], 267-282). The binomial model's only parameter β is the probability with which each visual element can be noticed and registered by the perceptual system. Here we analyzed the response times (RT) which were ignored in the previous report since there were no instructions concerning response speed. The relationship between the mean RT and the absolute difference |ΔN| between numbers of elements in two sets was described by a linear regression, the slope of which became flatter as the total number of elements N increased. Because the coefficients of regression between the mean RT and |ΔN| were more directly related to the binomial probability β rather than to the standard deviation of the best fitting cumulative normal distribution, it was regarded as evidence that the binomial model with a single parameter - probability β - is a viable alternative to the customary Thurstonian-Gaussian model.

On the usefulness of graph-theoretic properties in the study of perceived numerosity

Observers can quickly estimate the quantity of sets of visual elements. Many aspects of this ability have been studied and the underlying system has been called the Approximate Number Sense (Dehaene, 2011). Specific visual properties, such as size and clustering of the elements, can bias an estimate. For intermediate numerical quantities at low density (above five, but before texturization), human performance is predicted by a model based on the region of influence of elements (occupancy model: Allïk & Tuulmets, 1991). For random 2D configurations we computed ten indices based on graph theory, and we compared them with the occupancy model: independence number, domination, connected components, local clustering coefficient, global clustering coefficient, random walk, eigenvector centrality, maximum clique, total degree of connectivity, and total edge length. We made comparisons across a range of parameters, and we varied the size of the region of influence around each element. The analysis of the pattern of correlations suggests two main groups of graph-based measures. The first group is sensitive to the presence of local clustering of elements, the second seems more sensitive to density and the way information spreads in graphs. Empirical work on perception of numerosity may benefit from comparing, or controlling for, these properties.

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.

Electrophysiological Signatures of Numerosity Encoding in a Delayed Match-to-Sample Task

The number of elements in a small set of items is appraised in a fast and exact manner, a phenomenon called subitizing. In contrast, humans provide imprecise responses when comparing larger numerosities, with decreasing precision as the number of elements increases. Estimation is thought to rely on a dedicated system for the approximate representation of numerosity. While previous behavioral and neuroimaging studies associate subitizing to a domain-general system related to object tracking and identification, the nature of small numerosity processing is still debated. We investigated the neural processing of numerosity across subitizing and estimation ranges by examining electrophysiological activity during the memory retention period in a delayed numerical match-to-sample task. We also assessed potential differences in the neural signature of numerical magnitude in a fully non-symbolic or cross-format comparison. In line with behavioral performance, we observed modulation of parietal-occipital neural activity as a function of numerosity that differed in two ranges, with distinctive neural signatures of small numerosities showing clear similarities with those observed in visuospatial working memory tasks. We also found differences in neural activity related to numerical information in anticipation of single vs. cross-format comparison, suggesting a top-down modulation of numerical processing. Finally, behavioral results revealed enhanced performance in the mixed-format conditions and a significant correlation between task performance and symbolic mathematical skills. Overall, we provide evidence for distinct mechanisms related to small and large numerosity and differences in numerical encoding based on task demands.

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.

The Grouping-Induced Numerosity Illusion Is Attention-Dependent

Perceptual grouping and visual attention are two mechanisms that help to segregate visual input into meaningful objects. Here we report how perceptual grouping, which affects perceived numerosity, is reduced when visual attention is engaged in a concurrent visual task. We asked participants to judge the numerosity of clouds of dot-pairs connected by thin lines, known to cause underestimation of numerosity, while simultaneously performing a color conjunction task. Diverting attention to the concomitant visual distractor significantly reduced the grouping-induced numerosity biases. Moreover, while the magnitude of the illusion under free viewing covaried strongly with AQ-defined autistic traits, under conditions of divided attention the relationship was much reduced. These results suggest that divided attention modulates the perceptual grouping of elements by connectedness and that it is independent of the perceptual style of participants.

Grouping-Induced Numerosity Biases Vary with Autistic-Like Personality Traits

Individuals with autism spectrum disorder are thought to have a more local than global perceptual style. We used a novel paradigm to investigate how grouping-induced response biases in numerosity judgments depend on autistic-like personality traits in neurotypical adults. Participants judged the numerosity of clouds of dot-pairs connected by thin lines, known to cause underestimation of numerosity. The underestimation bias correlated strongly with autism-spectrum quotient (r = 0.72, Bayes factor > 100), being weaker for participants with high autistic traits. As connecting dots probably activates global grouping mechanisms, causing dot-pairs to be processed as an integrated whole rather than as individual dots, the results suggest that these grouping mechanisms may be weaker in individuals self-reporting high levels of autistic-like traits.

Proximity model of perceived numerosity

The occupancy model (OM) was proposed to explain how the spatial arrangement of dots in sparse random patterns affects their perceived numerosity. The model's central thesis maintained that each dot seemingly fills or occupies its surrounding area within a fixed radius r_o and the total area collectively occupied by all the dots determines their apparent number. Because the perceptual system is not adapted for the precise estimation of area, it looks likely that the OM is just a convenient computational algorithm that does not necessarily correspond to the processes that actually take place in the perceptual system. As an alternative, the proximity model (PM) was proposed, which instead relies on a binomial function with the probability β characterizing the perceptual salience with which each element can be registered by the perceptual system. It was also assumed that the magnitude of β is proportional to the distance between a dot and its nearest neighbor. A simulation experiment demonstrated that the occupancy area computed according to the OM can almost perfectly be replicated by the mean nearest neighbor distance. It was concluded that proximity between elements is a critical factor in determining their perceived numerosity, but the exact algorithm that is used for the measure of proximities is yet to be established.

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.

Dysfunctional perceptual antecedent can justify the social orienting deficit in autism spectrum disorder: an eye-tracking study

Purpose

Children with Autism Spectrum Disorder (ASD) show reduced attention to social stimuli. The reasons for these impairments are still being debated by researchers. The aim of this study is to analyse if reduced attention towards social stimuli is determined by initial underlying difficulties in the control of visual attention. Among the variables that could produce these difficulties, the authors considered geometric complexity and typology of geometric figures.

Design/methodology/approach

To test this hypothesis, in this paper, an eye-tracker paradigm was used for assessing visual exploration and recognition memory towards geometric figures (curved vs rectilinear) with two levels of geometric complexity (low and high) in 17 children with ASD matched with 17 children with typical development (TD).

Findings

The results showed that the ASD group seemed indifferent to both the geometric complexity and the typology of figures (curved and rectilinear), whereas the TD group showed higher performances with highly complex and curved geometric figures than with low complex and rectilinear geometric figures.

Research limitations/implications

Because of the chosen research approach, the research results may lack generalizability. Therefore, researchers are encouraged to test the proposed hypotheses further.

Practical implications

This paper includes implications upon the presence of an unspecified visual attention deficit that is present from the early stages of the processing of stimuli.

Social implications

The understanding of this deficit from the early stages of the processing of stimuli can help educators to intervene at an early stage when disturbances in social relationships are starting.

Originality/value

This study contributes to understanding the presence of dysfunctional perceptual antecedents that could determine general difficulties in paying attention to social stimuli in ASD subjects.

Excessive visual crowding effects in developmental dyscalculia

Visual crowding refers to the inability to identify objects when surrounded by other similar items. Crowding-like mechanisms are thought to play a key role in numerical perception by determining the sensory mechanisms through which ensembles are perceived. Enhanced visual crowding might hence prevent the normal development of a system involved in segregating and perceiving discrete numbers of items and ultimately the acquisition of more abstract numerical skills. Here, we investigated whether excessive crowding occurs in developmental dyscalculia (DD), a neurodevelopmental disorder characterized by difficulty in learning the most basic numerical and arithmetical concepts, and whether it is found independently of associated major reading and attentional difficulties. We measured spatial crowding in two groups of adult individuals with DD and control subjects. In separate experiments, participants were asked to discriminate the orientation of a Gabor patch either in isolation or under spatial crowding. Orientation discrimination thresholds were comparable across groups when stimuli were shown in isolation, yet they were much higher for the DD group with respect to the control group when the target was crowded by closely neighbouring flanking gratings. The difficulty in discriminating orientation (as reflected by the combination of accuracy and reaction times) in the DD compared to the control group persisted over several larger target flanker distances. Finally, we found that the degree of such spatial crowding correlated with impairments in mathematical abilities even when controlling for visual attention and reading skills. These results suggest that excessive crowding effects might be a characteristic of DD, independent of other associated neurodevelopmental disorders.

Feature Counting Under Dichoptic Viewing in Anisometropic and Strabismic Amblyopia

Purpose

While using their amblyopic eye, individuals with strabismic amblyopia count inaccurately and underestimate the number of features. These deficits are attributed to limitations in high-level cortical functions and attention. In the current study, we examined whether feature counting is affected in strabismic and anisometropic amblyopia during dichoptic viewing, a setup that can better capture binocular function disruptions.

Methods

Through a mirror stereoscope, Gabor patches were presented for 200 msec (Experiment 1) or 350 msec (Experiment 2) in both the left eye and the right eye of observers, who were required to combine the percepts and report the total number of patches. Counting performance and errors were compared across amblyopic groups and normal-sighted observers. The contribution and relation of each eye to performance was also evaluated.

Results

Anisometropic and strabismic amblyopia groups counted inaccurately and underestimated the number of features compared to the normal-sighted group. In both amblyopic groups, the amblyopic eye contributed less in comparison to the fellow eye. The strabismic group exhibited worse performance, and a more pronounced difference in eye contribution, in comparison to the anisometropic group.

Conclusions

Overall, our results support the view of higher-level cortical and binocular function deficits in amblyopia.

Translational Relevance

The current study bridges the gap between research on high-cortical function deficits and clinical binocular function disruptions in amblyopia, which can help us better understand the neural mechanism of amblyopia and inform clinical therapeutic tasks and strategies.

Frequency-based Dissociation of Symbolic and Nonsymbolic Numerical Processing during Numerical Comparison

Recent evidence suggests that during numerical calculation, symbolic and nonsymbolic processing are functionally distinct operations. Nevertheless, both roughly recruit the same brain areas (spatially overlapping networks in the parietal cortex) and happen at the same time (roughly 250 msec poststimulus onset). We tested the hypothesis that symbolic and nonsymbolic processing are segregated by means of functionally relevant networks in different frequency ranges: high gamma (above 50 Hz) for symbolic processing and lower beta (12–17 Hz) for nonsymbolic processing. EEG signals were quantified as participants compared either symbolic numbers or nonsymbolic quantities. Larger EEG gamma-band power was observed for more difficult symbolic comparisons (ratio of 0.8 between the two numbers) than for easier comparisons (ratio of 0.2) over frontocentral regions. Similarly, beta-band power was larger for more difficult nonsymbolic comparisons than for easier ones over parietal areas. These results confirm the existence of a functional dissociation in EEG oscillatory dynamics during numerical processing that is compatible with the notion of distinct linguistic processing of symbolic numbers and approximation of nonsymbolic numerical information.

Interaction of disparity size and depth structure on perceived numerosity in a three-dimensional space

The number of elements in two stereo-surfaces parallelly overlapped in depth is overestimated compared to that in a single flat surface, even when both have the same number of elements. Using stereoscopic pairs of elements, we evaluated two hypotheses on the overestimation: one that a higher-order process, forming a background surface, increases the number of perceived elements, and the other that the number of elements potentially occluded by the elements on a front surface is taken accounted for. The data from four experiments showed that (a) when binocular disparity between (or among) stereoscopic elements was small, the overestimation occurred for the stimuli we used-a two-surface-overlapping stimulus, where the likelihood for the process to operate was manipulated by changing the averaged luminance of each surface, a volumetric stimulus, where the likelihood for the background surface to be formed would decrease, and a two-non-overlapping-surface stimulus, where the surfaces in depth were not overlapped-, and (b) when binocular disparity was large, the overestimation occurred for the two-surfaces-overlapping stimulus, when the averaged luminance of the two surfaces were the same, and for the volumetric stimulus, but diminished for the surface-overlapping stimulus, when the averaged luminance differed between the surfaces and for the surfaces-non-overlapping stimulus. These results cannot be explained either hypothesis only. We explain the results by postulating that the sensory system processing disparities of elements interferes with that estimating the number of elements, resulting in an overestimation of the elements in a stereo-stimulus, and the disparity range within which the interference occurs may depend on the stimulus depth structure.

The neural signature of numerosity by separating numerical and continuous magnitude extraction in visual cortex with frequency-tagged EEG

The ability to handle approximate quantities, or number sense, has been recurrently linked to mathematical skills, although the nature of the mechanism allowing to extract numerical information (i.e., numerosity) from environmental stimuli is still debated. A set of objects is indeed not only characterized by its numerosity but also by other features, such as the summed area occupied by the elements, which often covary with numerosity. These intrinsic relations between numerosity and nonnumerical magnitudes led some authors to argue that numerosity is not independently processed but extracted through a weighting of continuous magnitudes. This view cannot be properly tested through classic behavioral and neuroimaging approaches due to these intrinsic correlations. The current study used a frequency-tagging EEG approach to separately measure responses to numerosity as well as to continuous magnitudes. We recorded occipital responses to numerosity, total area, and convex hull changes but not to density and dot size. We additionally applied a model predicting primary visual cortex responses to the set of stimuli. The model output was closely aligned with our electrophysiological data, since it predicted discrimination only for numerosity, total area, and convex hull. Our findings thus demonstrate that numerosity can be independently processed at an early stage in the visual cortex, even when completely isolated from other magnitude changes. The similar implicit discrimination for numerosity as for some continuous magnitudes, which correspond to basic visual percepts, shows that both can be extracted independently, hence substantiating the nature of numerosity as a primary feature of the visual scene.

Higher attentional costs for numerosity estimation at high densities

Humans can estimate numerosity over a large range, but the precision with which they do so varies considerably over that range. For very small sets, within the subitizing range of up to about four items, estimation is rapid and errorless. For intermediate numerosities, errors vary directly with the numerosity, following Weber’s law, but for very high numerosities, with very dense patterns, thresholds continue to rise with the square root of numerosity. This suggests that three different mechanisms operate over the number range. In this study we provide further evidence for three distinct numerosity mechanisms, by studying their dependence on attentional resources. We measured discrimination thresholds over a wide range of numerosities, while manipulating attentional load with both visual and auditory dual tasks. The results show that attentional effects on thresholds vary over the number range. Both visual and auditory attentional loads strongly affect subitizing, much more than for larger numerosities. Attentional costs remain stable over the estimation range, then rise again for very dense patterns. These results reinforce the idea that numerosity is processed by three separates but probably overlapping systems.

Evaluating Perceptual Bias During Geometric Scaling of Scatterplots

Scatterplots are frequently scaled to fit display areas in multi-view and multi-device data analysis environments. A common method used for scaling is to enlarge or shrink the entire scatterplot together with the inside points synchronously and proportionally. This process is called geometric scaling. However, geometric scaling of scatterplots may cause a perceptual bias, that is, the perceived and physical values of visual features may be dissociated with respect to geometric scaling. For example, if a scatterplot is projected from a laptop to a large projector screen, then observers may feel that the scatterplot shown on the projector has fewer points than that viewed on the laptop. This paper presents an evaluation study on the perceptual bias of visual features in scatterplots caused by geometric scaling. The study focuses on three fundamental visual features (i.e., numerosity, correlation, and cluster separation) and three hypotheses that are formulated on the basis of our experience. We carefully design three controlled experiments by using well-prepared synthetic data and recruit participants to complete the experiments on the basis of their subjective experience. With a detailed analysis of the experimental results, we obtain a set of instructive findings. First, geometric scaling causes a bias that has a linear relationship with the scale ratio. Second, no significant difference exists between the biases measured from normally and uniformly distributed scatterplots. Third, changing the point radius can correct the bias to a certain extent. These findings can be used to inspire the design decisions of scatterplots in various scenarios.

Three-systems for visual numerosity: A single case study

Humans possess the remarkable capacity to assess the numerosity of a set of items over a wide range of conditions, from a handful of items to hundreds of them. Recent evidence is starting to show that judgments over such a large range is possible because of the presence of three mechanisms, each tailored to specific stimulation conditions. Previous evidence in favour of this theory comes from the fact that discrimination thresholds and estimation reaction times are not constants across numerosity levels. Likewise, attention is capable of dissociating the three mechanisms: when healthy adult observers are asked to perform concurrently a taxing task, the judgments of low numerosities (<4 dots) or of high numerosities is affected greatly, not so however for intermediate numerosities. Here we bring evidence from a neuropsychological perspective. To this end we measured perceptual performance in PA, a 41 year-old patient who suffers simultanagnosia after a hypoxic brain injury. PA showed a profound deficit in attentively tracking objects over space and time (multiple object tracking), even in very simple conditions where controls made no errors. PA also showed a massive deficit on sensory thresholds when comparing dot-arrays containing extremely low (3 dots) or extremely high (64, 128 dots) numerosities as well as in comparing dot-distances. Surprisingly, PA discrimination thresholds were relatively spared for intermediate numerosity (12 and 16 dots). Overall his deficit on the numerosity task results in a U-shape function across numerosity which, combined with the attentional deficit and the inability to judge dot-distances, confirms previously suggested three-systems for numerosity judgments.

Density discrimination with occlusions in 3D clutter

We examined how well human observers can discriminate the density of surfaces in two halves of a rotating three-dimensional cluttered sphere. The observer's task was to compare the density of the front versus back half or the left versus right half. We measured how the bias and sensitivity in judging the denser half depended on the level of occlusion and on the area and density of the surfaces in the clutter. When occlusion level was low, observers in the front-back task were biased to judge the back as denser, and when occlusion level was high they were biased to judge the front as denser. Weber fractions decreased as density increased for both the front-back and left-right tasks, consistent with previous findings for two-dimensional density discrimination. Weber fractions did not vary significantly with area for the front-back task, but increased with area for the left-right task, and we attribute this difference to occlusions that have different effects in the two tasks. We also ran model observers that compared the image occupancies of the two halves against a known expected difference. As the occlusion level increased, this expected difference followed a similar trend as the biases of the human observers, with a roughly constant offset between them. Weber fractions for human and model observers followed some similar trends, but there were discrepancies as well that can be partly explained by the information available to human versus model observers in carrying out their respective tasks.

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.

Spontaneous perception of numerosity in pre-school children

There is strong evidence that humans can make rough estimates of the numerosity of a set of items, almost from birth. However, as numerosity covaries with many non-numerical variables, the idea of a direct number sense has been challenged. Here we applied two different psychophysical paradigms to demonstrate the spontaneous perception of numerosity in a cohort of young pre-school children. The results of both tasks showed that even at that early developmental stage, humans spontaneously base the perceptual choice on numerosity, rather than on area or density. Precision in one of these tasks predicted mathematical abilities. The results reinforce strongly the idea of a primary number sense and provide further evidence linking mathematical skills to the sensory precision of the spontaneous number sense, rather than to mechanisms involved in handling explicit numerosity judgements or extensive exposure to mathematical teaching.

Different reaction-times for subitizing, estimation, and texture

Humans can estimate and encode numerosity over a large range, from very few items to several hundreds. Two distinct mechanisms have been proposed: subitizing, for numbers up to four and estimation for larger numerosities. We have recently extended this idea by suggesting that for very densely packed arrays, when items are less segregable, a third "texture" mechanism comes into play. In this study, we provide further evidence for the existence of a third regime for numerosity. Reaction times were very low in the subitizing range, rising rapidly for numerosities greater than four. However, for tightly packed displays of very high numerosities, reaction times became faster. These results reinforce the idea of three regimes in the processing of numerosity, subitizing, estimation, and texture.

Spontaneous representation of numerosity in typical and dyscalculic development

Animals including humans are endowed with a remarkable capacity to estimate rapidly the number of items in a scene. Some have questioned whether this ability reflects a genuine sense of number, or whether numerosity is derived indirectly from other covarying attributes, such as density and area. In previous work we have demonstrated that adult observers are more sensitive to changes in numerosity than to area or density, particularly changes that leave numerosity constant, pointing to a spontaneous sensitivity to numerosity, not attributable to area and density. Here we extend this line of research with a novel technique where participants reproduce the size and density of a dot-array. They were given no explicit instructions of what to match, but could regulate freely all combinations of area and density by trackpad. If the task is mediated by matching separately area and texture-density, the errors in the two attributes have to be independent. Contrarily to this prediction, we found that errors in area and density were negatively correlated, suggesting that subjects matched numerosity, rather than area and density. We employed this technique to investigate processing of number in adolescents with typical and low math abilities (dyscalculia). Interestingly, we found that dyscalculics also reproduced numerosity rather than area or density. However, compared to typicals, dyscalculics had longer reaction times, a tendency to rely also on area, and their performance did not improve over sessions. Taken together, the data demonstrate that numerosity emerges as the most spontaneous and sensitive dimension, supporting the existence of a dedicated number sense and confirm numerosity atypicalities in dyscalculia.

Two's company, three's a crowd: Individuation is necessary for object recognition

Object recognition is essential for navigating the real world. Despite decades of research on this topic, the processing steps necessary for recognition remain unclear. In this study, we examined the necessity and role of individuation, the ability to select a small number of spatially distinct objects irrespective of their identity, in the recognition process. More specifically, we tested if the ability to rapidly individuate and enumerate a small number of objects (subitizing) can be impaired by crowding. Crowding is flanker-induced interference that specifically impedes the recognition process. We found that subitizing is impaired when objects are close to each other (Experiment 1), and if the target objects are surrounded by irrelevant but perceptually similar flankers (Experiments 2-4). This impairment cannot be attributed to confusion between targets and flankers, wherein flankers are inadvertently included in or targets are excluded from enumeration (Experiments 3-4). Importantly, the flanker induced interference was comparable in both subitizing and crowding tasks (Experiment 4), suggesting that individuation and identification share a common processing pathway. We conclude that individuation is an essential stage in the object recognition pipeline and argue for a cohesive proposal that both crowding and subitizing are due to limitations of selective attention.

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.

Regular Distribution Inhibits Generic Numerosity Processing

This study investigated the role of pattern regularity in approximate numerical processing. Experiment 1 demonstrated that the change in stimulus size has a distinct effect on the adaptation aftereffect for random and regular patterns. For regular patterns, adapting to large patterns and being tested with small patterns caused stronger aftereffects than the reverse treatment, in which the participants adapted to small patterns and were tested with large patterns. For random patterns, this effect was absent. Experiment 2 revealed a distinct connectedness effect on the numerosity processing of random and regular patterns. For random patterns, reference stimuli were perceived to contain fewer items when the dots were connected by lines than when they were not connected, and the number of items in the connected reference was further underestimated when the participants adapted to unconnected patterns with the same number of dots. For regular patterns, this effect was absent. Distinct mechanisms were thus suggested for the numerosity coding of random and regular patterns. For random patterns, the change in primary texture features would be abstracted from numerosity processing, while connectedness could affect this coding by affecting the processing of numerical unit individuation. For regular patterns, generic numerosity processing is inhibited, and numerical judgments appear to be inferred from the visual processing results of texture features such as dot size or the distance between adjacent dots.

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’.

Simultaneous density contrast and binocular integration

Most research on texture density has utilized textures rendered as two-dimensional (2D) planar surfaces, consistent with the conventional definition of density as the number of texture elements per unit area. How the brain represents texture density information in the three-dimensional (3D) world is not yet clear. Here we tested whether binocular information affects density processing using simultaneous density contrast (SDC), in which the perceived density of a texture region is changed by a surround of different density. We considered the effect on SDC of two types of binocular information: the stereoscopic depth relationships and the interocular relationships between the center and surround textures. Observers compared the perceived density of two random dot patterns, one with a surround (test stimulus) and one without (match), using a 2AFC staircase procedure. In Experiment 1 we manipulated the stereo-depth of the surround plane systematically from near to far, relative to the center plane. SDC was reduced when the difference in stereo-depth between test center and surround increased. In Experiment 2 we spread the surround dots randomly across a stereo-depth volume from small to large volume sizes, and found that SDC was slightly reduced with volume size. The decrease of SDC in both experiments was observed with dense surrounds only, but not with sparse surrounds. In the last experiment we presented center and surround in the same depth plane but dichopticly, monopticly, and binocularly. A strong interocular transfer of SDC was found in the dichoptic condition. Together these results show that texture density processing is sensitive to binocularity.

Repulsive Serial Effects in Visual Numerosity Judgments

We investigated how the approximate perceived numerosity of ensembles of visual elements is modulated by the numerosity of previously viewed ensembles depending on whether the first ensemble is held in visual working memory or not. We show that the numerosity of the previously seen ensemble has a repulsive effect, that is, a stimulus with high numerosity induces an underestimation of the following one and vice versa. This repulsive effect is present regardless of whether the first stimulus is memorized or not. While subtle changes of the experimental paradigm can have major consequences for the nature of interstimulus dependencies in perception, generally speaking the fact that we found such effects in a visual numerosity estimation task confirms that the process by which human observers produce estimates of the number of elements bears analogies to the processes that lead to the perception of visual dimensions such as orientation.