Topology-defined units in numerosity perception

An object numbering task reveals an underestimation of complexity for typically structured scenes

Our visual environments are composed of an abundance of individual objects. The efficiency with which we can parse such rich environments is remarkable. Previous work suggests that this efficiency is partly explained by grouping mechanisms, which allow the visual system to process the objects that surround us as meaningful groups rather than individual entities. Here, we show that the grouping of objects in typically and meaningfully structured environments directly relates to a reduction of perceived complexity. In an object numerosity discrimination task, we showed participants pairs of schematic scene miniatures, in which objects were structured in typical or atypical ways and asked them to judge which scene consisted of more individual objects. Critically, participants underestimated the number of objects in typically structured compared with atypically structured scenes, suggesting that grouping based on typical object configurations reduces the perceived numerical complexity of a scene. In two control experiments, we show that this overestimation also occurs when the objects are presented on textured backgrounds, and that it is specific to upright scenes, indicating that it is not related to basic visual feature differences between typically and atypically structured scenes. Together, our results suggest that our visual surroundings appear less complex to the visual system than the number of objects in them makes us believe.

Children's Understanding of Topological Relations

A core aim of developmental cognitive science is to uncover the basic building blocks of human thought. For instance, work revealing that even young children, adults without formal education, and distant animal species are sensitive to basic Euclidean properties indicates that humans may be endowed with some primitive understanding of Euclidean geometry. But what about other forms of geometry? Here, we explore children's sensitivity to topological spatial forms. We show that children, like adults, spontaneously distinguish and match items in accordance with their topological relations. As well, we show that children's judgments about object similarity are remarkably consistent with adults', indicating stability in object concepts throughout the lifespan. Finally, we compare children's sensitivity to various topological forms with their sensitivity to geometric properties like curvature, perpendicularity, and symmetry, and find that while there is some variability in performance across all the features tested, overall performance for geometric vs. topological is comparable. Collectively, these findings suggest that even young children have an intuitive understanding of topological relations and suggest that topological relations may be among the building blocks of human visuospatial representation.

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.

Perceiving Topological Relations

There are many ways to describe and represent the visuospatial world. A space can be described by its euclidean properties-the size of objects, the angles of boundaries, the distances between them. A space can also be described in nonspatial terms: One could explain the layout of a city by the order of its streets. Somewhere in between, topological representations-such as those commonly depicted in public-transit maps-capture coarse relational structure without precise euclidean detail, offering a relatively efficient, low-dimensional way of capturing spatial content. Here, we ask whether human adults quickly and automatically perceive such relations. In six experiments, we show that differences in simple topological features influence a range of visual tasks from object matching to number estimation to visual search. We discuss the possibility that topological relations are a kind of visual primitive that supports visuospatial representation.

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.

Deficits of the "Good" Eye in Amblyopia: Processing Geometric Properties

Purpose

Although fellow eyes of amblyopia are typically considered normal, recent studies have revealed impairments in certain aspects of vision. However, it remains unclear at which level of object processing these impairments occur. This study aims to investigate the functional level of visual perception impairment in the fellow eye of children and adults with amblyopia using the geometric functional hierarchy discrimination task based on Klein Mathematics methodology.

Methods

Seventy-six patients with amblyopia (40 children and 36 adults) and 77 age-matched healthy controls (40 children and 37 adults) were recruited for this study. The participants completed four sets of geometric hierarchies (in ascending order of stability: Euclidean, affine, projective, and topology) and one set of color discrimination tasks. They were instructed to rapidly and accurately select a distinct shape from the four quadrants.

Results

The participants' performance was evaluated using the inverse efficiency (IE) score (IE = response time (RT)/accuracy). The results of IEs show that the fellow eye of children with amblyopia exhibits normal topological processing, yet displays higher IEs in other geometric properties and color processing, suggesting impairments in these specific discrimination abilities. However, adults with amblyopia did not show deficits on any discrimination types compared with adult controls.

Conclusions

The lack of compromised topological processing suggests that amblyopia may not have inflicted any damage to the subcortical visual pathways. Furthermore, these deficits observed in the fellow eye tend to diminish significantly during adulthood, implying that amblyopia may potentially hinder the maturation process of the fellow eye.

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.

Number sense deficits in children with developmental dyscalculia, dyslexia, co-occurring disorder and their typically developing peers

The aim of this study was to explore a number sense deficits in children with developmental dyscalculia, dyslexia, co-occurring disorder and their typically developing peers. A non-symbolic quantity comparison task was used in this study to examine whether children with dyscalculia have number sense deficits. Children aged 10-11 years old from nine primary schools in Taif city, Saudi Arabia, were selected to participate in this study. The children were divided into the dyscalculia group (n = 62), the dyslexia group (n = 60), and co-occurring disorder group (n = 65), and the typically developing peers group (n = 100).4 groups (dyscalculia, dyslexia, co-occurring disorder and typically developing peers group) × 2 stimulus ratio (6:7; 8:12). There were significant differences in non-symbolic quantity comparison tasks between children with dyslexia, co-occurring disorder, and typically developing peers. These results indicate that children with dyscalculia do have number sense deficiencies, but number sense deficiencies are not specific to children with dyscalculia.

The interplay of motor adaptation and groupitizing in numerosity perception: Insights from visual motion adaptation and proprioceptive motor adaptation

Groupitizing is a well-established strategy in numerosity perception that enhances speed and sensory precision. Building on the ATOM theory, Anobile proposed the sensorimotor numerosity system, which posits a strong link between number and action. Previous studies using motor adaptation technology have shown that high-frequency motor adaptation leads to underestimation of numerosity perception, while low-frequency adaptation leads to overestimation. However, the impact of motor adaptation on groupitizing, and whether visual motion adaptation produces similar effects, remain unclear. In this study, we investigate the persistence of the advantage of groupitizing after motor adaptation and explore the effects of visual motion adaptation. Surprisingly, our findings reveal that proprioceptive motor adaptation weakens the advantage of groupitizing, indicating a robust effect of motor adaptation even when groupitizing is employed. Moreover, we observe a bidirectional relationship, as groupitizing also weakens the adaptation effect. These results highlight the complex interplay between motor adaptation and groupitizing in numerosity perception. Furthermore, our study provides evidence that visual motion adaptation also has an adaptation effect, but does not fully replicate the effects of proprioceptive motor adaptation on groupitizing. In conclusion, our research underscores the importance of groupitizing as a valuable strategy in numerosity perception, and sheds light on the influence of motion adaptation on this strategy.

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.

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.

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

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

The neural correlates of individual differences in numerosity perception: A voxel-based morphometry study

Numerosity perception is a fundamental cognitive function in humans and animals. Using an individual difference approach with a comprehensive dataset (N = 249), we performed a voxel-based morphometry analysis to unravel the neuroanatomical substrates associated with individual differences in numerosity perception sensitivity, measured by a classical non-symbolic numerical judgment task. Results showed that greater gray matter volume (GMV) in the left cerebellum, right temporal pole, and right parahippocampal was positively correlated to higher perceptual sensitivity to numerosity. In contrast, the GMV in the left intraparietal sulcus, and bilateral precentral/postcentral gyrus was negatively correlated to the sensitivity of numerosity perception. These findings indicate that a wide range of brain structures, rather than a specific anatomical structure or circuit, forms the neuroanatomical basis of numerosity perception, lending support to the emerging network view of the neural representation of numerosity. This work contributes to a more comprehensive understanding of how the brain processes numerical information.

•Unveils neuroanatomical basis of numerosity perception •Discovers positive and negative greater GMV correlations •Links GMV in a wide range of brain regions to numerical sensitivity •Supports the network view of the neural representation of numerosity perception

Numerosity Comparison in Three Dimensions in the Case of Low Numerical Values

This study investigated the perception of numbers in humans in 3D stimuli. Recent research has shown that number processing relies on "number sense" for small values, in line with Weber's law. While previous studies have reported 3D numerosity overestimation mainly in higher numerical values, our experiment examined whether this phenomenon occurs at lower numerical values. We also explored whether the Weber ratio follows Weber's law when comparing 2D and 3D stimuli in terms of the number of elements. Observers were presented with pairs of stimuli on a monitor and were asked to identify the stimulus with a larger number of elements. Using the constant method, we calculated the point of subjective equality (PSE), just noticeable difference (JND), and Weber ratios from the collected data. As a result, it was confirmed that the phenomenon of over-estimation of 3D numerical values occurs even when the numerical values are small. Additionally, we observed that the Weber fraction adhered to Weber's law within the measured range. These findings contribute to the existing body of research, supporting the existence of distinct mechanisms for perceiving numerosity and density.

Systems-level decoding reveals the cognitive and behavioral profile of the human intraparietal sulcus

INTRODUCTION

The human intraparietal sulcus (IPS) covers large portions of the posterior cortical surface and has been implicated in a variety of cognitive functions. It is, however, unclear how cognitive functions dissociate between the IPS's heterogeneous subdivisions, particularly in perspective to their connectivity profile.

METHODS

We applied a neuroinformatics driven system-level decoding on three cytoarchitectural distinct subdivisions (hIP1, hIP2, hIP3) per hemisphere, with the aim to disentangle the cognitive profile of the IPS in conjunction with functionally connected cortical regions.

RESULTS

The system-level decoding revealed nine functional systems based on meta-analytical associations of IPS subdivisions and their cortical coactivations: Two systems-working memory and numeric cognition-which are centered on all IPS subdivisions, and seven systems-attention, language, grasping, recognition memory, rotation, detection of motions/shapes and navigation-with varying degrees of dissociation across subdivisions and hemispheres. By probing the spatial overlap between systems-level co-activations of the IPS and seven canonical intrinsic resting state networks, we observed a trend toward more co-activation between hIP1 and the front parietal network, between hIP2 and hIP3 and the dorsal attention network, and between hIP3 and the visual and somatomotor network.

DISCUSSION

Our results confirm previous findings on the IPS's role in cognition but also point to previously unknown differentiation along the IPS, which present viable starting points for future work. We also present the systems-level decoding as promising approach toward functional decoding of the human connectome.

Perceptual confidence of visual stimulus features is associated with duration perception

It has been shown that the perceived duration of an object in the subsecond range is closely associated with its nontemporal perceptual properties, the mechanism under which remains unclear. Previous studies have revealed a modulatory effect of early visual feature processing on the apparent duration. Here, we further examined the relationship between perceptual confidence and subjective time by asking participants to simultaneously perform temporal and nontemporal perceptual judgments. The results revealed a significant effect on confidence levels. When participants' confidence in judging the coherent motion direction or relative dot numerosity increases, their perceived duration of the stimulus also appears longer. These results are discussed in the context of perceptual evidence accumulation and evaluation for the decision-making of perceptual properties. They suggest a profound contribution of object processing to the computation of subjective time and provide further insights into the mechanism of event timing.

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.

Attention drives human numerosity-selective responses

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

The complexity of simple counting: ERP findings reveal early perceptual and late numerical processes in different arrangements

The counting process can only be fully understood when taking into account the visual characteristics of the sets counted. Comparing behavioral data as well as event-related brain potentials (ERPs) evoked by different task-irrelevant arrangements of dots during an exact enumeration task, we aimed to investigate the effect of illusory contour detection on the counting process while other grouping cues like proximity were controlled and dot sparsity did not provide a cue to the numerosity of sets. Adult participants (N = 37) enumerated dots (8-12) in irregular and two different types of regular arrangements which differed in the shape of their illusory dot lattices. Enumeration speed was affected by both arrangement and magnitude. The type of arrangement influenced an early ERP negativity peaking at about 270 ms after stimulus onset, whereas numerosity only affected later ERP components (> 300 ms). We also observed that without perceptual cues, magnitude was constructed at a later stage of cognitive processing. We suggest that chunking is a prerequisite for more fluent counting which influences automatic processing (< 300 ms) during enumeration. We conclude that the procedure of exact enumeration depends on the interaction of several perceptual and numerical processes that are influenced by magnitude and arrangement.

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.

A connectome-based neuromarker of nonverbal number acuity and arithmetic skills

The approximate number system (ANS) is vital for survival and reproduction in animals and is crucial for constructing abstract mathematical abilities in humans. Most previous neuroimaging studies focused on identifying discrete brain regions responsible for the ANS and characterizing their functions in numerosity perception. However, a neuromarker to characterize an individual's ANS acuity is lacking, especially one based on whole-brain functional connectivity (FC). Here, based on the resting-state functional magnetic resonance imaging (rs-fMRI) data obtained from a large sample, we identified a distributed brain network (i.e. a numerosity network) using a connectome-based predictive modeling (CPM) analysis. The summed FC strength within the numerosity network reliably predicted individual differences in ANS acuity regarding behavior, as measured using a nonsymbolic number-comparison task. Furthermore, in an independent dataset of the Human Connectome Project (HCP), we found that the summed FC strength within the numerosity network also specifically predicted individual differences in arithmetic skills, but not domain-general cognitive abilities. Therefore, our findings revealed that the identified numerosity network could serve as an applicable neuroimaging-based biomarker of nonverbal number acuity and arithmetic skills.

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.

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.

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.

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.

Topological change induces an interference effect in visual working memory

The “irrelevant-change distracting effect” refers to the effect of changes in irrelevant features on the performance of the target feature, which has frequently been used to study information processing in visual working memory (VWM). In the current study, we reported a novel interference effect in VWM: the topological-change interference effect (TCIE). In a series of six experiments, we examined the influence of topological and nontopological changes as irrelevant features on VWM using a color change detection paradigm. The results revealed that only topological changes, although task irrelevant, could produce a significant interference effect. In contrast, nontopological changes did not produce any evident interference effect. Moreover, the TCIE was a stable and lasting effect, regardless of changes in locations, reporting methods, particular stimulus figures, the other salient feature dimensions and delay interval times. Therefore, our results support the notion that topological invariance that defines perceptual objects plays an essential role in maintaining representations in VWM.

Late Development of Early Visual Perception: No Topology-Priority in Peripheral Vision Until Age 10

Topological property (TP) is a basic geometric attribute of objects, which is preserved over continuous and one-to-one transformations and considered to be processed in early vision. This study investigated the global TP perception of 773 children aged 6-14, as compared to 179 adults. The results revealed that adults and children aged 10 or over show a TP priority trend in both central and peripheral vision, that is, less time is required to discriminate TP differences than non-TP differences. Children aged 6-8 show a TP priority trend for central stimuli, but not in their peripheral vision. The TP priority effect in peripheral vision does not emerge until age ˜10 years, and the development of central and peripheral vision seems to be different.

Processing symbolic magnitude information conveyed by number words and by scalar adjectives

Humans not only process and compare magnitude information such as size, duration, and number perceptually, but they also communicate about these properties using language. In this respect, a relevant class of lexical items are so-called scalar adjectives like “big,” “long,” “loud,” and so on which refer to magnitude information. It has been proposed that humans use an amodal and abstract representation format shared by different dimensions, called the generalised magnitude system (GMS). In this paper, we test the hypothesis that scalar adjectives are symbolic references to GMS representations, and, therefore, GMS gets involved in processing their meaning. Previously, a parallel hypothesis on the relation between number symbols and GMS representations has been tested with the size congruity paradigm. The results of these experiments showed interference between the processing of number symbols and the processing of physical (font-) size. In the first three experiments of the present study (total N = 150), we used the size congruity paradigm and the same/different task to look at the potential interaction between physical size magnitude and numerical magnitude expressed by number words. In the subsequent three experiments (total N = 149), we looked at a parallel potential interaction between physical size magnitude and scalar adjective meaning. In the size congruity paradigm, we observed interference between the processing of the numerical value of number words and the meaning of scalar adjectives, on the one hand, and physical (font-) size, on the other hand, when participants had to judge the number words or the adjectives (while ignoring physical size). No interference was obtained for the reverse situation, i.e., when participants judged the physical font size (while ignoring numerical value or meaning). The results of the same/different task for both number words and scalar adjectives strongly suggested that the interference that was observed in the size congruity paradigm was likely due to a response conflict at the decision stage of processing rather than due to the recruitment of GMS representations. Taken together, it can be concluded that the size congruity paradigm does not provide evidence in support the hypothesis that GMS representations are used in the processing of number words or scalar adjectives. Nonetheless, the hypothesis we put forward about scalar adjectives is still is a promising potential line of research. We make a number of suggestions for how this hypothesis can be explored in future studies.

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.

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.

Enumerating the forest before the trees: The time courses of estimation-based and individuation-based numerical processing

Ensemble perception refers to the ability to report attributes of a group of objects, rather than focusing on only one or a few individuals. An everyday example of ensemble perception is the ability to estimate the numerosity of a large number of items. The time course of ensemble processing, including that of numerical estimation, remains a matter of debate, with some studies arguing for rapid, "preattentive" processing and other studies suggesting that ensemble perception improves with longer presentation durations. We used a forward-simultaneous masking procedure that effectively controls stimulus durations to directly measure the temporal dynamics of ensemble estimation and compared it with more precise enumeration of individual objects. Our main finding was that object individuation within the subitizing range (one to four items) took about 100-150 ms to reach its typical capacity limits, whereas estimation (six or more items) showed a temporal resolution of 50 ms or less. Estimation accuracy did not improve over time. Instead, there was an increasing tendency, with longer effective durations, to underestimate the number of targets for larger set sizes (11-35 items). Overall, the time course of enumeration for one or a few single items was dramatically different from that of estimating numerosity of six or more items. These results are consistent with the idea that the temporal resolution of ensemble processing may be as rapid as, or even faster than, individuation of individual items, and support a basic distinction between the mechanisms underlying exact enumeration of small sets (one to four items) from estimation.

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.

Adaptation to visual numerosity changes neural numerosity selectivity

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

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.

Subitizing object parts reveals a second stage of individuation

Humans can efficiently individuate a small number of objects. This subitizing ability is thought to be a consequence of limited attentional resources. However, how and what is selected during the individuation process remain outstanding questions. We investigated these in four experiments by examining if parts of objects are enumerated as efficiently as distinct objects in the presence and absence of distractor objects. We found that distractor presence reduced subitizing efficiency. Crucially, parts connected to multiple objects were enumerated less efficiently than independent objects or parts connected to a single object. These results argue against direct individuation of parts and show that objecthood plays a fundamental role in individuation. Objects are selected first and their components are selected in subsequent steps. This reveals that individuation operates sequentially over multiple levels.

Possible Objects: Topological Approaches to Individuation

We think of the world around us as divided into physical objects like toasters and daisies, rather than solely as a smear of properties like yellow and smooth. How do we single out these objects? One theory of object concepts uses part-of relations and relations of connectedness. According to this proposal, an object is a connected spatial item of maximal extent: Any other connected item that overlaps (i.e., shares a part with) the object must be a part of that object. This article reports four experiments that test this proposal. Participants see descriptions or diagrams of spatial items that vary across trials in their relative positions. In separate experiments, participants decide whether the items are physical objects, whether they are wholes, or how many objects are present. All experiments find support for connectedness as a contributor to object status, but they find little support for maximality. The results suggest that maximality is not a necessary feature of wholes or of objects.

Effect of Finger Gnosis on Young Chinese Children's Addition Skills

Evidence has revealed an association between finger gnosis and arithmetic skills in young Western children, however, it is unknown whether such an association can be generalized to Chinese children and what mechanism may underlie this relationship. This study examines whether finger gnosis is associated with addition skills in young Chinese children and, if so, what numerical skills could explain this correlation. A total of 102 Chinese children aged 5–6 years were asked to complete finger gnosis and addition tasks in Study 1. Results showed that finger gnosis was significantly associated with addition performance. However, no significant correlation was found between finger gnosis and the use of finger counting in solving addition problems. Moreover, girls’ finger gnosis was better than boys’, and children with musical training demonstrated better finger gnosis than those without. In Study 2, 16 children with high finger gnosis and 20 children with low finger gnosis were selected from the children in Study 1 and asked to perform enumeration, order judgment, number sense, and number line estimation. Children with high finger gnosis performed better in number line estimation than their counterparts with low finger gnosis. Moreover, the number line estimation fully mediated the relationship between finger gnosis and addition performance. Together, these studies provide evidence of a correlation between finger gnosis and addition skills. They also highlight the importance of number line estimation in bridging this association.

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.

Visual working memory representation as a topological defined perceptual object

The question of what the basic unit is of visual working memory remains one of the most fundamental and controversial issues. In the current study, we proposed a unique perspective based on early topological perception to describe the nature of representation in visual working memory. In a series of updating change-detection tasks, the repetition-benefit effect on color memory was not affected when items in the second memory array underwent massive changes of nontopological features from the first memory array. However, when the topological properties of an item changed, the repetition-benefit effect was destroyed, suggesting that the item was perceived as a new object impairing the original memory. Hence, our results suggest that a perceptual object defined by its topological invariance might be a unique perspective from which to describe representations of visual working memory.

Similarity Grouping as Feature-Based Selection

Across the natural world as well as the artificial worlds of maps, diagrams, and data visualizations, feature similarity (e.g., color and shape) links spatially separate areas into sets. Despite a century of study, it is yet unclear what mechanism underlies this gestalt similarity grouping. One recent proposal is that similarity grouping-for example, seeing a red, vertical, or square group-is just global selection of those features. Although parsimonious, this account makes the counterintuitive prediction that similarity grouping is strictly serial: A green group cannot be constructed at the same time as a red group. We tested this prediction with a novel measure-a grouping illusion within number-estimation tasks that should work only if participants simultaneously construct groups-and found the strongest evidence yet in favor of serial feature-based attention ( Ns = 14, 12, and 12 for Experiment 1, Experiment 2, and Experiment 3, respectively).