Numerosity Perception Research Articles

The Mechanisms and Neural Signature of Time-averaged Numerosity Perception.

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

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.

Contextual coherence increases perceived numerosity independent of semantic content.

Number perception emerges from multiple stages of visual processing. Understanding how systematic biases in number perception occur within a hierarchy of increasingly complex feature representations helps uncover the multistage processing underlying our visual number sense. Recent work demonstrated that reducing coherence of low-level visual attributes, such as color and orientation, systematically reduces perceived number. Here, we ask when in the visual processing hierarchy coherence affects numerosity perception and specifically whether the coherence effect is exclusive to low-level visual features or instead whether it can be driven by contextual or semantic relationships. We tested adults in an ordinal numerical comparison task with contextual coherence mathematically manipulated using a statistical model of visual object co-occurrence. Across several experiments, we found that arrays with high contextual coherence were perceived as numerically larger than arrays with low contextual coherence. This contextual coherence effect was not attenuated even when we reduced objects to texforms (unrecognizable images that preserve midlevel visual features) or removed semantic content from the images through box scrambling and diffeomorphic warping. Together, these results suggest that visual coherence derived from natural statistics of object co-occurrence systematically alters perceived numerosity at low-level visual processing, even before later stages at which items can be explicitly categorized and identified. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

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.

The role of motor effort on the sensorimotor number system

The integration of numerical information with motor processes has emerged as a fascinating area of investigation in both animal and human cognition. The interest in a sensorimotor number system has recently generated neurophysiological and psychophysical evidence which combine to highlight the importance of motor functions in the encoding of numerical information. Nevertheless, several key questions remain, such as the influence of non-numerical motor parameters over numerical perception. Here we tested the role of physical effort, a parameter positively correlated with the number of actions, in modulating the link between hand-actions and visual numerosity perception. Effort was manipulated during sensorimotor adaptation as well as during a new actions-estimation paradigm. The results of Experiment 1 shows that physical effort in the absence of actions (passive effort) is not sufficient to activate the sensorimotor number system, indicating that self-produced actions are instead necessary. Further experiments demonstrated that effort is marginally integrated during motor adaptation (Experiment 2) but discarded when estimating the number of self-produced hand actions (Experiment 3). Overall, the results indicate that the sensorimotor number system is largely fed by the number of discrete actions rather than the amount of effort but also indicates that effort (under specific circumstances) might be integrated. These findings provide novel insights into the sensorimotor numerical integration, paving the way for future investigations, such as on its functional role.

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.

Weaker number sense accounts for impaired numerosity perception in dyscalculia: Behavioral and computational evidence.

Impaired numerosity perception in developmental dyscalculia (low "number acuity") has been interpreted as evidence of reduced representational precision in the neurocognitive system supporting non-symbolic number sense. However, recent studies suggest that poor numerosity judgments might stem from stronger interference from non-numerical visual information, in line with alternative accounts that highlight impairments in executive functions and visuospatial abilities in the etiology of dyscalculia. To resolve this debate, we used a psychophysical method designed to disentangle the contribution of numerical and non-numerical features to explicit numerosity judgments in a dot comparison task and we assessed the relative saliency of numerosity in a spontaneous categorization task. Children with dyscalculia were compared to control children with average mathematical skills matched for age, IQ, and visuospatial memory. In the comparison task, the lower accuracy of dyscalculics compared to controls was linked to weaker encoding of numerosity, but not to the strength of non-numerical biases. Similarly, in the spontaneous categorization task, children with dyscalculia showed a weaker number-based categorization compared to the control group, with no evidence of a stronger influence of non-numerical information on category choice. Simulations with a neurocomputational model of numerosity perception showed that the reduction of representational resources affected the progressive refinement of number acuity, with little effect on non-numerical bias in numerosity judgments. Together, these results suggest that impaired numerosity perception in dyscalculia cannot be explained by increased interference from non-numerical visual cues, thereby supporting the hypothesis of a core number sense deficit. RESEARCH HIGHLIGHTS: A strongly debated issue is whether impaired numerosity perception in dyscalculia stems from a deficit in number sense or from poor executive and visuospatial functions. Dyscalculic children show reduced precision in visual numerosity judgments and weaker number-based spontaneous categorization, but no increasing reliance on continuous visual properties. Simulations with deep neural networks demonstrate that reduced neural/computational resources affect the developmental trajectory of number acuity and account for impaired numerosity judgments. Our findings show that weaker number acuity in developmental dyscalculia is not necessarily related to increased interference from non-numerical visual cues.

Exploring Sex Differences in Numerosity Perception Using Dynamic Visual Stimuli: Insights into Cognitive Processing and STEM

Exploring Sex Differences in Numerosity Perception Using Dynamic Visual Stimuli: Insights into Cognitive Processing and STEM

Silvia De Marchi (1929) on numerical estimation: A translation and commentary.

Vittorio Benussi (1878-1927) is known for numerous studies on optical illusions, visual and haptic perception, spatial and time perception. In Padova, he had a brilliant student who carefully worked on the topic of how people estimate numerosity, Silvia De Marchi (1897-1936). Her writings have never been translated into English before. Here we comment on her work and life, characterized also by the challenges faced by women in academia. The studies on perception of numerosity from her thesis were published as an article in 1929. We provide a translation from Italian, a redrawing of its 23 illustrations and of the graphs. It shows an original experimental approach and an anticipation of what later became known as magnitude estimation.

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.

Disparate processing of numerosity and associated continuous magnitudes in rats.

The studies of number sense in different species are severely hampered by the inevitable entanglement of non-numerical attributes inherent in nonsymbolic stimuli representing numerosity, resulting in contrasting theories of numerosity processing. Here, we developed an algorithm and associated analytical methods to generate stimuli that not only minimized the impact of non-numerical magnitudes in numerosity perception but also allowed their quantification. We trained number-naïve rats with these stimuli as sound pulses representing two or three numbers and demonstrated that their numerical discrimination ability mainly relied on numerosity. Also, studying the learning process revealed that rats used numerosity before using magnitudes for choices. This numerical processing could be impaired specifically by silencing the posterior parietal cortex. Furthermore, modeling this capacity by neural networks shed light on the separation of numerosity and magnitudes extraction. Our study helps dissect the relationship between magnitude and numerosity processing, and the above different findings together affirm the independent existence of innate number and magnitudes sense in rats.

Inter-individual, hemispheric and sex variability of brain activations during numerosity processing

Numerosity perception is a fundamental and innate cognitive function shared by both humans and many animal species. Previous research has primarily focused on exploring the spatial and functional consistency of neural activations that were associated with the processing of numerosity information. However, the inter-individual variability of brain activations of numerosity perception remains unclear. In the present study, with a large-sample functional magnetic resonance imaging (fMRI) dataset (n = 460), we aimed to localize the functional regions related to numerosity perceptions and explore the inter-individual, hemispheric, and sex differences within these brain regions. Fifteen subject-specific activated regions, including the anterior intraparietal sulcus (aIPS), posterior intraparietal sulcus (pIPS), insula, inferior frontal gyrus (IFG), inferior temporal gyrus (ITG), premotor area (PM), middle occipital gyrus (MOG) and anterior cingulate cortex (ACC), were delineated in each individual and then used to create a functional probabilistic atlas to quantify individual variability in brain activations of numerosity processing. Though the activation percentages of most regions were higher than 60%, the intersections of most regions across individuals were considerably lower, falling below 50%, indicating substantial variations in brain activations related to numerosity processing among individuals. Furthermore, significant hemispheric and sex differences in activation location, extent, and magnitude were also found in these regions. Most activated regions in the right hemisphere had larger activation volumes and activation magnitudes, and were located more lateral and anterior than their counterparts in the left hemisphere. In addition, in most of these regions, males displayed stronger activations than females. Our findings demonstrate large inter-individual, hemispheric, and sex differences in brain activations related to numerosity processing, and our probabilistic atlas can serve as a robust functional and spatial reference for mapping the numerosity-related neural networks.

Neurocognitive Assessment of Mathematics-Related Capacities in Neurosurgical Patients.

A precise neuropsychological assessment is of the utmost importance for neurosurgical patients undergoing the surgical excision of cerebral lesions. The assessment of mathematical abilities is usually limited to arithmetical operations while other fundamental visuo-spatial aspects closely linked to mathematics proficiency, such as the perception of numerical quantities and geometrical reasoning, are completely neglected. We evaluated these abilities with two objective and reproducible psychophysical tests, measuring numerosity perception and non-symbolic geometry, respectively. We tested sixteen neuro-oncological patients before the operation and six after the operation with classical neuropsychological tests and with two psychophysical tests. The scores of the classical neuropsychological tests were very heterogeneous, possibly due to the distinct location and histology of the tumors that might have spared (or not) brain areas subserving these abilities or allowed for plastic reorganization. Performance in the two non-symbolic tests reflected, on average, the presumed functional role of the lesioned areas, with participants with parietal and frontal lesions performing worse on these tests than patients with occipital and temporal lesions. Single-case analyses not only revealed some interesting exceptions to the group-level results (e.g., patients with parietal lesions performing well in the numerosity test), but also indicated that performance in the two tests was independent of non-verbal reasoning and visuo-spatial working memory. Our results highlight the importance of assessing non-symbolic numerical and geometrical abilities to complement typical neuropsychological batteries. However, they also suggest an avoidance of reliance on an excessively rigid localizationist approach when evaluating the neuropsychological profile of oncological patients.

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

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

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 unifiedrepresentation.

Serial dependence in visual perception: A meta-analysis and review.

Positive sequential dependencies are phenomena in which actions, perception, decisions, and memory of features or objects are systematically biased toward visual experiences from the recent past. Among many labels, serial dependencies have been referred to as priming, sequential dependencies, sequential effects, or serial effects. Despite extensive research on the topic, the field still lacks an operational definition of what counts as serial dependence. In this meta-analysis, we review the vast literature on serial dependence and quantitatively assess its key diagnostic characteristics across several different domains of visual perception. The meta-analyses fully characterize serial dependence in orientation, face, and numerosity perception. They show that serial dependence is defined by four main kinds of tuning: serial dependence decays with time (temporal-tuning), it depends on relative spatial location (spatial-tuning), it occurs only between similar features and objects (feature-tuning), and it is modulated by attention (attentional-tuning). We also review studies of serial dependence that report single observer data, highlighting the importance of individual differences in serial dependence. Finally, we discuss a range of outstanding questions and novel research avenues that are prompted by the meta-analyses. Together, the meta-analyses provide a full characterization of serial dependence as an operationally defined family of visual phenomena, and they outline several of the key diagnostic criteria for serial dependence that should serve as guideposts for future research.

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.

The role of non-numerical information in the perception of temporal numerosity.

Numerosity perception refers to the ability to make rapid but approximate estimates of the quantity of elements in a set (spatial numerosity) or presented sequentially (temporal numerosity). Whether numerosity is directly perceived or indirectly recomputed from non-numerical features is a highly debated issue. In the spatial domain, area and density have been suggested as the main parameters through which numerosity would be recomputed. In the temporal domain, stimuli duration and temporal frequency could be similarly exploited to retrieve numerosity. By adapting a psychophysical technique previously exploited in the spatial domain, we investigated whether temporal visual numerosity is directly perceived. Adult participants observed sequences of visual impulses sampled from a stimulus space spanning several levels of temporal frequency and duration (and hence numerosity), and then reproduced the sequence as accurately as possible via a series of keypresses. Crucially, participants were not asked to reproduce any particular property (such as number of impulses) but were free to choose any available cue (such as total duration, or temporal frequency). The results indicate that while the overall sequence duration was barely considered, numerosity and temporal frequency were both spontaneously used as the main cues to reproduce the sequences, with a slight but significant dominance of numerosity. Overall, the results are in line with previous literature suggesting that numerosity is directly encoded, even for temporal sequences, but a non-numerical feature (temporal frequency) is also used in reproducing sequences.

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.

Numerosity depends on normalized contrast energy: Review and square-root law model

The perceived numerosity of many randomly-located items of fixed contrast depends on the integrated contrast energy (CE) of the display. We show here that a model based on √(CE), normalized by contrast amplitude, can fit numerosity judgment data in various tasks and over a wide range of numerosities. The model shows that judged numerosity increases linearly with √(N), where N is the number of displayed items above the subitization range, and can explain: 1) the general underestimation in absolute judgement of numerosity; 2) the contrast independence (constancy) of numerosity judgment in segregated displays, i.e., judged numerosities are not affected by item contrast; 2) a contrast-dependent illusion where the numerosity of higher-contrast items is further underestimated when intermingled with lower-contrast items; and 3) both the threshold and sensitivity of numerosity discrimination between displays of N and M items. The nearly perfect fit of numerosity judgment data by a square-root law over a wide range of numerosities, including the range typically described by Weber’s law, but excluding subitization, suggests that normalized contrast energy might be the prevailing sensory code underlying numerosity perception.