Small numbers are sensed directly, high numbers constructed from size and density

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.

Visual adaptation reveals multichannel coding for numerosity

Visual numerosity is represented automatically and rapidly, but much remains unknown about the computations underlying this perceptual experience. For example, it is unclear whether numerosity is represented with an opponent channel or multichannel coding system. Within an opponent channel system, all numerical values are represented via the relative activity of two pools of neurons (i.e., one pool with a preference for small numerical values and one pool with a preference for large numerical values). However, within a multichannel coding system, all numerical values are represented directly, with separate pools of neurons for each (discriminable) numerical value. Using an adaptation paradigm, we assessed whether the visual perception of number is better characterized by an opponent channel or multichannel system. Critically, these systems make distinct predictions regarding the pattern of aftereffects exhibited when an observer is adapted to an intermediate numerical value. Opponent channel coding predicts no aftereffects because both pools of neurons adapt equally. By contrast, multichannel coding predicts repulsive aftereffects, wherein numerical values smaller than the adapter are underestimated and those larger than the adapter are overestimated. Consistent with multichannel coding, visual adaptation to an intermediate value (50 dots) yielded repulsive aftereffects, such that participants underestimated stimuli ranging from 10-50 dots, but overestimated stimuli ranging from 50-250 dots. These findings provide novel evidence that the visual perception of number is supported by a multichannel, not opponent channel, coding system, and raise important questions regarding the contributions of different cortical regions, such as the ventral and lateral intraparietal areas, to the representation of number.

The malleable impact of non-numeric features in visual number perception

Non-numeric stimulus features frequently influence observers' number judgments: when judging the number of items in a display, we will often (mis)perceive the set with a larger cumulative surface area as more numerous. These "congruency effects" are often used as evidence for how vision extracts numeric information and have been invoked in arguments surrounding whether non-numeric cues (e.g., cumulative area, density, etc.) are combined for number perception. We test whether congruency effects for one such cue - cumulative area - provide evidence that it is necessarily used and integrated in number perception, or if its influence on number is malleable. In Experiment 1, we replicate and extend prior work showing that the presence of feedback eliminates congruency effects between number and cumulative area, suggesting that the role of cumulative area in number perception is malleable rather than obligatory. In Experiment 2, we test whether this malleable influence is because of use of prior experiences about how number naturalistically correlates with cumulative area, or the result of response competition, with number and cumulative area actively competing for the same behavioral decision. We preserve cumulative area as a visual cue but eliminate response competition with number by replacing one side of the dot array with its corresponding Hindu-Arabic numeral. Independent of the presence or absence of feedback, we do not observe congruency effects in Experiment 2. These experiments suggest that cumulative area is not necessarily integrated in number perception nor a reflection of a rational use of naturalistic correlations, but rather congruency effects between cumulative area and number emerge as a consequence of response competition. Our findings help to elucidate the mechanism through which non-numeric cues and number interact, and provide an explanation for why congruency effects are only sometimes observed across studies.

Visual timing-tuned responses in human association cortices and response dynamics in early visual cortex

Quantifying the timing (duration and frequency) of brief visual events is vital to human perception, multisensory integration and action planning. Tuned neural responses to visual event timing have been found in association cortices, in areas implicated in these processes. Here we ask how these timing-tuned responses are related to the responses of early visual cortex, which monotonically increase with event duration and frequency. Using 7-Tesla functional magnetic resonance imaging and neural model-based analyses, we find a gradual transition from monotonically increasing to timing-tuned neural responses beginning in the medial temporal area (MT/V5). Therefore, across successive stages of visual processing, timing-tuned response components gradually become dominant over inherent sensory response modulation by event timing. This additional timing-tuned response component is independent of retinotopic location. We propose that this hierarchical emergence of timing-tuned responses from sensory processing areas quantifies sensory event timing while abstracting temporal representations from spatial properties of their inputs.

Early visual cortical responses increase with event duration and frequency, while later timing-tuned responses quantify event timing. Here, the authors show timing tuning gradually emerges up the visual hierarchy, and separates temporal and spatial event features.

Characterizing ontogeny of quantity discrimination in zebrafish

A sense of non-symbolic numerical magnitudes is widespread in the animal kingdom and has been documented in adult zebrafish. Here, we investigated the ontogeny of this ability using a group size preference (GSP) task in juvenile zebrafish. Fish showed GSP from 21 days post-fertilization and reliably chose the larger group when presented with discriminations of between 1 versus 3, 2 versus 5 and 2 versus 3 conspecifics but not 2 versus 4 conspecifics. When the ratio between the number of conspecifics in each group was maintained at 1 : 2, fish could discriminate between 1 versus 2 individuals and 3 versus 6, but again, not when given a choice between 2 versus 4 individuals. These findings are in agreement with studies in other species, suggesting the systems involved in quantity representation do not operate separately from other cognitive mechanisms. Rather they suggest quantity processing in fishes may be the result of an interplay between attentional, cognitive and memory-related mechanisms as in humans and other animals. Our results emphasize the potential of the use of zebrafish to explore the genetic and neural processes underlying the ontogeny and function of number cognition.

The relationship between numerosity perception and mathematics ability in adults: the moderating role of dots number

Background

It has been proposed that numerosity perception is the cognitive underpinning of mathematics ability. However, the existence of the association between numerosity perception and mathematics ability is still under debate, especially in adults. The present study examined the relationship between numerosity perception and mathematics ability and the moderating role of dots number (i.e., the numerosity of items in dot set) in adults.

Methods

Sixty-four adult participants from Anshun University completed behavioral measures that tested numerosity perception of small numbers and large numbers, mathematics ability, inhibition ability, visual-spatial memory, and set-switching ability.

Results

We found that numerosity perception of small numbers correlated significantly with mathematics ability after controlling the influence of inhibition ability, visual-spatial memory, and set-switching ability, but numerosity perception of large numbers was not related to mathematics ability in adults.

Conclusions

These findings suggest that the dots number moderates the relationship between numerosity perception and mathematics ability in adults and may contribute to explaining the contradictory findings in the previous literature about the link between numerosity perception and mathematics ability.

Numerical cognition: Unitary or diversified system(s)?

Many researchers, including Clarke and Beck, describe the human numerical system as unitary. We offer an alternative view - the coexistence of several systems; namely, multiple systems (general magnitude, parallel individuation, and symbolic) existing in parallel, ready to be activated depending on the task/need. Based on this alternative view, we present an account for the representation of rational numbers.

Groupitizing modifies neural coding of numerosity

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

Numerosity adaptation partly depends on the allocation of implicit numerosity-contingent visuo-spatial attention

Like other perceptual attributes, numerosity is susceptible to adaptation. Nevertheless, it has never been fully investigated whether adaptation to numerosity is fully perceptual in nature or if it stems from the mixed influence of perception and attention. In the present work, we addressed this point throughout three separate experiments aiming at investigating the potential role played by visuo-spatial attentional mechanisms in shaping numerosity perception and adaptation. In Experiments 1 and 2, we showed that the magnitude of numerosity adaptation can be strongly influenced by the distribution of numerosity-contingent visuo-spatial attentional resources during the adaptation period. Results from Experiment 1 revealed a robust reduction of adaptation magnitude whenever a second numerical stimulus was presented in a diametrically opposite location from that of the adaptor, despite this second adapter being neutral as matched in numerosity with the following stimulus displayed in that location. In Experiment 2, we showed that this reduction in adaptation did not occur in cases where the second stimulus was not numerical, suggesting that attentional resources specifically related to numerosity information accounts for the results of Experiment 1. Finally, in Experiment 3, we showed that uninformative visuo-spatial cues shape numerosity discrimination judgments both at baseline and during adaptation. Taken together, our results seem to indicate that visuo-spatial attention plays a relevant role in numerosity perception and that adaptation to numerosity is actively influenced by this cognitive process.

The Number Sense Represents (Rational) Numbers

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

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

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

Development of Binomial Sequential Sampling Plans for Sugarcane Aphid (Hemiptera: Aphididae) in Commercial Grain Sorghum

The sugarcane aphid (Melanaphis sacchari Zehntner) is a significant economic pest of grain sorghum (Sorghum bicolor (L.) Moench) in the Southern United States. Current nominal and research-based economic thresholds are based on estimates of mean aphids per leaf. Because enumerating aphids per leaf is potentially time consuming, binomial sequential sampling plans for M. sacchari were developed that allow users to quickly classify the economic status of field populations and determine when an economic threshold has been exceeded. During 2016 and 2017, counts of M. sacchari were recorded from 281 sampling events in 140 sorghum fields located in six states (Oklahoma, Kansas, Texas, Arkansas, Louisiana, Mississippi) . Regression analysis was used to describe the relationships between the mean M. sacchari density per two-leaf sample and proportion of plants infested with one or more aphids. Tally thresholds of T50 and T100 aphids per two-leaf sample were selected based on goodness of fit and practicality. Stop lines for both tally thresholds were developed for selected economic thresholds using Wald's sequential probability ratio test. Model validations using an additional 48 fields demonstrated that reliable classification decisions could be made with an average of 11 samples regardless of location. This sampling system, when adopted, can allow users to easily and rapidly determine when M. sacchari infestations need to be treated.

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.

Three-systems for visual numerosity: A single case study

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

Spatial-numerical associations: Shared symbolic and non-symbolic numerical representations

During the last decades, there have been a large number of studies into the number-related abilities of humans. As a result, we know that humans and non-human animals have a system known as the approximate number system that allows them to distinguish between collections based on their number of items, separately from any counting procedures. Dehaene and others have argued for a model on which this system uses representations for numbers that are spatial in nature and are shared by our symbolic and non-symbolic processing of numbers. However, there is a conflicting theoretical perspective in which there are no representations of numbers underlying the approximate number system, but only quantity-related representations. This perspective would then suggest that there are no shared representations between symbolic and non-symbolic processing. We review the evidence on spatial biases resulting from the activation of numerical representations, for both non-symbolic and symbolic tests. These biases may help decide between the theoretical differences; shared representations are expected to lead to similar biases regardless of the format, whereas different representations more naturally explain differences in biases, and thus behaviour. The evidence is not yet decisive, as the behavioural evidence is split: we expect bisection tasks to eventually favour shared representations, whereas studies on the spatial-numerical association of response codes (SNARC) effect currently favour different representations. We discuss how this impasse may be resolved, in particular, by combining these behavioural studies with relevant neuroimaging data. If this approach is carried forward, then it may help decide which of these two theoretical perspectives on number representations is correct.

Repulsive Serial Effects in Visual Numerosity Judgments

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