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

Abstract

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.

Introduction

Although verbal numeric estimates for large numbers of dots are typically underestimates (Kaufman, Lord, Reese, & Volkmann, 1949), humans show a clear sensitivity to variations in visual number, and this ability seems to reflect the importance of relative number in evaluating both threats and sources of benefit in the environment. Adaptation to dense texture has been shown to affect the perceived numerosity of texture (Durgin, 1995), in a manner that cannot be explained by Anstis’ (1974) spatial frequency adaptation of texture (Durgin & Huk, 1997; Durgin & Proffitt, 1996). Although more recent studies have argued that number itself is adaptable (Burr & Ross, 2008), other evidence suggests that the capacity of the true number system may be quite limited (Durgin & Portley, 2023; Portley & Durgin, 2019; Tsouli et al., 2021; Zimmerman, 2018). For relatively numerous collections, there is good reason to suspect that texture models that integrate local measures of density (e.g., local samples, Solomon & Morgan, 2018) over area might be preferable. For example, Zimmermann and Fink (2016) showed that aftereffects of size affected perceived number consistent with such an integration model (for numerosities ranging from 13 to 100). If making a patch of dots look smaller by means of size adaptation makes it look less numerous without affecting perceived density, as Zimmermann and Fink observed, that seems most consistent with the estimation of visual number resulting from the integration of density over perceived area. 

Two prior studies have sought to dissociate number and density adaptation. The first was published in response to Burr and Ross (2008). Durgin (2008) varied adapter size while using adapting areas, numerosities, and densities that were all higher than the test patterns. The relatively smaller, less numerous, but denser adapter produced a greater reduction in perceived number than a larger, more numerous adapter, that was not as dense, which is consistent with a greater influence of density than of size or number. This result appeared to show that density rather than number was the relevant dimension that was adapted. Ultimately, adherents of number adaptation have seemed to retreat from claims about dense displays (e.g., Anobile, Turi, Cicchini, & Burr, 2014), but have not otherwise accounted for the original observation that seemed to confirm that density was the adapted dimension. 

However, DeSimone, Kim, & Murray (2020) later argued that the prediction of Durgin's (2008) experiment was ambiguous (perhaps a lesser/nearer adapting number simply produces more adaptation than a farther number?). They, instead, used a larger, more numerous adapter that was less dense than the critical test display. They argued that the lowered density of the adapter ought to predict an increase in perceived number (if density was crucial), whereas in their critical condition they found a decrease that was consistent with the prediction of number adaptation according to their analysis. 

There are two difficulties with interpreting the result of DeSimone et al. (2020). First, they did not seek to use a symmetrical design in which density and number could compete in both directions. Indeed, they did not even employ symmetrical adaption at the simplest level: Their adapter/test density difference was only by a factor of 1.4, whereas the adapter/test number difference was by a factor of 2. Both these facts are important because one past point of controversy has been over whether adaptation to perceived density is even bidirectional. Durgin (2001) suggested that both contrast adaptation and density adaptation were unidirectional (downward only). However, following on the evidence from Burr and Ross (2008) for bidirectional adaptation for number, Sun, Kingdom, & Baker (2017) parametrically tested for bidirectional effects of texture density. Importantly, Sun et al. used textures that maintained a fairly constant local density by maintaining a minimum interdot distance calculated to force the dots to be fairly evenly spaced (see also Durgin, 1995). In contrast, DeSimone et al. (2020) used a fairly small minimum interdot distance in their dot-patch generation algorithm (as did Durgin, 2001), which allows local density to be quite variable, and this may have inadvertently produced adaptation to (locally) high densities when ostensibly seeking to test adaptation to low densities. If the neutral local expectations of the visual system are for low densities, local peaks in density due to variations in local density across multiple “low” density adapters could easily produce the apparent asymmetry in global density adaptation discussed by Durgin (2001), which in turn could account for DeSimone et al.’s result. 

Second, although DeSimone et al. (2020) graphically represented their stimuli in terms of both size and density parameters, they did not measure perceived size following adaptation. Instead, they plotted their results in a two-dimensional space after having only measured one dimension (perceived relative number). Arguably, this, too, leaves their result uninterpretable: Because their adapter was larger than their test fields, and they only measured perceived number, it is theoretically possible that any upward shift in perceived density that may have been caused by their somewhat sparser adapter texture was simply overshadowed in number comparisons by a larger downward shift in perceived size due to adaptation to the large size of the adapter. Thus further investigation is merited of even this one condition. 

In addition to previously demonstrated effects of size adaptation on number perception (Zimmermann & Fink, 2016). Hisakata, Nishida, & Johnston (2016) have also observed that adaptation to dense textures actually produces size reductions for outline circles, so there may even be density-related size reduction effects that add to apparent asymmetries when perceived number is used, on its own (implicitly or explicitly), to assess perceived density. The effect of adapter patch size on perceived patch size has simply not been measured in previous studies. 

The present study sought to do a more direct test of the relationship between size adaptation, density adaptation, and number comparison while using displays with well-defined (low variance) densities. To investigate these issues, we constructed an experiment in which we manipulated both the size and the density of the adapting patches, and measured both apparent changes in number and apparent changes in size in order to indirectly estimate apparent changes in density. By using dot textures with an even distribution of dots, we expected to be able to replicate Sun et al.’s (2017) demonstrations of bidirectional density adaptation indirectly (using variations in adapter size to render density logically independent of the number of dots in the adapting fields) and to clarify whether apparent number adaptation might reflect texture density adaptation in combination with size adaptation. 

Open practices statement

The design and analysis plan for this study was pre-registered on aspredicted.org following pilot testing of some conditions: https://aspredicted.org/XCN_KKG. The complete raw data files are available on OSF, as well as the analysis file used for the present paper, and the source code for the experiment: https://osf.io/tay5u/?view_only=6620aed6edca4aa5a13ab0d00c6cfe65. 

Methods

Participants

The study was approved by a local IRB and treatment of human participants adhered to the Declaration of Helsinki. A total of 48 Swarthmore undergraduate students, aged 18 to 22 (19 men, 28 women, and one non-binary person) participated. This was a sample of convenience. Demographic information, including for 14 pilot study participants, was as follows: 40% identified as White, 35% as Asian or Asian American, 24% as Hispanic, Latino, or Spanish origin, 18% as Black, African, or African American, 1% as Indigenous, and 1% as Afro-Latina. When asked about the social class of their parents, 6% indicated low income, 24% working class, 23% middle class, 32% upper middle class, and 16% upper class. 

Design

The design of the study included a 2 × 2 manipulation of adapter size (large or small: 400 vs. 50 deg²) and adapter density (dense or sparse: 2.56 vs. 0.32 dots/deg²), with an additional nuisance variable of side of adaptation (left or right), all between-subjects; see Figure 1. This meant that the large dense adapter contained 1024 dots, and the small sparse adapter contained only 16 dots, whereas both the large sparse adapter and the small dense adapter contained 128 dots. Note that the small adapters were presented in multiple locations sequentially over time so as to adapt the same retinotopic region as the large adapters, as shown in Figure 2 and discussed below. 

Figure 1.

 

The four different adaptation conditions are represented here by single images depicting the size and density of the adapting stimuli. The ratios of areas (400: 50 deg²) and of densities (2.56: 0.32 dots/deg²) are both 8:1, with the result that the small dense texture and the large sparse texture each contained the same number of dots (128). The large-dense adapter contained 1024 dots, and the small sparse adapter consisted of just 16 dots. All the tested standards were intermediate in size and density between these adapters. See below for information about the timing.

Figure 2.

 

The large outer circle shows the location of the large adaptation texture (diameter = 22.6°; 1005 pixels) when it was to the right of fixation. Top-up (readaptation before each test trial) consisted of three flashes in the large-adapter conditions. The smaller circles (diameter = 8°; 356 pixels) depict the 21 sequentially-presented positions of the small adapter regions (numbered 0–20 in order of presentation) when they were on the right, starting and ending at the center of the large adapter region, and moving 80° around the circle otherwise. Note that each numeric label (only shown here for illustration) is contained within three of the small circles, matching the three-flash adaptation top-up cycle of the large adapter. For participants being adapted on the left side, the motion was a mirror image (i.e., predominantly counterclockwise).

There were three distinct tasks that always occurred in blocks of a fixed order. Each task used a staircase method in which one of several standards was presented in the adapted side and a variable comparison stimulus was presented on the unadapted side to measure the subjectively matching value (points of subjective equality [PSE]) along the dimension (number or area) being assessed. The first task was a number comparison task between circular patches of dots of the same size. The second task was a size comparison task between outline circles (2-pixels thick). The third task was a size comparison task between circular patches of dots of the same number (or, for most participants, additionally, of half the number). 

In each task the area of the standard circle could be either 100 or 200 deg². In the number task, the density of the standard was either 0.64 or 1.28 dots/deg². In the dot-patch size task, the density of the standard was always 1.28 dots/deg². Thus, the tested number standards contained either 64 dots, 128 dots (in two cases), or 256 dots, while the tested dot-patch size standards contained either 128 or 256 dots. Sample test stimuli for the various tasks are shown in Figure 3. 

Figure 3.

 

Images representing the three comparison tasks (the standard is in the right side of each image, with a comparison on the left that differs by 25% in the relevant dimension). Note that all test stimuli are intermediate in size and density between the four types of adapters. The upper four images show the four standards for number comparison which vary in size (100 or 200 deg²) and density (0.64 or 1.28 dot/deg²) and thus in number (64, 128, 128, 256). The third row of images shows the two standards for the circle-size (area) task. The bottom row shows the two standards for the dot-patch-size (area) task. Fine print near fixation reminded participants both of the task (size or number) and to maintain fixation.

Further details for each task are as follows: 

• Task 1 details: Measurement of perceived number. There were four different standard stimuli that could be presented on the adapted side, with variable comparison stimuli presented on the non-adapted side. The four standards crossed the two test sizes (100 and 200 deg²) and densities (0.64 and 1.28 dots/deg²), and thus contained 64, 128, or 256 dots (see Figure 3). The standard densities were intermediate between the adapted densities. The task was to judge which side had more dots. Initial comparison densities depended on the expected direction of adaptation observed during pilot studies. For dense adapters, the initial comparison densities were in a ratio of 0.5 or 1.0 to the standard. For sparse adapters, the starting ratios were 0.735 or 1.47. Patch size of the comparison field was matched to that of the standard. Based on responses given, the comparison value to be presented on a future trial was adjusted up or down by a multiplier of 1.1665. This value was used so that the two staircases would finely sample an interleaved set of values. Each of the 8 staircases controlled 15 trials, for a total of 120 randomly-interleaved trials.
• Task 2 details: Measurement of perceived size (outline circles). To test whether adaptation to even a small dense texture could produce the kind of size adaptation reported by Hisakata et al. (2016), we included a test of the effect of the various adaptation stimuli on the perceived size of outline circles. There were two different standard size stimuli (the two standard sizes of 100 or 200 deg²) that could be presented on the adapted side, with variable comparison stimuli presented on the non-adapted side. Three staircases were associated with each circle size. The task was to judge which circle was larger. Based on pilot experiments suggesting that these effects would be small, the staircases were collectively initially unbiased, with the unadapted side having a ratio to the standard area of 1, 1.03⁻⁸ (∼0.79), or 1.03⁸ (∼1.27). The step size applied in response to each judgment was a change by a ratio of 1.03³ (∼1.09). Each staircase contributed 12 trials to the final psychometric function. All 72 trials were randomly interleaved.
• Task 3: Measurement of perceived patch size (random dot patches). The same standard stimulus sizes as in Tasks 1 and 2 could be presented on the adapted side (100 and 200 deg²), always with a density of 1.28 dots/deg² (for 128 or 256 dots). To additionally test whether the relative number of dots affected estimates of relative size, the number of dots in the comparison circle (on the unadapted side) was either equal to or exactly half of the number in the standard region. Thus, four measurements were made, crossing the two standard sizes with two different values of relative number. Three staircases for each size/relative-number pairing started with the initial values based on the state arrived at from the second task for that size, and using the same step size. In this case 120 trials (10 for each staircase) were randomly interleaved.
• Note that for 6 participants, a programming error meant that that comparison field was always identical in number with the standard area during the patch-size comparison task. This was not seen as a concern, because the equal-number condition was intended to be used as the baseline comparison condition for modeling.

Instructions and procedure

Participants were all naive about the purpose of the experiment. Initial oral instructions emphasized five aspects of the experiment: (1) that there would be three different tasks (each was described briefly), (2) that each judgment was to be based on perception, (3) that although the judgments were simple, they would often be difficult, meaning that participants should make the best judgment they could, (4) that the comparison stimuli in each task would be briefly flashed so that it was important to attend to the center spot (fixation mark) to facilitate comparison, and finally, once the other instructions were understood, (5) the inclusion of periods of stimuli that were not to be responded to (i.e., adapters) were described, and it was made clear that maintenance of gaze on the fixation mark during all parts of the experiment was required. 

Participants were additionally warned that although the longest subsections of the experimental procedure would only last about 10–15 minutes, it would typically feel like much longer. This was to prepare them for the challenge of staying on task. Participants were allowed to listen to music (without lyrics) during the experiment to help to maintain attention. 

Following these oral instructions, participants read written instructions that amplified the points above and then signed a consent form. The experimenter then started the experimental program, and invited participants to follow the instructions on the screen, but to feel free to ask any questions. Participants were then left alone in the experimental room to complete the experiment. The program itself provided them with further instruction and practice with each task prior to commencing each of the three tasks in turn. 

The total time taken to complete the experiment was about 45 minutes in the small-adapter conditions and about 35 minutes in the large-adapter conditions. Participants also filled out a paper form requesting information about demographics either before or after the experiment. 

Stimulus specifications

The experiment was programmed using Psychtoolbox (Brainard & Vision, 1997) in Matlab using MacOS. Four experimental rooms were used, each had a display screen with a spatial resolution of 2560 × 1440 pixels with a horizontal dimension of 0.6 m, viewed from a distance of approximately 57 cm, which meant that the screen subtended a rectangle of approximately 55.5° × 33°. The dot textures were random textures of black and white dots on a gray background that was set for each display so that the addition of equal numbers of black and white dots did not change the net luminance on the screen. The room lights were on throughout the experiment. 

The large and small adapter patches were, specifically, 22.6° (1005 pixels) and 8° (356 pixels) in diameter, to fill areas of 400 deg² or 50 deg², respectively. The dense and sparse adapters had dot densities of 2.56 or 0.32 dots/deg². A re-adaptation cycle prior to each trial consisted of 3 flashes of a large adapter or 21 flashes of a small adapter. During initial adaptation for each task, there were with 10 preliminary cycles of adaptation 

Texture properties

The dots for each display were 8 pixels in diameter (∼0.18°) randomly scattered such that the minimum distance between dots was constrained to maintain a constant Ratio of Regularity (Durgin, 1995). The Ratio of Regularity expresses the reciprocal of the ratio between the size of the minimum interdot distance actually enforced during location randomization while generating random dot positions and the theoretical maximum of the minimum possible for a given density of dots (i.e., max = sqrt(1/density)). For all dot fields in this experiment the ratio of regularity was 0.5. Note that, in order to maintain the intended densities, dots were scattered no closer than half the minimum interdot distance to the notional boundary of the texture, meaning that lower density patches were slightly smaller than higher density patches since they had larger minimum interdot distances. 

All test displays were generated on the fly. Adaptation textures were pre-generated for each participant. This was because, for the large, dense adaptation fields, novel stimulus generation time was significant. To obviate this problem, a set of 10 different adapter coordinate locations were randomized at the beginning of the experiment while the participants were reading instructions. During top-up adaptation, random subsets of this set of ten were used. Each time one of these 10 pre-randomized sets of coordinates was drawn as an adapter, the assignment of black or white to each location was newly-randomized. 

Timing and positioning of stimuli

To minimize strategic use of eye-movements, test stimuli to be compared were presented for 200 ms simultaneously on either side of fixation. Whereas the adapted regions were separated from fixation by 2° horizontally either to the left or to the right, the test circles were positioned so that their inner edges would be about 4° from fixation and were each jittered in their vertical and horizontal positions randomly by up to ± 1°, so that vertical alignment could not be substituted for size comparison. 

The timing and positioning of large adapters was relatively straightforward. Prior to each trial, there were 3 200-ms presentations of the large adapter (each followed by 500 ms of blank screen), positioned always to one side of fixation, centered 13.3° horizontally from fixation so that the nearest edge of the large adapter was 2° from fixation. 

The smaller adapters were presented sequentially so as to “paint” the large adapted region a little at a time. Temporally, this occurred as a series of 21 presentations of a small adapter. Each presentation was 200 ms, to match the large adapters, but all 21 presentations occurred consecutively with no pause in between, to minimize the duration. The adapter appeared to jump about the screen. The sequence is depicted in Figure 2, which shows that 18 positions around the circle at 20° increments were adapted by rotating the image around the adaptation region by 80° increments four times, with one presentation at the center before, in the middle and at the end of the 21 presentations. This meant that most of the large adapted region was adapted 3 times during this adaptation period by the small adapter, consistent with our goal of adapting a large retinotopic region to either a high or low density, using small adapter patches. There was a 500 ms delay prior to the test stimulus. Note that each top-up (readaptation prior to each trial) for the small adapters therefore took 4.7 s, whereas each top-up in the large adapter condition took only 2.6 s. Because there were over 300 top-ups during the three tasks, the small-adapter versions of the experiment typically took about 10 minutes longer than the large-adapter conditions. 

Parameter estimation and data exclusion

Logistic psychometric functions were applied to the 10 different measurements taken across the three tasks (4 for number, 2 for circle size, 4 for patch size) for each participant using least-squares fits in custom software in R. All fits were made to log-transformed values. Points of subjective equality (PSE) were expressed as the difference between the 50% point of the psychometric function and the standard in log space (i.e., the natural log of the ratio between them, which is negative if the PSE is less than the true value), and the just-noticeable difference (JND) was defined as the difference between the 50% and the 75% points (note that the log transform means that the JND is essentially a Weber fraction). Based on preregistered criteria, participant data for a given task was dropped from analysis if one of their JNDs for that task was more than 2 standard deviations from the mean. This resulted in five exclusions in the number comparison task, two of these were also excluded in the circle size comparison task. Note that a second preregistered exclusion criterion was based on whether PSEs differed from other PSEs in the same condition by more than 3 SD. Before computing this, an average PSE was computed for each participant, for each task. No data were excluded based on this criterion. 

For the patch size task, six participants were mistakenly tested only with patches that had the same number of dots, and these were retained for analyses using only data from trials with equal dot numbers. Because two participants seemed to have misunderstood the third task (their psychometric functions were inverted), we eliminated them before computing the mean and standard deviations of the JNDs. Two further participants were then excluded due to the preregistered JND criterion, leaving 44 participants included. 

For the nonequal number patch size condition of Task 3, six participants were excluded because they were not shown such stimuli, two were excluded because of inverted psychometric functions, and two were excluded for high JNDs, leaving 38 participants for that analysis. 

Results and discussion

The grand mean PSE for each task is plotted in Figure 4 as a function of adapter size and density, along with the 95% confidence intervals. The data represent the natural log of the ratio between the matching comparison stimulus (presented on the unadapted side) and the standard (presented in the adapted region) at the point of apparent equivalence. Thus a negative aftereffect represents a reduction in perceived number or in perceived area. Our ultimate goal is to model our data in terms of perceived density, which, by hypothesis, can be inferred from size and number judgments, and is shown in Figure 5. We first discuss how each of the measured dimensions (number and area) varied as a function of adapter density and adapter size. The test stimuli are all intermediate to the adapters in size and density. 

Figure 4.

 

Grand mean aftereffects for each adaptation condition in each task. Error bars represent 95% confidence intervals when (conservatively) considering the mean aftereffect value for each participant. Blue bars represent measured aftereffects to dense adapting textures, and green bars represent aftereffects to sparse adapting texture. Aftereffects to large adapters are represented by darker shades of color. All test patches and circles were intermediate in area between the large and small adapter sizes, and dot patches were intermediate in density, but similar in number (64, 128, or 256) to the small, dense adapter (128 dots), and the large sparse adapter (128 dots).

Figure 5.

 

Modeled density adaptation averaged across all test stimuli. Note that all tested stimuli were of densities that were intermediate to the densities of the adapters, so all density effects should be away from the adapters (down after adaptation to dense adapters, upward after adaptation to sparse adapters, as observed). Estimates of density aftereffects were achieved by statistically modeling the difference between the size-task aftereffects and the number-task aftereffects (all in log space) using LMER models. See test for details. Error bars represent 1 standard error of the mean in each direction, as modeled by LMERs.

Aftereffects on perceived numerosity

A preregistered linear mixed regression (LMER) was used to test whether adapter size and density each affected perceived number with adapter number represented by the interaction of adapter size and adapter density. We used the Satterthwaite approximation for degrees of freedom (Luke, 2017). Although the interaction of adapter size and adapter density showed a small effect, β = −0.27, t(39) = 2.06, p = 0.046, the principal predictor of number adaptation was adapter density, β = −0.74, t(39) = 8.15, p < 0.0001. 

In an exploratory analysis, we contrasted predictions of density adaptation with number adaptation by separately modeling with LMER the effect of adapting to a small dense texture (of 128 dots) and the effect of adapting to a large sparse texture (of 128 dots). Consistent with the observations of Sun et al. (2017) for density, dense textures and sparse textures of 128 dots produced entirely different directions of effect on perceived number. Unlike the observations of DeSimone et al. (2020) that sought to dissociate number and density adaptation, these new observations cannot be explained by size adaptation, given that the large sparse adapting texture produced an overall increase in perceived number, intercept = 0.19, t(9.0) = 2.75, p = 0.022, rather than the decrease predicted by size adaptation alone. Conversely, the small dense adapting texture produced a strong reduction in perceived number rather than the increase predicted by size adaptation alone, intercept = −0.39, t(11.5) = 9.64, p < 0.0001. 

The predictions of number adaptation, per se, depend on the relationship between the adapter number and the test number. We will return to a more detailed discussion of how number adaptation seems clearly contra-indicated toward the end of the Results section where we consider the case of the large sparse stationary adapter (of 128 dots) in relation to the denser, but, less numerous, 64-dot test standard. The main conclusion of the statistical modeling presented above is simply that adapter density was far and away the strongest predictor of the aftereffects of number. The reliable interaction effect between density and area, can be understood (as suggested by Figure 5) as resulting from stronger density adaptation (in each direction) when the adapter was larger rather than smaller, and this might simply reflect that the smaller adapters took 4 seconds to paint the adapted area, so that weaker adaptation may have resulted from the decay of the local adaptation during the readaptation process. 

Aftereffects on perceived outline circle size

Hisakata et al. (2016) reported that adaptation to dense texture reduced perceived density, but also reduced the perceived size of circles and lines. There was no evidence of an interaction between size and density effects in our initial model of perceived size (p = 0.46), so we modeled our outline-circle-size aftereffect data using an LMER with adapter density and adapter size as predictors, and found effects of both adapter size, β = −0.07, t(43) = 2.66, p = 0.011, and adapter density, β = −0.10, t(43) = 3.74, p = 0.0005. We have thus replicated the observation of a reduction in perceived outline circle size that is related to adaptation to dense texture (independent of texture size). 

Aftereffects on perceived dot-patch size

As is evident in Figure 4, the effect of adaptation on the perceived size of texture patches appears even stronger than that on outline circles. Again, there was no evidence of an interaction between the size and density of the adapters (p = 0.81), but a model using adapter density and adapter size as predictors found a particularly strong effect of adapter size, β = −0.22, t(41) = 6.38, p < 0.0001, as well as a significant effect of adapter density, β = −0.10, t(41) = 3.01, p = 0.004. Note that the magnitude of the effect of adapter density on the perceived size of dot patches was similar to that observed for outline circles (β = −0.10 in both models). 

An exploratory analysis tested for interactions between adapter-size aftereffects and task (outline circles vs. patches of texture) and adapter-density size aftereffects and task. This analysis confirmed that the effect of adapter size was larger for the patch size task than for the circle-size task, β = −0.14, t(41.1) = 5.40, p < 0.0001 (whereas no difference was found for the size aftereffect of adapter density, p = 0.84). This result is consistent with the idea that the adapter size aftereffect is stimulus contingent, extending to dot patches more strongly than to outline circles. 

Our pre-registration included a test of the effect of perceived number/density difference in measuring the size effect, which we modeled using size aftereffect data when the number of dots in the comparison size patch was half that in the test patch. An LMER across all adaptation conditions, showed that this produced a shift by 0.12 (in log space) in the size match (i.e., a patch of fewer dots seemed smaller), t(68.0) = 5.70, p < 0.0001. About half of this effect (i.e., ∼0.06) can be accounted for by correcting for the smaller actual boundary imposed by our scatter algorithm when number was cut in half. Because the perceived relative number would have varied across adaptation conditions during the size-comparison tasks, it is possible that differences in perceived number influenced the size perception task, overestimating the size effects. However, apparent number effects would typically have been much less than a 50% reduction (our manipulated amount, which corresponds to a density shift of −0.69 expressed as a log ratio), so the estimated correction would be smaller still. Moreover, because the size effects observed were mostly due to size adaptation rather than density adaptation, we choose not to try to correct for a small possible effect of perceived (rather than actual) dot number. 

A bidirectional density aftereffect

Assuming that judgments of perceived number are affected both by adaptation to density and adaptation to size means that a first approximation of how density was affected can be obtained by measuring the aftereffect of the number task relative to the aftereffect in the size task. This was done using LMERs that used task as a predictor for each of the four adaptation conditions, with the number aftereffect tested relative to the size aftereffect (both expressed as log ratios). The resulting density effects (with standard error bars) are plotted in Figure 5. In each case, the density aftereffect is reliably different from zero in the appropriate direction, all with p values less than 0.0013 (files containing analyses are available on OSF). 

As is evident in Figure 5, the inferred density aftereffects show fairly symmetrical bi-directional effects of adapting to high and low densities. Reasonably, the aftereffects were somewhat stronger for the larger adapting patches than they were for the smaller, jumping patches, yet strong for both. 

Modeling specific tested standards

The analyses above have averaged across all of the test stimuli, which were all intermediate in density and area. As an exploratory analysis to represent the generality of the findings to specific test stimuli relevant to the consideration of possible number adaptation, we considered separately the two test patches that were used in both the size task and the number task. These both had a density of 1.28 dots/deg but one was 100 deg² and the other was 200 deg², and so the larger one had twice as many dots (256) as the large sparse adapter (128). We used the above strategy to compute simultaneously both the apparent size and apparent density of these particular stimuli following each type of adaptation, and used the graphical format used by DeSimone et al. (2020) to express how these two test patches were affected by the four adapters we tested. 

Figure 6 plots the actual density and size of the two standard stimuli using a white circle and a white square. The four adapter densities and sizes are depicted by large green and blue circles (light for small, dark for large) plotted in the same area × density space. Note that iso-numerosity lines run diagonally in this log/log plot, so that the small test patch (empty small circle in the plot), the large sparse adapter (large light blue circle), and the small dense adapter (large dark green circle), which all contained 128 dots, all fall along the main diagonal, with a slope of −1 in the plot. The size and density matches (apparent size and density) resulting from each adapter are shown in the color corresponding to the adapting stimulus, and in the shape and size corresponding to the symbols for the standard stimuli. Note both of these stimuli show evidence that the sparse adapters produced upward shifts in perceived density, whereas the dense adapters produced downward shifts, consistent with the averages shown in Figure 5. Most tellingly, the small light blue symbols both deviate down and to the left in this space (down in measured number and measured area), but the adapters for these were a moving field of just 128 dots, which is equal to the number of the standard with circular symbol shape and less than the number of the standard with the square symbol shape, and so should not produce adaptation of number per se. Thus these deviations seem to clearly implicate a combined effect of density and area adaption, not number adaptation, per se. 

Figure 6.

 

Apparent size and density of the two test stimuli used both for the number task and the patch-size task are plotted for each adapter (adapter values are indicated by large colored circles at the four corners of the space) in log space. Error bars represent 2 standard errors of the means in each direction, as modeled by LMERs. Note that all points along the indicated diagonal represent stimuli with 128 dots. The deviations of the small light blue symbols (small, dense adapter) from their true value (white symbols) are similar in direction in this 2D space to those of the small dark blue symbols. This indicates the common effect of adapting to high density in each case, though the number of dots in the small adapter was either similar to or less than the tested number). This shows that both the large and small dense adapters produced reductions in perceived number by moving them down (in density) and to the left (in size) in this space (signaling downward effects on perceived number), despite that the small dense adapter was, in fact, smaller in number (128 dots) than the test patch represented by the square symbols (256 dots).

Possible limitations of the method

It might be argued that the small moving number adapters were simply more numerous than we have described because the presumptive number system might integrate number over time and space. We admit that this is a possible line of argument with respect to the small adapters, but the subjective impression of these adaptors was of a small patch that jumped around on the screen, with the actual changes in texture not even being noticeable due to motion capture (Ramachandran & Cavanaugh, 1987). That is, the sequence of adapters appeared similar to a single (round) adapter that simply jumped in position around the screen while remaining constant in number. We therefore believe that the goal of “painting” the to-be-adapted region with the desired density (and patch size) was achieved by the method, but future work can explore this further. 

The clearest evidence against a number-adaptation account

In support of the idea that our study supercedes that of DeSimone et al., (2020) rather limited study, we call attention to the effects of adaptation to a stationary large sparse adapter (plotted as dark green in Figures 4, 5, and 6). This adapter was stationary, so this condition can be compared to DeSimone's method directly. DeSimone et al. performed a single test to discriminate number adaptation from density, and they specifically used a large sparse adapter to argue against a density interpretation, claiming that the higher density of their test stimulus predicted an upward direction for density adaptation, whereas the lower number predicted downward adaptation from number adaptation. In the present study, the low density of the large, sparse adapter (0.32 dots/deg²) more clearly predicts an upward effect in density for all four test stimuli (whether of 0.64 dots/deg² or 1.28 dots/deg²), and unambiguously predicts downward adaptation (if number were the adapting dimension) only for the sparser of the two smaller test pattern (which contained 64 dots). Thus this last stimulus is a replication of DeSimone et al. 

Recall that DeSimone et al. (2020) (1) used clumpy adapters, which may be less effective in producing upward density adaptation, (2) used a greater adapting ratio difference for number than for density, (3) did not consider the possibility of size adaptation, nor measure perceived size, and (4) did not preregister their design. Now, consider the critical test case of 64 dots in the present study (i.e., half the number, but presented at twice the density of the large sparse stationary adapter of 128 dots). This ought to have gone “down” in perceived number according to the logic of number adaptation. In fact it showed a reliable upward shift in perceived number (mean PSE = 81.6 dots), t(9) = 2.81, p = 0.02, contrary to the standardly understood prediction of number adaptation to which DeSimone et al. appealed. This effect is shown in Figure 7. When size adaptation from this same adapter is taken into account (downward, as expected), the elevated PSE for number is therefore even more surprising (from the number adaptation perspective). These data together imply a rather strong upward effect on apparent density, as indicated in Figure 7, just as predicted by upward density adaptation. 

Figure 7.

 

Measured number aftereffect, estimated size aftereffect, and implied density aftereffect for the matches made to 64 dots presented in a circle of 100 deg² (0.64 dots/deg²), following adaptation to 128 dots in a circle of 400 deg² (0.32 dots/deg²). Perceived numerosity should have been reduced according to prediction of number adaptation; moreover size adaptation (measured) would only have augmented that direction of effect; but perceived numerosity (measured) was actually increased. Only upward density adaptation (inferred) predicts this outcome.

General discussion

In the present study we sought to measure density aftereffects indirectly by measuring the effects of adaptation on both the perceived number of dots and the perceived size of patches of dots. Whereas it might be argued that the comparison of number is more psychologically direct (easier to explain to participants for example), (Durgin 1995; Durgin 2001; Durgin 2008) has argued that aftereffects of perceived number were based on a local correlate of number that corresponded to texture density (in the sense of units or “dots” per unit of area; not in the sense of pixels covered), but not to spatial frequency (because element size is fixed). Dissociating number and density is challenging (Leibovich, Katzin, Harel, & Henik, 2017). By adding a case where density can be high while numerosity is low—accomplished by using a jumping adapter patch so that a small patch of texture (whether dense or sparse) could adapt a large retinotopic region—we have distinguished between the size of the adapted retinotopic region (always large) and the size of the adapter (large or small). This allowed us to include in our design not only large stationary adaptors that are higher in number and lower in density than some of the tested standards but also to include small adapters that were higher in density but lower in number than some of the tested standards. 

The claim that density adaptation underlies number adaptation (for large numbers, at least) should not be mistaken for the claim that global density information is more accessible than number information. As Dakin, Tibber, Greenwood, Kingdom, & Morgan (2011) put it, density and number are often “entangled” in explicit judgments, but that entanglement may reflect the fact that density is not usually of intrinsic interest, and may be less accessible to consciousness, while still being an analytic primitive, computed locally, from which relative number comparisons can be derived for large collections. There are many intermediate variables that the visual system uses to compute high-level perceptual properties that are difficult to access. For example, angular declination has been established as a primary cue to ground distance (Ooi, Wu, & He, 2001; Wallach & O'Leary, 1982), but explicit judgments of angular declination show contamination from ground distance to the visible horizon, suggesting that participants do not have direct cognitive access to the angular declination variable used to measure distance (Keezing & Durgin, 2018). It may be that the perception of texture density (which is known to be affected by patch size; Dakin et al., 2011) is similarly, a rather indirect measure of what is actually an earlier variable that can be used to compute number. Similarly, the McCullough effect (an orientation-contingent color aftereffect; McCollough, 1965) has been shown to depend on retinal color signals rather than the post-constancy perceived color of the adapter (Thompson & Latchford, 1986). The intuition that density is more challenging to judge (see Morgan, Raphael, Tibber, & Dakin, 2014) is why we chose to ask participants to compare variables that seemed most accessible: relative number and relative size. 

Anobile, Cicchini, & Burr (2014) tested discrimination of number and of density, and concluded that the two appeared to have different psychophysical properties (although only two data points were reported for density). But the dot displays they used, once again, used a small fixed minimum interdot distance, which meant that local density was highly variable across the displays until they got fairly dense. If density is coded locally, and number is derived by integrating local estimates of density, then contrasting global judgments of density with global judgments of number will, of course, show very different patterns. But this doesn't show that density information isn't preliminary to number; it just may show that the global density is not normally computed, and alternatives (like number, distance or size) are selected when access is required. Morgan et al. (2014), using highly irregular shapes (rather than circles or squares), found that number comparisons become quite variable (large Weber fractions on the order of 0.4 rather than 0.2), suggesting that a well-defined area is quite helpful in number comparison. 

Anobile et al. (2015) have argued that a switch from number to density occurs at a specific value of density that varies with eccentricity (which they suggested is determined by crowding). However, the square-root of N function described by Anobile et al. as a signature of “density,” bears a striking resemblance to the central limit theorem. If number comparisons are accomplished by sampling (as has been shown by Solomon and Morgan, 2018; see also Durgin, Aubry, Balisanyuka-Smith, & Yavuz, 2022), then random samples that contain more data (i.e., larger numbers of dots) should reduce uncertainty concerning mean as a function of the square root of N. This has nothing to do with average distance and everything to do with sampling over a limited area (i.e., density in the sense of local dots/deg²). Simulations confirm that the “density” function (1/√N) for Weber fractions reported by Anobile et al., (2014) can be produced by randomly sampling half the display (e.g., by simply limiting consideration to one hemifield of each of two dot patches) in displays statistically similar to those they used (theoretically, any sample area would do). Performance for lower numerosities that exceeds that predicted by random sampling may reflect guided (intelligent) sampling. Thus the notional switch from “number to density” discovered by Anobile et al. 2014, Anobile et al. 2015, may actually be a switch from guided sampling of the display, to random sampling as the displays become higher in number and random sampling becomes more efficient. Both forms of sampling could involve using local density information to help estimate total number (Dakin et al., 2011). 

Studying bidirectional adaptation of density, per se, probably requires using more uniformly dense fields of dots as adapters (see Figures 1 and 3), as was pioneered by Sun et al., (2017), rather than the clumpy random displays used by DeSimone et al. (2020). This is because density adaptation might be quite local, so that adaptation to locally dense clumps of dots within a “sparse” adapter over many repeated trials may resulting in a downward bias over a large area, as observed by DeSimone et al. Although Durgin (1995) found that dense adapters need not be particularly uniform to produce strong downward distortions of perceived number (for numbers above 20), he also found that low-numerosity images (i.e., 20 dots) appeared more spread out (as if local peaks in density were adapted away) following such adaptation. Our current suggestion is that density is coded quite locally and that this coding is adaptable. This state of affairs may also account for the interesting recent observation by Yousif, Clarke, & Brannon (2023) that testing a subset of the precise dot locations that were adapted seems to produce greater underestimation/adaptation than selecting exclusively empty locations for test. This might occur because the random selection of pre-adapted locations versus unadapted locations naturally samples more from densely-adapted regions of the adapting texture vs more sparsely adapted regions. The strong evidence observed here of bidirectional adaptation, on the other hand, is inconsistent with the idea that density adaptations and their aftereffects on number are solely unidirectional, as Yousif et al. have speculated. 

Here we varied the density and size of adapters, and found that taking into account both size comparison data and number comparison data implied the existence of a bidirectional adaption of density information that informed number judgments. Importantly, the adapting number (16, 128, or 1024 dots) was much less important than the adapting density in producing the aftereffects observed here. This replicates and extends the observation of Durgin (2008) as well as those of Hisakata et al. (2016), Zimmermann and Fink (2016), and Sun et al. (2017), to support a model of number perception for large numerosities that integrates local density information across space (see also Dakin et al., 2011, and Durgin, 1995). 

Prior studies that have ignored possible size adaptation effects (e.g., DeSimone et al., 2020), may have prematurely supported the idea that number adaptation did not involve density adaptation—either because unmeasured aftereffects of perceived size in that study could have been sufficient to explain their results, or because the types of dot patterns they used were ill-suited to producing upward adaptation of density. This is not to say that number can only be computed from density and size, but it does suggest that density information (spatial sampling) is relevant for large collections (including for numbers as low as 64 in the present study). The present results confirm that bidirectional density adaptation and patch size adaptation both affect perceived number. Although Burr and Ross (2008) dismissed prior findings of aftereffects on perceived numerosity (Durgin, 1995) due to differences in density of adapting texture elements as being distinct from the phenomenon they reported, the present evidence suggests that the phenomena are likely the same. Although it remains possible that true number adaptation may occur for small numbers (e.g., Tsouli et al., 2021), the very strong effects reported by Durgin (1995) and by Burr and Ross (2008) appear likely to be primarily due to local, retinotopic density adaptation (and perhaps also to size adaptation), rather than to global number adaptation. 

In addition to providing additional evidence of bidirectional density adaptation (Sun et al., 2017) and replicating that adaptation to dense texture reduces the perceived size of outline circles (Hisakata et al., 2016), we have demonstrated that the perceived size of dot patches can be subject to adaptation from other dot patches that are larger or smaller than the test stimulus, and concluded that both size adaptation and density adaptation contributed to (indeed determined) aftereffects on perceived numerosity. This contributes to our understanding of how aftereffects of size can affect judgments of number (Zimmerman & Fink, 2016). The demonstration by Durgin (2008) that a smaller, denser adapter produced greater “number” adaptation than a larger, more numerous adapter is now accompanied by a second demonstration that a larger, more numerous, but less dense adapter was able to increase the perceived number of a test stimulus by increasing its perceived density. These results imply that density adaptation (and size adaptation) are deeply implicated in both upward and downward adaptation of perceived numerosity. 

Acknowledgments

This study was preregistered on AsPredicted.org. A review copy of the pre-registration is available at the following link: https://aspredicted.org/XCN_KKG 

The full raw data set as well as a record of the conducted analyses, and the source code for the experiment is available on OSF at the following link: https://osf.io/tay5u/?view_only=6620aed6edca4aa5a13ab0d00c6cfe65. 

Commercial relationships: none. 

Corresponding author: Frank H. Durgin. 

Email: [e-mail]. 

Address: Department of Psychology, Swarthmore College, 500 College Avenue, Swarthmore, PA 19081, USA.