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# Mixture model investigation of the inner–outer asymmetry in visual crowding reveals a heavier weight towards the visual periphery

## Abstract

Crowding, the failure to identify a peripheral item in clutter, is an essential bottleneck in visual information processing. A hallmark characteristic of crowding is the inner–outer asymmetry in which the outer flanker (more eccentric) produces stronger interference than the inner one (closer to the fovea). We tested the contribution of the inner-outer asymmetry to the pattern of crowding errors in a typical radial crowding display in which both flankers are presented simultaneously on the horizontal meridian. In two experiments, observers were asked to estimate the orientation of a Gabor target. Instead of the target, observers reported the outer flanker much more frequently than the inner one. When the target was the outer Gabor, crowding was reduced. Furthermore, when there were four flankers, two on each side of the target, observers misreported the outer flanker adjacent to the target, not the outermost flanker. Model comparisons suggested that orientation crowding reflects sampling over a weighted sum of the represented features, in which the outer flanker is more heavily weighted compared to the inner one. Our findings reveal a counterintuitive phenomenon: in a radial arrangement of orientation crowding, within a region of selection, the outer item dominates appearance more than the inner one.

## Introduction

Crowding refers to our inability to identify an object (primarily in the peripheral visual field) because it is presented with nearby objects1,2,3,4,5,6. Crowding is considered to be an impediment to reading7, face recognition8, eye movements9,10, visual search3 as well as deficits like macular degeneration11, amblyopia12 and dyslexia58,59,60. Research has characterized crowding and distinguished it from other spatial interferences, such as masking6,13, lateral interaction14 and surround suppression15. Yet, the underlying processes of crowding are still unclear3,5,16,17. Researchers have associated crowding with a variety of factors: the reduction in spatial resolution in the periphery due to increased receptive field size of cells at early visual areas18,19,20, decreased cortical magnification21 and reduced attentional resolution22,23,24,25.

A hallmark characteristic of crowding is the inner-outer asymmetry: an outer flanker (more peripheral) produces stronger interference than an inner one (closer to the fovea)1,4,15,22,23,26,30,34 (but see33,57). The asymmetry effect has been demonstrated with different stimulus types and tasks, such as letter recognition34, face recognition27 and Gabor orientation discrimination15. Nevertheless, the mechanisms underlying this phenomenon remain a matter of debate.

Researchers have proposed various causes of the inner-outer asymmetry. According to the cortical magnification account, the smaller cortical distance between the outer flanker and the target leads to stronger interference between these two items21,35. Other accounts postulate that increased receptive field size in the periphery biases sampling rate toward the outer flanker26 or leads to sparse selection in the visual periphery30. Finally, the attentional account attributes display layout and experimental task in biasing spatial attention outwards, leading to the selection of the outer flanker22,23. Yet, none of the proposed accounts directly link the inner-outer asymmetry with recent models of crowding.

Proposed models of crowding were often classified to either pooling models6,36,37,38,39,40,41 or substitution models17,42,43,44. According to pooling models, under crowding, due to inappropriate integration field size in the periphery, observers simultaneously detect and pool excessive information of low-level features, including those that belong to the flankers6,19,36,37,38,39. In the more simple form of pooling models, crowding reflects an averaging of target and flanker features. For example, simple averaging errors were observed in judging target orientation39 and target position36. However, simple pooling models cannot explain the well-documented finding showing that under crowding conditions observers often misreport a flanker as the target17,42,43,44. Such misreport errors led to the proposal of substitution models, according to which increased location uncertainty in the periphery45,46,47 renders observers failure to spatially differentiate between the target and the flankers.

In recent years, researchers were able to reconcile these conflicting findings by proposing more advanced pooling models that can predict both averaging and misreporting errors (reviewed by Rosenholtz et al.48). For example, several studies have proposed a population coding model that codes target and flanker features as a weighted sum within a receptive field and can explain both averaging and misreporting errors of orientation37,40,49, color and spatial frequency49. Investigations of these models often use estimation reports in which observers estimated the target feature in a continuous space; for example, reporting the orientation of the target by adjusting the orientation of a probe. In such tasks, observers often reported the flanker instead of the target42,49,50. By fitting a probabilistic mixture model with a misreport component (probability of reporting a flanker value), researchers were able to independently assess the precision of reported item and the contribution of the flankers to the distribution of the estimation errors42,49. This misreport mixture model acts similarly to the population coding model and can, therefore, serve to determine the relative activation of each item in the population49. However, whether the relative contribution, and hence activation, of the outer flanker is different from the inner one is still unclear.

Here, we addressed this issue by investigating the inner–outer asymmetry in a radial Gabor orientation crowding display with an orientation estimation task. We separately assessed the contribution of the inner and outer flankers to the pattern of crowding errors by comparing between various mixture models. Our findings show that a misreport mixture with a separate misreport component for the inner and the outer flanker outperformed all other tested models. Interestingly, we revealed that the misreport rate was much higher for the outer flanker than the inner one.

## Experiment 1

Observers performed an orientation estimation task of peripheral sinusoidal gratings (Gabor patches) in which the target (7° eccentricity) appeared alone (uncrowded conditions) or flanked (crowded conditions) by either two (two-flanker condition) or four (four-flanker condition) Gabors. Target and flankers were arranged on the horizontal meridian, either to the left or to the right of fixation. The centre-to-centre distance between two adjacent items was 1.5° (Fig. 1A). In crowded conditions, the target was always in the middle of the string of Gabors, such that in the two-flanker condition there was one inner flanker and one outer flanker, whereas in the four-flanker condition there were two inner flankers and two outer flankers with respect to the target (Fig. 1B).

## Method

### Apparatus

The apparatus was the same as described in “Experiment 1”.

### Stimuli and procedure

Figure 1A,C illustrate a trial sequence and stimulus condition in “Experiment 2”. Stimuli and procedure were the same as in “Experiment 1”, except for the location of the target. In this experiment, the target appeared on the horizontal meridian with 8.5° eccentricity, either in the right or left hemifield. In the two-flanker condition, the target was the outer item, whereas in the four-flanker condition, the target was located near the outer flanker. Here too, the centre-to-centre distance between two adjacent items was 1.5°. In the two-flanker condition, flanker eccentricities were 5.5° and 7° for the second inner (2I) and first inner (1I) flankers respectively. In the four-flanker condition, flanker eccentricities were 4°, 5.5°, 7° and 10° for the third inner (3I), second inner (2I), first inner (1I) and first outer (1O) flankers respectively.

### Design

The design was the same as described in “Experiment 1”.

### Models and analyses

The statistical analyses were the same as described in “Experiment 1”.

## Results

Figure 5A illustrates the error distribution for each condition. Here again, we first analyzed the bias of the errors by calculating mean error for each subject in each condition. Mean error was close to zero in the uncrowded condition (M = − 1.01, SD = 2.29), the two-flanker condition (M = − 0.41, SD = 2.75) and the four-flanker condition (M = 0.34, SD = 4.03). As in “Experiment 1”, we calculated precision as the inverse of the variance of the errors for each observer in each condition (Fig. 5B). One-way ANOVA analyses on precision and crowding condition as within subjects factor revealed a significant main effect of crowding conditions on precision, F(2,26) = 44.30, p = 0.000, partial η2 = 0.77, indicating higher precision in the uncrowded condition compared to the two-flanker condition, t(13) = 6.13, p = 0.000, Cohen's d = 1.45, and in the two-flanker condition compared to the four-flanker condition, t(13) = 4.19, p = 0.001, Cohen's d = 1.60.

### Probabilistic models

In the uncrowded condition, the standard mixture model described well the distribution of the errors (Fig. 5A). For each flanker condition, we calculated the Akaike information criterion with correction (AICc) for each observer, in order to choose the best model among the two relevant models. Figure 5C presents mean AICc for the relevant models in each crowding condition. In the two-flanker condition, there was no best-fitting model, meaning that no significant differences were found between the standard misreport model and the standard mixture model, t(13) = 1.30, p = 0.216, and between the independent two-misreport model and the standard misreport model, t(13) = 1.25, p = 0.232. This result is explained by small crowding interference in this condition. However, in the four-flanker condition, the standard misreport model outperformed the standard mixture model, t(13) = 4.06, p = 0.001, Cohen's d = 0.69, and the independent four-misreport outperformed the standard misreport model, t(13) = 3.80, p = 0.002, Cohen's d = 0.91. In this condition, misreport rates were higher for some flankers compared to others. As in “Experiment 1”, we tested the fitting of models with mean bias towards the outer flanker (Supplementary Analysis). Again, adding the mean bias component did not improve model fitting (Fig. 5C and Figure SI2), which indicates that the misreport models are sufficient to explain our results.

Next, we analyzed the fitted parameters of the best performing model (smallest AICc) in each condition. Figure 6 depicts the mean fitted variabilities (σ) and guess rates (γ) of the best fitted models in each condition. In order to examine the effect of crowding on performance, we conducted a one-way ANOVA with crowding condition (uncrowded vs. two-flanker vs. four-flanker) as a within subject factor and variability (σ), guess rate (γ) and target reporting rate (PT) as dependent variables. Significant differences were found between the crowding conditions for γ, F(2,26) = 9.86, p = 0.001, partial η2 = 0.43, σ, F(2,26) = 15.71, p = 0.001, partial η2 = 0.55, and PT, F(2,26) = 71.66, p < 0.001, partial η2 = 0.85. Specifically, the probability to report on the target decreased as the number of flankers increased, with significant differences between the uncrowded condition and the crowded conditions (two-flanker: t(13) = 3.76, p = 0.002, Cohen's d = 1.35, four-flanker: t(13) = 21.17, p < 0.001, Cohen's d = 7.17), and between the two crowded conditions, t(13) = 6.36, p = 0.001, Cohen's d = 2.26.

We assessed the contribution of each flanker to crowding errors by comparing the misreport rates of the different flankers. Figure 7 depicts the report distribution around the value of each flanker and the probability of reporting each presented item. In the two-misreport model (the two-flanker condition), in which subjects were asked to report on the outer flanker, we found insignificant effect for flankers' positions, (t(13) = − 0.35, p = 0.730). In the four-misreport model (four-flanker condition), we conducted a one-way ANOVA on misreport rate with flanker position (third inner, second inner, first inner, and first outer) as within subject factor. A significant effect for flanker position was obtained in the four-flanker condition in which the target was located near the outer flanker, F(3,39) = 12.94, p = 0.000, partial η2 = 0.50. Here, planned comparisons revealed that the misreport rate of the first (adjacent) outer flanker (1O) (M = 0.44, SD = 0.25) was significantly higher compared to the averaged misreport rate of the other flankers (M = 0.06, SD = 0.09), t(13) = 4.41, p = 0.001, Cohen's d = 2.03. These findings are consistent with Experiment’s 1 results, and support the view that observers often misreport the first (adjacent) outer flanker as the target.

The results of “Experiment 2” show that the reduction of overall precision in orientation estimation task under crowding is due to both reporting a flanker instead of the target and increased variability (σ) over the representation of the target. Here again, we found that observers reported the adjacent flanker that was more eccentric rather than the target. However, when the target was the outer item, crowding was reduced and there was no significant difference in the misreport rate (PT) between the flankers. Furthermore, as shown in previous studies6,53, in both “Experiment 1” and “Experiment 2” we found that the frequency of responses around the target decreases with an increase in the number of flankers.

Finally, we examined the effect of flanker position on crowding interference. To do so, we conducted an independent t-test for each crowded condition and compared the target reporting rate (PT) and standard deviation (σ) between “Experiment 1” and “Experiment 2”. PT analyses revealed that in the uncrowded condition, target reporting rate did not differ significantly, t(25) =− 0.39, p = 0.701, meaning that observers were unaffected by item eccentricity per se. However, significant differences were found between the two experiments in the two crowded conditions. In the two-flanker condition, when observers reported the outer item (“Experiment 2”) crowding was smaller than when observers reported the item in the middle (“Experiment 1”), t(25) = − 2.69, p = 0.013, Cohen's d = 1.04. Although we increased the target eccentricity, crowding interference was reduced when observers reported the outer item (“Experiment 2”), suggesting that our findings cannot be explained by target eccentricity alone. In the four-flanker condition, when the outer target was flanked by an outer flanker (“Experiment 2”), crowding was stronger than when the central target was flanked by two outer flankers (“Experiment 1”), t(25) = 2.40, p = 0.024, Cohen's d = 0.92 (see54 for a similar finding). Although the four-flanker condition in “Experiment 2” might involve a higher degree of uncertainty compared to the four-flanker condition in “Experiment 1”, the fact that observers misreported the adjacent outer flanker here too is consist with the observed findings in “Experiment 1”. Finally, no differences were observed for σ between “Experiment 1” and “Experiment 2”, with t-values bigger than 0.30 in each crowding condition.

## Discussion

Using an orientation estimation task and probabilistic models designed to assess the contribution of each flanker to crowding errors, we showed that the hallmark inner-outer asymmetry in crowding reflects misreports of the outer flanker as the target. First, as in previous studies17,42,49, our results showed that orientation crowding errors reflect reporting a flanker instead of the target (misreport errors). Second, we found that in a radial crowding display (“Experiment 1”), in which a target is flanked on both sides, the misreport rate of the outer flanker was significantly higher compared with the misreport rate of the inner one. Third, when two flankers appeared on each side of the target (four-flanker condition), the misreport rate of the outer flanker adjacent to the target was higher than the misreport rate of the most eccentric flanker. Finally, when the target was the outer item (“Experiment 2”, two-flanker condition) crowding errors were substantially reduced.

Our misreport mixture models provide a stochastic approximation of previously proposed population coding models37,40,49. These models explain crowding by a weighted summation of population coding within a receptive field. These models rely on an integration process, and hence they can be considered pooling models48. Here, the misreport model with independent flanker components outperformed models with either no misreport component or a single misreport component to all flankers. Thus, our modeling explains misreport errors due to pooling over feature representations.

Simple pooling models of crowding predict averaging of target and flanker features36,39. Within the context of the inner–outer asymmetry, simple pooling models assert weighted average with more weight to the outer flanker and, therefore, predict mean bias towards the outer flanker32. Here, we show that adding a mean bias as a free parameter to the misreport model does not improve model fitting. This finding is inconsistent with previous accounts that suggests that crowding errors reflect a combination of misreport and averaging errors41,50. Our study, therefore, supports a single pooling mechanism account of crowding.

Importantly, our results show that the inner-outer asymmetry can be explained by a weighted summation in which the peripheral (outer) item receives heavier weight than the foveal (inner) item. In terms of population coding models, heavier weight over the peripheral item perhaps can be implemented by assuming that summation weights scale with receptive field size, such that heavier weights are assigned to more peripheral items (larger receptive field). A similar principle was proposed with a Bayesian inference framework and with some assumption regarding the growth of receptive field size in the visual periphery26. In particular, the number of receptive fields that cover only the outer item is larger than the number of receptive fields that cover only the inner item. Here we suggest that larger receptive fields in the periphery render errors towards the outer flanker.

Our results are inconsistent with the cortical magnification account. According to this view, crowding asymmetry results from cortical mapping in which the outer flanker is closer to the target than the inner one6,35. This explanation predicts that within the same display (i.e. same cortical distances), crowding interference will be the same regardless of whether observers are reporting on the middle or the outer item. Thus, the reduction of crowding interference in the two-flanker condition of “Experiment 2” compared to “Experiment 1” suggests that cortical magnification alone is insufficient to explain the inner-outer asymmetry of crowding26.

In this study we investigated the inner-outer asymmetry along the horizontal meridian by using a symmetrical display. Although previous studies found the asymmetry effect along the horizontal, but not along the vertical meridian23, it is still unclear whether and how this effect is reflected in a typical symmetrical crowding display across the visual field. Furthermore, our results are limited to orientation errors involved in crowding asymmetry. Since recent studies found dissociation of crowding errors across dimensions, such as orientation, color and spatial frequency49, and between color and motion55, ‏further work is needed to determine whether the current findings apply to other feature dimensions. Moreover, the use of Gabor patches also limited our ability to distinguish between object level and feature level. Due to previous studies that pointed to different processes involved at these levels33,49,56, further work should use a different type of stimulus, which would allow a clear distinction between those levels. Finally, this study does not directly test the predictions of simple pooling models of crowding, which predict averaging. Future studies should compare the weighted summations (misreport) model, as proposed here, with weighted averaging models to reconcile between the different pooling models.

## Conclusions

Our findings reveal that in a typical radial crowding display, observers confuse the target with the nearby outer flanker, but not vice versa. This confusion occurs regardless of the number of flankers and reported item position. Probabilistic mixture models explain that these results as due to heavier weights toward the visual periphery. These findings are consistent with recent models that explain crowding in terms of population coding with weighted summation within receptive fields. Our results demonstrate a counterintuitive phenomenon: within a region of selection, the more eccentric item dominates appearance.

## Data availability

The data and analysis codes are available via the Open Science Framework and can be accessed at https://osf.io/pvknd/.

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## Acknowledgements

We would like to thank Hadas Nathan Gamliel for her invaluable help with data collection.

## Funding

This work was supported by The Israel Science Foundation Grant Nos. 1980/18 (to A. Yashar).

## Author information

Authors

### Contributions

A.Y. conceptualized this study. A.S. collected the data. A.S. and A.Y. contributed to the research design, statistical analysis, interpretation of the data and writing.

### Corresponding author

Correspondence to Amit Yashar.

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### Competing interests

The authors declare no competing interests.

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Shechter, A., Yashar, A. Mixture model investigation of the inner–outer asymmetry in visual crowding reveals a heavier weight towards the visual periphery. Sci Rep 11, 2116 (2021). https://doi.org/10.1038/s41598-021-81533-9

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• DOI: https://doi.org/10.1038/s41598-021-81533-9

• ### Mixture-modeling approach reveals global and local processes in visual crowding

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