Ebbinghaus Illusion Distortions in Amblyopia: Impairments of Visual Size Processing and Interocular Integration

Abstract

Purpose

The perceived size of an object can be manipulated by its surroundings. In the current study, we wanted to investigate whether amblyopic patients have normal object size perception and how their perception is affected by abnormal interocular interactions.

Methods

Sixteen adult amblyopes and 16 controls participated in this experiment. We measured the Ebbinghaus illusion magnitude with large and small inducers under one binocular, two monocular, and four dichoptic viewing configurations. Exploratory factor analysis was used to extract common structures and underlying mechanisms in different conditions.

Results

The mean Ebbinghaus illusion magnitude in amblyopia was smaller with large inducers (F_1,30 = 5.874, P = 0.021) and larger with small inducers (F_1,30 = 5.814, P = 0.022) than in controls. For the small-inducer geometry, the interaction effect between groups and viewing configurations (F_6,180 = 4.472, P < 0.001) was significant. Compared to the control population, the exploratory factor analysis in amblyopia extracted three more factors mainly accounting for dichoptic viewing configurations.

Conclusions

Amblyopes tended to overestimate the object size relative to controls in the Ebbinghaus illusion, thereby amplifying it in the small-inducer geometry and diminishing it in the large-inducer geometry. Our factor analysis revealed a limited interocular transfer of the Ebbinghaus illusion in amblyopia, suggesting a defective mechanism in local binocular integration circuitry.

Amblyopia (lazy eye) refers to the loss of best-corrected visual acuity (BCVA) in one or both eyes due to abnormal visual experience in early life, such as strabismus, uncorrected anisometropia, uncorrected high refractive error, or deprivation of visual input.¹ This abnormal visual experience can affect development of the visual system and distort processing of the visual information. Apart from the loss of BCVA, increasing evidence shows that patients with amblyopia present abnormalities at various visual information processing stages, such as reduced contrast sensitivity,^2,3 spatial localization deficits,^4,5 visual crowding,⁶ interocular delay,⁷ increased temporal synchrony threshold,⁸ and reduced motion extrapolation.⁹

Accurately perceiving the size of objects is an important visual feature that guides our daily actions, such as precisely picking up a cup of a certain size. Our visual system uses contextual cues to modulate the observed size so that we can reasonably perceive the real-world size of an object.^10–14 This function develops during visual development and becomes relatively stable after a certain transitional age.^15,16 Compared to adults, children as young as 8 years old are less sensitive to the contextual environment.^17,18 As the visual system matures, the contextual visual information plays an increasing role in modulating the observed size throughout visual processing.^15,19 For patients with amblyopia, whose visual system development is disrupted during the unmatured critical period, it remains unknown how their brains utilize the contextual environment to transform observed size into real-world size.

Visual illusions are an effective tool to study visual perception and can help understand the neural mechanism of visual processing.^20,21 The Ebbinghaus illusion is a classical illusion in which the perceived size of a central disc is modulated by the size of contextual surrounding inducers.²² In this illusion, a central disc is perceived as larger when surrounded by small inducers and is perceived as smaller when surrounded by large inducers. Previous studies have revealed that the Ebbinghaus illusion is conditioned by the development of the visual system in the normal population.^16,23 The amplitude of the illusion is reduced in children younger than 10 years old, whose visual system is still in the unmatured critical period.¹⁶ Thus, the Ebbinghaus illusion could be a marker of abnormal development of the visual system regarding size perception and contextual interactions in patients with amblyopia.

In the brain, the perception of the size of objects begins in the primary visual cortex (V1).²⁴ The activation intensity in V1 varies according to the perceived object size.^13,25,26 Thus, the developmental dysfunction of V1 could impair the perception of object size, and this impairment may vary depending on the visual context.^27,28 Moreover, V1 is the first cortical site substantiating interocular transfer of visual information between the two eyes.^29,30 Studies of visual illusions in dichoptic viewing configurations provide evidence that the interocular transfer of contextual modulations occurs in V1.^31,32 Specifically, Song et al.³¹ showed that, in healthy individuals, the Ebbinghaus illusion can be induced by dichoptically presenting the central target and the surrounding inducers. On the other hand, patients with amblyopia present abnormal interocular interactions, characterized by an abnormal structure and function of V1.^33–36 Therefore, we asked whether the interocular transfer of contextual modulations supporting size perception in V1 may be impaired in amblyopia.

In the current study, we investigated visual size information processing in amblyopia using the Ebbinghaus illusion. We studied the Ebbinghaus illusion under large- and small-inducer geometries in adults with unilateral amblyopia and control participants. To explore the interocular interactions in size perception, we analyzed the differences in the Ebbinghaus illusion magnitude under binocular, monocular, and dichoptic viewing configurations, then we performed an exploratory factor analysis to identify the underlying mechanisms of visual size processing.

Methods

Participants

Sixteen adults with amblyopia (eight females; mean age, 26.4 ± 4.9 years) and 16 controls (seven females; mean age, 24.7 ± 3.0 years) with normal or corrected-to-normal visual acuity participated in this study. Amblyopia was defined as unilateral/bilateral BCVA below age-specific norms, or at least a two-chart-line interocular BCVA difference.¹ We measured the suppression by the Bagolini striated glasses test and stereoacuity by the Titmus stereopsis test. The Table shows the clinical details for the amblyopic participants. The Porta test was used to determine the dominant eye of the control participants. All participants were instructed to wear spectacles to fully correct their refractive error throughout the experiment. Participants were naïve to the purpose of this study. This study followed the tenets of the Declaration of Helsinki and was approved by the Ethics Committee of West China Hospital of Sichuan University. Written consent was obtained from all participants.

Table.

Clinical Details of Amblyopic Subjects

Subject	Age (Y)/Sex	Type	Eye	BCVA (logMAR)	Refraction	Squint	Suppression	Stereo	Patched	Surgery
A1	21/M	Anisometropic	FE (OS)	0	−0.50 × 180°	No	Strong	NA	From 5–6 y	No
			AE (OD)	1	+3.50/+1.75 × 85°
A2	23/F	Mixed	FE (OD)	−0.2	Plano	ET 8 PD	Weak	NA	From 2–8 y	No
			AE (OS)	0.1	+1.25/ +0.25 × 85°	L/R 6 PD
A3	28/F	Anisometropic	FE (OD)	0	−4.50	No	Strong	NA	No	No
			AE (OS)	0.7	+3.25/−1.00 × 40°
A4	22/M	Anisometropic	FE (OS)	0	−3.25/−2.00 × 180°	No	Central	NA	At 13 y	No
			AE (OD)	0.5	+6.50/ −3.00 × 10°
A5	21/M	Anisometropic	FE (OD)	−0.1	Plano	No	Weak	NA	No	No
			AE (OS)	1	+8.00/−2.00 × 30°
A6	37/F	Anisometropic	FE (OD)	0	+0.75	No	Central	NA	From 5–7 y	No
			AE (OS)	0.2	+4.75
A7	19/F	Anisometropic	FE (OS)	0	+3.00/−0.50 × 180°	No	Weak	200′′	No	No
			AE (OD)	0.2	+7.50
A8	29/F	Anisometropic	FE (OS)	0.1	−6.00	No	Weak	200′′	No	No
			AE (OD)	0.2	+3.00/−5.00 × 180°
A9	25/M	Mixed	FE (OS)	0	+1.00/−0.75 × 65°	ET 5 PD	Strong	NA	2 h/d from 10–11 y	No
			AE (OD)	0.3	+2.50/−1.00 × 100°
A10	26/M	Anisometropic	FE (OD)	0	Plano	No	Central	NA	5 d/wk at 7 y	No
			AE (OS)	0.5	+5.00/−1.50 × 180°
A11	25/M	Anisometropic	FE (OS)	−0.1	−2.75	No	Central	NA	2 h/d from 5–6 y	No
			AE (OD)	1	+2.25/−0.75 × 145°
A12	31/F	Anisometropic	FE (OD)	0	−2.25/−0.50 × 170°	No	No	NA	4 h/d from 16–18 y	No
			AE (OS)	0.7	+3.00/+1.50 × 100°
A13	28/F	Anisometropic	FE (OD)	0	Plano	No	Weak	NA	No	No
			AE (OS)	0.2	+5.50/−0.50 × 180°
A14	34/M	Anisometropic	FE (OD)	0	+2.50	No	No	200′′	No	No
			AE (OS)	0.4	+4.25/−0.50 × 70°
A15	25/M	Mixed	FE (OD)	0	−6.25	EX 30 PD R/L 6 PD	Strong	NA	8 h/d from 7–11 y	No
			AE (OS)	1	−2.00/−0.50 × 180°
A16	29/F	Anisometropic	FE (OS)	0	−0.50 × 140°	No	Weak	NA	No	No
			AE (OD)	0.4	+2.00

ET, esotropia; EX, exotropia; L, left; NA, not applicable; OD, right eye; OS, left eye; PD, prism diopters; R, right.

Apparatus

All stimuli were generated on a ThinkPad laptop (AMD Ryzen 7 PRO 5850U; Lenovo, Beijing, China) with MATLAB 2018b (MathWorks, Natick, MA, USA) and Psychtoolbox 3.0.9 extension.³⁷ Stimuli were displayed on gamma-corrected, head-mounted, dual-screen goggles (GOOVIS Pro; NED Optics, Shenzhen, China), with a field of view of 46° × 26°, resolution of 2880 × 1620 pixels, maximal luminance of 150 cd/m², and a refresh rate of 60 Hz in each eye. The left and right eyepieces of the goggles presented images to the left and right eye, respectively. Participants viewed the stimuli in a dark room, with a chin and forehead rest to fix their head. A keyboard was connected to the computer, serving as a response console.

Stimuli

Our stimuli consisted of a typical white Ebbinghaus stimulus³⁸ and a reference disc, randomly located side by side (4° from the vertical meridian) on a gray background. The Ebbinghaus stimulus (Fig. 1A) consisted of a central test disc and surrounding inducer discs. The physical size of the test disc was fixed at 2°. The physical size of the reference disc varied across trials with eight different reference/test ratios (0.7, 0.8, 0.9,1, 1.1, 1.2, 1.3, and 1.4), which were broadly consistent with the ones used in previous studies related to the Ebbinghaus illusion.^32,39 There were two inducer sizes: large and small (Fig. 1A). The large inducers consisted of six large discs with a physical size of 3° and an eccentricity of 3.3° from the test center. The small inducers consisted of 11 small discs with a physical size of 0.6° and an eccentricity of 1.7° from the test center. The reference and the Ebbinghaus stimuli were randomly presented in the left or right side of the visual field to avoid a lateralization effect. The whole stimulus was displayed for 1000 ms. An orange binocular fixation dot was presented at the center throughout the experiment. A rectangular frame was presented surrounding the stimuli in each eye to help the participants maintain a stable fusion during the experiment.

Figure 1.

Stimuli and viewing configurations design. (A) Ebbinghaus stimuli with large or small inducers. (B) Illustration of the experimental stimuli viewing configurations (taking Ebbinghaus stimulus with large inducers as an example). There are seven viewing configurations: BI, binocular viewing; MF, monocular FE/DE viewing; MA, monocular AE/NDE viewing; DRTF, dichoptic reference and test to FE/DE viewing; DRTA, dichoptic reference and test to AE/NDE viewing; DITF, dichoptic inducers and test to FE/DE viewing; and DITA, dichoptic inducers and test to AE/NDE viewing.

Design

We tested the Ebbinghaus illusion in seven viewing configurations (Fig. 1B): (1) BI, binocular viewing where the Ebbinghaus stimulus and the reference disc were presented to both eyes; (2) MF, monocular fellow eye (FE) (dominant eye [DE] for controls) viewing where both the Ebbinghaus stimulus and the reference disc were presented to the FE (or DE); (3) MA, monocular amblyopic eye (AE) (non-dominant eye [NDE] for controls) viewing where both the Ebbinghaus stimulus and reference disc were presented to the AE (or NDE); (4) DRTF, dichoptic reference and test to FE (or DE) viewing where the test and inducer discs were presented to the FE (or DE) and the reference disc was presented to the AE (or NDE); (5) DRTA, dichoptic reference and test to the AE (or NDE) viewing where the test and inducer discs were presented to the AE (or NDE) and the reference disc was presented to the FE (or DE); (6) DITF, dichoptic inducers and test to the FE (or DE) viewing where the test disc and reference disc were presented to the FE (or DE) and the inducers discs were presented to the AE (or NDE); and (7) DITA, dichoptic inducers and test to the AE (or NDE) viewing where the test disc and reference disc were presented to the AE (or NDE) and the inducers discs were presented to the FE (or DE).

One viewing configuration was tested in each block. Both large and small inducers were tested in the same block, with 10 repetitions for each reference/test ratio, for a total of 160 trials per block. The seven viewing configurations were repeated twice and were tested in random order. Each block took about 8 minutes to complete, for a total of around 2 hours for the 14 testing blocks.

Procedures

Before the experiment began, participants could first adjust the distance between the left and right eyepieces of the goggles to ensure visibility of the full field. Then, they performed an alignment task to avoid image misalignment. During this task, two perpendicular line segments (“┐” and “└”) were dichoptically presented to the two eyes, and the participants were asked to adjust these two segments until their vertexes aligned and formed a crossing sign (“+”) by using the keyboard. All participants were able to achieve this alignment task. The corresponding coordinates of the two segments were then used to display the stimuli in the two eyes. In each trial, the stimulus was displayed for 1000 ms. After the stimulus disappeared, the participants were instructed to judge whether the left or right central disc (test or reference disc) was smaller, which they indicated by pressing the corresponding key. They had unlimited time to respond. The keypress then initiated the next trial. No participants reported any incomplete perception or diplopia of the stimulus throughout the experiment.

Data Analysis

Psychometric function fits were performed with MATLAB R2023b. Psychometric functions were fit individually for each stimulus and viewing configuration by a logistic function forced between 0 and 1. The estimated midpoint of the function defined the point of subjective equality (PSE), at which participants perceived a comparable size between the test and reference discs (Figs. 2A, 2B). Example psychometric functions from one representative participant in the large and small configurations are plotted in Figures 2C and 2D, respectively.

Figure 2.

The psychometric functions of the Ebbinghaus illusion. The PSE was defined as the reference/test ratio with a 50% response rate of the test disc. We labeled the PSE of the BI viewing configuration with large inducers and small inducers of control subject in panels A and B. The distance of PSE from one (i.e., physical equality ratio) was defined as the illusion magnitude. We plotted the psychometric functions of the Ebbinghaus illusion across seven viewing configurations, and the large inducers and small inducers of one representative control subject are shown in panels C and D.

The Ebbinghaus illusion magnitude, which quantifies the strength of the illusion, was defined mathematically as

$$\begin{array}{ccc}& & Ebbinghausillusionmagnitude\hfill \\ & & =\left|\frac{Perceivedsize-Physicalsize}{Physicalsize}\right|\hfill \end{array}$$

where Physical size is the diameter of test disc, which was fixed at 2°, and Perceived size was calculated as PSE × Physical size. (Note, therefore, that the Ebbinghaus illusion magnitude can be simplified as the absolute value of PSE shift from 1: |PSE – 1|). We also estimated the slope of the fitted psychometric function for further analysis.

Further statistical analysis was performed with R 4.3.1 (R Foundation for Statistical Computing, Vienna, Austria), using the packages ARTool, EnvStats, psych, and GPArotation. The difference of Ebbinghaus illusion magnitude was analyzed between and within groups by a repeated-measures aligned rank transformation analysis of variance (ANOVA).⁴⁰ For the post hoc analyses, an independent two-sample permutation test was used for between-group pairwise comparisons, and a paired-sample permutation test was used for within-group pairwise comparisons. Bonferroni correction was applied for the post hoc analyses.

In the factor analysis, the eigenvalues of each factor were estimated using the fa function of the psych package. The Guttman criterion and Scree test were used to identify the number of factors. The factor models were fitted using maximum likelihood, and the loadings for each configuration were calculated by varimax rotation. We set the level of 0.50 as a factor loading threshold at which the factor could explain the configuration. Correlations between the Ebbinghaus illusion magnitude with large and small inducers were analyzed by Spearman's test. The level of significance was set at P < 0.05.

Results

The psychometric functions of a representative control subject under the seven viewing configurations in the large and small inducer geometries are plotted in Figures 2C and 2D, respectively. All individuals’ psychometric functions for the amblyopic and control groups are plotted in Supplementary Figures S1 and S2, respectively. Logistic functions were accurately fitted in all configurations and for all participants (mean R² = 0.981). Also, the confidence intervals (CIs) of each PSE were calculated with 400 bootstrap replications. The widths of the CIs did not significantly differ between the large- and small-inducer geometry groups (0.072 ± 0.030 vs. 0.072 ± 0.025; P = 0.959), and both were around 10% of the reference/test ratio range.

The Ebbinghaus illusion magnitude, characterized by the absolute value of PSE shift from 1, were significantly different from zero in most tested configurations (for all, P ≤ 0.022) (Supplementary Fig. S3), except the DITA for large inducers in the amblyopic group (P = 0.248) and the DITF for small inducers in the amblyopic (P = 0.628) and control (P = 0.222) groups. This indicates that participants experienced Ebbinghaus illusion in most viewing configurations in our experiment. The sign of the magnitude solely indicates the direction of the illusion: The positive sign signifies that the test disc was perceived larger than its physical size, whereas the negative sign indicates that the test disc was perceived smaller.

Magnitude of the Ebbinghaus Illusion: Controls Versus Amblyopes

The left side of each panel in Figure 3 illustrates the mean Ebbinghaus illusion magnitude for the large-inducer geometry for the two groups. For the large-inducer geometry, we found that the mean Ebbinghaus illusion magnitude of amblyopes was smaller than that of controls (F_1,30 = 5.874, P = 0.021, partial η² = 0.164) (Fig. 3, left pairs of boxes). In addition, the effect of viewing configurations (F_6,180 = 9.930, P < 0.001, partial η² = 0.249) was significant. However, the interaction between groups and viewing configurations was not significant (F_6,180 = 1.586, P = 0.154, partial η² = 0.050). To better analyze the differences in Ebbinghaus illusion magnitudes between groups across the viewing configurations, we used an independent two-sample permutation test for post hoc analysis with Bonferroni correction. Most of the viewing configurations (BI, MA, DRTF, DITF, and DITA) showed a significantly smaller illusion magnitude (P ≤ 0.047) in the amblyopic group in comparison with the control group.

Figure 3.

Comparisons of Ebbinghaus illusion magnitudes between the amblyopic and control groups. (A–G) Magnitudes are compared between the amblyopic group (light colored boxes) and the control group (dark colored boxes) groups with large or small inducers (left or right side of each panel) under seven viewing configurations. Each dot represents individual results under different viewing configurations; large dots indicate the geometry with large inducers and small dots indicate the geometry with small inducers. The left y-axis indicates the Ebbinghaus illusion magnitude, and the right y-axis indicates the perceived size of the test disc (degree). Boxes show interquartile ranges (IQRs) with the median; whiskers extend to the minimum and maximum values within 1.5 × IQR. *P < 0.05, **P < 0.01, ***P < 0.001.

For the small-inducer geometry, we found that the mean Ebbinghaus illusion magnitude of amblyopes was larger than that of controls (F_1,30 = 5.814, P = 0.022, partial η² = 0.162) (Fig. 3, right pairs of boxes), which is the opposite of what we found for large inducers. In addition, the effect of viewing configurations (F_6,180 = 23.573, P < 0.001, partial η² = 0.440) and the interaction between groups and viewing configurations (F_6,180 = 4.472, P < 0.001, partial η² = 0.130) were significant. However, the post hoc analysis performed on each configuration individually revealed that the Ebbinghaus illusion magnitude of amblyopes was significantly larger than that of controls only in the MA configuration (P < 0.001) (Fig. 3C) and DRTF configuration (P = 0.009) (Fig. 3D).

Slopes of the Psychometric Functions

Figure 4 shows the slopes of the psychometric functions across the seven viewing configurations in the amblyopic and control groups. For the large-inducer geometry, we found a significant interaction effect of groups and viewing configurations (F_6,180 = 2.592, P = 0.020, partial η² = 0.080), suggesting that the slopes of amblyopes were shallower than those of controls. The results of the post hoc analysis revealed that the slopes of MA (P = 0.006), DITF (P = 0.020), and DITA (P < 0.001) were significantly shallower for the amblyopes compared to the controls (Fig. 4A).

Figure 4.

Differences in the slopes of the fitted psychometric function between amblyopic and control groups. (A, B) Slopes are compared between the amblyopic group (light colored boxes) and the control group (dark colored boxes) with large-inducer (A) or small-inducer (B) geometries under seven viewing configurations. Each dot represents individual results under different viewing configurations; large dots indicate the geometry with large inducers and small dots indicate the geometry with small inducers. Boxes show IQRs with the median; whiskers extend to the minimum and maximum values within 1.5 × IQR. *P < 0.05, **P < 0.01, ***P < 0.001.

For the small-inducer geometry, the interaction effect between groups and viewing configurations was also significant (F_6,180 = 2.213, P = 0.044, partial η² = 0.069). The post hoc analysis revealed that the slopes of MA (P = 0.030) and DITA (P < 0.001) in amblyopes were significantly shallower than those of the controls (Fig. 4B).

Magnitude of the Ebbinghaus Illusion: Monocular Versus Dichoptic Viewings

To assess whether the Ebbinghaus illusion could transfer between the eyes, we further analyzed the magnitude difference between the monocular and dichoptic viewing configurations. To simplify the data analysis, we first combined the data of the two monocular configurations MF and MA into the monocular Mon viewing configuration, and we also combined the dichoptic viewing configurations into two groups: dichoptic reference and test DRT (the data combination of DRTF and DRTA), and dichoptic inducers and test DIT (the data combination of DITF and DITA) in both control and amblyopic groups. Our statistical results demonstrated that the Ebbinghaus illusion magnitude of the DIT was significantly smaller than that of the Mon and DRT configurations in both groups and both inducer sizes (P < 0.01), but we did not find a significant difference between Mon and DRT (P ≥ 0.557) (Fig. 5). In general, the Ebbinghaus illusion magnitude was smaller in both groups when the test and inducers discs were dichoptically presented.

Figure 5.

Difference in Ebbinghaus illusion magnitude among the three combined viewing configurations (Mon, DRT, and DIT) within groups. (A–D) The combined viewing configurations with large inducers are plotted in A (light color boxes indicate amblyopes) and B (dark color boxes indicate controls), and those with small inducers are plotted in C (for amblyopes) and D (for controls). The left y-axis indicates the Ebbinghaus illusion magnitude, and the right y-axis indicates the perceived size of the test disc (degree). Distinct colors represent the data for the three combined viewing configurations, such that blue, turquoise, and green represent Mon, DRT, and DIT viewing configurations, respectively. Boxes show IQRs with the median; whiskers extend to the minimum and maximum values within 1.5 × IQR. *P < 0.05, **P < 0.01, ***P < 0.001.

Exploratory Factor Analysis of the Seven Viewing Configurations

To identify the mechanisms underlying the interocular transfer of the Ebbinghaus illusion and compare them between controls and amblyopes, we performed a factor analysis on the 14 non-independent observed variables (2 inducer sizes × 7 viewing configurations) in each group.⁴¹ First, we performed a parallel analysis with the method of principal axis to obtain the eigenvalues of factors for both control and amblyopic groups. According to the Guttman criterion and Scree test, we determined that two factors would be sufficient to describe the data (Supplementary Fig. S4). We then applied a varimax rotation to make the factors more interpretable.⁴²

In controls, the two-factor model accounted for 83.1% of the variance in the 14 dimensions. The first factor explained 44.0% of the variance, and it was focused on the seven viewing configurations of large-inducer geometry with loading values larger than 0.870 (Fig. 6A, dark blue points). The second factor explained 39.2% of the variance. It was focused on the seven viewing configurations of small-inducer geometry and the loadings were larger than 0.571 (Fig. 6A, violet points). Overall, the factor analysis of the control group data identified two factors, which suggests two different mechanisms in the Ebbinghaus illusion—one for the large-inducer geometry and the other for the small-inducer geometry. We additionally found that there was no significant correlation of Ebbinghaus illusion magnitude between large- and small-inducer geometries for all viewing configurations in both groups by Spearman's test (P > 0.061) (Supplementary Figs. S5, S6).

Figure 6.

Factor tunings. (A–C) The two-factor tunings of the two-factor model for the control (dark circle) and amblyopic (light circle) groups are plotted in A and B. The five-factor tunings of the five-factor model for amblyopic group are plotted in C. The absolute loadings larger than 0.5 indicate that the corresponding viewing configuration is dominated by the factor. Gray-shaded areas indicate the absolute loadings less than 0.5, which suggests that the configuration was less correlated with that factor.

In amblyopes, although the Guttman criterion and the scree plot suggested the implication of two factors, the two-factor model did not present any clear pattern as a function of the different viewing and stimulus configurations (Fig. 6B). In addition, the two-factor model only accounted for 54.9% of the variance, with 26.9% for the first factor and 28.1% for the second factor, which was much less than in the controls. This suggests that more factors are required to explain the amblyopic dataset. Therefore, we increased the number of factors up to five, based on the criterion of 75% of total variance explained (Supplementary Table S1).^41,43,44

The five-factor model accounted for 78.6% of amblyopes’ variance. The factor-tuning curves are plotted in Figure 6C. The first factor accounted for 27.9% of the variance, and the loadings at BI, MF, MA, DRTF, and DRTA in the large-inducer geometry were larger than 0.568 (Fig. 6C). The second factor accounted for 24.2% of the variance, and the loadings at BI, MF, MA, DRTA, and DITF in the small inducers were larger than 0.519 (Fig. 6C). The third factor accounted for 9.5% of the variance and was primarily responsible for DITF and DITA in the large-inducer geometry (loadings ≥ 0.701) (Fig. 6C). The fourth factor accounted for 8.6% of the total variance and was mainly loaded onto the DITA data in the small-inducer geometry (loading = 0.765) (Fig. 6C). Last, the fifth factor was concentrated on DRTF (loadings = −0.673) and DRTA (loadings = 0.545) data in small inducers (Fig. 6C), contributing 8.3% of total variance. Overall, in this five-factor model in amblyopes, the first and the second factors somehow describe the same two mechanisms (large and small inducers) that we found in the two-factor model of controls. The other three factors, totally accounting for 26.4% of the variance, mainly underly the correlation of data under the different dichoptic viewing configurations according to their loadings. This indicates that, although the fundamental mechanisms underlying the illusion with small and large inducers may be shared between the two populations, amblyopes exhibit unique patterns of integration or interaction under dichoptic viewing configurations.

Discussion

The aim of our study was to investigate how contextual modulation transfers interocularly to modulate object size perception in amblyopia. We measured the magnitude of the Ebbinghaus illusion under different inducer sizes and viewing configurations (binocular, monocular, and dichoptic) in controls and amblyopes. Our results show that participants with amblyopia do exhibit the Ebbinghaus illusion; however, compared to controls, the magnitude of the illusion is reduced with large inducers and amplified with small inducers. In both control and amblyopic groups, the dichoptic transfer of the Ebbinghaus illusion was seamless when the inducers and test were presented to the same eye; however, it was significantly weaker when the inducers and test were presented to different eyes. Our factor analysis revealed two mechanisms to characterize the Ebbinghaus illusion with small and large inducers, respectively, in controls; however, five factors were required to account for the variance of the inducer geometry and viewing configurations in amblyopia.

The reduced Ebbinghaus illusion magnitude with large inducers and the amplified illusion magnitude with small inducers in amblyopia both suggest a global increased perceptual size of the test disc surrounded by inducers compared to controls. It has been proven that the perceptual size depends on the cortical magnification in V1^24,45,46 and that the functional surface area of V1 could predict the magnitude of the Ebbinghaus illusion.⁴⁷ Hussain et al.⁴⁸ estimated the cortical magnification based on positional error and found a smaller cortical magnification in amblyopia. Furthermore, Clavagnier et al.⁴⁹ found that the population receptive field size in V1 was increased for the amblyopic eye. Large population receptive fields are associated with small cortical magnification⁵⁰ or, alternatively, close intercortical distance.⁵¹ In psychophysical studies, the perceptual size has been found to increase when the distance between the inducers and the target is reduced or when the eccentricity of the entire stimulus is increased, both of which lead to a closer cortical distance between the inducers and the target representations in V1.⁵² Therefore, our findings, showing that perceptual size is increased in amblyopia (Fig. 3), could be explained by reduced cortical magnification or increased population receptive field size in amblyopia.^53,54

In binocular viewing, information for the two eyes is separately input in V1 layer 4, then first combined in V1 upper layers, and then processed higher in the hierarchy.^55,56 Our results and previous studies using dichoptic presentation showed a reduced illusion magnitude in dichoptic configurations, indicating a partial interocular transfer of the Ebbinghaus illusion.^31,32 This suggests a contribution of both low- and high-level processing stages in the Ebbinghaus illusion.

Notably, not all of the dichoptic viewing configurations induced the Ebbinghaus illusion in both control and amblyopic groups. Specifically, both groups did not exhibit the Ebbinghaus illusion in DITF with small inducers (Supplementary Fig. S3). This absence of Ebbinghaus illusion may be due to the stimulus parameters of stimuli size and distance effect.⁵⁷ However, we did not observe a significant Ebbinghaus illusion magnitude in DITA with large inducers (Supplementary Fig. S3) in amblyopia either, suggesting impaired interocular integrations.^58,59

Indeed, it was our expectation that amblyopic suppression could attenuate or amplify the illusion magnitude in the dichoptic condition. To further explore the interocular transfer of Ebbinghaus perception, we combined our four dichoptic configurations into two dichoptic viewings: DRT and DIT, and we compared the difference in Ebbinghaus magnitude between monocular and dichoptic viewing in both groups. First, the comparative analysis revealed no statistical difference in the Ebbinghaus illusion between DRT and Mon in either the amblyopic group or the control group (Fig. 5). This suggests that the perceptual comparison between reference and test discs involves complete interocular transfer at later cortical stages, such as parietal cortex.^10,60 In contrast, the Ebbinghaus illusion under DIT showed significant attenuation compared to Mon and DRT (Fig. 5) in both groups, indicating that the local surround modulation mechanisms are predominantly mediated by monocular processing pathways.^30–32 This attenuation was even greater in the amblyopic group, to the point that the illusion was barely present in the DIT configuration (Figs. 5A, 5C), suggesting that there is very limited, if any, local dichoptic integration. Such defective interocular interactions and binocular integration are well documented in amblyopia at both the behavioral level^61,62 and the neuronal level^63,64 in V1. These defective interactions were also characterized by the flatter slope of the psychometric functions in dichoptic viewing (Fig. 4).

To obtain a deeper understanding of the common mechanisms underlying the two inducer geometries of the Ebbinghaus illusion and the seven viewing configurations, we performed an exploratory factor analysis.⁴¹ The essence of exploratory factor analysis is to extract common structures underlying different designs. In the large-inducer geometry, the ANOVA found a main effect of viewing configurations and groups, but no interaction between them. However, the factor analysis showed different latent structures for viewing configurations in the two groups. In controls, all of the viewing configurations loaded onto a single factor (Fig. 6A), suggesting that the different viewing configurations might share a common basic channel of the Ebbinghaus illusion. However, in amblyopes, a single factor was not enough to explain all the viewing configurations for large-inducer geometry (Fig. 6B). An extra third factor was observed, accounting for the DITF and DITA configurations, i.e. the configurations in which the test and inducers were presented to different eyes (Fig. 6C).

In the small-inducer geometry, a significant interaction effect between groups and viewing configurations was found in the Ebbinghaus illusion magnitude. The interaction effect was further supported by the factor analysis, which showed different latent structures for viewing configurations in control and amblyopic groups. As for the small-inducer geometry, one factor accounted for all viewing configurations in controls (Fig. 6A), but this was not sufficient in amblyopes (Fig. 6B). To better extract the underlying common structure of viewing configurations in amblyopes, another two more factors were required (Fig. 6C). The DITA configuration loaded onto the fourth factor, but its opposite configuration (DITF) loaded onto the second factor. Additionally, the DRTF configuration loaded onto the fifth factor, implying the involvement of a different channel or processing mechanism. This might be associated with the observation that, in DRTF, amblyopes showed a significantly increased perceived test size compared to controls (Fig. 3). The other configurations (BI, MF, DRTA, and DITF) were grouped together with MA in the second factor. This factor analysis did not isolate MA from BI and MF, suggesting that the binocular and monocular viewing configurations may share a similar structural basis.

On the other hand, the post hoc analysis of the ANOVA showed significant differences between the control and amblyopic groups in the MA configuration. In that configuration, the small inducers might be close enough to the central test disc to be integrated into the local monocular cortical circuit in V1.^65,66 In amblyopia, size processing may be impaired, as suggested by the flatter slope (Fig. 4), which could be due to the spatial visual deficit of AE, introducing more internal noise and less effective connections in amblyopic V1.^67–69

One potential limitation of our study is that the reference/test disc ratio levels were not symmetrically distributed around the PSE estimates. Although they were chosen based on the literature,^32,39 this asymmetry may have biased the PSE estimation by producing a near-floor effect in the lower range (i.e., for the large-inducer geometry). This could also introduce artificial correlations in the exploratory factor analysis in this lower range, which could partly explain the specific factors observed for the large-inducer geometry. As a check, we computed the 95% CIs of the PSE via bootstrap resampling and found no significant difference in CI widths between the large-inducer and small-inducer geometry. This suggests that the asymmetry in the tested range probably did not affect the validity of our results.

In conclusion, our results show that the Ebbinghaus illusion can transfer interocularly in both groups; however, the amblyopes presented increased size perception during monocular viewing which may be due to the reduced cortical magnification^48,49,52 and reduced interocular transfer in the dichoptic viewing configurations due to their abnormal interocular interactions.^70–73