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Eye Movements, Strabismus, Amblyopia and Neuro-ophthalmology  |   April 2016
Deficient Binocular Combination of Second-Order Stimuli in Amblyopia
Author Affiliations & Notes
  • Jiawei Zhou
    School of Ophthalmology and Optometry and Eye Hospital Wenzhou Medical University, Wenzhou, Zhejiang, People's Republic of China
    McGill Vision Research, Department of Ophthalmology, McGill University, Montreal, Quebec, Canada
  • Rong Liu
    CAS Key Laboratory of Brain Function and Disease, and School of Life Sciences, University of Science and Technology of China, Hefei, Anhui, People's Republic of China
  • Lixia Feng
    Department of Ophthalmology, First Affiliated Hospital, Anhui Medical University, Hefei, Anhui, People's Republic of China
  • Yifeng Zhou
    CAS Key Laboratory of Brain Function and Disease, and School of Life Sciences, University of Science and Technology of China, Hefei, Anhui, People's Republic of China
  • Robert F. Hess
    McGill Vision Research, Department of Ophthalmology, McGill University, Montreal, Quebec, Canada
  • Correspondence: Jiawei Zhou, Wenzhou Medical University, 270 Xueyuan Road, Wenzhou, Zhejiang, 325003, China; jiawei.zhou@mcgill.ca
Investigative Ophthalmology & Visual Science April 2016, Vol.57, 1635-1642. doi:10.1167/iovs.15-18253
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      Jiawei Zhou, Rong Liu, Lixia Feng, Yifeng Zhou, Robert F. Hess; Deficient Binocular Combination of Second-Order Stimuli in Amblyopia. Invest. Ophthalmol. Vis. Sci. 2016;57(4):1635-1642. doi: 10.1167/iovs.15-18253.

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Abstract

Purpose: Sensory imbalances in humans with amblyopia have been well documented using luminance-modulated (first-order) stimuli. However, little is known regarding whether there is a deficient binocular combination in amblyopes for stimuli defined by modulations in contrast (second-order stimuli). To address this, we asked two questions: Does a sensory imbalance also exist in the binocular combination of second-order stimuli, and if so, is it more severe than that expected on the basis of the imbalance for first-order stimuli?

Methods: The sensory imbalances of 14 adult amblyopes (mean age: 30.5 ± 11.5 years; 5 with strabismus and 9 without) were measured using a dichoptic phase combination task. Three types of second-order dichoptic stimulus pairs were used in the study, where the carriers in the two eyes were either correlated, anticorrelated, or uncorrelated. Results were compared with those obtained using first-order stimuli in all observers.

Results: We found that second-order binocular combination in amblyopes was not affected by the interocular carrier correlations. The amblyopic eye's contribution to the binocularly fused percept was much less than that of the nonamblyopic eye. The resulting sensory imbalance in binocular combination for second-order images was comparable to that for first-order images in 8 of the observers but was more severe in the other 6 amblyopes.

Conclusions: These results indicate that amblyopia does not disrupt the normal architecture of binocular combination for second-order signals; however, there is an additional deficit in binocular combination of second-order image in some amblyopes that cannot be fully accounted for by the known first-order sensory imbalance.

Amblyopia is a common disorder caused by abnormal visual development during childhood.13 This makes amblyopia a unique model for investigating what factors are critical in early development. Normally, people with amblyopia have reduced visual function when viewing stimuli with their amblyopic eye, even when the appropriate optical correction is used. These deficits can include reduced visual acuity, reduced contrast sensitivity function, and spatiotemporal distortions.38 Recently, imbalances in binocular function have been put forward as a possible primary factor in the disorder.911 This has led to new binocular training therapies.1216 Such a sensory imbalance has been demonstrated in several paradigms, for example, local phase combination,9,17 global motion coherence,18 and global orientation coherence.19 Even at low spatial frequencies where the two eyes have equivalent monocular visual thresholds, the amblyopic eye still makes less of a contribution to the binocularly fused percept than the nonamblyopic eye.9,11,1719 Moreover, the sensory imbalance tends to be more severe for tasks involving extrastriate pathways.19 The magnitude of the imbalance in the three paradigms mentioned above is greater for the global motion coherence task than for the global orientation coherence task, and both of these have greater effects than seen in the local phase combination task. 
Most of the studies mentioned above were conducted with luminance-modulated (first-order) visual stimuli, as opposed to second-order stimuli, which are defined by modulations in contrast,20,21 texture,2226 or orientation.27 First-order stimuli are thought to be processed directly via linear spatiotemporal filters, while the filter-rectify-filter model for the processing of second-order stimuli has three stages: linear filtering to extract the first-order response, followed by nonlinear rectification, and a second stage of linear filtering.23,28 Previous evidence shows that first-order information is processed by the neurons in striate cortex,29 while the extrastriate cortex is involved in the processing of second-order information.30 Although the amblyopic deficit was once thought to be limited to the striate cortex,31 there is now growing evidence that the amblyopic deficit may also involve the extrastriate cortex, as there are deficits for detecting second-order stimuli that cannot be fully accounted for by the known first-order deficit.3234 
Here we address two questions: Does a sensory imbalance also exist in the binocular combination of second-order stimuli, and if so, is it more severe than that expected on the basis of the imbalance for first order? We do this using a binocular phase combination paradigm35 that we recently modified to investigate the pattern of binocular combination of second-order phase (contrast-modulated) in normal adults.36 There we showed that the binocular combination of second-order stimuli was the same whether the carriers in the two eyes were correlated, anticorrelated, or uncorrelated; that the tuning function of the binocularly perceived phase versus interocular modulation ratio was quite linear compared with that of the first-order condition; and that the two eyes contributed equally to binocular combination in normal adults. We showed that second-order modulation and phase combination are also contrast-gain controlled, in that the signals from the two eyes are weighted by the carriers' contrast and linear summed thereafter.37 In the current study we use the same design that we used on those normal adults36 to assess the extent of any abnormality in binocular combination for second-order stimuli in amblyopes. An illustration of the visual stimuli we used for the test is provided in Figure 1, and the details of stimuli and the curve fits are provided in the Appendix. 
Figure 1
 
Dichoptic stimuli in the binocular phase combination test. Four dichoptic pairs were used: second-order gratings with correlated (A), anticorrelated (B), or uncorrelated (C) carriers in the two eyes and first-order gratings (D). For each dichoptic pair, gratings with a fixed modulation depth (contrast) of 100% and phase shift of 22.5° in one direction were presented to the amblyopic eye; and gratings with a signal strength ratio of δ (0, 0.1, 0.2, 0.4, 0.8, 1) and an opposite phase shift by the same magnitude were presented to the nonamblyopic eye.
Figure 1
 
Dichoptic stimuli in the binocular phase combination test. Four dichoptic pairs were used: second-order gratings with correlated (A), anticorrelated (B), or uncorrelated (C) carriers in the two eyes and first-order gratings (D). For each dichoptic pair, gratings with a fixed modulation depth (contrast) of 100% and phase shift of 22.5° in one direction were presented to the amblyopic eye; and gratings with a signal strength ratio of δ (0, 0.1, 0.2, 0.4, 0.8, 1) and an opposite phase shift by the same magnitude were presented to the nonamblyopic eye.
Methods
Participants
Fourteen adult amblyopes (age: 17–52 years; mean age, 30.5 ± 11.5 years) without (A1–A9) or with (S10–S14) strabismus participated. Clinical details for all subjects are provided in the Table, in which refractive errors refer to the subjects' refractive correction, which in most cases corresponded to their current spectacles. Squint was measured with the Amblyoscope (Clement Clarke International, England, UK). Stereopsis was measured by Titmus Stereo test (Stereo Optical Co., Inc., Chicago, IL, USA). All of our subjects were able to fixate constantly with either eye. This was verified by the stable alignment prior to the balance measurements, which was monitored before each trial (see Procedures). 
Table
 
Clinical Details of the Participants
Table
 
Clinical Details of the Participants
All subjects were naive as to the purpose of the experiment. A written informed consent was obtained from each of them after explanation of the nature and possible consequences of the study. This study complied with the Declaration of Helsinki and was approved by the Institutional Review Boards of University of Science and Technology of China and McGill University. 
Apparatus
All stimuli were generated and controlled by an Apple Mac computer (Apple, Inc., Cupertino, CA, USA) running Matlab (MathWorks, Natick, MA, USA) with Psychtoolbox 3.0.9.38,39 A pair of head-mounted goggles (Z800 pro goggles; eMagin Corp., Washington, DC, USA) was used to dichoptically present images to the two eyes. The goggles had a simulated viewing distance of 3.6 m, a spatial resolution of 800 × 600 pixels, a refresh rate of 60 Hz, and a mean luminance of 160 cd/m2 in each eye. The mini-OLED screens of the goggles are linear in their luminance response40 and exhibit pixel independence in image presentation.41 Thus we would not expect any nonlinear effects due to the display equipment42 in our study. 
Procedures
A binocular phase combination paradigm35 was used to assess each eye's contribution to the binocularly fused percept. Each trial contained two tasks. Firstly, a task was performed to confirm the alignment of the images presented to the two eyes. A display was presented that consisted of binocular fixation crosses (3.78 × 3.78 degree2) and four monocular dots (0.378° diameter); two of these were in the first and third quadrants in the left eye, and the other two were in the second and fourth quadrants in the right eye. This was presented in the center of the larger high-contrast frame together with four white diagonal lines. Observers were instructed to move the image in their amblyopic eye using the arrow keys to horizontally and vertically align the images from two eyes. After achieving stable fusional alignment, observers were asked to press the space key. The corresponding coordinate between two eyes was then used in the second task. 
Next, observers were asked to adjust the position of a 1-pixel black horizontal reference line to indicate the perceived phase of the cyclopean sine-wave grating. This was defined as the location of the center of the lower contrast stripe of the second-order grating or the dark stripe of the first-order grating. Horizontal sine-wave gratings (modulating either the luminance for first-order stimuli or the carrier contrast for second order) with equal and opposite phase shifts of 22.5° (relative to the center of the screen) were dichoptically presented to the two eyes. The signal strength (i.e., the modulation depth of second-order gratings or the contrast of first-order gratings) was fixed as 100% in the amblyopic eye and varied with a ratio δ (0, 0.1, 0.2, 0.4, 0.8 or 1) in the fellow eye. For each ratio, the phase of the binocular cyclopean image was measured with the adjustment method. Two configurations were used in measuring the perceived phase of the binocular cyclopean image to cancel any positional bias: (1) The phase shift was +22.5° in the amblyopic eye and −22.5° in the fellow eye; (2) the phase shift was −22.5° in the amblyopic eye and +22.5° in the fellow eye. The perceived phase at each interocular signal ratio (δ) was quantified by half the difference between the measured perceived phases in these two configurations. In the measurement, the two configurations at the six interocular signal strength ratios were repeated at least four times for each observer. The function of perceived phase versus interocular signal ratios (PvR function) was then derived for each stimulus type. 
The reference line was presented with an initial position randomly (−9 to 10 pixels) assigned relative to the center of the frame in each trial. It was adjusted by the observers with a fixed step size of 1 pixel, corresponding to 4° phase angle of the grating. The gratings, frames, and reference lines were presented continually in the two eyes during the phase adjustment. After completing the task, observers were asked to press the space key to confirm the location. The next trial started 1 second later. A typical trial lasted approximately 10 seconds. Voluntary breaks were allowed during the test. Practice trials were provided prior to data collection. 
Results
Figure 2 shows the binocularly perceived phase versus interocular modulation depth ratio functions (PvR functions) for 14 amblyopes and their average (in different parts of figure) for the three second-order dichoptic carrier types (i.e., correlated, anticorrelated, and uncorrelated carriers in the two eyes). The patterns of the functions are essentially the same in the three second-order dichoptic pairs. A repeated-measures within-subject analysis of variance (ANOVA) also showed that the binocularly perceived phase of the cyclopean image depended significantly on the modulation ratio (F(5,65) = 268.95, P < 0.001), and that this effect did not differ for the three dichoptic carrier types (no significant interaction: F(10,130) = 1.58, P = 0.12). 
Figure 2
 
Binocularly perceived phase as a function of interocular modulation ratio for the three dichoptic carrier types. Results from 14 amblyopes and their average are shown in separate parts of the figure. The vertical axes represent perceived phase of the cyclopean grating; the horizontal axes represent interocular modulation ratio (nonamblyopic eye/amblyopic eye); the symbols represent data for the three carrier types: ○, second-order stimuli with correlated carriers in the two eyes; △, second-order stimuli with anticorrelated carriers, and □, second-order stimuli with uncorrelated carriers. The horizontal lines indicate a perceived phase of zero (where the eyes are contributing equally to the percept, i.e., the zero-crossing pint). Error bars represent standard errors.
Figure 2
 
Binocularly perceived phase as a function of interocular modulation ratio for the three dichoptic carrier types. Results from 14 amblyopes and their average are shown in separate parts of the figure. The vertical axes represent perceived phase of the cyclopean grating; the horizontal axes represent interocular modulation ratio (nonamblyopic eye/amblyopic eye); the symbols represent data for the three carrier types: ○, second-order stimuli with correlated carriers in the two eyes; △, second-order stimuli with anticorrelated carriers, and □, second-order stimuli with uncorrelated carriers. The horizontal lines indicate a perceived phase of zero (where the eyes are contributing equally to the percept, i.e., the zero-crossing pint). Error bars represent standard errors.
The zero-crossing points of the PvR functions (i.e., the balance point, the interocular modulation ratio that results in the two eyes contributing equally to the cyclopean percept) of all the observers occurred at ratios much lower than 1. Compared with the previous report with the same paradigm on normal adults36 (which showed that the zero-crossing points of the functions were not significantly different from 1), our findings in this study indicate that the two eyes in amblyopes are quite imbalanced. Their nonamblyopic eye needs less modulation depth to balance the amblyopic eye's contribution to the binocular combination of these second-order stimuli. 
This imbalance in the binocular combination of second-order stimuli in amblyopia leads to our second question: Is this imbalance more severe than expected on the basis of the imbalance for first-order stimuli? To answer this, we conducted a first-order PvR function measurement on each subject (Fig. 1D) to compare the balance points between first- and second-order combinations. Given the finding that the second-order binocular combination was quite similar regardless of whether the carriers in the two eyes were correlated or uncorrelated, we averaged results for the three carrier types to simplify further analysis. The averaged second-order PvR of each subject is presented in Figure 3 as filled circles, while the corresponding first-order PvR is shown as unfilled circles. According to Equation 5 in the Appendix, the balance point (bp) and the nonlinearity (γ) jointly define the shape of the PvR, we thus quantitatively assessed any difference in the PvR for different stimuli by calculating these two critical parameters based on the contrast-gain control model17,37; we also conducted an F-test for nested models to statistically compare the difference in balance point (bp) in the different PvRs (see Curve Fits in Appendix). These fits are drawn in Figure 3
Figure 3
 
Binocularly perceived phase as a function of interocular signal ratio for second-order and first-order stimuli. Results from 14 amblyopes and their average are shown in separate parts of the figure. The vertical axes represent perceived phase of the cyclopean grating; the horizontal axes represent interocular modulation ratio (nonamblyopic eye/amblyopic eye); •, data for second-order stimuli (averaged over carrier types); ○, data for first-order stimuli. The curves show fits from the contrast-gain control model. The horizontal lines indicate a perceived phase of zero (where the eyes are contributing equally to the percept). Error bars represent standard errors (most are smaller than the marker size).
Figure 3
 
Binocularly perceived phase as a function of interocular signal ratio for second-order and first-order stimuli. Results from 14 amblyopes and their average are shown in separate parts of the figure. The vertical axes represent perceived phase of the cyclopean grating; the horizontal axes represent interocular modulation ratio (nonamblyopic eye/amblyopic eye); •, data for second-order stimuli (averaged over carrier types); ○, data for first-order stimuli. The curves show fits from the contrast-gain control model. The horizontal lines indicate a perceived phase of zero (where the eyes are contributing equally to the percept). Error bars represent standard errors (most are smaller than the marker size).
According to the F-test for nested models, the full model (with different balance points in first- and second-order binocular combinations) failed to generate better fits (P > 0.05) than the reduced model (with the same balance points in first- and second-order binocular combinations), and the latter successfully accounted for 92.5% to 99.3% of the variance for observers A4, A8, A9, S10 through S14, and the average observer. These results indicate that the balance points in first- and second-order binocular combination were not significantly different in these observers. For the other six observers (A1–A3 and A5–A7), the balance point of the second-order PvR was less than that of the first-order PvR. The F-test for nested models also showed that the full model generated better fitting (P < 0.05) than the reduced model and the former successfully accounted for 95.9% to 99.0% of the variance for these six observers. 
Finally, we assessed whether this dichotomy in response can be attributed to either the present clinical characteristics of the patients or their past clinical history. Figure 4 plots the two categories of response (i.e., second- to first-order balance point ratio = 1 or < 1) across the relevant clinical dimensions of strabismus, acuity deficit, stereopsis, and treatment histories. The patients with additional second-order deficits tended to have less severe acuity loss and better stereopsis and not to have strabismus. 
Figure 4
 
The relationship between the first- and second-order binocular imbalances plotted across the clinical dimensions of acuity deficit, strabismus, stereopsis, and patching treatment. Here, no stereopsis means stereo acuity ≥ 800s, and severe amblyopia means visual acuity of the amblyopia eye ≥ 0.5 (in logMAR). BP2 refers to the balance point of the second-order PvR, and BP1 refers to the balance point of the first-order PvR.
Figure 4
 
The relationship between the first- and second-order binocular imbalances plotted across the clinical dimensions of acuity deficit, strabismus, stereopsis, and patching treatment. Here, no stereopsis means stereo acuity ≥ 800s, and severe amblyopia means visual acuity of the amblyopia eye ≥ 0.5 (in logMAR). BP2 refers to the balance point of the second-order PvR, and BP1 refers to the balance point of the first-order PvR.
Discussion
In this study, we set out to assess the deficit in the binocular combination of second-order signals in amblyopia. We found that the perceived phase of binocularly combined second-order signals was dependent on the interocular modulation ratio; that it did not matter whether the carriers in the two eyes were correlated, anticorrelated, or uncorrelated; and that the zero-crossing point of the PvR curve (i.e., the balance point) was less than 1 for all the amblyopic observers, requiring a greater modulation in the amblyopic than the nonamblyopic eye to achieve a balanced percept. In fitting the PvR functions using the contrast gain-control model,17,37 we quantified the difference in sensory eye dominance between the binocular combination of second-order and first-order gratings. The fits showed that the sensory imbalance in these two conditions were consistent in eight observers but different in six other observers, suggesting an additional deficit for second-order combination. 
The similarity of the PvR curves for the three dichoptic carrier types indicates that the second-order signal must be extracted before the binocular combination; otherwise, the binocular combination of carriers (by either linear summation or correlation) would produce quite different results in the three dichoptic carrier types. This result is consistent with previous studies in normal adults for measuring binocular combination of second-order phase,36 second-order stereopsis,43 and second-order modulation depth summation at44 and above37 threshold. Our current findings in amblyopes together with previous reports in normals suggest that the normal architecture of binocular combination for second-order signals is intact in amblyopia, as has previously been shown for first-order signals.18,45 
For the first-order condition, it is well documented that binocular combination of first-order gratings is contrast-gain controlled.9,11,17,19,35,4648 Recently, we have shown that second-order binocular combination could also be explained by the contrast-gain control theory.37 Thus for normal adults, since the carrier contrasts are the same in the two eyes, they will have equal weight in the binocular summation of second-order signals.36,37 For amblyopes, however, owing to attenuation/suppression in binocular viewing,9 the carrier contrasts in the amblyopic eye will be less effective17 and therefore would be expected to be given less weight in the binocular sum. 
This speculation is consistent with the sensory imbalance we found in binocular combination of second-order binocular gratings in all our amblyopes. However, we should note that 8 of the 15 observers had similar balance points in both the first- and second-order conditions while the other 6 observers had second-order sensory imbalances greater than those found for first order. Theoretically, the imbalance in second-order processing might arise in one of two ways: in the carrier contrast-related processing, that is, the contrast-gain control processing, or in the modulation depth-related processing. According to our current understanding of binocular combination,37 second-order binocular combination has a similar contrast-gain control to that of first-order combination. Thus the additional second-order deficits we found might be modulation depth related. In particular, it probably occurred at either the modulation depth binocular summation stage or in the early second-order envelope extracting stage. Since the second-order gratings we used here had a low spatial frequency (i.e., 0.29 cyc/deg) where the amblyopic eye's modulation sensitivity is not affected,49 the former explanation is more likely. 
We have shown that some amblyopes do exhibit an additional binocular deficit for second-order stimuli. At this stage we do not know if this occurs in the majority of cases or whether it is seen only in a minority of cases. The only clinical factor that bore a relationship with our results was that all those who exhibited additional second-order deficits were anisometropic, rather than strabismic. As a consequence they also had, on average, milder acuity loss and better stereopsis. However, our sample size was small, and it was biased toward anisometropes (70%). Another reason why our sample was biased toward milder cases of amblyopia was that more severe cases had suppression that was sufficiently large10 to limit our testing of the binocular balance for second-order stimuli. A larger-sample study needs to be undertaken using a more abbreviated approach to identify the incidence and clinical profile of amblyopes with this additional deficit involving the integration of second-order stimuli. Nevertheless, our results suggest that at the very least in some cases of mild anisometropic amblyopia, a binocularly based therapy15 might need to involve second-order as well as first-order stimuli to be truly effective. 
In conclusion, our results indicate that amblyopia does not disrupt the normal architecture of binocular combination for second-order signals; however, there are additional deficits for the binocular combination of second-order image features in some amblyopes (6/14) that cannot be fully accounted for by their first-order sensory imbalance. These probably occur at the binocular modulation depth summation stage. 
Acknowledgments
The authors thank Alex Baldwin for his very kind help in correcting our English. 
Supported by National Natural Science Foundation of China Grant NSFC 81500754 (JZ), Canadian Institutes of Health Research Grants MOP-53346, CCI-125686, and MT-10818 (RFH), and National Natural Science Foundation of China Grants NSFC 81261120562 (YZ) and NSFC 81300796 (LF). The funding organizations had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. 
Disclosure: J. Zhou, None; R. Liu, None; L. Feng, None; Y. Zhou, None; R.F. Hess, None 
Appendix: Stimuli and Curve Fits
Stimuli
The sine-wave gratings had a period of 2 cycles (180 pixels), which subtended 6.8° of visual angle (i.e., 0.29 cyc/deg). A high-contrast frame (width 0.378°; length 20.43°) with four white diagonal lines (width 0.378°; length 9.63°) was presented surrounding the grating in each eye to help observers maintain fusion. The reference lines used to measure the perceived phase were 1 pixel tall and were presented horizontally at the two sides of the gratings. 
For each second-order dichoptic pair, the gratings in the two eyes were defined as:   where L0 is the background luminance; g1(y) and g2(y) are the two-dimensional binary noise carriers in the two eyes; for the dichoptic pair with correlated carriers (Fig. 1A), g1(y) = g2(y); with anticorrelated carriers (Fig. 1B), g1(y) = −g2(y); with uncorrelated carriers (Fig. 1C), g1(y) ≠ g2(y); Cg = 20% is the contrast of carrier; f = 0.29 cyc/deg is the spatial frequency of the envelope sine-wave gratings; M0 = 100% is the base modulation depth in the amblyopic eye; and δ is the interocular modulation depth ratio, δ = [0, 0.1, 0.2, 0.4, 0.8, 1.0]. The two dichoptic gratings in the test had equal and opposite phase shifts of θ/2, where θ = 45°.  
For the first-order dichoptic pair (Fig. 1D), stimuli were defined as:   where C0 = 100% is the contrast in the amblyopic eye and f = 0.29 cyc/deg is the spatial frequency of the gratings. The other variables have the same roles as above.  
Curve Fits
The phase versus ratio (PvR) functions for different dichoptic pairs were fitted with a modified contrast-gain control model from Huang et al.17:  in which φ is the measured perceived phase when the interocular signal strength ratio is δ; θ is the interocular phase difference; and the two free parameters, bp and γ, represent the effective signal ratio at balance point (i.e., φ = 0°) and the nonlinear factor in the binocular combination, respectively. These two parameters jointly define the shape of the PvR function. Note that here we use a factor bp to replace the attenuation/suppression factor in Huang et al.,17 which is convenient in showing the zero-crossing point of the function (i.e., the balance point, the interocular signal ratio that results in the two eyes contributing equally to the cyclopean percept). Our model is, however, mathematically identical with theirs. According to our recent studies, the second-order stimuli in the two eyes are linearly summed when the carrier contrasts are the same in the two eyes.37 We thus set the nonlinearity as 0 (i.e., γ = 0) for second-order binocular combination.  
All fits were conducted in Matlab (MathWorks) using nonlinear least squares method to minimized ∑(φtheoryφobserved)2. The goodness-of-fit was tested by computing the r2 value:    
An F-test for nested models was used to compare the balance point between the first- and second-order binocular combinations for each observer. In particular, we compare the full model where the two PvRs have different balance points (bp1bp2) with the reduced model where the two PvRs have identical balance points (bp1 = bp2). For two nested models with kfull and kreduced parameters, the F statistic is defined as:  in which df1 = kfullkreduced, and df2 = Nkfull, for N data points. If these two models generate significantly different fits (i.e., P < 0.05), we choose the full model; otherwise, we take the prediction of the reduced model.  
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Figure 1
 
Dichoptic stimuli in the binocular phase combination test. Four dichoptic pairs were used: second-order gratings with correlated (A), anticorrelated (B), or uncorrelated (C) carriers in the two eyes and first-order gratings (D). For each dichoptic pair, gratings with a fixed modulation depth (contrast) of 100% and phase shift of 22.5° in one direction were presented to the amblyopic eye; and gratings with a signal strength ratio of δ (0, 0.1, 0.2, 0.4, 0.8, 1) and an opposite phase shift by the same magnitude were presented to the nonamblyopic eye.
Figure 1
 
Dichoptic stimuli in the binocular phase combination test. Four dichoptic pairs were used: second-order gratings with correlated (A), anticorrelated (B), or uncorrelated (C) carriers in the two eyes and first-order gratings (D). For each dichoptic pair, gratings with a fixed modulation depth (contrast) of 100% and phase shift of 22.5° in one direction were presented to the amblyopic eye; and gratings with a signal strength ratio of δ (0, 0.1, 0.2, 0.4, 0.8, 1) and an opposite phase shift by the same magnitude were presented to the nonamblyopic eye.
Figure 2
 
Binocularly perceived phase as a function of interocular modulation ratio for the three dichoptic carrier types. Results from 14 amblyopes and their average are shown in separate parts of the figure. The vertical axes represent perceived phase of the cyclopean grating; the horizontal axes represent interocular modulation ratio (nonamblyopic eye/amblyopic eye); the symbols represent data for the three carrier types: ○, second-order stimuli with correlated carriers in the two eyes; △, second-order stimuli with anticorrelated carriers, and □, second-order stimuli with uncorrelated carriers. The horizontal lines indicate a perceived phase of zero (where the eyes are contributing equally to the percept, i.e., the zero-crossing pint). Error bars represent standard errors.
Figure 2
 
Binocularly perceived phase as a function of interocular modulation ratio for the three dichoptic carrier types. Results from 14 amblyopes and their average are shown in separate parts of the figure. The vertical axes represent perceived phase of the cyclopean grating; the horizontal axes represent interocular modulation ratio (nonamblyopic eye/amblyopic eye); the symbols represent data for the three carrier types: ○, second-order stimuli with correlated carriers in the two eyes; △, second-order stimuli with anticorrelated carriers, and □, second-order stimuli with uncorrelated carriers. The horizontal lines indicate a perceived phase of zero (where the eyes are contributing equally to the percept, i.e., the zero-crossing pint). Error bars represent standard errors.
Figure 3
 
Binocularly perceived phase as a function of interocular signal ratio for second-order and first-order stimuli. Results from 14 amblyopes and their average are shown in separate parts of the figure. The vertical axes represent perceived phase of the cyclopean grating; the horizontal axes represent interocular modulation ratio (nonamblyopic eye/amblyopic eye); •, data for second-order stimuli (averaged over carrier types); ○, data for first-order stimuli. The curves show fits from the contrast-gain control model. The horizontal lines indicate a perceived phase of zero (where the eyes are contributing equally to the percept). Error bars represent standard errors (most are smaller than the marker size).
Figure 3
 
Binocularly perceived phase as a function of interocular signal ratio for second-order and first-order stimuli. Results from 14 amblyopes and their average are shown in separate parts of the figure. The vertical axes represent perceived phase of the cyclopean grating; the horizontal axes represent interocular modulation ratio (nonamblyopic eye/amblyopic eye); •, data for second-order stimuli (averaged over carrier types); ○, data for first-order stimuli. The curves show fits from the contrast-gain control model. The horizontal lines indicate a perceived phase of zero (where the eyes are contributing equally to the percept). Error bars represent standard errors (most are smaller than the marker size).
Figure 4
 
The relationship between the first- and second-order binocular imbalances plotted across the clinical dimensions of acuity deficit, strabismus, stereopsis, and patching treatment. Here, no stereopsis means stereo acuity ≥ 800s, and severe amblyopia means visual acuity of the amblyopia eye ≥ 0.5 (in logMAR). BP2 refers to the balance point of the second-order PvR, and BP1 refers to the balance point of the first-order PvR.
Figure 4
 
The relationship between the first- and second-order binocular imbalances plotted across the clinical dimensions of acuity deficit, strabismus, stereopsis, and patching treatment. Here, no stereopsis means stereo acuity ≥ 800s, and severe amblyopia means visual acuity of the amblyopia eye ≥ 0.5 (in logMAR). BP2 refers to the balance point of the second-order PvR, and BP1 refers to the balance point of the first-order PvR.
Table
 
Clinical Details of the Participants
Table
 
Clinical Details of the Participants
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