Using a suprathreshold binocular summation paradigm developed by J. Ding and G. Sperling (2006, 2007) for normal observers, we investigated suprathreshold cyclopean perception in anisometropic amblyopia. In this paradigm, two suprathreshold sinewave gratings of the same spatial frequency but different spatial phases are presented to the left and right eyes of the observer. The perceived phase of the binocularly combined cyclopean image is measured as a function of the contrast ratio between the images in the two eyes. We found that both eyes contributed equally in normal subjects. However, stimulus of equal contrast was weighted much less in the amblyopic eye relative to the fellow eye in binocular combination. For the five amblyopes, the effective contrast of the amblyopic eye in binocular combination is equal to about 11%–28% of the same contrast presented to the fellow eye, much less than the ratio of contrast sensitivity between the two eyes (0.73–1.42). The results from the current study have many important implications in amblyopia research and treatment.

Subjects | Age | Gender | Acuity (MAR) | Stereo (″) | Refraction | Type | |
---|---|---|---|---|---|---|---|

CCK | 22 | M | AE | 5.00 | 400 | +4.75/+0.75 × 80 | Aniso |

FE | 1.00 | +1.00 | |||||

JFM | 24 | M | AE | 5.00 | 400 | +7.00 | Aniso |

FE | 0.74 | +1.50 | |||||

LYM | 28 | M | AE | 2.00 | 800 | 1.5 | Aniso |

FE | 0.50 | +1.50/+1.00 × 95 | |||||

WF | 20 | M | AE | 2.00 | 140 | +2.75 | Aniso |

FE | 0.67 | Plano | |||||

WHB | 26 | M | AE | 1.63 | 800 | +1.00 | Aniso |

FE | 0.83 | −2.00 |

^{2}sinusoidal luminance modulations (“sine-wave gratings”) were used to measure contrast sensitivity functions.

^{1}The gratings were presented in the center of the display at a viewing distance of 2.28 m. To minimize edge effects, a 0.5 deg half-Gaussian ramp was added to each side of the stimulus to blend the stimuli to the background. Sine-wave gratings of different spatial frequencies were used to measure contrast thresholds in different conditions.

^{2}and viewed at a distance of 68 cm. The luminance profiles of the gratings to the left and right eyes can be described by the following equations:

*L*

_{0}= 31.2 cd/m

^{2}is the background luminance,

*C*

_{ L}and

*C*

_{ R}are the grating contrasts in the left and right eyes,

*θ*

_{ L}and

*θ*

_{ R}are the phases of the gratings in the left and right eyes, and

*f*= 0.68 c/d is the spatial frequency of the gratings. Each eye was shown exactly two cycles of the sine wave gratings.

^{2}), high-contrast frame with clearly marked white diagonals ( Figure 1). Both binocular fixation crosses (0.167 × 1.67 deg

^{2}) and monocular fixation dots (0.167 deg diameter) in the 1

^{st}and 3

^{rd}quadrants in the left eye and 2

^{nd}and 4

^{th}quadrants of the right eye were also used. Subjects were instructed to start a new trial only after they had achieved stable vergence.

*C*

_{ n+1}= 0.90

*C*

_{ n}) and one wrong response resulted in an increase in contrast (

*C*

_{ n+1}= 1.10

*C*

_{ n}), converging to a performance level of 79.3% correct (Levitt, 1971). One hundred trials were used to measure contrast threshold at each spatial frequency. A reversal results when the staircase changes from increasing to decreasing contrast or vice versa. Following the standard practice, we averaged the contrasts of an even number of reversals to estimate the contrast threshold after excluding the first three or four reversals.

*ϕ,*of the cyclopean sine wave grating as a function of the base contrast (

*C*

_{0}), the contrast ratio between the sine-wave gratings in the two eyes (

*δ*), the phase shift between the two monocular sine-wave gratings (

*θ*), and the dichoptic configuration. Four stimulus configurations were used ( Figure 2): (a)

*C*

_{ L}=

*C*

_{0},

*C*

_{ R}=

*δC*

_{0},

*θ*

_{ L}= −

*θ*

_{ R}=

*θ*/2, (b)

*C*

_{ R}=

*C*

_{0},

*C*

_{ L}=

*δC*

_{0},

*θ*

_{ L}= −

*θ*

_{ R}=

*θ*/2, (c)

*C*

_{ L}=

*C*

_{0},

*C*

_{ R}=

*δC*

_{0},

*θ*

_{ L}= −

*θ*

_{ R}= −

*θ*/2, (d)

*C*

_{ R}=

*C*

_{0},

*C*

_{ L}=

*δC*

_{0},

*θ*

_{ L}= −

*θ*

_{ R}= −

*θ*/2.

*C*

_{0}= 0.05, 0.10, 0.20 and 0.40), six contrast ratios (

*δ*= 0, 0.3, 0.5, 0.71, 0.86, 1.0), and three phase difference (

*θ*= 45°, 90° and 135°) were tested on one normal subject (ZJW). A subset of their conditions, three base contrasts (

*C*

_{0}= 0.16, 0.32 and 0.64), six contrast ratios (

*δ*= 0, 0.3, 0.5, 0.71, 0.86, 1.0), and three phases (

*θ*= 45°, 90° and 135°), was used for the other normal subject (CS). There were therefore 6 × 4 × 3 × 4 = 288 and 6 × 3 × 3 × 4 = 216 conditions for ZJW and CS, respectively.

*C*

_{0}= 0.16, 0.32 and 0.64), six contrast ratios (

*δ*= 0, 0.3, 0.5, 0.71, 0.86 and 1) and three phases (

*θ*= 45°, 90° and 135°) were tested. There were a total of 6 × 3 × 3 × 4 = 216 conditions. One of the amblyopic observers, CCK, was tested with a wider range of contrast ratios in an additional experiment.

*f*is the spatial frequency of the sine-wave grating,

*c*

_{t}is the predicted contrast sensitivity threshold, and

*a*

_{1},

*b*

_{1},

*c*

_{1},

*a*

_{2},

*b*

_{2}, and

*c*

_{2}are parameters.

*a*and

*c*[

*ϕ*

_{ A}=

*ϕ*

_{ a}−

*ϕ*

_{ c}], and configurations

*b*and

*d*[

*ϕ*

_{ F}=

*ϕ*

_{ b}−

*ϕ*

_{ d}] to construct two “phase versus contrast ratio (PvR)” functions for each base contrast (

*C*

_{0}) and phase shift (

*θ*) condition. Although the two sets of PvR functions are expected to be identical for normal observers,

^{2}it is necessary to keep them separate for amblyopic observers ( 2).

*C*

_{0}is presented to the left eye and a sine-wave grating with contrast

*δC*

_{0}is presented to the right eye (Figures 2a and 2c), the phase difference between the cyclopean images of configurations

*a*and

*c*is:

*C*

_{0}is presented to the right eye and a sine-wave grating with contrast

*δC*

_{0}is presented to the left eye (Figures 2b and 2d), the phase difference between the cyclopean images of configurations

*b*and

*d*is:

*γ*.

*η*< 1. When a sine-wave grating with base contrast

*C*

_{0}is presented to the amblyopic eye and a sine-wave grating with contrast

*δC*

_{0}is presented to the fellow eye ( Figures 2a and 2c), the phase difference between the cyclopean images of configurations

*a*and

*c*is:

*C*

_{0}is presented to the fellow eye and a sine-wave grating with contrast

*δC*

_{0}is presented to the amblyopic eye ( Figures 2b and 2d), the phase difference between the cyclopean images of configurations

*b*and

*d*is:

*γ*and

*η*. When the effective contrasts of the two eyes are equal (

*δ*=

*η*in Equation 4, or

*δ*= 1/

*η*in Equation 5), the perceived phase of the cyclopean image is equal to zero (

*ϕ*

_{ A}= 0;

*ϕ*

_{ F}= 0). We define

*η*as the effective contrast ratio of the amblyopic eye.

*C*

_{0}is presented in the amblyopic eye and a sine-wave grating with contrast

*δC*

_{0}is presented in fellow eye ( Figures 2a and 2c), the perceived phase shift of the cyclopean image is:

*C*

_{0}is presented to the fellow eye and a sine-wave grating with contrast

*δC*

_{0}is presented to the amblyopic eye ( Figures 2b and 2d), the perceived phase shift of the cyclopean image is:

*γ*was fit to the data of the normal observers. For the amblyopic observers, the attenuation model, with two parameters

*η*and,

*γ*was fit to the data to compute the effective contrast ratios in the amblyopic eye. A re-sampling procedure (Maloney, 1990) was used to estimate the standard deviations of the model parameters. The attenuation model and the Ding-Sperling model were compared statistically. In addition, the inhibition model and all its reduced forms (2) were also fit to the data. The full inhibition model and its reduced forms were compared statistically.

*fminsearch*in Matlab (Mathworks Inc.) that minimized ∑(

*ϕ*

_{theory}−

*ϕ*

_{observed})

^{2}. The goodness-of-fit was evaluated by the

*r*

^{2}statistic:

*F*-test for nested models was used to statistically compare the models. For two nested models with

*k*

_{full}and

*k*

_{reduced}parameters, the

*F*statistic is define as:

*df*

_{1}=

*k*

_{full}−

*k*

_{reduced}, and

*df*

_{2}=

*N*−

*k*

_{full};

*N*is the number of predicted data points.

*ϕ*

_{ A}and

*ϕ*

_{ F}) of the cyclopean images are plotted as functions of the contrast ratios between the two eyes. The two sets of PvR functions,

*ϕ*

_{ A}and

*ϕ*

_{ F}, are virtually identical, reflecting equal contributions of the two eyes in binocular combination.

*γ*accounted for 96.3% and 96.7% of the total variances for subjects ZJW and CS. The parameters of the best fitting model are listed in Table 2. The predictions of the best fitting model are plotted as smooth curves in Figure 4. The results are consistent with Ding and Sperling (2006, 2007), providing additional validation of their contrast-gain control model of binocular combination for normal observers. For these observers, the fits of the Ding-Sperling model are statistically equivalent to the attenuation model (

*p*> 0.30).

Normal | Amblyopic observers | |||||||
---|---|---|---|---|---|---|---|---|

CS | ZJW | CCK1 | JFM | LYM | WF | WHB | CCK2* | |

γ | 0.69 ± 0.03 | 1.58 ± 0.05 | 0.47 ± 0.04 | 1.46 ± 0.01 | 0.94 ± 0.01 | 0.59 ± 0.05 | 0.53 ± 0.01 | 0.63 ± 0.01 |

η | 1.00 | 1.00 | 0.28 ± 0.01 | 0.11 ± 0.01 | 0.21 ± 0.01 | 0.21 ± 0.01 | 0.25 ± 0.01 | 0.24 ± 0.01 |

r ^{2} | 0.963 | 0.967 | 0.971 | 0.993 | 0.979 | 0.996 | 0.961 | 0.989 |

F(1,105) | 1.95 | 1.20 | 2119.00 | 6203.00 | 4758.10 | 9637.00 | 2425.60 | 8276.20 |

*γ*and

*η*accounted for 97.1%, 99.3%, 97.9%, 99.6%, 96.1%, and 98.9% of the variance for CCK1, JFM, LYM, WF, WHB, and CCK2 respectively. For all amblyopic observers, the attenuation model provided significantly better fit to the data than the original Ding-Sperling model (all

*p*< 0.0001).

*p*< 0.01).

*α*=

*β,*provided statistically equivalent fits to the data as the full inhibition model ( Table 3). For all five observers, the model in which the fellow eye only exerts stronger gain on the gain-control in the fellow eye but not on the gain-control on the amblyopic eye (

*α*= 1) is inferior to the full inhibition model (all

*p*< 0.01). For observers JFM, LYM and WHB, the model in which the fellow eye only exerts stronger gain-control on the amblyopic eye but not on the gain of the gain-control in the fellow eye (

*β*= 1) is also inferior to the full inhibition model (all

*p*< 0.01). The overall pattern of results suggest that the model in which

*α*=

*β*is the best fitting inhibition model. This model accounts for 97.9%, 99.3%, 98.3%, 99.7%, 97.2%, and 99.2% of the variance for CCK1, JFM, LYM, WF, WHB, and CCK2, respectively.

CCK1 | JFM | LYM | WF | WHB | CCK2* | |
---|---|---|---|---|---|---|

γ | 0.73 | 1.46 | 1.03 | 0.59 | 0.33 | 0.415 |

α | 5.30 | 228.42 | 23.77 | 11.98 | 6.87 | 9.24 |

β | 5.30 | 228.42 | 23.77 | 11.98 | 6.87 | 9.24 |

r ^{2} | 0.979 | 0.993 | 0.983 | 0.997 | 0.972 | 0.992 |

F _{ α = β}(1,104) | 1.11 | 0.07 | 2.89 | 0.18 | 1.99 | 2.50 |

F _{ β = 1}(1,104) | 0.05 | 51.30 | 22.47 | 0.31 | 23.19 | 1.83 |

F _{ α = 1}(1,104) | 530.50 | 6626.80 | 2047.10 | 5847.40 | 685.05 | 2732.30 |

*α*=

*β*inhibition model are mathematically equivalent in predicting the perceived phase of the cyclopean image ( 2), we cannot distinguish the two potential mechanisms underlying abnormal binocular combination in amblyopia based on the data in this study. We can however treat the effective contrast ratio as an empirical measure of the imbalance of the contributions of the two eyes in binocular combination in amblyopic vision.

*α*=

*β*), is however mathematically equivalent to the attenuation model in determining the phase of the cyclopean images (2). Although we are aware that Baker et al. (2008) concluded that strabismic amblyopia attenuates the signal and increases internal noise in the amblyopic eye, we are at present uncommitted to either the attenuation or asymmetric inhibition theory in anisometropic amblyopia. We are conducting new studies to further test the two hypotheses.

_{L}) and right (Lum

_{R}) eye inputs defined by Equations 1 and 2 is:

*ɛ*

_{L}and

*ɛ*

_{R}are the contrast energies presented to the two eyes and can be simply expressed as

*ɛ*

_{L}=

*ρ*(C

_{L})

^{γ}and

*ɛ*

_{R}=

*ρ*(C

_{R})

^{γ},

*ρ*is the gain-control efficiency of the signal sine-wave grating;

*γ*is the exponent of the non-linearity. The contrast energies

*ɛ*

_{L}and

*ɛ*

_{R}are generally much greater than 1.0 even for the lowest contrast stimulus. We thus dropped the term “1” in Equation A1 in all the subsequent model development. For the four configurations illustrated in Figure 2, the phase shifts in the cyclopean images are:

*γ*. Strictly speaking,

*ϕ*

_{ A}and

*ϕ*

_{ F}are not the perceived phase in any one of the four configurations in Figure 2. Rather, they are the difference between the phases of the cyclopean images of each pair of configurations. They have the following properties:

- when the sine-wave in one eye is absent (
*δ*= 0), the perceived phase of the cyclopean image is equal to that of the sine-wave in the other eye (*ϕ*_{ A}=*θ*;*ϕ*_{ F}=*θ*); and - when the amplitudes of the sine-waves in the two eyes are equal (
*δ*= 1), the perceived phase of the cyclopean image is equal to zero (*ϕ*_{ A}= 0;*ϕ*_{ F}= 0).

*A*and

*F*stand for the amblyopic and fellow eyes, respectively. Again, we dropped the term “1” in Equation B1 in all the subsequent model development. We considered two ways to elaborate the Ding-Sperling model:

- Attenuation in the amblyopic eye ( Figure 3b): The effective contrast in the amblyopic eye is equal to its physical contrast multiplied by a factor
*η*< 1, i.e.,*ɛ*_{ A}=*ρ*(*ηC*_{ A})^{ γ},*ɛ*_{ F 1}=*ɛ*_{ F 2}=*ρC*_{ F}^{ γ}, and - Increased inhibition from the fellow eye ( Figure 3c): All the contrast-gain control terms originated from the fellow eye are multiplied by factors greater than 1 in Equation B1, i.e.,
*ɛ*_{ A}=*ρ*(*C*_{ A})^{ γ},*ɛ*_{ F 1}=*αρC*_{ F}^{ γ},*ɛ*_{ F 2}=*βρC*_{ F}^{ γ}, with*α*> 1 and/or*β*> 1.

*C*

_{ A}and

*C*

_{ F}represent image contrasts in the amblyopic and fellow eyes, respectively. We describe the two models in turn.

*C*

_{0}is presented to the amblyopic eye and a sine-wave grating with contrast

*δC*

_{0}is presented to the fellow eye (Figures 2a and 2c), the perceived phase shift of the cyclopean image is:

*C*

_{0}is presented to the fellow eye and a sine-wave grating with contrast

*δC*

_{0}is presented to the amblyopic eye (Figures 2b and 2d), the perceived phase shift of the cyclopean image is:

- when the sine-wave in one eye is absent (
*δ*= 0), the perceived phase of the cyclopean image is equal to that of the sine-wave in the other eye (*ϕ*_{ A}=*θ*;*ϕ*_{ F}=*θ*); and - when the effective contrasts of the two eyes are equal (
*δ*=*η*in Equation B4, or*δ*= 1/*η*in Equation B5), the perceived phase of the cyclopean image is equal to zero (*ϕ*_{ A}= 0;*ϕ*_{ F}= 0).

*γ*and

*η*.

*C*

_{0}is presented to the amblyopic eye and a sine-wave grating with contrast

*δC*

_{0}is presented to the fellow eye ( Figures 2a and 2c), the perceived phase shift of the cyclopean image is:

*C*

_{0}is presented to the fellow eye and a sine-wave grating with contrast

*δC*

_{0}is presented to the amblyopic eye ( Figures 2b and 2d), the perceived phase shift of the cyclopean image is:

*α, β,*and

*γ*. We next consider three special cases:

*ɛ*

_{ F 1}) is equal to the gain of the contrast gain in the fellow eye (

*ɛ*

_{ F 2}), or equivalently,

*α*=

*β*. From Equations B6 and B7, we have

*α*=

*η*

^{−(1+ γ)}, Equations B8 and B9 are equivalent to Equations B4 and B5. In other words, attenuating the contrast in the amblyopic eye is mathematically equivalent to increasing the inhibition from the fellow eye in determining the phase of the cyclopean images if the strengths of the increased inhibitions are the same in the two gain-control paths.

*ɛ*

_{ F 1}) on the amblyopic eye, not the gain of the gain-control in the fellow eye (

*ɛ*

_{ F 2}), i.e.,

*β*= 1. From Equations B6 and B7, we have:

*ɛ*

_{ F 2}), not gain-control on the amblyopic eye (

*ɛ*

_{ F 1}), i.e.,

*α*= 1. From Equations B6 and B7, we have:

^{1}Vertical gratings instead of horizontal gratings were used to measure contrast sensitivity functions in a separate project. We report the procedure and results here because contrast sensitivities to sinewave gratings in cardinal orientations are almost the same at most spatial frequencies in anisometropic amblyopia (Koskela & Hyvärinen, 1986).

^{2}For normal observers, Ding and Sperling (2006) obtained a single measure of the perceived phase of the cyclopean image by combining the results from all four configurations [

*ϕ*=