The perceived binocular visual direction of a fused disparity stimulus with an interocular contrast difference is biased toward the direction signaled by the eye presented with the higher contrast image (J. S. Mansfield & G. E. Legge, 1996). Does the amplitude of binocular saccadic eye movements have a similar bias? We examined saccades to fused disparate Gabor patches with interocular contrast differences. The effect of these contrast differences on saccadic amplitudes was compared to the perceptual biases in binocular direction obtained in a vernier acuity task. Saccades to unequal contrast targets landed between the end points for equal contrast and monocular targets. For three of our eight subjects, the saccadic bias equaled the perceptual effect. For the other subjects, however, saccades were affected to a lesser extent. Three models for binocular combination were used to evaluate these responses: A maximum-likelihood model failed to predict our results, whereas a model with contrast-dependent weighting of direction estimates by two monocular channels and a gain control model of binocular contrast summation gave a better approximation to our data. Both models showed that for the perceptual system, the influence of the eye that was presented with the higher contrast image was more dominant in the binocular combination than expected from the stimulus contrast ratio. The oculomotor system, however, was close to following linear summation.

*A*is the amplitude,

*f*the spatial frequency,

*θ*the orientation of the sinusoidal grating,

*σ*the standard deviation of the Gaussian envelope, and

*M*the mean luminance of the background. The contrast of the stimulus was defined by the ratio of the amplitude of the grating to the mean luminance level. The Gabor carrier was in cosine phase to center a dark bar in the Gaussian envelope. Binocular disparity was produced with position shifts of the entire Gabor patch.

_{ueq}= PSE

_{mon}), whereas a value of 0 would be obtained if unequal contrast targets were seen at the same position as equal contrast targets (PSE

_{ueq}= PSE

_{eq}). A CIB between 0 and 1 occurs if unequal contrast targets were perceived at a location in between the extreme positions of equal contrast and monocular targets (|PSE

_{eq}| < |PSE

_{ueq}| < |PSE

_{mon}|).

*t*tests), except for the right shifts of subject AS. For most subjects, the perceived directions of the monocular stimuli were biased toward the center point between the two monocular stimulus positions. Additionally, two subjects showed large biases that could not be explained by their ocular dominance: For AS, most PSEs (including 100:0) were shifted to the left. Subject UN only exhibited an obvious bias for the equal contrast condition. The thresholds were relatively high in all conditions and for all subjects (just noticeable differences between 0.15° and 0.56°).

*SE*) for perceived left shifts and 0.70 ± 0.05 for perceived right shifts. The difference between these means was not significant (

*t*test:

*p*= .44; Wilcoxon rank sum test for medians:

*p*= .5).

Subject | Left shift (40:100) | Right shift (100:40) |
---|---|---|

AS | 0.58 | 0.88* |

JG | 0.49 | 0.45 |

JW | 0.83 | 0.70 |

LC | 0.50 | 0.75 |

NJ | 0.47 | 0.64 |

PB | 0.68 | 0.76 |

TH | 0.82 | 0.87 |

UN | 0.82 | 0.62 |

*p*< .05 for all tests. Differences in means were analyzed using

*t*tests for independent samples when we could verify that the underlying data were described by a normal distribution. Normality was tested using the Kolmogorov–Smirnov and Lilliefors test with further inspection of normal probability plots. If normality was in doubt, additionally differences in medians were tested with the Wilcoxon rank sum test.

*F*tests were used to determine the significantly better fit. In most cases, a single Gaussian was sufficient to describe the underlying distribution ( Figure 7a). For some data sets, however, the CIB histograms showed bimodality with one distribution consisting of suppression trials that clustered around 1 and another distribution with lower mean ( Figure 7b). Trials in the suppression distribution were rejected. In addition, all unimodal distributions with a CIB above 0.9 were excluded from further data analysis because they most likely were composed of suppression responses. A limit of 0.9 was chosen because perceptual CIBs never exceeded that value. Ideally, suppression distributions should have a mean of 1, but it could also be higher or lower due to variance.

*t*tests). Thus, it can be ruled out that interocular contrast differences influenced the oculomotor behavior in the absence of disparities.

*t*tests) for almost all data sets of all subjects. The standard deviations of saccadic end points were on average 0.45°, varying between subjects but not systematically between conditions. For a more detailed analysis, the raw amplitude values were normalized by calculating the saccadic CIB ( Equation 3). Mean saccadic CIBs, after exclusion of suppression trials (see curve fitting section), were compared between the four stimulus conditions (short- vs. long-jump, leftward vs. rightward saccades) using the pooled data of all subjects. Because no significant differences between these conditions were found, the data were combined ( Figure 8). The dotted lines mark the hypothetical CIB limits of 0 (equal contrast position) and 1 (monocular suppression). A substantial fraction of trials lies outside this range due to variance, but the peak of the distribution is centered between 0 and 1. This means that saccadic amplitudes were influenced by the interocular contrast difference. The mean CIB values (±

*SD*) for each subject are shown in Figure 9 together with the means for the equal and monocular contrast condition, which were always 0 and 1 due to the normalization process. Differences between the means for equal and unequal contrast condition and between the means for monocular and unequal contrast condition were always significant. The CIB varied between the subjects.

*t*test, sign test for nonnormally distributed data sets). This was, however, only true for the data after the exclusion of suppression trials (blue bars). Because suppression can only be detected for the saccade data, a difference between perceptual and saccadic CIBs could result from an unbalanced exclusion of suppression trials. Because of this, data containing suppression trials are plotted for comparison (blue and red bars together). The group mean for data including suppression trials is not different from 100%. However, three individual subjects still show a significantly smaller saccadic than perceptual CIB, including JW, who did not exhibit suppression at all. Thus, the relation between perceptual and saccadic CIB seemed to be different for individual subjects whether suppression was accounted for.

*w*

_{L}and

*w*

_{R}are the left (L) and right (R) eye weights that are normalized to sum up to 1. The weights represent reliabilities and are dependent on the variances (standard deviations

*σ*) of the single cue estimates as defined by:

*w*

_{L}+

*w*

_{R}= 1 into Equation 4 to achieve

*C*

_{L}

^{n},

*C*

_{R}

^{n}) of left and right eyes' images and then summed to obtain the estimate of the binocular visual direction (

*C*

_{L}

^{ n}and

*C*

_{R}

^{ n}are required to sum up to 1 and are defined by the following equations:

*C*

_{L}and

*C*

_{R}are the absolute contrasts of the left and right eyes' images. Because the contrasts of both images are scaled by the same factor, the contrast ratio (

*C*

_{L}/

*C*

_{R}) stays constant. For contrasts of 40% and 100%, the predicted weights were approximately 0.286 and 0.714 and (unlike the weights for MLE) were constant across subjects. The same data as for Model 1 were used (because Equation 7 is analogous to Equation 4), where the observed weights were derived from Equation 6.

*linear*models that could not sufficiently predict our results, we also applied our data to a recently proposed nonlinear binocular summation model (Ding & Sperling, 2006). Models 1 and 2 failed to describe the data by either under- or overestimating the influence of the eye presented with the higher contrast image. In the third model, the strength of the influence of the eye seeing the higher contrast image is an adjustable parameter (

*γ*). We determined this parameter

*γ*from our observed data and compared the results for vernier and saccade task.

*positions*(phases) of the two eyes' images were weighted by their respective contrasts and then the left and right eyes' position estimates were summed ( Figure 12a). In Model 3, the

*amplitude*of each eye's Gabor sine wave carrier is weighted by its contrast ( Figure 13a). Then, the weighted sine waves of left and right eyes are summed, which results in another sine wave with an intermediate amplitude and phase (position). Assuming that the carrier frequency is the same for left and right eyes, the phase of the combined sine wave is given by

*A*and

*B*are the amplitudes and

*θ*

_{ A},

*θ*

_{ B}, and

*θ*

_{ C}are the phases of the sine waves. For example, when sine waves

*A*and

*B*have phases of 0° and 90° and equal contrast, their combined phase (

*θ*

_{ C}) would be 45°. For the same phases of 0° and 90°, but unequal contrasts of 40% for sine wave

*A*and 100% for sine wave

*B,*the phase of the combined sine wave would be approximately 68°. This illustrates that weighting each eye's sine wave amplitude will bias the phase of the combined sine wave resulting from linear summation. Hence, in contrast to the monocular combination model (Model 2), here the phase of the fused percept is derived in a binocular channel that sums the two weighted sine waves.

*γ*) that causes the model to be nonlinear. The model is described by the following equation:

*θ*the real phase difference between left and right eyes' sine waves,

*δ*the contrast ratio (0 ≤

*δ*≤ 1), and

*γ*a subject-dependent parameter. The term

*γ*+ 1 scales the contrast ratio. If

*γ*= 0, the model reduces to the simple linear summation of two sine waves described in Equation 9. As

*γ*increases (

*γ*> 0), the eye seeing the higher contrast image gains more and more influence in the binocular combination process, i.e., the contrast ratio becomes exaggerated. If

*γ*decreases below 0, the eye seeing the lower contrast image gains more and more weight in the binocular combination process (the contrast-induced perceived phase shift decreases), until both eyes' images are weighted equally (

*γ*= −1). For further details of the model, see Ding and Sperling.

*θ*) originated from the disparity (2.5° of visual angle = 153° of phase angle). However, because the perceived separation between left and right eyes' images was smaller than the physical separation ( Experiment 1, Figure 3), we used the perceived separation obtained from the vernier data as input value for

*θ*for each subject. In doing so, we have to assume that the perceived phase of the stimuli changed without being accompanied by a change in perceived spatial frequency.

*γ*was determined graphically: Model predictions were plotted (

*δ*) for a substantial number of

*γ*values together with our observed data. Each

*γ*plot resulted in a separate prediction curve. The

*γ*for a particular data point was determined by finding the curve, on which that data point fell. This curve corresponded to a certain

*γ*value.

*γ*are plotted for each subject in Figure 13b. For the vernier data ( Figure 13b, top), the

*γ*values are positive or close to 0 (median: 0.27). This means that the subjects either linearly summed the two sine waves (

*γ*= 0) or scaled the contrast ratio so that it became larger (

*γ*> 0). The latter means that the eye presented with the higher contrast image dominated the binocular combination more than expected simply from the contrast ratio. A

*γ*of .27 indicates that the physical contrast ratio of 40:100 was treated by the system as a ratio of approximately 31:100.

*γ*values were negative. However, subject AS had a much higher

*γ*than all other subjects. Most likely, the data analysis did not exclude all of his suppression trials so that his data were still confounded by suppression. To avoid overemphasis of AS' anomalous behavior, we used median values. The group average resulted in a value close to 0. This was true for data including (blue bars, median: −0.11) and excluding suppression trials (red bars, median: −0.13). The negative

*γ*value indicates that the system effectively treats the physical contrast ratio (40:100) as a contrast ratio of approximately 44:100. Hence, the results resemble the differences between perception and saccades found with the monocular combination model (although differences in medians did not reach significance due to the small sample size).

*γ*close to zero). For perception, however, the binocular combination was on average dominated more by the eye presented with the higher contrast than expected from the contrast ratio (

*γ*= .27). Thus, differential contrast scaling is required in the two systems. Psychophysical measurements revealed that some scaling for contrast perception takes place in the visual system. This results in a reduction of apparent contrast with respect to the physically presented contrast (Gottesman, Rubin, & Legge, 1981; Kulikowski, 1976). The contrast threshold determines the difference between perceived and physical contrast (Kulikowski, 1976). However, because thresholds for high contrast stimuli similar to those used in the present study increase approximately linearly with pedestal contrast (Bird, Henning, & Wichmann, 2002), this effect cannot account for the scaling required for the model.