Subjective Fixation Disparity Affected by Dynamic Asymmetry, Resting Vergence, and Nonius Bias

Abstract

Purpose.: This study was undertaken to investigate how subjectively measured fixation disparity can be explained by (1) the convergent–divergent asymmetry of vergence dynamics (called dynamic asymmetry) for a disparity vergence step stimulus of 1° (60 arc min), (2) the dark vergence, and (3) the nonius bias.

Methods.: Fixation disparity, dark vergence, and nonius bias were measured subjectively using nonius lines. Dynamic vergence step responses (both convergent and divergent) were measured objectively.

Results.: In 20 subjects (mean age, 24.5 ± 4.3 years, visual acuity, ≥1.0; all emmetropic except for one with myopia, wearing contact lenses), multiple regression analyses showed that 39% of the variance in subjective fixation disparity was due to the characteristic factors of physiological vergence: dynamic asymmetry (calculated from convergent and divergent velocities), and dark vergence. An additional 23% of variance was due to the subjective nonius bias (i.e., the physical nonius offset required for perceived alignment of binocularly [nondichoptically] presented nonius lines). Together, these factors explained 62% of the interindividual differences in subjectively measured fixation disparity, demonstrating the influence of oculomotor and perceptual factors.

Conclusions.: Clinically relevant subjective fixation disparity originates from distinct physiological sources. Dynamic asymmetry in vergence dynamics, resting vergence, and nonius bias were found to affect fixation disparity directly, not only via changes in vergence dynamics.

For stable single vision, the fusional system keeps the images of the two eyes on corresponding points as close as possible. However, fusion is not always exact. A difference may occur between the physical position of the target and the point actually fixated by the eyes (i.e., the intersection point of the two visual axes during binocular fusion); this difference is called fixation disparity. The two visual axes may intersect in the plane of the fixation target or in front of or behind it; these conditions are known as zero, eso, or exo fixation disparity, respectively. Fixation disparity varies reliably among subjects with normal binocular vision and typically amounts to a few minutes of arc—most often less than 10 arc min and nearly always less than 25 arc min, ^1,2 when measured subjectively with dichoptic nonius lines. With such a small amount of fixation disparity, the image still lies within Panum's area. Thus, observers do not experience diplopia or changes in the quality of the image. ^1,2  

There are numerous disputes regarding the origin of fixation disparity. One classic concept maintains that fixation disparity may be a condition of stress on the vergence system, ² where larger amounts of fixation disparity or steeper forced fixation disparity curves indicate a less adaptive vergence system. ³ In feedback control theory–based models ^{2,4 –7} fixation disparity is a necessary error to stimulate the fusional vergence system. In the simplest formulation of proportional controllers or leaky integral controllers, ^4,7 including only disparity stimuli, fixation disparity (FD) is the difference between vergence stimulus (VS) and vergence response (VR): FD increases proportionally to VS (relative to V_bias, i.e., the resting vergence position) and depends on the gain factor G of the neural integrators. G describes the responsiveness to disparity stimuli and is represented by the vergence velocity:  

 

 

Control theory–based models incorporate just one direction of disparity vergence step stimuli (i.e., either convergent or divergent; relative to baseline). To provide predictions for each direction, the model must be applied separately with vergence gain factors that may be different for the two directions. ^8,9  

Patel et al. ^10,11 described a neural network model that directly incorporates two vergence directions: two opponent pathways for convergence and divergence with sensory motor gains G_con and G_div, respectively. Accordingly, fixation disparity is predicted to be proportional to the asymmetry in convergent and divergent dynamic responsiveness ^{10 –12} :  

 

 

The gain factors were represented by peak vergence velocities: G_con = V_con and G_div = V_div. Thus, an eso (exo) fixation disparity results if the convergent velocity is larger (smaller) than the divergent velocity. ¹¹ In cases with zero fixation disparity, the gain of dynamic vergence responses in both directions should be equal. Patel et al. ¹¹ provided evidence for this prediction from an intraindividual approach. A linear relation between fixation disparity and dynamic asymmetry was found in each of five observers when the load on the vergence system was increased by crossed disparity. 

Since Jones, ¹³ Fredenburg and Harwerth ¹² have reported individual differences in dynamic asymmetry, Jaschinski et al. ¹⁴ used an interindividual approach to predict fixation disparity from dynamic asymmetry (as suggested by Patel et al. ¹¹ ) in a group of 16 subjects. They explained approximately 50% of the variance in subjectively measured fixation disparity at a viewing distance of 60 cm with a 1° disparity step stimulus. The dynamic asymmetry explained a larger proportion of variance in fixation disparity than convergence (approximately 30%) or divergence (approximately 20%) velocity alone. 

To further explain interindividual variability in fixation disparity, the following additional factors could be relevant:

1.  
   Resting vergence can be measured in two ways: as either tonic vergence or heterophoria. Tonic vergence is a position of vergence without any stimulus for vergence or accommodation; if tested in the darkness, it is referred to as dark vergence. Francis and Owens ¹⁵ and Jaschinski ^16,17 observed that fixation disparity is correlated with dark vergence. Fixation disparity is small at a viewing distance corresponding to dark vergence. ^{15 –17} Heterophoria is a resting vergence state without a fusion stimulus (open loop condition) but including an accommodative stimulus. Heterophoria is often reported as being related to fixation disparity. ^{18 –20} Studies differ in reports of the strength of this correlation, which may depend on viewing distance (Jampolsky et al. ²⁰ ). There are also cases with different directions of heterophoria and fixation disparity (see Palmer and van Noorden, ²¹ Jampolsky et al., ²⁰ and Ogle ¹⁹ ). However, authors agree that large amounts of fixation disparity are associated with large amounts of horizontal heterophoria. This correlation was also confirmed by objective measures. ²²
2.  
   Fixation disparity also correlates with binocular nonius bias. ²³ The nonius bias is the physical offset of the nonius lines that is adjusted by the observer to perceive them as aligned when both nonius lines are presented to both eyes (binocular nonius bias). The nonius bias increases with the vertical offset of the nonius lines and results from spatial irregularities in optical or retinal structures. ²³

There is no research that simultaneously addresses all the following factors: (1) the dynamic asymmetry of vergence response, (2) resting vergence (tonic vergence or heterophoria), and (3) nonius bias. Therefore, it was the purpose of the present study to investigate the extent to which the interindividual variability of subjective fixation disparity can be explained by these three factors. 

Methods

Subjects

The sample of 20 subjects had an age of 24.5 ± 4.3 years (mean ± SD; range 16 to 34 years). Subjects were screened to include those with normal monocular and binocular vision with a minimal visual acuity of 1.0 (in decimal units) in each eye at a distance (5 m) and at the test viewing distance (60 cm). The subjects were emmetropic (both sphere and cylinder in a range of ±0.5 D), except for one subject (−2.0 D myopia in both eyes), who wore contact lenses during testing. All subjects had binocular single vision and good stereovision at 5 m (Polatest; Carl Zeiss Meditec, Inc., Oberkochen, Germany) and 40 cm (TNO; Lameris Ootech, Nieuwegein, The Netherlands), in both the crossed (67.5 ± 44.5 arc sec) and uncrossed (82.5 ± 49.5 arc sec) direction. The experiments were undertaken with the written consent of each subject. The procedures of the present study were approved by the ethics review board of the Institut für Arbeitsforschung and complied with the Declaration of Helsinki. 

Stimulus and Apparatus

In a mirror stereoscope (60-cm viewing distance, mean 5.7° absolute vergence angle), we measured fixation disparity and nonius bias subjectively with a nonius technique presenting targets on two LCD screens (one for each eye). ^14,24 Dynamic vergence step responses (both convergent and divergent) and heterophoria (during monocular calibration before and after each recording) were measured objectively using an eye-tracking system (EyeLink II; SR Research; Mississauga, Ontario, Canada), at 500 Hz, in the pupillary tracking mode. ^22,24 The vergence stimuli were within the 40° horizontal gaze-tracking range. Despite the high physical precision of the system, its practical reliability is limited by the stability of head position: test–retest correlations of vergence velocity among different sessions were between 0.62 and 0.91; regarding the difference between velocity measures between two sessions, the range of ±1.96 SD was ±2.23 deg/s and ±1.02 deg/s for the convergent and divergent velocity, respectively, as found in Bland-Altman plots (not shown). Thus, individual differences in vergence velocity could be measured reliably. 

The whole experiment was run in a separate room with dim lighting. Because of the mirrors placed close to the eyes, the edges of the monitors were not visible to the subject. Thus, there were no direct fusion targets helping the subject to fuse targets other than the stimuli generated on the screens. 

In measurements of subjective fixation disparity, the stationary fusion target (black on a white background with a luminance of approximately 8 cd/m²) contained a frame (300 arc min width × 230 arc min height; 12 arc min stroke width) with a central fixation cross (30 × 30 arc min; stroke width 6 arc min). Monocular nonius lines (45 arc min long; 8 arc min stroke width; vertical separation of 50 arc min) for the right and left eye were flashed for 100 ms inside the frame, above and below the fixation cross, respectively. 

We determined the nonius offset d required for subjective alignment, which allows calculation of fixation disparity (FD):  

 

with the individual interpupillary distance PD and the viewing distance s (0.6 m). To determine d, the adaptive psychometric procedure Best-PEST ²⁵ was used. The nonius lines were flashed 20 times for 100 ms at 3-second intervals, with varying amounts of nonius offset, whereas the subjects responded as to whether the upper nonius line was perceived left or right relative to the lower line. The 20 trials of measuring fixation disparity were randomly interleaved by 20 trials in which the dichoptic separation of the nonius lines was not active (i.e., both eyes viewed the upper and lower nonius lines). This is not a measure of vergence but rather a measure of the nonius bias. ²² The run with all 40 trials took approximately 2 minutes. 

For objective measurements, a nine-point monocular, purpose-made calibration was performed. ²⁴ A bite bar was not used, but the subject's head was stabilized with a chin and forehead rest, pads for the cheeks, and a headband. The same target (as for fixation disparity measurement) was presented as a disparity step stimulus of 1° = 60 arc min, convergent or divergent relative to the absolute vergence angle of approximately 5.7° (in a range of 5.2° to 6.15° for the range of interpupillary distances of 54.5–64.5 mm). Disparity was introduced by relative lateral displacement of the images for the left and right eyes. The vergence state reached at certain moments in time during the response to the disparity stimulus was simultaneously estimated subjectively with dichoptic nonius lines: the nonius lines were flashed for 100 ms at a defined delay after the onset of the disparity step stimulus (Fig. 1): 0, 100, 200, 300, 400 ms, relative to the onset of the step stimulus. This allows for a subjective estimation of velocity; the present report, however, presents only the more precise objective velocity measures. Separate runs were made with each of the various length of nonius delay. ²⁴ One run for vergence response (in both convergence and divergence directions) measurement took approximately 5 minutes, for each length of nonius delay. 

Figure 1.

 

(A) Time scheme of a single trial showing a convergent disparity step stimulus of 1° and the time points when the nonius lines were presented. (B) Sequence of one convergent and one divergent disparity step stimulus.

One run comprised 20 convergent and 20 divergent step stimuli (randomly interleaved) presented for 2 seconds, after which a zero disparity was presented again so that the vergence response returned to a baseline vergence (BV) state. This vergence state sometimes differed from the vergence stimulus angle of 5.7° because of the fixation disparity between the step responses. The BV state could be used to calculate the magnitudes of the effective disparity stimuli, which were 1° − BV and 1° + BV for convergent and divergent stimuli, respectively. Previous studies ^11,14 considered this effective disparity stimulus when calculating dynamic asymmetry, by weighting the gain factors: G _con = V _con/(1° − BV) and G _div = V _div/(1° + BV). This, however, tends to artificially increase the correlation between fixation disparity and dynamic asymmetry (equation 2), because fixation disparity itself is correlated with baseline vergence. Therefore, in the present study, we simply assumed G _con = V _con and G _div = V _div, which leads to a more conservative estimation of the correlation of fixation disparity with dynamic asymmetry, because dynamic asymmetry tends to be underestimated. 

Dark Vergence Measurements

Dark vergence was measured with an adaptive psychometric procedure, Best-PEST ²⁵ (similarly to fixation disparity). A small red square (17 arc min width × 17 arc min height; seen with the right eye) and a red line (155 arc min length; 3 arc min stroke width; seen with the left eye) were flashed for 100 ms on a dark screen at a 100-cm viewing distance in a completely dark room. Testing was started after brief adaptation with the lighting switched off in a windowless room. The average of two measurements was taken for further analyses (in meter angles). The tests took approximately 5 minutes. 

Velocity Calculations

We calculated velocity profiles for each epoch using a two-point central difference algorithm ²⁶ incorporating a central difference of ±1 sampling interval of 4 ms. After 10 Hz low-pass filtering, the smoothed data were scanned for the maximum objective velocity within a time interval of 100 to 500 ms after the onset of the disparity vergence step stimulus. The average of all maximum velocities was then calculated from all useful epochs. 

Calculation of Heterophoria

During monocular calibrations, one eye fixated the calibration target, and the other eye was not provided with a target. Therefore, the binocular recordings showed a vergence angle without a fusion stimulus, a condition known as heterophoria. ²² Two heterophoria measurements were taken from each central calibration point, one while each eye was fixating. Because of the high correlation of both measures (r = 0.99), they were averaged. Individual heterophoria was described as exophoria (uncrossed visual axes; minus sign), esophoria (crossed visual axes; plus sign) or orthophoria (visual axes intersect perfectly at the visual target; zero heterophoria). Therefore, heterophoria was obtained at the same viewing distance (60 cm) and in the same experimental conditions as vergence velocity and fixation disparity. 

Experimental Design

All subjects participated in three repeated sessions on separate days. Each session comprised vergence disparity step response measurements for five lengths of nonius delays, including both convergent and divergent directions and three fixation disparity measurements. The session always started with a vergence response measurement (randomly varied amounts of nonius delay). Fixation disparity was measured at the end of each session after a rest of approximately 10 to 15 minutes. Each session took approximately 1.5 hours, including rests. Periods of near vision did not occur and thus did not induce vergence adaptation. 

Results

There were nonrandom, systematic changes in dynamic asymmetry (A) from session to session, shown by a correlation of differences between sessions in the series of 3 days (A1, A2, A3), since the correlation between (A3 – A1) and (A2 – A1) was significant (r = 0.69; P < 0.001, n = 20). This is a partial correlation analysis made to remove the common variance due to A1 included in these two differences. Thus, these changes in dynamic asymmetry were reliably observed in sessions 2 and 3, however, they were partly in opposite directions (positive or negative) in different observers. This finding is in conflict with the purpose of the study in which we sought to explain interindividual differences in subjective fixation disparity as clinically measured with short test procedures not affecting the measured value itself. Therefore, the present analyses used data from session 1 reflecting the natural vergence state, which can be assumed not (or only minimally) to be affected by training effects of the test procedure itself. 

In a first approach, we tested whether interindividual variability in subjective fixation disparity could be explained by dynamic asymmetry, dark vergence (quantified as a resting vergence measurement), and nonius bias. Analysis showed that the data of one subject (S11) violated the normal distribution for nonius bias (7 arc min). This was not a measurement error, because repeated measurements yielded similar results. For statistical analyses, we had to exclude this subject, because inclusion would have violated the assumption of normal distribution. Thus, we included 19 subjects in the subsequent analyses. 

The relation of all three factors with subjective fixation disparity was analyzed in two steps. First, we tested the relation separately for each factor (Fig. 2, Table 1). The correlations ranged from 0.41 to 0.69 and were statistically significant (P < 0.05; n = 19). The strongest correlation was observed for nonius bias. Fixation disparity was not significantly correlated with each velocity alone (r = 0.29 for convergence and r = 0.16 for divergence, respectively, n = 19), rather the balance between both was relevant. 

Figure 2.

 

Correlation between subjective fixation disparity and (A) dynamic asymmetry for velocities of dynamic convergence and divergence step responses, (B) dark vergence, and (C) subjectively measured nonius bias. (▵) Data removed from the analyses (subject S11 had an extreme nonius bias and was not included in the correlations or the corresponding multiple regression analysis). (□) Data for subject S19, who had an extreme exophoria (3.8°), but was still included in the correlations (see also Figs. 3 and 4).

Table 1.

 

Results of Simple Correlation and Multiple Linear Regression

Factors	Subjective Fixation Disparity (n = 19)
	Simple Correlation		Multiple Linear Regression (r 2 = 0.62, P = 0.002)
	r	P (1-Tail)	Parameter	t	P (1-Tail)
Intercept			−3.04	−1.97	0.034
Dynamic asymmetry	0.41	0.039	12.96	1.79	0.047
Dark vergence	0.45	0.025	3.03	1.78	0.048
Nonius bias	0.69	0.0005	0.85	3.01	0.004

 Data explain the interindividual variability of subjective fixation disparity by combining the following three factors: dynamic asymmetry, dark vergence, and nonius bias.

Second, using multiple linear regression analyses, we tested all three factors as predictors of subjective fixation disparity. Multiple linear regression analyses require that the factors included are independent. We generally found low and insignificant intercorrelations: dark vergence versus nonius bias r = 0.31 (P = 0.10; n = 19), dynamic asymmetry versus nonius bias r = 0.25 (P = 0.15; n = 19), dark vergence versus dynamic asymmetry r = 0.03 (P = 0.44; n = 19). 

The complete multiple regression model in Table 1 shows that the three factors had significant coefficients and explained 62% (r = 0.79, P = 0.002; n = 19) of the interindividual differences in subjective fixation disparity. We calculated a further multiple regression model to determine the extent to which the subjective fixation disparity could be explained by the characteristic factors of physiological vergence alone and found that 39% (r = 0.63, P = 0.02; n = 19) was due to dynamic vergence and dark vergence. An additional 23% of variance was due to subjective nonius bias, an aspect of the subjective measurement procedure. 

In a second approach, we tested the hypothesis that interindividual variability of subjective fixation disparity could be explained by dynamic asymmetry, heterophoria (as another measurement of resting vergence), and nonius bias. A check of the normal distribution showed that the data of one additional subject led to a violation of the normal distribution for heterophoria (S19: 3.8° exophoria). This also was not a result of measurement errors, but rather the particular condition of this observer, as confirmed by repeated measurements. For statistical analyses, we had to exclude this subject, to meet the assumption of normal distribution. Thus, 18 subjects were included in the subsequent analyses. 

We observed nonsignificant correlations between dynamic asymmetry versus nonius bias (r = 0.27, P = 0.14; n = 18) and heterophoria versus nonius bias (r = 0.35, P = 0.08; n = 18), but heterophoria and dynamic asymmetry (r = 0.51, P = 0.02; n = 18) correlated significantly. Thus, the three factors were not independent, and therefore multiple linear regression analysis was not useful. However, heterophoria alone correlated significantly with subjective fixation disparity: r = 0.72 (P = 0.0004, n = 18; Fig. 3). 

Figure 3.

 

Correlation between subjective fixation disparity and heterophoria. (▵) Data for two subjects not included in the correlation. Subjects S19 and S11 were outliers with respect to heterophoria and nonius bias, respectively.

Discussion

Subjective fixation disparity correlated with each of the three factors investigated: dynamic asymmetry, dark vergence, and nonius bias. The combination of these three factors in a multiple regression analysis explained approximately 62% of the interindividual variability in subjective fixation disparity. In a previous study, ¹⁴ we explained approximately 50% of interindividual variance in fixation disparity using dynamic asymmetry alone. In the present study, only 17% (n = 19) of the variance was explained. This is an obvious difference; however, in the previous study, ¹⁴ the range of fixation disparity varied from approximately 15 arc min eso to 20 arc min exo (partly due to the selection of subjects with large fixation disparities), while in the current random sample the range of observed fixation disparities was smaller (from 5 arc min exo to 5 arc min eso). Accordingly, the range of dynamic asymmetry was also much smaller in the present study, so that the overall correlations tended to be smaller. The multiple regression analysis explained a larger proportion of variance than each factor alone, since the three factors influencing subjective fixation disparity showed smaller intercorrelations. Because data of 1 of our 20 subjects had to be excluded as outlier, our findings refer to a sample of subjects with normally distributed data, but not to any observer. 

Despite the significant correlation between heterophoria and subjective fixation disparity (Fig. 3), heterophoria was not included in multiple regression analyses, because there was a significant correlation between heterophoria and dynamic asymmetry (r = 0.51). Interestingly, Kim et al. ²⁷ also found a strong correlation between heterophoria and dynamic asymmetry (r = 0.9), defined as a ratio of convergent-to-divergent peak velocity (V _con/V _div). Recalculating this correlation according to their definition, our data also showed a high correlation (r = 0.54, P = 0.01; Fig. 4). Interestingly, the dynamic asymmetry did not correlate significantly to dark vergence, the other possible measure for the resting vergence (r = 0.008, n = 18; r = 0.03, n = 19). The possible reason for a correlation of the asymmetry with heterophoria but not with dark vergence may lie in the role of accommodative functions that are included in heterophoria but not in dark vergence. A full account for accommodation would require measurement of the accommodative response and AC/A ratio and inclusion of these factors in a multiple regression analysis; such accommodative data were not available in the present study. In a similar study by Jaschinski, ¹⁷ such accommodative components tended to be small and nonsignificant in multiple regression analyses for fixation disparity; however, these factors may still play a certain role. These findings suggest that the correlation between dynamic asymmetry and heterophoria, found by Kim et al., ²⁷ and in the present study, appeared due to accommodative components that are components of heterophoria. 

Figure 4.

 

Correlation between heterophoria and dynamic asymmetry, calculated as suggested by (A) Patel et al. ¹¹ and (B) Kim et al. ²⁷ (▵) Data for two subjects not included in the correlations. Subjects S19 and S11 were outliers with respect to heterophoria and nonius bias, respectively.

The present study represents a novel investigation in that (1) we extracted dynamic asymmetry from objective measures of vergence velocity (while subjective estimations were used in our previous study ¹⁴ ) and (2) we additionally included resting vergence (dark vergence) and nonius bias into combined analyses of these three factors. Regarding the theoretical framework used to explain fixation disparity, Patel et al. ¹¹ have also mentioned the possible influences on fixation disparity of vergence adaptation, proximal cues, viewing distance, heterophoria, and dark vergence. They supposed that when observing fixation disparity “under conditions that eliminate (or keep fixed) the aforementioned parameters (i.e., in the absence of adaptation, for stimuli without proximal cues, when accommodation input and viewing distance are kept constant), that these are modulatory effects, rather than being the basic neural origin of fixation disparity. These factors may affect fixation disparity indirectly via changes in vergence dynamics” (Patel et al. ¹¹ ). So far, however, the results of the present experiment suggest that clinically relevant subjective fixation disparity can originate from a combination of independent physiological sources that directly affect fixation disparity, not only via changes in vergence dynamics. Nevertheless, the relative contributions of these modulating factors in determining vergence dynamics and fixation disparity should be investigated in more detail. 

The present study has the following limitations. Only a single step stimulus amplitude of 60 arc min at a single 60-cm viewing distance was included; the use of a wider range of disparity vergence step stimuli and distances could provide a data set for a more critical testing of the model. Other alternative vergence control type models may be tested within an interindividual approach to explain the physiological origin of fixation disparity. The conclusions refer to a sample of subjects with normal distribution of vergence parameters. Additional physiological explanations may be needed for subjects with deviating vergence states or clinical conditions. 

Footnotes

 Supported by DAAD and Deutsche Forschungsgemeinschaft JA747/4.

Footnotes

 Disclosure: A. Švede, None; J. Hoormann, None; S. Jainta, None; W. Jaschinski, None