INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

How are the eyes oriented when we look around in a rich visual environment? Images can be fused only within a certain range of retinal disparities (angular difference between the left and right eyes' image-locations of a single target). Thus when we like to inspect a small object, the lines of sight of both eyes are made to intersect in this object by a horizontal vergence eye movement. Even if the two lines of sight intersect, both eyes still can be rotated about their lines of sight; in opposite directions (cyclovergence) or in the same direction (cycloversion). This affects the horizontal and vertical disparities of the more eccentric parts of the fixated object as well as its background. Cyclovergence is known to promote retinal correspondence (Howard and Zacher 1991; van Rijn et al. 1992, 1994a) for nearly parallel gaze lines when visual stimuli are rotated in opposite directions about the eyes' lines of sight. This type of cyclovergence, to which we refer as visually driven cyclovergence, is slow and related to vertical shear and cyclodisparities (Howard and Kaneko 1994). Cyclovergence movements also occur when the eyes converge. These cyclovergence eye movements obey the extended Listing's law (L2) and are virtually independent of the visual environment (Minken and van Gisbergen 1994; van den Berg et al. 1997). L2 has been studied mainly with sparse visual stimuli and for fixation. Our study deals with the possible interaction between L2-related and visually driven cyclovergence during pursuit in a rich visual environment. Before we further explain L2, we briefly introduce the relation between viewing direction and torsion that is known as Listing's law.

According to Listing's law, the torsion component of the eye orientation is constrained as follows: all axes about which the eye can rotate from a single reference orientation to any other orientation lie in a plane. Such a plane is called a velocity or displacement plane (Tweed and Vilis 1990). The unique reference direction that is normal to the displacement plane is called the primary direction. The matching displacement plane is called Listing's plane. Modern studies use the rotation vector format (Haustein 1989) for the description of eye rotation. Briefly, the format specifies the axis direction and the amount of rotation about that axis that is required to carry the eye from the reference orientation into the specified eye orientation. In this format Listing's law corresponds to: rx = 0; i.e., there is no component of rotation about the axis perpendicular to Listing's plane.

Unfortunately this description ignores the complications that arise for nonparallel gaze directions that occur during fixation of nearby targets (Donders 1869; Nakayama 1983). We need to take into account this complexity when we ask whether the torsion eye movements promote retinal correspondence in near vision. Recently Listing's law has been extended to describe the three-dimensional orientations of the two eyes (Minken and van Gisbergen 1994; Mok et al. 1992; van Rijn and van den Berg 1993) in such conditions. The extended Listing's law or L2 states that: the displacement plane of each eye turns by an amount µ * D when the eyes are converged D (Mok et al. 1992). As Tweed (1997) writes: "this means that when you converge your eyes so that the angle between your lines of sight is 40°, the planes swing out like saloon doors, pivoting µ * 40° about the ocular centers."

This intuitively appealing description does not specify explicitly how the eyes are oriented toward a particular target because it is not immediately clear to which pair of points in the rotated Listing's planes this fixation corresponds (this pair can be found, however, by a geometric construction). van Rijn and van den Berg (1993) formulated the extended Listing's law in another way (LRB model) that provides such an explicit description. The LRB model is fed with Helmholtz angles (Fig. 1) that specify the target position, and it produces two rotation vectors (one for each eye). The LRB model states that the torsional difference between the rotation vectors of left and right eye (cyclovergence) is proportional to the product of the Helmholtz elevation of- and the Helmholtz horizontal vergence within the plane of regard. The plane of regard is the plane that contains the two lines of sight (of the left and the right eye). This form of L2 is practical for researchers who describe their stimuli in a Helmholtz coordinate system, as commonly used in the field of stereovision.

View larger version (26K): [in this window] [in a new window]   Fig. 1. Helmholtz coordinate system. Orientations of the 2 eyes relative to the ego-center (E) expressed in Helmholtz angles. x axis is chosen such that it coincides with the version primary direction.  denotes vertical version,  denotes horizontal version, and  denotes horizontal vergence. Helmholtz coordinate system has 2 eye-fixed rotation axes (the vertical and the torsion axis) and 1 head-fixed axis (horizontal). In this coordinate system, torsion (, not drawn) is about the line of sight.

The LRB model and L2 are related. The LRB model predicts that the rotation vectors of left and right eyes are located in planes when the targets are configured on an iso-vergence surface. These planes are not parallel. The angle between them corresponds to L2 with µ = 0.5. Both models predict that when we look upward (negative vertical version) to a target that is nearer than infinity (negative horizontal vergence), the cyclovergence part of the rotation vector is positive (intorsion). When we look down, the cyclovergence is negative (extorsion). The amount of cyclovergence increases in proportion to the horizontal vergence angle.

The description of eye orientation in terms of rotation vectors is efficient. For the description of retinal disparities, however, it is useful to describe L2 in terms of Helmholtz angles. In this coordinate system, torsion is a rotation about the line of sight rather than about an axis fixed in the head as for the rotation vectors. If µ is between 0.0 and 0.25, Helmholtz (HH) cyclovergence is positive when one looks down while converging and is negative when one looks up (Fig. 2A). If µ = 0.25, HH cyclovergence does not differ much from zero (Fig. 2B). In this case, targets in the plane of regard always stimulate corresponding retinal meridia, irrespective of the fixated location. Finally, if µ > 0.25, HH cyclovergence is negative when one looks down while converging, and cyclovergence is positive when one looks up (Fig. 2C). Thus whereas different levels of µ correspond to qualitatively similar patterns of cyclovergence for near vision when expressed in rotation vectors, this is not the case when cyclovergence is expressed in Helmholtz coordinates. Because Helmholtz torsion of the two eyes bear a direct relation to retinal disparity, keeping in mind this difference is important for the analysis of the potential visual benefit of L2.

View larger version (21K): [in this window] [in a new window]   Fig. 2. Helmholtz torsion of the 2 eyes as a function of µ and vertical version. In this example, horizontal version () equals 0, horizontal vergence is negative (fixation of a near target), vertical version () is either negative, 0 or positive. A: torsion angles predicted by L2 with µ <0.25, B: torsion angles predicted by L2 with µ = 0.25. C: torsion angles predicted by L2 with µ > 0.25. Upward gaze is associated with extorsion for µ < 0.25, no cyclovergence for µ = 0.25 and intorsion for µ > 0.25.

The reason for Listing's law (as well its recent extension) has remained a mystery. It has been proposed that for µ = 0.25, L2 helps to keep in register the images of the local surface around the fixation point that is perpendicular to the plane of regard (van Rijn and van den Berg 1993). Tweed (1997) pointed out, however, that this cannot be true because vertical meridians are not aligned when the horizontal meridians of the eyes are located in a single plane (Helmholtz 1867; Ogle 1950). Hence a vertical line on that perpendicular surface is not imaged on corresponding points when horizontal lines are. Second, L2 by itself cannot always bring the eye's images (i.e., local patches around the fixation point) in register in dynamical situations. Let us consider the following example. An observer is looking down at a table (Fig. 3A1). If he moves his head forward and maintains fixation at a point on the table (Fig. 3, B1 and C1), the eyes need to move downward while converging (Fig. 3, A2-C2). Depending on the µ of this observer, the eyes will show HH extorsion (µ > 0.25), no HH cyclovergence (µ = 0.25), or HH intorsion (µ < 0.25). In Fig. 3C, 1-3, it is analyzed what kind of cyclovergence would help to reduce the cyclodisparity of the table's top surface. Shown are projections of the cross on the table on the eye sockets at the three instants during the head translation as depicted in Fig. 3A. Because of perspective, the projections of the cross are sheared horizontally relative to each other during the head translation. The intorsion of the eyes if µ < 0.25 would help to decrease the large cyclodisparity of the vertical lines at the expense of an increase of the cyclodisparity of the horizontal lines (Fig. 2A). Yet such an eye movement will maintain both the horizontal and the vertical line within the fusion range for a longer period. In contrast, the extorsion that would occur if µ > 0.25 would increase the cyclodisparity of both lines. Thus certain combinations of µ and slant may reduce cyclodisparity, whereas other combinations may increase cyclodisparities during forward motion.

View larger version (41K): [in this window] [in a new window]   Fig. 3. Cyclodisparities during head translations. A1-C1: head translation above a table while the eyes fixate a cross on this table. A2-C2: vertical version and vergence required to fixate the cross on the table. L, left eye; r, right eye; c, ego center; F, fixation point. Version primary direction (- - -) intersects the table in p. Angle lfr, horizontal vergence; angle fcp, vertical version. A3-C3: perspective projections of the cross in the left and right eye socket during the head movement of A1-C1. As indicated by the arrows, the movement from A to C causes an intorsion of the vertical lines. Minimization of whole-field cyclodisparity would require intorsion of the eyes by half the amount of the intorsion of the vertical lines.

This begs the question to what extent cyclovergence is evoked visually in situations in which L2 does not reduce cyclodisparity (as pointed out in Fig. 3, when L2 is not perfect). To answer that question one needs to know whether L2 that was described for fixation (Minken and van Gisbergen 1994; Mok et al. 1992; van Rijn and van den Berg 1993) holds for pursuit and whether visual stimuli during pursuit are as effective in evoking cyclovergence as during fixation (Howard and Zacher 1991; van Rijn et al. 1992, 1994a). Specifically, we investigated visually driven cyclovergence during fixation (as in the Howard and Zacher experiment) and during pursuit in a static or moving environment. The latter case consists of a simulated head movement, in which the images show the changing perspective of an eye that is moving through space. Our study thus aims to further probe the conditions under which L2 is valid and how visual and L2-related cyclovergence combine.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Subjects

Six male subjects (age 24-32 yr, subject IH is the first author) participated in the experiments. Subjects IH, JB, MF, and HW were experienced in wearing scleral coils for eye movement recording. Subjects EP and JR were naive paid subjects and had no experience with the dual search-coil method. After the experiments, we found that subject EP showed different behavior than the other subjects. Therefore we sent him to the eye hospital for investigation. His stereopsis was optimal (Titmus stereo test). The vertical fusion range was normal. EP had a small exophoria (<5°) at near. In the midsagittal plane, EP had hypertropia (the right eye has an negative vertical offset relative to the right eye). The inferior oblique of the left eye showed under action.

Eye-orientation measurements

Three-dimensional (3D) eye orientations were measured with the dual search-coil technique (Skalar Eye position meter 3020, Delft, The Netherlands) (Collewijn et al. 1975, 1985; Robinson 1963). Horizontal, vertical, and torsion eye orientations were measured at a sampling rate of 125 Hz. To investigate the torsion signal at high resolution, it was split into two signals. The first signal was amplified four times and fed through an offset-compensator (Collewijn 1977). The offset-compensator resets a signal to 0.0 V within a period of two ms if it exceeds 1.0 V or 1.0 V (Fig. 4). Before digitization, the eight signals (horizontal, vertical, torsion, and amplified "offset compensated" torsion of the 2 eyes), were fed through a low-pass analogue filter with a cutoff frequency of 62.5 Hz. The offset-compensated signal (which lacks information about the offset) and the offset of the original torsion signal were used to reconstruct the torsion eye position signal off-line. As a result of this manipulation, the resolution of the torsion signal (600 mV/°) was 4.0 times higher than the resolution of both the horizontal (150 mV/°) and the vertical signals. By this method, we measured with a standard 12 bits analogue AD converter both large shifts and very slow small changes in torsion eye orientation. This was necessary due to L2 and coil misalignment; torsion angles of an individual eye ranged between 20 and +20°. Data were stored on disk for off-line analysis.

View larger version (21K): [in this window] [in a new window]   Fig. 4. Offset-compensated and -corrected signals. A: output of the offset compensator. Two-channel offset compensator was fed with the torsion signal (0.150 V/°) of the left and right eye. Offset compensator resets a signal within 2 ms to 0 V if its magnitude exceeds 1.0 V. This corresponds to ±6.67° (±1.0/0.150). Thus if the absolute value of the torsion signal exceeds 6.67°, it is reset to 0°. We added the offset of the original torsion signal to the corrected signal because the offset compensated signal lacks an overall offset. B: torsion signal after offset correction.

Procedure

To prevent the subject from making head movements, an adjustable (3 axes) bite-board was used. Each experimental session started with careful positioning of the subject's head. We asked the subject to position the head in such a way that the interocular axis was horizontal and approximately parallel to the screen. This was checked with two horizontally placed hairlines on each side of the magnetic field cubicle. We were satisfied when for each eye in a side view, the two hairlines were aligned and cut right through the pupil. The 3D location of the center of rotation of each eye relative to the screen then was measured by a computerized trigonometric method (van den Berg 1996).

Experiments were done on three different days because the scleral coil method limits the duration of experimental sessions to ~30 min. To position the subject's head in the same position and orientation in subsequent sessions, we used a laser pointer attached to the bite-board. We inspected the torsion signal while we placed the coils on the subject's eyes. To this end we used a plastic suction device to avoid distortion of the magnetic field. Subjects were instructed to look straight ahead during the placement of the coils to limit torsion offset due to L2.

The experiment contained three types of trials, cyclovergence trials (trial duration: 32 s), primary direction trials (duration: 32 s), and calibrations (duration: 2 s). Each experimental session consisted of 19 (experiment 1), 18 (experiment 2), or 24 cyclovergence trials (experiment 3).

In cyclovergence trials, subjects fixated or made various pursuit eye movements, and cyclovergence occurred as a result of L2, was evoked visually, or was evoked by both methods at the same time.

During a primary direction trial, subjects were asked to fixate at their own pace nine dots of a rectangular grid in random order (20 × 20°). The pattern was presented dichoptically at a simulated distance of 19 m (see STIMULI). In this way, we measured the orientation of the Listing's planes of the two eyes with almost parallel gaze.

During calibration, the subject gazed at a single target at simulated distance of 19 m straight ahead, i.e., in the direction perpendicular to the revolving magnetic field. This allows one to measure the horizontal, vertical and torsion offset angles. This coil misalignment could change over time due to coil slippage. Therefore each primary direction measurement and each cyclovergence trial was preceded by a calibration trial.

Data analysis

GENERAL DATA ANALYSIS: FROM COIL VOLTAGES TO LISTING'S COORDINATES AND HELMHOLTZ ANGLES. The Skalar eye position meter provides coil voltages. Figure 5 shows a diagram of the transformations and manipulations, which were used to transform coil voltages to Helmholtz angles and Listing's coordinates. We need both Helmholtz angles and rotation vectors (Listing's coordinates) because we use the LRB scheme to estimate µ. Preceding each experiment, the coils were calibrated to determine the sensitivity of and the relative orientation between the direction and the torsion coils (Bruno and van den Berg 1997b). Coil voltages were transformed to Fick angles by a linearization and a correction for coil nonorthogonality. This nonorthogonality error correction was done by the method described in Bruno and van den Berg (1997a). Subsequently the Fick angles were transformed to rotation vectors (Haustein 1989). By a 3D counter rotation (Haslwanter 1995), the rotation vectors were corrected for the coil misalignment, which was determined in the preceding calibration trial.

View larger version (71K): [in this window] [in a new window]   Fig. 5. From coil voltages to Listing's and Helmholtz coordinates. Each experimental session contained 3 types of trials. Signals from the primary direction measurement and the calibration were used to transform the cyclovergence measurement signals from coil voltages to Listing's and Helmholtz coordinates (fat arrow). Signals from all measurements were transformed to Fick angles (A), linearized (C), and corrected for the nonorthogonality of the direction and the torsion coils (D). We used the offset compensator in the cyclovergence trials. Before linearization and correction for nonorthogonality, we corrected the torsion signals of the cyclovergence trial for the effects of the offset compensator (B). Calibration trial was used to determine the coil misalignment (E). Signals of the primary direction measurement and the cyclovergence measurement were corrected for coil misalignment (F) using the correction factors (D). From the signals of the primary direction measurement, we determined the version primary direction (G). Version primary direction was used to transform the signal of the cyclovergence measurements to Listing's coordinates (H). Subsequently, these signals were transformed to Helmholtz angles (I).

STIMULI. Stimuli were generated by a Graphics workstation (SGI ONYX). The computer presented the perspective view appropriate for each eye of a simulated 3D scene. The locations of the eyes relative to the screen as determined directly before the experiment were used to compute these images. The scene consisted of line objects of random orientation, length, and position. These objects were located within a cone-like volume, of which the apex was located at the ego-center of the subject. Thus a circular part of the screen was filled with a bunch of lines, that could recede in depth (Fig. 6, A-C). We varied the diameter of the ground plane of the cone, its height (= simulated depth), the number of lines, and the distance between the lines and the subject.

View larger version (27K): [in this window] [in a new window]   Fig. 6. Stimulus. A-C: top, side, and front view of the stimulus. In experiment 1, we varied the radius of the stimulus, the distance of the simulated bunch of lines (horizontal vergence angle), the simulated cyclovergence amplitude, number of lines, depth in the stimulus, and size of the fixation marker.

View larger version (48K): [in this window] [in a new window]   Fig. 7. Stimulus of experiment 2 (A) and experiment 3 (B). A: in experiment 2, the fixation marker traveled through a bunch of lines [vertical near, diagonal (from near down to far up), horizontal (above a ground plane), vertical far, diagonal (from near up to far down) and circular]. B: in experiment 3, we used the same 6 movements (shown are 3 movements). Now the bunch of lines was attached to the moving fixation marker. Whole stimulus translated (it did not rotate) through space. In the horizontal movement stimulus the stimulus consisted of a bunch of lines (instead of the ground plane of experiment 2).

SPECIFIC DATA ANALYSIS. Experiment 1.   Coil voltages were off-line transformed to Helmholtz angles using the procedure described in the general data analysis (Fig. 5). Eye orientations were expressed in HH coordinates. Thus the cyclovergence was simulated around the line of sight. A computer program removed blinks and saccades based on velocity (10°/s) and duration criteria (>12 ms). Gaps in the cyclovergence signals were interpolated using a second-order interpolation. All trials also were inspected by eye. Linear regression analysis of the cyclovergence response was used to correct it for drift and offset. Subsequently the cyclovergence response was transformed by a fast Fourier routine. We computed the visually evoked cyclovergence gains and phase lag from the component that was related to the stimulus frequency of 0.125 Hz.

View larger version (17K): [in this window] [in a new window]   Fig. 8. Subtraction method: determination of the visually driven cyclovergence gains during pursuit. Thin lines denote real cyclovergence signals. Fat line denotes the simulated cyclovergence. A-C: cyclovergence signals of a +trial, a 0trial, and a trial. D: if L2-related cyclovergence is equal for the +trial, the 0trial, and the trial, the residual cyclovergence should equal 0. E: we computed (+trial  trial)/2 to determine the visually driven cyclovergence gains

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Calibrations

Each cyclovergence or Listing's plane (or primary direction) measurement was preceded by a calibration. The calibration was used to compensate for coil misalignment due to slip and offset. Despite the careful placement of the coils, offsets often remain. How stable was the coil attached to the eye? In general, the standard deviation of the torsion component of all calibration trials of an experimental session was lower than 2°. Torsion offsets ranged from 18 to 22°. In the majority of the trials, we found offsets that ranged from 10 to 10°. The time interval between two calibrations was ~50 s (8 s to accustom the subject to the new stimulus, 32-s trial duration and 10-s between a trial and the subsequent calibration). Inspection of the calibration trials showed that the torsion offset is stable over time. Thus the coils were well attached to the eye.

Listing's planes

From our binocular data, we determined the orientations of the displacement planes for each eye separately and the displacement plane of the averaged rotation vectors: the versional displacement plane. We started by extracting the primary directions from the version signals and proceeded with reporting the differences in orientation between the primary directions of the two eyes.

The interocular axis of the subject was positioned approximately parallel to the y axis of the coil frame. Figure 9, C and D, shows the orientations of the version Listing's plane with respect to the Skalar coil frame. Rotations about the z axis ranged (Fig. 9B) from 2.5 to +7.0° in experiment 2 and from 3 to +3° in experiment 3. Within subjects, differences between experiments 2 and 3 were <1° for four subjects (MF, JB, HW, and IH) and ~5° for two subjects (EP and JR). Large upward rotations of the versional primary position (Fig. 9A) were found for subjects MF (23° for experiment 2 and 20° for experiment 3) and HW (17° for experiment 2). Rotation about the y axis ranged from 4 to +5° for the other four subjects. Within subjects, differences between experiments 2 and 3 were largest for subjects HW (10°) and JR (4°) and 3° for subjects MF, JB, EP, and IH. In summary, orientations of version Listing's plane varied between subjects and experiments. Between subjects, we found a large variation in rotation about the y axis. Variation in rotation about the z axis was much smaller.

View larger version (28K): [in this window] [in a new window]   Fig. 9. Orientation of the version Listing's plane with respect to the coil frame and the orientation of the left and right eye Listing's plane in Listing's coordinates. Clockwise rotation about the y axis (A) and z axis (B) are denoted with a negative angles. C and D: orientation of the version Listing's plane with respect to the coil frame. Light bars represent rotations of the version Listings plane about the z axis of the coil frame. Dark bars represent the rotation of Listing's plane about the y axis of the coil frame. E: alpha [the angle between the left and right eye Listing plane (Listing's coordinates)]. Alpha is negative in this illustration. F: alpha obtained from experiment 2 (black bars) and experiment 3 (white bars)

In some previous studies in our laboratory (Bruno and van den Berg 1997a, normal subjects; van de Berg and van Rijn 1995, patients), it was found that the primary directions of the two eyes are not parallel when looking at targets at optical infinity. We confirm this observation. Figure 9F shows the angle (alpha) between the left and right eye's Listing planes (these planes are normal to the primary directions) when looking at infinity. Alpha is negative when the left-eye plane is rotated counter clockwise and the right plane is rotated clockwise (in a top view as in Fig. 9E). There were large differences between subjects. Subject EP and HW had the largest values for alpha (12 to 18°), corresponding to excess cyclovergence [even with respect to van Rijn and van den Berg's (1993) model for the actual amount of horizontal convergence]. Alpha's of the other subjects ranged from 6 to +3°. Also Listing's planes measured on different days could vary in relative orientation. The modified model (van den Berg et al. 1995), to which we will refer as LBR, compensates for nonparallel Listing's planes. In the paragraph describing the results of experiments 2 and 3, we will estimate µ by both the LRB and the LBR model.

Experiment 1. Visually driven cyclovergence during fixation

We had a number of reasons to do this experiment. First, we wished to check whether our stimuli were as effective as described previously (Howard and Zacher 1991; van Rijn et al. 1992, 1994a). Second, we wished to explore further the stimulus factors that may affect the cyclovergence in response to changing cyclodisparities. This could help us understand the dichotomy between the results of Howard and Zacher and van Rijn et al. Finally, we wished to explore the effect of a factor that was not controlled for explicitly in those older studies: the horizontal vergence angle of the eyes.

Whole-field opposite cyclorotations of left and right eye images are known to evoke the percept of a slanted stimulus (Collewijn et al. 1991; Howard and Kaneko 1994) if a visual reference like an object placed before the screen is visible. In the absence of a visual reference, the stimulus cyclovergence goes unnoticed and no changes in slant are perceived. In the visual periphery, the borders of the coil frame were dimly visible, but except for JB subjects reported that they perceived a stable vertically oriented stimulus. Subject JB sometimes became aware that the slant of the stimulus had changed but never did he perceive rotation of the stimulus about a horizontal axis. Thus we conclude that perceived slant was stable across conditions and was not related to the eye movements reported in the following text.

A typical example of the cyclovergence response to simulated cyclovergence is shown in Fig. 10. Amplitude of the simulated cyclovergence is defined as the peak-to-peak value divided by 2 [note: Howard and Zacher (1991) report the peak-to-peak value]. Figure 11A shows visually driven cyclovergence gains as a function of simulated cyclovergence amplitude. Gains ranged from 0.08 (IH) to 0.70 (MF). Varying the amplitude of the simulated cyclovergence had a large effect for subjects MF and HW and a small effect for subjects IH and JB. Gains decreased with increasing amplitude of the simulated cyclovergence. In general, MF has the highest and IH has the lowest gains. If the radius of the stimulus increases from 28 to 32°, the area of the stimulus increases with a factor of 1.5. This has no effect on the visually driven cyclovergence gains (Fig. 11B). Except for MF (radius 30°) gains ranged from 0.12 (IH) to 0.41 (MF). Again MF has the highest and IH has the lowest gains. This result is in agreement with results of Kertesz and Sullivan (1978) and Howard et al. (1994). In their experiments, gain began to fall off for stimuli that had a radius <20°. The same holds for the "number of lines in the stimulus" and "the size of the fixation marker" conditions (Fig. 11, C and D). These two parameters did not affect visually driven cyclovergence.

View larger version (15K): [in this window] [in a new window]   Fig. 10. Visually driven cyclovergence signal vs. time. This figure shows data of subject MF. Stimulus was flat and had a radius of 35°. Number of lines in the stimulus was 400 and vergence angle was 12°. Cyclovergence signal was corrected for drift and offset. Both the simulated cyclovergence and visually driven cyclovergence are expressed in Helmholtz angles. Thus the angles represent the difference in rotations about the lines of sight. Positive angles correspond to intorsion. Amplitude of the visually driven cyclovergence is smaller than the amplitude of the simulated cyclovergence. In this condition, the cyclovergence response has a phase lag of ~30°.

View larger version (24K): [in this window] [in a new window]   Fig. 11. Cyclovergence gains of experiment 1. In each trial, 1 of the 6 parameters was varied relative to a default stimulus. Specifications of the default stimulus were: simulated cyclovergence amplitude = 4°, radius = 35°, density = 400 lines, fixation marker = 0.85°, horizontal vergence = 2°, and depth (the linear dimension of the stimulus along the x axis) was 1% of the distance between the observers eyes and the far-end of the stimulus (the plane in which the fixation marker was located).

In the experiments of Howard and Zacher (1991) and van Rijn et al. (1992, 1994a), the horizontal vergence angle was unspecified. It seems likely that the experiments were done with almost parallel gaze. We systematically varied the horizontal vergence angle by shifting the fixation target in depth within the stimulus. All subjects produced the demanded vergence angle. As Fig. 11F shows, horizontal vergence angle does not affect the cyclovergence gain at all in one subject (IH). For the other three subjects, there is a slight trend discernable for the cyclovergence gain to increase for larger convergence.

Figure 11E depicts cyclovergence gain versus depth in the stimulus. Nearby lines cause huge horizontal and vertical disparities, yet varying depth in the stimulus does not affect the cyclovergence gains systematically.

Figure 12 depicts the phase lag as a function of amplitude, radius, number of lines, size of the fixation marker, horizontal vergence angle and depth. In general phase lag ranged from 22 to 57°. The majority of the phase lags are ~40°. Except for the amplitude condition, none the varied parameters had any systematic effect on the phase lags. The phase lag of subject MF decreases as rotation amplitude increases.

View larger version (23K): [in this window] [in a new window]   Fig. 12. Phase lag of visually driven cyclovergence of experiment 1. In each trial 1 of the 6 parameters was varied relative to a default stimulus. Specifications of the default stimulus were: simulated cyclovergence amplitude = 4°, radius = 35°, density = 400 lines, fixation marker = 0.85°, horizontal vergence = 2°, and depth was 1% of the distance between the observers eyes and the far-end of the stimulus (the plane in which the fixation marker was located).

To prevent the effect of a stationary frame on cyclovergence we used a circular aperture. However, the frame of the almost invisible Skalar box was very dimly visible

To examine for the possible effect of the almost invisible skalar box (because the subjects wore red green anaglyphes) on the cyclovergence gains, each experiment included a control trial. The visual stimulus of the control trial was similar to the other stimuli except for its shape on the screen. The outline of the control stimulus was rectangular. The outline was fixed to the screen so the stimulus looked like a large bunch of lines seen through a rectangular aperture. If the dimly visible skalar box was affecting the cyclovergence gains, the less eccentric and sharp borders of the control stimulus should have reduced the gain even more. Gains of the visually evoked cyclovergence were 0.22 (HW), 0.16 (IH), 0.25 (JB), and 0.48 (MF). These gains are comparable with the gains found for all other stimuli in experiment 1 (Fig. 11). Apparently the sharp stationary borders of the control stimulus did not act as a visual reference. Thus we think that the dimly visible skalar box did not affect cyclovergence gains at all. Perhaps this is not convincing for subjects with a low gain, it is for subject MF.

Summarizing, except for simulated cyclovergence amplitude and less so horizontal vergence angle, none of the varied parameters had a systematic influence on cyclovergence gain. The most important observations are 1) that cyclovergence gains are as low as in the experiment of van Rijn et al. If a subject has a low gain, he has a low gain in all conditions (IH). The order of the subjects with respect to their gains is almost the same in each figure; 2) that the cyclovergence gains are subject dependent; and 3) that phase lags were nearly constant at a level of 40° at 0.125 Hz.

Experiments 2 and 3

SLANT PERCEPTS. Trials having simulated cyclovergence and trials without were mixed. After the last experimental session, we asked each subject whether he had been aware of torsional motion of the images in two-thirds of the trials. None of the subjects had noticed the difference between zero-condition trials and simulated cyclovergence trials. None of the subjects reported changes in perceived slant of the stimulus during the experiment.

L2 during smooth pursuit in depth

The target motion through a 3D scene evoked smooth-pursuit eye movements often with large changes in elevation and horizontal vergence. We start with a description of the cyclovergence evoked in ⁰trials in which no cyclovergence of the images was presented. The visual environment (a bunch of lines) was fixed with respect to the head (experiment 2) or with respect to the moving fixation dot (experiment 3). Pursuit was usually smooth and accurate, but this did not always prevent loss of fusion. During trials A (nearby target, pure vertical movement), C (diagonal movement), D (horizontal), and E (diagonal movement), some subjects complained that they occasionally lost fusion. Figure 13 shows a typical example of a trial in which subject MF lost fusion (experiment 3, diagonal movement from far down to near up). HH horizontal vergence dropped when the HH horizontal vergence of the target was large (about 5 to 6°). This occurred in each period of the diagonal movement. The majority of the subjects followed the dot smoothly. However, MF made many small saccades (Fig. 13B, little arrows). Because of L2, the vertical saccades affected cyclovergence. During the first saccade (down), the eyes make an intorsion movement. During the second saccade (up), the eyes make an extorsion movement. This relation holds for all other saccades in the figure. Notice that HH cyclovergence ranged between 1.5 and 2° for this huge vertical version movement.

View larger version (24K): [in this window] [in a new window]   Fig. 13. Examples of pursuit (experiment 3). Subjects followed the fixation dot perfectly in the majority of the trials. However, A shows an example of vergence drop. This is data of subject MF (experiment 3, diagonal movement from far down to near up). MF does not follow the dot in the intervals between the dashed lines. B: example of subject MF. In this trial (experiment 3, circular movement, no simulated cyclovergence (0trial)). MF made many small saccades (little arrows). Other subjects followed the dot more smoothly. Because of L2, the vertical saccades affected cyclovergence. During the first saccade (down), the eyes make an intorsion movement. During the second saccade (up), the eyes make an extorsion movement. This relation holds for all other saccades in the figure. Notice that HH cyclovergence ranges between 1.5 and 2° for this huge vertical version movement.

Figure 14A shows eye orientations (Listing's coordinates) of subject EP while following a dot that moved vertically with an amplitude of 0.125 m at a simulated distance 0.33 m in front of his head. This movement evoked nearly 30° of vertical version and 4° changes in horizontal vergence. The cyclovergence panel of Fig. 14A contains two lines. The thin line represents the measured cyclovergence, which is not corrected for offset or drift. The fat line represents cyclovergence predicted (LRB) based on the vertical version and the horizontal vergence. The shape of the measured cyclovergence signal corresponds to the predicted cyclovergence signal, but the amplitude is smaller. Figure 14B shows an example of pursuit by subject JB. In this trial, JB followed a dot that was moving diagonally with an amplitude of 0.35 m from up near to far down. The middle point of the trajectory was at eye height at a distance of 0.66 m in front of the subject. In this trial cyclovergence and predicted cyclovergence matched perfectly. Halfway through the trial subject JB blinks. The blink affects all eye-movement signals. After the blink, the eyes make an intorsion movement and return within 2 s to orientations comparable with these of the previous and the following period. In Fig. 14, both A and B, we observe a small periodical vertical vergence movement with an amplitude of ~0.6° (EP) and 0.25° (JB). These vertical vergence angles correspond to a relative difference between the vertical components of the left and right eye of ~3% (EP) and 1% (JB). The 1% difference of JB may be due to gain differences between the left and the right version channels. The 3% difference of EP may be related to the hypertropia (see Subjects).

View larger version (45K): [in this window] [in a new window]   Fig. 14. Vertical vergence, vertical version, horizontal vergence, cyclovergence, and predicted cyclovergence (LRB) against time. Numbers on the y axis are the tangent of half the rotation vector component. These numbers roughly correspond to the angle in degrees divided by 100.

We computed µ for each stimulus using the cyclovergence signal and the LRB model prediction. Because the computation of µ essentially involves a division by the average cyclovergence predicted by the model, this computation was sensible only when the predicted cyclovergence was larger than the noise. To determine whether a trial was suitable, we therefore adopted the criterion that the LRB model should predict cyclovergence angles that were three times larger than the standard deviation of the cyclovergence signal of a calibration trial. Sixty two of the 83 ⁰trials passed this criterion. There were no consistent effects of the pursuit trajectory on µ across subjects. Figure 15 therefore depicts µ averaged over trials per subject. There was a large variation in µ between subjects. For subjects MF, HW, and EP, we find high values for µ. Moreover, µ varied between experiments 2 and 3 in some subjects (MF and HW).

View larger version (33K): [in this window] [in a new window]   Fig. 15. L2-related cyclovergence. µ represents the prediction of the LRB model. µ-corrected represents the prediction of the LBR model. Predicted cyclovergence according to the LBR model depends on the angle of convergence added to the angle between the Listing planes for fixation of distant targets. A: µ (LRB) averaged over trials for 6 subjects., B: µ (LBR) averaged over trials for 6 subjects. C: µ (LBR) averaged over subjects for 6 different stimuli. Light bars represent results from experiment 2. Dark bars represent results from experiment 3. Error bars denote standard error of the mean.

We wondered to what extent this variation in µ was correlated to the variation across subjects of the relative orientation of the Listing planes of the two eyes. In the LRB model, it is assumed that Listing's planes are parallel when looking at infinity. Figure 9 shows that this is usually not the case. The LBR model compensates for nonparallel Listing's planes when looking at infinity. Essentially, µ of this model describes how much cyclovergence changes when fixation changes from distant to nearby viewing without the requirement that Listing planes should be parallel for parallel gaze. Figure 15B depicts µ (LBR) averaged over trials per subject. Now the variation in µ is much smaller.

Finally, Fig. 15C depicts µ (LBR) averaged over trials. We did not find a significant effect of stimulus on µ. The exception is stimulus D (pure horizontal vergence movement). In this case, only one measurement passed the noise criterion.

Visually driven cyclovergence gains

As explained in METHODS, we isolated the visually evoked cyclovergence by subtraction of trials that only differed in imposed visually driven cyclovergence. The assumptions are that L2-related cyclovergence matched in these trials and that the imposed cyclovergence consisted of a rotation about the visual axis. We could not control the latter directly (this would have required feedback of eye orientation to the stimulus generator). Because we simulated rotation about the axis through the eye and the fixation dot on the screen, we had to rely on correct pursuit of the dot. We investigated the validity of both assumptions.

First, we determined in (Helmholtz angles) the error of pursuit during each trial. Trials that had an average unsigned-difference between viewing direction and target direction larger than 2.5° were excluded from the analysis. This occurred in 13 of 246 trials. After removing these trials, average fixation error over all subjects and trials was smaller than 1.0°. We found this an acceptable deviation because the diameter of the fixation marker was 0.85°. Thus in the majority of the measurements pursuit was sufficiently accurate.

The residual cyclovergence provides a measure of the match of the L2-related cyclovergence in each triple of trials (Fig. 8). Ideally, the average of the cyclovergence of the ⁺trial and the trial equals the cyclovergence of the ⁰trial, and residual cyclovergence is zero. However, there was always some residual cyclovergence due to variations in pursuit and variation in the cyclovergence signal. We found that the amplitude of the averaged, visually driven cyclovergence [(⁺trial  trial)/2] was three or more times larger than the amplitude of the residual cyclovergence [(⁺trial + trial)/2  ⁰trial] at the stimulus frequency (0.125 Hz) in 52 of 83 triples. In those 52 triples, the visually driven cyclovergence was on average 6.2 times higher than the residual cyclovergence. Thus in the majority of trials we could reliably measure the visually driven cyclovergence during smooth pursuit in 3D.

Either the simulated cyclovergence can be in the same direction as or opposite to L2-related cyclovergence (depending on the µ). Thus we checked whether the direction of the simulated cyclovergence relative to L2-related cyclovergence affected the visual gains. In the analysis we determined the amplitude of the L2-related HH cyclovergence (from the ⁰trial). This component was added to the amplitude (4°) of the simulated cyclovergence (present in both ⁺trial and trial). Depending on the value for µ, the amplitude of the changing whole field cyclodisparities (caused by L2 and simulated cyclovergence) could be larger or smaller than 4°. The visual gain was determined by dividing the visually driven cyclovergence by the amplitude of the changing whole field cyclodisparities (due to L2 and simulated cyclovergence). The visually driven cyclovergence is defined by the difference between the cyclovergence signals of the ⁺trial or trial and the ⁰trial. Figure 16A shows that the amplitude of the changing whole field cyclodisparity does not affect the gain of the visually evoked cyclovergence. This implies that the relative direction between L2-related and visually driven cyclovergence does not affect the visual gains. Based on this observation we further computed the gains of the visually driven cyclovergence as shown in Fig. 8 and described in Specific data analysis, experiments 2 and 3.

View larger version (44K): [in this window] [in a new window]   Fig. 16. Visually driven cyclovergence. A: gains of the visually driven cyclovergence vs. whole-field cyclodisparity caused by both imperfect L2 (µ  0.25) and simulated cyclovergence. B: visually driven cyclovergence gains relative to L2 averaged over trials for 6 subjects. C: visually driven cyclovergence gains relative to L2 averaged over subjects for 6 different stimuli. D: phase lags of the visually driven cyclovergence relative to L2 averaged over trials for 6 subjects. E: phase lags of the visually driven cyclovergence relative to L2 averaged over subjects for 6 different stimuli. White bars represent results from experiment 2. Black bars represent results from experiment 3. Dashed bars represent results of experiments 2 and 3 averaged. Error bars denote standard error of the mean.

There was no systematic effect across subjects of the pursuit trajectory on the gain of the visually driven cyclovergence. Figure 16B shows gains of the visually driven cyclovergence averaged over trials for each subject. Visual cyclovergence gains ranged from 0.1 to 0.3. Similar to the gains for L2-related cyclovergence, gains of the visually driven cyclovergence did not depend on the stimulus (Fig. 16C). Again, we find low gains for the pure horizontal vergence stimulus. We also find low gains for stimulus B (vertical movement while converging). We do not see a systematic difference between experiments 2 and 3. Figure 16, D and E, shows the phase lags, which ranged from 30 to 50°. The majority of the subjects have a phase lag of ~40°. In summary, visually driven cyclovergence gains are low (~0.2) and phase lags are ~40°.

µ and Helmholtz cyclovergence

Tweed (1997) and van Rijn and van den Berg (1993) showed that if µ = 0.25, cyclovergence in Helmholtz angles should be close to zero. If µ has a larger or smaller value, HH cyclovergence should deviate from zero. To provide insight in how µ is related to HH cyclovergence angles, we plotted the standard deviation of the Helmholtz cyclovergence signals versus |µ_LRB  0.25| (Fig. 17A) for the ⁰trials. We ask if the standard deviation of the HH cyclovergence increases with |µ_LRB  0.25| increasingly different from zero. Figure 17A shows indeed, that there is a linear relation between |µ_LRB  0.25| and the deviation of the HH cyclovergence. In the majority of the conditions, the standard deviation of the cyclovergence is small (<2°). The standard deviation of the cyclovergence measured during fixation (van Rijn et al. 1994b) is represented by - - -. Thus the standard deviation of the HH cyclovergence during fixation is much smaller than during pursuit.