Three-dimensional kinematics of saccadic and pursuit eye movements in humans: relationship between Donders’ and Listing’s laws

Abstract

For Listing’s law to be obeyed during eye movements, the “half-angle rule” must be satisfied: the eye velocity axis must tilt away from Listing’s plane by half the angle of eye position eccentricity from primary position. We aimed to determine if this rule is satisfied during horizontal and vertical pursuit compared with saccades. Three-dimensional (3-d) eye rotation data were acquired from five normal head-fixed humans using the search coil technique. Saccades were recorded in response to 40° horizontal or vertical steps in target position, at different elevations and azimuths. Pursuit was recorded while tracking a target moving horizontally or vertically at 20°/s, with peak-to-peak amplitude of 40°, at the same elevations and azimuths. First- and second-order surfaces were fitted to 3-d eye position data from periods of fixation. In all subjects, eye positions did not lie on a planar surface, but on a twisted surface in 3-d space. The tilt-angle coefficient (TAC) during saccades and pursuit was calculated as the ratio of the angle of eye velocity axis tilt to the angle of eye position eccentricity. During horizontal saccades and pursuit, mean TACs were 0.58 and 0.64, respectively. During vertical saccades and pursuit, mean TACs were 0.35 and 0.43, respectively, and lower than their horizontal counterparts (p<0.05). These findings suggest that Listing’s law is not perfectly satisfied during saccades or pursuit. On the basis of model simulations, we propose that the discrepancy in horizontal and vertical TACs causes eye positions to lie on a twisted rather than a planar surface.

1. Introduction

Although the eye could assume an infinite number of torsional positions for any given gaze direction, it is well established that, during visual fixation of a distant target with the head upright, the eye adopts a unique torsional position for each gaze direction (Donders, 1847); this kinematic principle is known as Donders’ law. Listing’s law, an extension of Donders’ law, states that the eye only assumes positions that can be reached from a reference position by single rotations about position axes that lie within a two-dimensional plane in three-dimensional (3-d) space (von Helmholtz, 1866). If the reference gaze direction is orthogonal to this plane, the reference position is known as primary position and the plane is known as Listing’s plane (von Helmholtz, 1866). For eye position to remain in Listing’s plane during an eye movement, the half-angle rule must be satisfied: the instantaneous eye velocity axis must tilt away from Listing’s plane by half the angle of eye position eccentricity from primary position (Tweed & Vilis, 1987, 1990). Eye velocity axis tilt can be quantified by the tilt-angle coefficient (TAC), the ratio of the angle of eye velocity axis tilt to the angle of eye position eccentricity (Palla et al., 1999; Angelaki et al., 2003; Walker et al., 2004; Adeyemo & Angelaki, 2005; Crane et al., 2005; Klier et al., 2006; Tian et al., 2006; Thurtell et al., 2008a); a TAC of 0.5 indicates that the half-angle rule is satisfied, whereas a TAC of less or more than 0.5 indicates that the eye velocity axis tilts by less or more than half the angle of eye position eccentricity, respectively.

It is well established that eye position remains approximately confined to Listing’s plane during fixations, saccades, and pursuit in head-fixed monkeys and humans (von Helmholtz, 1866; Ferman et al., 1987; Tweed & Vilis, 1990; Haslwanter et al., 1991; Tweed et al., 1992). However, it has been postulated that the half-angle rule is not perfectly satisfied, since eye positions lie on a twisted rather than a planar surface in 3-d space (Glenn & Vilis, 1992; Desouza et al., 1997; Bergamin et al., 2004), and there are low-amplitude transient torsional deviations away from this surface during blinks, saccades, and pursuit (Tweed et al., 1994; Straumann et al., 1995, 1996; Raphan, 1998). The half-angle rule is approximately satisfied during horizontal saccades and pursuit, although most studies that have measured the TAC have found that it is higher than 0.5, especially for pursuit. The average TAC in previous studies ranged from 0.42 to 0.57 for horizontal saccades and 0.5 to 0.66 for horizontal pursuit (Table 1; Tweed & Vilis, 1990; Tweed et al., 1992; Bruno & van den Berg, 1997; Palla et al., 1999; Angelaki et al., 2003; Walker et al., 2004; Adeyemo & Angelaki, 2005; Crane et al., 2005; Klier et al., 2006; Tian et al., 2006; Thurtell et al., 2008a). During vertical saccades, the half-angle rule is also not perfectly satisfied, although in this case the TAC is less than 0.5; in previous studies, the average TAC ranged from 0.28 to 0.45 (Table 1; Bruno & van den Berg, 1997; Crane et al., 2005; Thurtell et al., 2008a). It is unclear if the half-angle rule is satisfied during vertical pursuit; eye position does remain roughly confined to Listing’s plane (Haslwanter et al., 1991; Tweed et al., 1992), but TACs have not been reported. In the current study, we aimed to determine the TAC during horizontal and vertical pursuit compared with saccades in the same human subjects. We also aimed to determine how variations in TAC might relate to the shape of second-order surfaces fitted to 3-d eye position data, using a model simulation. Preliminary findings have been presented in abstract form (Thurtell et al., 2008b) and, for the saccade data, as conference proceedings (Thurtell et al., 2011).

Table 1.

Tilt-angle coefficients (TAC) reported in prior studies of saccadic and pursuit kinematics.

Study	Horizontal Saccade TAC	Vertical Saccade TAC	Horizontal Pursuit TAC	Vertical Pursuit TAC
Tweed & Vilis, 1990 [h,m]	~0.5	Not reported	-	-
Haslwanter et al., 1991 [m]	-	-	Not reported	Not reported
Glenn & Vilis, 1992 [h]	Not reported	Not reported	-	-
Tweed et al., 1992 [h]	-	-	~0.5	Not reported
Straumann et al., 1995 [h]	Not reported	Not reported	-	-
Straumann et al., 1996 [h]	Not reported	-	Not reported	-
Bruno & van den Berg, 1997 [h]	0.42 ± 0.13 (mean ± SD)*	0.28 ± 0.19 (mean ± SD)†	-	-
DeSouza et al., 1997 [h]	Not reported	Not reported	-	-
Palla et al., 1999 [h]	0.44 ± 0.11 (mean ± SD)	-	0.50 ± 0.09 (mean ± SD)	-
Angelaki et al., 2003 [m]	-	-	0.60 ± 0.07 (mean ± SD)	-
Bergamin et al., 2004 [h]	Not reported	Not reported	-	-
Walker et al., 2004 [h]	-	-	0.65 ± 0.03 (mean ± CI)	-
Adeyemo & Angelaki, 2005 [m]	-	-	0.57 ± 0.12 (mean ± SD)	-
Crane et al., 2005 [h]	0.50 ± 0.07 (mean ± SD)	0.45 ± 0.11 (mean ± SD)	-	-
Klier et al., 2006 [m]	-	-	0.66 ± 0.17 (mean ± SD)	-
Tian et al., 2006 [h]	-	-	0.64 ± 0.04 (mean ± CI)	-
Thurtell et al., 2008a [h]	0.57 ± 0.07 (mean ± SD)	0.30 ± 0.10 (mean ± SD)	-	-
Current Study [h]	0.57 ± 0.09 (mean ± SD)	0.34 ± 0.04 (mean ± SD)	0.66 ± 0.06 (mean ± SD)	0.44 ± 0.11 (mean ± SD)

h = human; m = monkey; CI = 95% confident interval; SD = standard deviation; TAC = tilt-angle coefficient (or equivalent measure); - = not studied

^*The ratio of the actual angle of velocity axis tilt to the angle of tilt predicted by the “half-angle” rule was 0.83 ± 0.25 (mean ± SD), giving a TAC of 0.42 ± 0.13.

^†The ratio of the actual angle of velocity axis tilt to the angle of tilt predicted by the “half-angle” rule was 0.56 ± 0.38 (mean ± SD), giving a TAC of 0.28 ± 0.19.

2. Methods

2.1. Subjects

Five normal human subjects (all male; ages 32–61 yrs; 46 ± 14 yrs, mean ± SD) were studied. No subject had a history of neurologic or ophthalmic disease, apart from refractive error. All subjects had best-corrected visual acuities of 20/20 or better and normal ocular motility. All gave written informed consent in accordance with the Declaration of Helsinki and the Human Subjects Institutional Review Board of the Cleveland Veterans Affairs Medical Center.

2.2. Recording system

3-d eye and head rotation data were acquired using the magnetic search coil technique (Robinson, 1963), with subjects seated in the center of 1.9 × 1.9 × 1.9 m magnetic field coils (CNC Engineering, Seattle, USA). Eye rotations were recorded using a dual search coil (Skalar, Delft, The Netherlands) placed on the left eye, after the application of a topical anesthetic (proparacaine hydrochloride 0.5% ophthalmic solution, Alcon Laboratories Inc, Fort Worth, USA). To monitor for inadvertent head movement, head rotations were recorded using a dual search coil secured to the forehead. The search coils were calibrated in vitro prior to each experimental session (Liao et al., 2008). Six signals were recovered from each search coil by phase detectors (CNC Engineering, Seattle, USA) and then passed through a low-pass filter (bandwidth: 0–120 Hz). The signals were digitized at 500 Hz, with 16-bit precision, and saved for off-line analysis. The standard deviation of the system noise was ±0.016°.

2.3. Experiment protocols and paradigms

During the experiments, the subject’s head was immobilized in space using a padded circumferential head support attached to the recording chair. The subject was instructed to look toward a visual target generated by a solid-state red laser that was rear-projected onto a tangent screen 110 cm in front of the subject. In vivo calibration data were acquired prior to each part of the experiment while the subject was fixating on a centrally positioned target; this was considered the reference position, and these data were used to correct for any offset of the search coils (Tweed et al., 1990).

FIXATIONS: Fixation data were acquired while subjects looked toward the target as it moved from 0° to 20°, in 5° steps, along horizontal, vertical, and diagonal radial trajectories. The laser returned to the center of the screen after moving along each radial trajectory; data from periods where the subject was fixating the center target were used to correct for small slips of the search coil on the eye (Steffen et al., 2000).

SACCADES: Saccades were recorded while subjects looked toward the target as it made 40° (±20°) steps in position. Horizontal saccades were recorded at five elevations: 20° up, 10° up, 0°, 10° down, and 20° down. Vertical saccades were recorded at five azimuths: 20° left, 10° left, 0°, 10° right, and 20° right. Fourteen saccades were recorded in each direction at each of the eccentricities.

PURSUIT: Pursuit was recorded while subjects looked toward the target as it moved horizontally or vertically at constant velocity (20°/s), with peak-to-peak amplitude of 40°. Horizontal and vertical pursuit was recorded at the same elevations and azimuths as for saccades. Fourteen half-cycles of pursuit were recorded in each direction at each of the eccentricities.

2.4. Data analysis

The data were analyzed in the S-plus (version 8.0, TIBCO Software, Palo Alto, CA) and MatLab (version 7.3.0.267, The MathWorks Inc, Natick, MA) programming environments. The rotation matrices representing head and eye rotation in 3-d space were computed, and the data were scaled using the gain values calculated from the in vitro calibration (Liao et al., 2008). Offset head and eye rotation matrices were computed from in vivo calibration data, to correct for any search coil offset relative to reference position (Tweed et al., 1990). The rotation matrix representing eye-in-head rotation was then computed from the head and eye rotation matrices (Haslwanter, 1995). All subsequent analysis was on eye-in-head position data in axis-angle form (Schnabolk & Raphan, 1994), which gives the orientation of the position axis in 3-d space and the angle of rotation about this axis, and eye-in-head velocity data in velocity vector form (Haslwanter, 1995).

A first-order surface (plane) was fitted to the fixation data by least-squares regression, after using data from periods of central fixation to correct for small slips of the search coil on the eye (Steffen et al., 2000). The orientation of this plane in 3-d space was determined, so that all subsequent data could be rotated into Listing’s coordinates for analysis (Tweed et al., 1990). A second-order surface was also fitted to the fixation data, such that:

$$\mathrm{x}={\mathrm{a}}_1+{\mathrm{a}}_2\mathrm{y}+{\mathrm{a}}_3\mathrm{z}+{\mathrm{a}}_4{\mathrm{y}}^2+{\mathrm{a}}_5\text{yz}+{\mathrm{a}}_6{\mathrm{z}}^2$$

where x, y, and z are the torsional, vertical, and horizontal components of eye position, respectively, and a₁ … a₆ are coefficients. The twist score, a quantitative measure of the amount and direction of surface twist, was given by the a₅ coefficient (Glenn & Vilis, 1992; Desouza et al., 1997). The standard deviation of the fixation data away from the first- and second-order fits was also calculated.

The orientation of the eye velocity axis was determined, for each data point, by calculating the angle of eye velocity axis tilt relative to Listing’s plane. Since the eye velocity axis tilts relative to the z-axis in the pitch (z-x) plane during horizontal eye movements and relative to the y-axis in the yaw (y-x) plane during vertical eye movements, the angle of eye velocity axis tilt was calculated as the arctangent of the ratio of torsional to horizontal or vertical eye velocity. The tilt-angle coefficient (TAC) was calculated as the magnitude of the slope of the relationship between the angle of eye velocity axis tilt and the orthogonal component of eye position (vertical eye position for horizontal eye movements and horizontal eye position for vertical eye movements), as determined by linear least-squares regression to the data from each paradigm. Pursuit data were desaccaded prior to the calculation of TAC (Wyatt, 1998). Data showing evidence of blinks or search coil slip on the eye were discarded. Statistical analysis was performed in S-plus. A paired t-test was used to determine if there was a significant difference between two populations, with = 0.05.

3. Results

3.1. Fixation data spatial plots and twist scores

First- and second-order surfaces were fitted to fixation data, as described above. The standard deviations of data away from these fits and the twist scores, determined from the second-order fits, are given for each subject in Table 2. Across all subjects, the standard deviations of data away from the first- and second-order fits were 0.41 ± 0.11° (mean ± SD) and 0.32 ± 0.10°, respectively. The second-order fits gave a positively twisted surface rather than a planar surface in all subjects; data from a representative subject (subject #1) are shown in Fig. 1 (left panel; twist score 0.28; note that, for display purposes, data are not plotted in Listing’s coordinates). Across all subjects, the twist scores were 0.22 ± 0.12. Note that a twist score of 0 indicates that there is no surface twist (i.e., a planar surface). A positive twist score (>0) indicates surface twist, with positive torsion when the eye is up-right and down-left, and negative torsion when the eye is up-left and down-right. A negative twist score (<0) indicates surface twist, with positive torsion when the eye is up-left and down-right, and negative torsion when the eye is up-right and down-left.

Table 2.

Standard deviations (SD) from first- and second-order surface fits, and twist scores from second-order surface fits, to fixation data.

Subject	SD from First-Order Fit (°)	SD from Second-Order Fit (°)	Twist Score
1	0.27	0.20	0.28
2	0.49	0.28	0.37
3	0.45	0.36	0.25
4	0.52	0.46	0.10
5	0.33	0.32	0.11
Mean ± SD	0.41 ± 0.11	0.32 ± 0.10	0.22 ± 0.12

Fig. 1.

Eye position data and the second-order surfaces fitted to these data are plotted in 3-d space during fixations (left panel), saccades (middle panel), and pursuit (right panel) for a representative subject. The data are not plotted in Listing’s coordinates, for display purposes. Note that the second-order fits gave positively twisted surfaces instead of planes. The twist scores from these fits were 0.28 for fixations, 0.30 for saccades, and 0.33 for pursuit. (Positive values indicate leftward, downward, and clockwise eye rotations.)

3.2. Saccade and pursuit spatial plots

Eye position data during saccades and pursuit are plotted in 3-d space, for a representative subject (subject #1), in Fig. 1 (middle and right panels). As for the fixation data from this subject (Fig. 1, left panel), the second-order fit to the saccade data gave a positively twisted surface (Fig. 1, middle panel; twist score 0.30). The second-order fit to the pursuit data from the same subject also gave a positively twisted surface (Fig. 1, right panel; twist score 0.33).

Eye velocity data during horizontal saccades and pursuit are plotted, for the same subject, in the pitch (z-x) plane in the upper and lower panels of Fig. 2, respectively. During horizontal saccades and pursuit, the direction and magnitude of eye velocity axis tilt was modulated by vertical eye position (Fig. 2). Note that the directions and magnitudes of eye velocity axis tilt were similar for the two paradigms. However, the axis tilt angles did not always remain constant throughout the trajectories; these minor variations were reproducible within subjects, and were dissimilar to the deviations seen with blinks and search coil slips.

Fig. 2.

Eye velocity data during horizontal saccades and pursuit at different elevations are plotted for a representative subject in the pitch (z–x) plane in the upper and lower panels, respectively. Note the differing axis scales for saccade and pursuit data. (Z, horizontal eye velocity; X, torsional eye velocity; (L), left; (CW), clockwise.)

Eye velocity data during vertical saccades and pursuit are plotted, for the same subject, in the yaw (y-x) plane in the upper and lower panels of Fig. 3, respectively. During vertical saccades and pursuit, the direction and magnitude of eye velocity axis tilt was modulated by horizontal eye position (Fig. 3). The directions and magnitudes of eye velocity axis tilt were similar for the two paradigms, but the axis tilts were smaller in magnitude when compared with those occurring during horizontal saccades and pursuit (see Fig. 2). Again, the axis tilt angles did not always remain constant throughout the trajectories, but any minor variations observed were reproducible within subjects.

Fig. 3.

Eye velocity data during vertical saccades and pursuit at different azimuths are plotted for a representative subject in the yaw (y–x) plane in the upper and lower panels, respectively. Note the differing axis scales for saccade and pursuit data. (Y, vertical eye velocity; X, torsional eye velocity; (D), down; (CW), clockwise.)

3.3. Saccade and pursuit tilt-angle coefficients

Eye velocity axis tilt angle data are plotted against instantaneous eye position data, for a representative subject (subject #1), in Fig. 4. A line was fitted to the data from each paradigm by linear least-squares regression (solid lines in Fig. 4), and the TAC was calculated as the magnitude of the slope of the fitted line (indicated at bottom-right in each panel of Fig. 4). For this subject, the TACs for horizontal and vertical saccades were 0.63 ± 0.027 (mean ± 95% CI) and 0.28 ± 0.012, respectively, whereas the TACs for horizontal and vertical pursuit were 0.66 ± 0.002 and 0.31 ± 0.003, respectively. Thus, in this subject, the TACs were greater in magnitude for horizontal than for vertical saccades and pursuit.

Fig. 4.

Axis tilt angles are plotted against instantaneous vertical eye position data for horizontal saccades (upper left panel) and pursuit (lower left panel), and against instantaneous horizontal eye position data for vertical saccades (upper right panel) and pursuit (lower right panel) for a representative subject. The linear least-squares regression fits to the data are indicated as lines in each of the panels. The gradient of that line, the tilt-angle coefficient (TAC), is indicated in the lower right corner of each panel.

A similar trend was observed across all subjects; the fitted lines from each of the subjects are plotted for the four paradigms in Fig. 5. The TACs for horizontal and vertical saccades, across all subjects, were 0.58 ± 0.13 (mean ± SD) and 0.35 ± 0.05, respectively, while the TACs for horizontal and vertical pursuit were 0.64 ± 0.06 and 0.43 ± 0.09, respectively. The TACs for horizontal saccades were significantly greater than those for vertical saccades (p<0.05), and the TACs for horizontal pursuit were significantly greater than those for vertical pursuit (p<0.05). Across all subjects, the average of the horizontal and vertical TACs for saccades and pursuit were 0.47 ± 0.06 and 0.54 ± 0.05, respectively; the average TACs were not significantly different from 0.5 for saccades or pursuit (p>0.05).

Fig. 5.

Linear least-squares regression fits to axis tilt angle versus instantaneous eye position data are plotted for horizontal saccades (upper left panel), horizontal pursuit (lower left panel), vertical saccades (upper right panel), and vertical pursuit (lower right panel), for all subjects. The average tilt-angle coefficient (TAC) is indicated in the lower right corner of each panel.

4. Discussion

4.1. Is the half-angle rule satisfied during saccades and pursuit?

Our results provide further evidence that the half-angle rule is not perfectly satisfied during saccades or pursuit in humans. Although the averages of the horizontal and vertical TACs were not significantly different from 0.5, the horizontal TACs were significantly greater than the vertical TACs for both saccades and pursuit. These findings are consistent with those of prior studies, in which the reported TACs for both horizontal saccades and pursuit were greater than those for vertical saccades (Table 1). However, our study is the first to report the TAC during vertical pursuit and the first to measure the TAC during all four conditions in the same subjects.

4.2. Relationship of eye velocity axis tilt to 3-d eye position

If the half-angle rule were satisfied during saccades and pursuit, 3-d eye positions during these movements would lie on a planar surface in 3-d space. It has previously been noted, however, that 3-d eye positions do not lie on a planar surface, but on a positively twisted surface (Ferman et al., 1987; Glenn & Vilis, 1992; Desouza et al., 1997; Bergamin et al., 2004). The surface twist is present for fixations and saccades (Desouza et al., 1997; Bergamin et al., 2004), and indicates a violation of Listing’s law; since a unique torsional eye position is adopted with each gaze direction, Donders’ law is still satisfied. We also found that, during fixations, 3-d eye positions did not lie on a planar surface, but on a positively twisted surface. The standard deviations of data away from the second-order surface fits were less than 0.5° in all subjects. Furthermore, the twist scores from the second-order surface fits were similar in magnitude and variability when compared with those reported in prior studies (Glenn & Vilis, 1992; Desouza et al., 1997; Bergamin et al., 2004).

While the surface twist might come about because the half-angle rule is not satisfied, it is unclear how the degree and direction of surface twist is related to the horizontal and vertical TACs. Furthermore, it has not been established if Donders’ law is violated when the half-angle rule is not satisfied or when there is a difference between the horizontal and vertical TACs for the same type of eye movement. To investigate these unresolved issues, we developed a model simulation (see Appendix). A 50 × 50° grid was divided into 121 positions (5° horizontal and vertical increments). These positions were randomly ordered, and constant-velocity eye movements were simulated between successive points (starting at position 0°, 0°). For each simulation step of 0.2°, instantaneous torsional velocity was calculated from eye position, horizontal and vertical eye velocity, and the specified TACs (see Appendix), and the instantaneous rotation vector was determined. As expected, if the horizontal TAC (TAC_h) was 0.5 and the vertical TAC (TAC_v) was 0.5, a planar eye position surface was predicted (Fig. 6, left panel), i.e., both Donders’ and Listing’s laws were satisfied. If the TAC_h was greater than 0.5, the TAC_v was less than 0.5, but the two TACs averaged to 0.5, Donders’ law was satisfied, but Listing’s law was not; eye positions did not lie on a planar surface, but on a positively twisted second-order surface (Fig. 6, middle panel). In the extreme case where one TAC was 1 and the other was 0, the simulation predicted either a Helmholtz surface (TAC_h 1, TAC_v 0, twist score 1; Fig. 6, right panel) or a Fick surface (TAC_h 0, TAC_v 1, twist score −1). Between these extremes, the predicted twist scores were linearly related to the difference between the two TACs, so long as the average of the TACs remained 0.5. In our subjects, the average of the TAC_h and TAC_v for both saccades and pursuit was ~0.5, but the difference between them was ~0.2; for this difference, the simulation predicted a twist score of ~0.2, which is close to the mean twist score (0.22) obtained from second-order surface fits to fixation data in our subjects and close to those reported in prior studies (Glenn & Vilis, 1992; Desouza et al., 1997; Bergamin et al., 2004).

Fig. 6.

Simulation predictions of second-order eye position surfaces with variations in horizontal tilt-angle coefficient (TAC_h) and vertical tilt-angle coefficient (TAC_v). If TAC_h = TAC_v = 0.5, the surface is planar (left panel). If TAC_h = 0.6 and TAC_v = 0.4, the surface is positively twisted, with a twist score of 0.20 (middle panel). In the extreme case where TAC_h = 1 and TAC_v = 0, a Helmholtz surface is produced, with a twist score of 1 (right panel).

4.3. Potential experimental artifacts

The potential for artifact, especially in the measurement of torsional eye position, is a concern with the magnetic search coil technique, because the search coil can slip about the line of sight on the eye (van Rijn et al., 1994; Straumann et al., 1995, 1996). We discarded all data showing evidence of search coil slip, which was typically manifest as abrupt inappropriate or inconsistent changes in torsional eye position. In addition, we used data from periods where the subject’s eye returned to the reference position to correct for small slips of the search coil on the eye during the acquisition of the fixation data (Steffen et al., 2000).

Several prior studies have suggested that the orientation of the search coil exiting wire might influence measurement of torsional eye position; the original dual search coil has the exiting wire positioned nasally, whereas a modified version has the exiting wire repositioned inferiorly to minimize mechanical interaction with the eyelids (Bergamin et al., 2002, 2004). One study found that the standard deviation of torsion in each gaze direction was less with the modified search coil (Bergamin et al., 2004), suggesting that Donders’ law was better satisfied, yet a similar degree of second-order surface twist was seen with the two coils and there was no difference in transient torsional deviations. Although the data from the original and modified search coils could not be directly compared (it is not possible to place two search coils on the same eye) or compared with data obtained simultaneously using a separate recording technique, it was proposed that data from the modified search coil is more accurate due to a lesser degree of eyelid interaction with the exiting wire (Bergamin et al., 2004).

In the current study, we used original search coils. Despite this, the standard deviations of data away from the first-order surface fits for our subjects were smaller than those obtained with the modified search coil in the prior study (see Fig. 4a in Bergamin et al., 2004); without incorporating the correction for small search coil slips, the standard deviations of data away from the first-order surface fits for our subjects (0.83 ± 0.39°) were similar to those reported for the modified search coil in the prior study (Bergamin et al., 2004). The twist scores from our subjects were also similar to those reported for both the original and modified search coils in that study (Bergamin et al., 2004). During the saccadic and pursuit trajectories in some subjects, however, there were minor variations in axis tilt angle (e.g., see trajectories for leftward horizontal saccades while looking 20° down in Fig. 2, upper panel); such variations might be partly responsible for the scatter of data seen in the plots of axis tilt angle versus eye position (see Fig. 4). These variations in axis tilt angle were highly reproducible within individual subjects and have been observed in prior studies (e.g., see Figs. 2, 8, and 10 in Tweed et al., 1990 and Fig. 6 in Bruno & Van den Berg, 1997). Given the consistency of our findings within and between subjects for both saccades and pursuit, it is unlikely that our main finding of differing TACs for horizontal and vertical eye movements was due to search coil slip or eyelid interaction with the exiting wire. Ultimately, simultaneous comparison of search coil and high-resolution video techniques will be required to determine the extent of experimental artifact occurring with search coil measurements of ocular torsion.

4.4. Mechanism underlying variations in eye velocity axis tilt

Findings from anatomic, neurophysiologic, imaging, and modeling studies suggest that eye position-dependent eye velocity axis tilts might arise due to extraocular muscle pulleys altering the tendon pulling directions in a manner depending on instantaneous eye position (Demer et al., 1995; Raphan, 1998; Quaia and Optican, 1998; Clark et al., 2000; Kono et al., 2002; Klier et al., 2006, 2011). It is unclear, however, why the horizontal TAC is greater than the vertical TAC. The horizontal rectus pulleys might be positioned more anteriorly in the orbit than the vertical rectus pulleys, thereby giving rise to a larger angle of axis tilt. Alternatively, the combined actions of the vertical recti and oblique muscles might result in torsional torque being partially negated and, thus, a smaller angle of axis tilt during vertical eye movements. However, the pulley locations are not static – they can change depending on instantaneous eye position and eye movement type (Kono et al., 2002; Miller, 2007; Demer et al., 2008) – and, thus, further studies are required to clarify how the disparate TACs arise.

4.5. Conclusions

In summary, we found that Donders’ law is obeyed during eye fixations. As in prior studies, we also found that 3-d eye positions do not lie in a plane, but on a slightly twisted second-order surface. For both saccades and pursuit, TACs are less for vertical than for horizontal eye movements. Although this indicates a violation of Listing’s law, Donders’ law may still be obeyed, with an eye position surface that is similar to that observed during fixations. Our simulation findings suggest that the relationship between the horizontal and vertical TACs is critical for determining both the direction and degree of 3-d eye position surface twist.

Highlights.

• Eye positions during fixation lie on a slightly twisted surface in 3-d space

• Tilt-angle coefficients are greater for horizontal than vertical saccades/pursuit

• The discrepancy in tilt-angle coefficients relates to the degree of surface twist

Appendix

To determine the effect of TAC on 3-d eye position, we simulated a sequence of eye movements between points on a 50 × 50° grid, spaced at 5° intervals and randomly ordered. The simulation began at the center point (0°, 0°); the torsion at this point was set to 0°. The line between adjacent (horizontal-vertical) points in the list was divided into 0.2° increments. A constant velocity eye movement (i.e., pursuit) was simulated along the line; the effective horizontal and vertical velocity components were calculated based on the respective amplitudes and number of steps:

$${\omega }_{H_j}=\raisebox{1ex}{$({\theta }_{H_{j+1}}-{\theta }_{H_j})$}\!\left/ \!\raisebox{-1ex}{$n$}\right.$$ Eq. (A.1)

$${\omega }_{V_j}=\raisebox{1ex}{$({\theta }_{V_{j+1}}-{\theta }_{V_j})$}\!\left/ \!\raisebox{-1ex}{$n$}\right.$$ Eq. (A.2)

where ω_Hj and ω_Vj are the horizontal and vertical components of angular velocity, ω_Hj and ω_Vj are the horizontal and vertical components of eye position, all at step j; and n is the number of 0.2° increments between θ_j and θ_j+1.

The horizontal (TAC_H) and vertical (TAC_V) tilt angle coefficients were then used to calculate the torsional velocity, ωT_j:

$${\phi }_{H_j}={\theta }_{V_j}*{\mathit{\text{TAC}}}_H$$ Eq. (A.3)

$${\phi }_{V_j}={\theta }_{H_j}*{\mathit{\text{TAC}}}_V$$ Eq. (A.4)

$${\omega }_{{\tau }_j}={\omega }_{H_j}*\text{tan}({\phi }_{H_j})-{\omega }_{V_j}*\text{tan}({\phi }_{V_j})$$ Eq. (A.5)

$${\omega }_j=[\begin{array}{ccc}{\omega }_{{\tau }_j}\hfill & {\omega }_{V_j}\hfill & {\omega }_{H_j}\hfill \end{array}]$$ Eq. (A.6)

where φ_Hj and φ_Vj represent the eye velocity axis tilt angles for horizontal and vertical eye movements, respectively. The rotation vector representing the next position was determined by considering the angular velocity vector ω_j as a rotation from r_j to r_j+1:

$$r_{j+1}=\raisebox{1ex}{$r\prime +r_j+r\prime \times r_j$}\!\left/ \!\raisebox{-1ex}{$(1-r\prime ·r_j),$}\right.r\prime =\text{tan}(\raisebox{1ex}{${\omega }_j$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$$ Eq. (A.7)

At the end of each simulated eye movement, the final rotation vector was used as the starting position for the next movement. The process was continued until the last fixation was reached. First- and second-order surfaces were then fitted to the full sequence of rotation vectors, as for the eye position data (see section 2.4), to determine the standard deviations and twist score.