Three-Dimensional Optokinetic Eye Movements in the C57BL/6J Mouse

Abstract

Purpose.: To study three-dimensional optokinetic eye movements of wild-type C57BL/6J mice, the most commonly used mouse in oculomotor physiology. Optokinetic eye movements are reflexive eye movements that use visual feedback to minimize image motion across the retina. These gaze-stabilizing reflexes are a prominent model system for studying motor control and learning. They are three dimensional and consist of a horizontal, vertical, and torsional component.

Methods.: Eye movements were evoked by sinusoidally rotating a virtual sphere of equally spaced dots at six frequencies (0.1–1 Hz), with a fixed amplitude of 5°. Markers were applied to the mouse eye and video oculography was used to record its movements in three dimensions. In addition, marker tracking was compared with conventional pupil tracking of horizontal optokinetic eye movements.

Results.: Gains recorded with marker and pupil tracking are not significantly different. Optokinetic eye movements in mice are equally well developed in all directions and have a uniform input–output relation for all stimuli, including stimuli that evoke purely torsional eye movements, with gains close to unity and minimal phase differences.

Conclusions.: Optokinetic eye movements of C57Bl6 mice largely compensate for image motion over the retina, regardless of stimulus orientation. All responses are frequency–velocity dependent: gains decrease and phase lags increase with increasing stimulus frequency. Mice show strong torsional responses, with high gains at low stimulus frequency.

Compensatory eye movements are a popular model system for connecting neurophysiology and behavior, specifically to study the neural correlates of behavioral plasticity. It is a system in which the sensory input can be fully defined. The output, reflexive compensatory eye movements, and electrophysiological activity, can be recorded and correlated with the sensory input. In addition, by manipulating those reflexive eye movements by using different combinations of sensory input, motor learning can be studied in a well-controlled environment. Oculomotor studies focus more and more on mice. ^1–3 Their small size and fast reproduction rates make them attractive, especially when combined with the availability of many techniques to generate and characterize mutants. Inducing mutations in the oculomotor system or more generally in the cerebellum is a highly useful tool for gaining more insight in the function of the oculomotor or cerebellar system. 

Compensatory eye movements combine visual and vestibular sensory information to provide an organism with a stable retinal image, either during head motion or during motion in the outside world. In afoveate species, image stabilization is achieved by the combination of two reflexes: the vestibulo-ocular reflex (VOR) and the optokinetic reflex (OKR). Foveate species also use smooth pursuit to track a moving target in the outside world. The VOR and OKR work in tandem, the VOR functions best at high-frequency head movements, whereas the OKR operates best at low angular velocities of the visual surround. ^3–7 The combined, synergistic action of these two reflexes results in a good ocular stability over the entire frequency range of natural head rotations. ⁸  

To establish the synergistic action of VOR and OKR, the oculomotor system must transform the visual and vestibular sensory input into a common coding in 3-D space. Head rotations occur in all directions. Therefore, the gaze stabilization reflexes must operate in three dimensions as well. Vestibular information, the change in rotational velocity, is already perceived as three dimensional, due to the architecture of the vestibular canal system. ^9,10 Each organ consists of three semicircular canals that are approximately orthogonal to each other, resulting in a 3-D reference frame with an earth-vertical axis and two horizontal axes, at angles of 45° and 135° measured clockwise from the naso-occipital axis, as viewed from above. ¹¹  

In the retina, information about image motion (retinal slip) is conveyed by direction-selective ganglion cells. Retinal slip signals are further processed at subsequent neuronal stages that are approximately collinear with the best response axes of the vestibular system. The segregation into direction-selective areas is also found anatomically in the accessory optic system, ^12,13 the inferior olive (cat, ¹⁴ rabbit, ¹⁵ and rat ¹⁶ ), and the flocculus (monkey, ¹⁷ rabbit, ^18–20 rat, ²¹ and mouse ²² ). 

So far, the properties of 3-D compensatory eye movements have been described in several species (e.g., primates, ²³ rabbits, ²⁴ and chameleons ²⁵ ). Data from mice, however, are still lacking. In this study, we recorded 3-D optokinetic eye movements in wild-type C57BL/6J mice, the most commonly used mouse strain in oculomotor neurophysiology. 

Materials and Methods

To investigate how the mouse eye moves in three dimensions we used a technique to record eye movements using an infrared video system and three artificial reflective markers that are painted on the cornea. This method merges two earlier methods ^26,27 for recording eye movements using video-oculography. Pupil tracking, the most commonly used video-oculographic method, is unable to capture the full 3-D orientation of the eye, because rotation around the pupillary axis (torsion) is not measured. On the other hand, scleral search coils ^28,29 are known to interfere too much with the eye movement dynamics. ²⁷  

Animal Preparation

In this study, five adult male mice of the C57BL/6J strain were used. They were housed with a 12 hour–12 hour light–dark cycle and unrestricted access to food and water. Experiments were performed during the light phase. All surgical procedures and experimental protocols were performed in accordance with the guidelines set by the Animal Welfare Committee of the Erasmus University and were in accordance with the European Communities Council Directive (86/609/EEC) and the ARVO Statement for the Use of Animals in Ophthalmic and Vision Research. 

Setup and Eye Movement Recording

The animals were prepared for head fixation and placed in a virtual reality setup that displays panoramic monochrome stimuli fully surrounding the animal, allowing a 360° field of view (Fig. 1A). Eye movements were recorded with an infrared video system (ETL-200 with marker tracking modifications; Iscan, Burlington, MA). Images of the eye were captured at 120 Hz with an infrared-sensitive CCD camera. From this image, x and y positions of each of the three makers were recorded in real time, in pixel positions, giving their location on the 512 × 256-pixel grid, with a resolution of one-third pixel horizontally and one-tenth pixel vertically. These positions were low-pass filtered with a cutoff frequency of 300 Hz (Cyberamp 380; Axon Instruments, Union City, CA), sampled at 1 kHz and stored for offline analysis. For pupil tracking, x and y positions of pupil and corneal reflection were recorded in the same way as the markers. Descriptions of all procedures and equipment have been published. ³⁰ Luminance was 17.5 cd/m² and Michelson contrast was 100% in all stimulus conditions, so that the contrast was sufficiently high for the gain versus contrast(/luminance) function of the OKR to be saturated. 

Figure 1.

 

The experimental setup. (A) Top view. A mouse was placed in the setup, with its left eye in the exact center. It was surrounded by three screens on which the stimulus was projected in such a way that it appeared as a virtual sphere from the perspective of the mouse. Reprinted with permission from van Alphen B, Winkelman BHJ, Frens MA. Age-and sex-related differences in contrast sensitivity in C57B1/6 mice. Invest Ophthalmol Vis Sci. 2009;50:2451–2458. © ARVO. (B) Side view with toy mouse for demonstration. The field of view of the mouse was kept unobstructed by recording the eye movements with an infrared camera that was placed under the setup. The eye was tracked through a hot mirror and illuminated by two infrared LEDs at the base of the mirror. (C) Stimuli were rotated about the five axes indicated with dashed lines. (D) Screenshot of a mouse eye with three markers, shown in the white box. The other two dots are corneal reflections of the illumination LEDs. Black crosses: the marker centroids. White box: Iscan region of interest. Note that up and down are reversed in the camera image, because the camera is orthogonal to the table surface and the hot mirror is under a 45° angle,

Marker Placement

Before each recording session, the animal was anesthetized briefly with a mixture of isofluorane and oxygen. Its left eye was topically anesthetized (benoxinate hydrochloride 0.4%, Minims). A small area of the cornea (0.5 mm²) was dried with a paper tip, after which the markers were placed using a blunt needle to apply three dots of titanium dioxide pigment (art. no. 650; Custom Tattoo Supplies Europe, Wervershoof, The Netherlands). These markers can be placed anywhere on the cornea, as long as they remain visible at all times. They are temporary, staying on the eye for 4 to 12 hours, after which they wash off without leaving a trace. After marker placement, the mice were head fixed in the restrainer, placed in the setup, and allowed to recover from anesthesia. Generally, the mice were fully awake and alert after 15 to 20 minutes. 

Optokinetic Stimuli

The stimulus was rendered using a high-performance graphics program (openGL; Silicon Graphics, Inc., Fremont, CA) and consisted of 1592 green dots, each 2° visual angle in diameter, that were equally spaced on a virtual sphere that had its center at the left eye, which in turn was positioned above the table axis, in the center of the triangular screen configuration. The dots were rotated about an axis that ran through the eye. We used sinusoidally moving stimuli with fixed amplitude of 5° and frequencies of 0.1, 0.2, 0.4, 0.6, 0.8, and 1 Hz. Peak velocities were therefore 3.14, 6.28, 12.57, 18.85, 25.13, and 31.42 degrees per second, respectively. The stimuli oscillated about the pitch (interaural), roll (naso-occipital), and yaw (earth vertical) axis, as well as the 45° and 135° horizontal axes (Fig. 1C). Note that the mouse is a lateral eyed animal with a viewing angle of 65° so stimuli about the pitch axis evoke torsional eye movements and roll stimuli evoke vertical eye movements. 

Calibration Procedure

Positions of the markers were recorded in head-fixed camera coordinates: x for the horizontal axis and y for the vertical axis. The optical axis of the camera was defined as the z-axis. The unique orientation of the eye can be computed when the 3-D position of the markers is known. However, only the x and y coordinates of the markers are recorded. When its is assumed that the eye is spherical and moves exclusively with rotation about a single center of rotation, all marker positions are bound to a sphere on which they travel when the eye rotates. Because the camera records a near orthogonal projection of the markers onto a flat surface, the 3-D coordinates of the markers can be straightforwardly computed when the projection of the center of rotation onto the camera image and the radius of the projected sphere are known. Before each experiment, a calibration procedure was performed to find these parameters. 

Procedure for Locating the Center of Rotation.

Four IR LEDs were placed in a square formation around the lens of the camera, equidistant from its optic axis. When the camera's optic axis passes through the center of curvature of a reflective, spherical object (such as a mouse eye), the reflections of the four LEDs recorded by the camera are also equidistant from the center pixel of the recorded image, retaining their square arrangement. This gives the virtual x- and y-coordinates of the center of the sphere over which the markers move. This method was described earlier. ²⁶  

Procedure for Determining the Radius.

A reference LED was placed on the camera, aligned in the vertical plane with its optical axis, creating a corneal reflection. By continuously oscillating, the camera, while translating the mouse both in its longitudinal direction and along the optic axis of the camera, until the recorded CR displacement was minimal, we moved the center of the corneal curvature on the rotation axis of the camera arm (and hence the center of the stimulus setup). Then the camera was placed multiple times in its two most extreme positions, at −13.06° and +13.06°. At each position the x-coordinate of the corneal reflection and the marker closest to the center of the pupil was recorded (c₁ and c₂, p₁ and p₂, respectively). The radius of rotation of the marker was then approximated by using the following formula:  

 

where  

 

This method of determining the radius of rotation of the pupil has been published elsewhere.²⁷ A limitation of the technique is that the derivation of the radius equation assumes that the marker is relatively close to the reference corneal reflection (in angular terms) when the camera is at the center of its swing range. During the marker placement procedure, it is important to place at least one marker as close to the center of the pupil as possible. 

Data Analysis

Marker positions were scaled to positions on a unit sphere and converted into rotation matrices, which in turn were used to compute the instantaneous angular eye velocity, expressed in head centric components (roll, pitch, and yaw). A full description of the mathematical procedure is provided in the Appendix and Supplementary Figure S1. All data were analyzed after the experiment by using custom routines (written in MatLab; The MathWorks Inc., Natick, MA). Experimenters were masked to the experimental conditions. Trials were randomized, mice were assigned a number, and the data analysis scripts were automated. Eye velocity was smoothed with a Gaussian filter (σ = 100 ms). To correct for the camera delay, the velocity signal was shifted 28 ms back in time. Fast phases were removed by using a velocity threshold of twice the stimulus velocity. The time periods that were cut out were extended from 50 ms before the first threshold crossing to 100 ms after the second threshold crossing. Average amplitude and phase values of sinusoidal fits were calculated by using multiple linear regression of eye velocity to in-phase and quadrature components of the stimulus³¹ of the form:  

 

where  

 

Here, x̂_t represents the eye velocity, f is the stimulus frequency, and t − t₀ is the time since stimulus onset. Amplitude (Ā) was then computed as  

 

and phase angle (θ̄) as  

 

where  

 

Gains were calculated as the ratio between the fitted eye velocity amplitude and stimulus velocity amplitude. The best response axis to each stimulus condition was defined as the axis corresponding to the first principal component of the eye velocity data matrix, which consisted of the head-fixed coordinate eye velocities about the roll, pitch and yaw axis (SPSS, ver. 16; SPSS Inc., Chicago, IL). 

Results

Pupil Tracking Compared with Marker Tracking

OKR responses to yaw stimuli were measured two times, once using marker tracking and once using pupil tracking. ²⁷ Stimuli consisted of sinusoidal oscillations about the yaw axis, at six different frequencies (0.1, 0.2, 0.4, 0.6, 0.8, and 1 Hz) with constant amplitude of 5°, resulting in stimulus velocities ranging from 3.14 to 31.42 deg/s. Both methods showed very similar gain (Fig. 2A) and phase plots (Fig. 2B), with high gains at low stimulus velocities, dropping off as stimulus frequency increases. Differences between the two recording methods were tested for significance with a repeated-measures ANOVA with two factors: one between-subjects factor with two levels (recording method with two levels: marker tracking and pupil tracking) and one within-subject factor (stimulus frequency with six levels). We found no main effect of recording method (between subject; F = 1.041, P = 0.365). There was no significant interaction effect: stimulus frequency × recording method (F = 1.322, P = 0.317). We found a main within-subject effect of stimulus frequency (F = 177.635, P < 0.001). These results indicate that marker tracking is a viable alternative for pupil tracking. Also, the average radius of the eye that was found during calibration was 1.52 mm, which places the diameter well within range of other studies. ^32–34  

Figure 2.

 

Marker tracking and pupil tracking were compared by using both methods to measure OKR responses to yaw stimuli. (A) Gains and (B) phases recorded with pupil tracking and marker tracking. Gains and phases were not significantly different (repeated-measures ANOVA, F = 1.041, P = 0.365). n = 5; error bars, SEM.

Best Axis Responses

The optokinetic reflex over the best response axis showed highly uniform response magnitudes for all stimulus conditions, being close to unity at low stimulus frequencies and decreasing with increased stimulus frequency. Similarly, phase lags (the lag between stimulus and resulting eye movement) were small at low stimulus velocities and increased as the stimulus oscillated faster. The five gain curves in Figure 3 were compared by repeated-measures ANOVA. The gain for roll stimuli was significantly lower than yaw (F = 56.44, P < 0.001), HA45 (F = 7.42, P < 0.05), and HA135 (F = 16.85, P < 0.01) stimuli. All other curves were not significantly different. These results show that, at low stimulus frequencies and velocities, the OKR achieves almost perfect compensation for image motion across the retina, regardless of whether the retinal image slip is horizontal (Fig. 3A), vertical (Fig. 3B), torsional (Fig. 3C) or a combination of both (Figs. 3-D, 3E). At higher stimulus frequencies/velocities this compensation deteriorates more or less uniformly for all stimulus directions. 

Figure 3.

 

The best axis responses for stimuli rotating about one of the five stimulation axes: (A) yaw; (B) roll; (C) pitch; (D) H45; and (E) H135. Top: gains at six frequencies; bottom: phase lags between stimulus and eye movement. n = 5; error bars, SEM. Responses show similar low-pass filter characteristics for all conditions. Gains for roll stimuli were significantly lower than yaw (***P < 0.001), HA45 (*P < 0.05), and HA135 (**P < 0.01) stimuli.

3-D Components of the Optokinetic Reflex

Optokinetic responses were decomposed into pitch, roll, and yaw components (Fig. 4). At all stimulus velocities, yaw stimulation resulted in an almost pure horizontal eye movement (i.e., movement about the yaw axis of the eye; see Fig. 1C for the five different stimulation axes). Likewise, roll stimulation resulted in a vertical eye movement (rotation about the roll axis). However, pitch stimulation resulted in a combined vertical and torsional eye movement. Stimulation about the HA45 and HA135 axis resulted in a combined torsional and vertical eye movement of similar magnitude, resulting in an eye movement that was practically on axis. 

Figure 4.

 

(A–E) Gains of the 3-D components (yaw, roll, pitch) of the optokinetic response to stimuli oscillating about one of the five axes (Fig. 1C) are shown. n = 5; error bars, SEM. Some curves were moved by 0.01 Hz (pitch) or −0.01 Hz (yaw) over the x-axis to avoid overlapping error bars. (F) Angle of eye movement (lines) for all horizontal stimulation axes. Each circle represents a gain at a particular stimulus frequency.

Discussion

The present experiments demonstrated for the first time that optokinetic eye movements in mice are well developed in all directions and hence have similar low-pass filter characteristics for all optokinetic stimuli (Fig. 3). At low frequency (0.1 Hz; velocity = 3.14 deg/s), gains were close to unity and phase lags were minimal (∼1°) for all stimulus conditions, which indicates that eye movement largely compensates for image motion over the retina, regardless of stimulus orientation. As stimulus peak velocity increased with frequency, OKR gains decreased and phase lags became larger. The gain and phase of the eye movement response are consistent with reports about horizontal optokinetic eye movements in mice, ^4,7,35,36 rabbit, ^24,37 and cat. ³⁸ Gain curves (Fig. 3) were not significantly different. The only exceptions were gains evoked by roll stimuli, which are significantly lower than those of other horizontal axes stimuli (pitch, HA45, and HA135). The reduced gain for roll stimuli has also been reported by Stahl et al. ³⁹ who found that yaw stimuli evokes gains of ∼0.75 while roll stimuli produces gains of ∼0.5, clearly lower. Also, Andreescu et al. ⁴⁰ reported gains of ∼0.5 for roll stimuli. Although that particular study did not use Yaw stimuli, gains for those can be found in their other work ⁴¹ and are ∼0.8. 

The 3-D eye movements showed some deviation from the stimulation axis, depending on the stimulus condition. Responses to roll stimuli were relatively on axis but had lower gains than the other conditions. Responses to pitch stimuli had high gains but were off axis, falling almost halfway between the pitch and the HA45 axis. Responses to HA45 stimuli had high gains but became more off axis with increasing gains. The HA135 response seemed privileged in comparison to other horizontal axes responses. It had a high gain and was almost on axis so the eye followed the stimulus well in both amplitude and direction (Fig. 4F). This was also described by Tan et al. ⁴² who found best responses for HA135 stimuli in the rabbit. 

The response to the HA45 stimulus was primarily torsional (i.e., an eye rotation about the optic axis of the eye) and reached unity at the lowest stimulus frequency, similar to the rabbit. ²⁴ In comparison, the response in humans to roll optokinetic stimulation (the compensation for which would require a similar torsional eye movement) is only 0.36. ⁴³  

A possible explanation for this can be found in the anatomy of the mouse retina, which differs in anatomy from the human retina. The human retina contains a fovea, a specialized area for vision in which photoreceptors are densely packed within an area that covers 1° of the visual field. To see the world at high spatial resolution, the foveas of both eyes are aimed with fast ballistic movements (saccades) at interesting regions in the field of view. Once a target is fixated, it can be tracked with smooth pursuit movements and the image is stabilized on the retina using compensatory eye movements. In mice, photoreceptors are distributed almost homogenously over the retina, which means that a mouse sees equally well with all parts of its retina; there is no fovea that has to be aimed to get a clearer view. Afoveate animals see optimally when they fully compensate for blurring caused by image motion across the retina, using compensatory eye movements. 

Much of the research on the control of 3-D eye movements focuses on primates. ²³ By using mice in these studies, with all the genetic tools that are available, new approaches can be used to tackle 3-D motor control problems. Of course there are several important differences between mice and primates; most notably the lack of a fovea and related pursuit and saccade systems. This simplicity can be a benefit, because mice face similar challenges as higher vertebrates when controlling compensatory eye movements and have to combine visual, vestibular, and ocular information to achieve optimal compensation without the confound of the participation of foveate ocular motor mechanisms. 

Under natural conditions, when the animal is unrestrained, the OKR works closely together with the vestibulo-ocular reflex (VOR) to keep a stable image on the retina. The OKR compensates for low-velocity retinal slip, whereas the VOR compensates for high-frequency head movements. Together, these two reflexes are able to keep a stable image on the retina for a large range of natural head movements. ⁸ Because the murine OKR works equally well for all stimulus directions used in these experiments, we hypothesize that the VOR of mice will function equally well for sinusoidal oscillations about the yaw, pitch, roll, HA45, and HA135 axes. This hypothesis can be tested with a 3-D motion platform, in combination with a setup as described in this article. 

We recorded 3-D eye movements by tracking the position of three markers that were painted on the eye (Fig. 1D). When recording horizontal optokinetic eye movements evoked by a stimulus that oscillates sinusoidally about the yaw axis, marker tracking and pupil tracking give similar results (Figs. 2A, 2B). Even though one or more markers can be directly in front of the pupil, it is unlikely that this affected acuity or OKR performance. In an earlier paper, ³⁰ we demonstrated that OKR gains decrease as stimuli becomes harder to see, but that a 40% reduction in pupil size did not have a significant effect on visual acuity and optokinetic performance. If visual acuity was affected by markers in front of the pupil, OKR gains recorded using marker tracking would be lower than OKR gains recorded with pupil tracking, but this was not the case, on average. 

The use of video-oculography to track eye movements further reduces the attractiveness of the search coil technique. ²⁹ Scleral search coils affect eye movements in several ways. Physically, they obstruct free eye rotation, a problem that becomes more apparent in smaller animals ²⁷ but that also has consequences for human eye movements. ^44,45 In addition, implanting one search coil in a mouse's eye is complicated. To record 3-D eye movements, a second coil must be implanted, increasing the complexity of the operation. This second coil may further obstruct free rotation of the eye. 

In this study, we used temporary markers that disappeared after 4 to 12 hours. Permanent markers would be more desirable. Keratography, the technique of tattooing the cornea, is successfully used in patients in whom the iris has been damaged or the eye discolored. A plastic surgeon can tattoo a new, artificial iris and pupil on the cornea that looks remarkably real. ⁴⁶ We tried tattooing markers on the cornea, but all those attempts failed, because the mouse cornea is very thin (0.1 mm), tough, and flexible. Successful tattooing of a marker on the cornea necessitates injecting the pigment into the stroma, the middle layer of the cornea that consists of connective tissue. Tattooing markers should be no problem in species with large eyes, such as rabbits, cats, and primates. 

Supplementary Materials

Supplementary Figure S1 - (PDF) 

Footnotes

 Supported by Grants NWO-ALW 813.07.002 (BA) and 016.048.306 NWO-VIDI (BW, MF).

Footnotes

 Disclosure: B. van Alphen, None; B.H.J. Winkelman, None; M.A. Frens, None

The authors thank Jos N. van der Geest for useful comments on the manuscript. 

Appendix

The rotation matrix R_eye describing the unique eye orientation relative to the reference position of the eye expressed in head-fixed coordinates can be computed from marker positions recorded by a camera in several ways. The first step consists of finding the matrix M containing the 3-D coordinates of the marker array:  

 

The x- and y-coordinates (m_x and m_y) of each marker on the unit sphere are determined by subtraction of the projected center of rotation and division by the radius of rotation. The z-coordinate can then be straightforwardly found by using Pythagoras' rule:  

 

The most compact formula that describes the transformation into R_eye involves a multiplication of M by the inverse of the marker array matrix in reference position (M_ref)²⁶:  

 

We provide a more extended version that shows the geometric interpretation in a series of steps, illustrating the relationship with pupil tracking and search coils. The rotation matrix R_eye is decomposed into a sequence of two passive rotations, a torsional rotation about the z-axis followed by a rotation about an axis in the horizontal/vertical plane:  

 

 

The horizontal/vertical rotation matrix R ₂ (ϕ) describes the direction of the circumcenter of the marker array (Supplementary Fig. S1A, black arrow) relative to the direction of the circumcenter when the eye is in reference position (Supplementary Fig. S1A, gray arrow). R ₂(ϕ) is similar to the rotation matrix calculated with pupil tracking techniques or a direction search coil. ⁴⁷ The torsional rotation matrix R ₁(ψ) also describes rotation about the axis pointing toward the circumcenter (Supplementary Fig. S1B) and is comparable to the rotation matrix of an orthogonalized torsion coil. Both matrices can be deduced from the marker positions. 

Once the 3-D coordinates of the markers are known, the surface normal vector n̂ of the marker array M, pointing toward its circumcenter, can be found by calculating the cross-product of two sides of the triangle:  

 

where  

 

The axis ω̂ (Supplementary Fig. S1A, black dotted line) of the horizontal/vertical rotation R₂(ϕ) is perpendicular to the projection of n̂ onto the camera image (n_proj):  

 

where  

 

The angle of rotation ϕ can be calculated from the magnitude of n_proj:  

 

The rotation matrix R₂(ϕ) can be found by using Rodrigues' rotation formula⁴⁸:  

 

To find the torsional rotation matrix R₁(ψ) the marker array M was first rotated back over axis ω̂, using the transpose of the previously found matrix R₂(ϕ), so that n̂ was now aligned with the z-axis (Supplementary Fig. S1B, gray arrow):  

 

In this position, the difference in orientation of each marker i relative to the marker of the array in an arbitrary reference position (with the same alignment of n̂ with the z-axis) gives the torsion angle ψ_i. R₁(ψ) can be calculated as follows:  

 

where  

 

The instantaneous angular eye velocity was determined by computing the rotation matrix yielding the rotation from the previously sampled eye orientation to the next eye orientation. The resultant rotation matrix was in turn converted into axis-angle format⁴⁹ and multiplied by half the sampling frequency. Because the eye velocity was computed relative to the camera, which in turn was positioned at 70° azimuth from straight ahead, a final multiplication with matrix R₃θ was necessary to align the coordinate system with the stimulus (roll, pitch, and yaw). This consisted of swapping the y- and z-components, rotation of the x- and y-components about the z-axis by angle θ = −20° and an inversion of the x-axis, which makes the coordinate system right-handed: