## Abstract

To investigate the role of noncommutative computations in the oculomotor system, three-dimensional (3D) eye movements were measured in seven healthy subjects using a memory-contingent vestibulooculomotor paradigm. Subjects had to fixate a luminous point target that appeared briefly at an eccentricity of 20° in one of four diagonal directions in otherwise complete darkness. After a fixation period of ∼1 s, the subject was moved through a sequence of two rotations about mutually orthogonal axes in one of two orders (30° yaw followed by 30° pitch and vice versa in upright and 30° yaw followed by 20° roll and vice versa in both upright and supine orientations). We found that the change in ocular torsion induced by consecutive rotations about the yaw and the pitch axis depended on the order of rotations as predicted by 3D rotation kinematics. Similarly, after rotations about the yaw and roll axis, torsion depended on the order of rotations but now due to the change in final head orientation relative to gravity. Quantitative analyses of these ocular responses revealed that the rotational vestibuloocular reflexes (VORs) in far vision closely matched the predictions of 3D rotation kinematics. We conclude that the brain uses an optimal VOR strategy with the restriction of a reduced torsional position gain. This restriction implies a limited oculomotor range in torsion and systematic tilts of the angular eye velocity as a function of gaze direction.

## INTRODUCTION

The vestibuloocular reflexes (VORs) stabilize gaze and, at the same time, adjust ocular orientation in space. To accomplish this task and to maintain visual spatial orientation constant, the brain must keep track of intervening head rotations and estimate the resultant change in head orientation relative to space. How the visual and oculomotor systems solve the underlying computations that involve noncommutative operations is still a matter of debate.

In an elegant study, Tweed and colleagues (1999) have shown that human subjects maintain fixation stable on a peripheral target after it is switched off, even during intervening rotations in the yaw and roll plane that result, depending on the order of the rotations, in different final head orientations. This observation strongly suggests that the VOR generates eye position holding commands that appropriately reflect the underlying three-dimensional (3D) kinematics, even under circumstances where the final head-in-space orientation results from intervening rotations about different axes in space. On the computational level, this finding implies that the brain controls ocular orientation by processing head angular velocity and position signals in a highly nonlinear way (Crawford et al. 2003; Tweed 1997; Tweed and Vilis 1987). It further implies that the neural commands to the oculomotor plant cannot be independently generated by functionally separate horizontal and vertical motor controllers because the underlying computations generally involve all three rotational degrees of freedom.

If the oculomotor system has learned during evolution to control eye position by complying with the kinematic intricacies of head movements in 3D space, it should appropriately work for any combination of head rotations because appropriate position control depends on the planes of intervening rotations. The experiments of Tweed and colleagues (1999) have shown that in the yaw-roll domain the vertical gaze direction depends on a noncommutative controller. In an analogous way, the order of rotations in the pitch-roll domain affects the horizontal gaze direction. Combinations of yaw and pitch rotations in different order, however, have little impact on the final gaze direction as they lead to discrepancies of less than ∼0.2° for targets within a range of 20° around straight ahead. In contrast to the gaze direction, the torsional orientation of the eyes will, however, change polarity and can differ by several degrees in amplitude. The reason for this is that such head movement sequences mimic a roll rotation, characterized by a fixed rotation sense relative to straight ahead. In conflict with these predictions, it has been suggested that the VOR compromises ocular stability by partially complying with Listing's law (Migliaccio et al. 2003; Misslisch et al. 1994, Misslisch et al. 1996; Palla et al. 1999). This law describes ocular torsion as a unique function of horizontal and vertical eye position (Helmholtz 1866; Tweed and Vilis 1990), implying that there could be no change in torsional polarity for the same gaze direction in yaw–pitch rotation sequences if the VOR complied to whatever degree with this kinematic constraint.

In this paper, we studied combinations of yaw and pitch rotations to probe the extent to which the VOR controls ocular orientation as a function of gaze direction. More specifically, we were interested in the question whether 3D rotation kinematics could explain the properties of the rotational VOR during combined head-in-space rotations in the yaw and pitch plane without presupposing any compliance with Listing's law. Because the head will change orientation relative to gravity after intervening rotations about orthogonal axes, we also tried to determine the role of the otolith signals by comparing the ocular torsion following combinations of yaw and roll rotations in upright and supine orientation. We found that all characteristics of the rotational VORs in far vision can be derived from the concept of an optimal VOR that per definition mirrors the 3D rotational kinematics of the head-in-space-motion.

## METHODS

### Subjects

Experiments were conducted on seven healthy subjects (3 men, 4 women), in the age of 30–47 yr after they gave informed consent to the experimental protocols. The general experimental procedures have been approved previously by the Ethic Committee of the Canton Zürich, Switzerland.

### 3D eye-movement recording

Three-dimensional eye movements were measured with the magnetic search coil technique using a three-field Angle Meter NT (Primelec, Regensdorf, Switzerland). Three pairs of cubic single wire frames of 70 cm length generated three orthogonal, digitally synchronized magnetic fields (51.2 kHz ± 400 Hz). The search coil signals were amplified and multiplexed before passing through the turntable slip rings. In a first step, the offset voltages in the measuring chain (search coil-preamplifiers-turntable slip rings-Angle Meter NT-Cambridge Electronics Device 1401 Plus-PC) were zeroed after having placed the coil into a metal tube in the center of the magnetic field to shield it from the magnetic flux. The 3D sensitivity of the coil system was measured by maximizing the coil output along each of the three orthogonal field directions. The ratio of these three numbers was used to equalize small gain differences of the system along these directions. Finally, the subject was seated inside the coil frame and the search coil was placed onto the anesthetized eye (0.4% Oxybuprocaine). In a second step, 3D eye position was calibrated by having the subject fixate nine target positions in close-to-primary and secondary gaze positions. Based on these fixations both the coil orientation on the eye and offset voltages were simultaneously computed (Klier et al. 2006). We also estimated the orientation of primary eye position (Listing's plane) (Misslisch and Hess 2000; Tweed and Vilis 1990) based on fixation data. In a few subjects, primary position deviated >1–2° from straight ahead. However, transformation of eye-position data into Listing's coordinates did not significantly affect results. All signals were sampled at a rate of 833.33 Hz and stored on the hard disk of a personal computer for off-line analysis with Matlab software (the Math Work).

### Experimental protocols

Subjects were seated comfortably on a three-axis servo-controlled motorized chair (Acutronic) with the head centered at the intersection of the three axes. The upper and lower torso was secured with a five-point aviation safety belt. To reduce passive movements of body during tilt, evacuation pillows filled the empty space under the arms and between the shoulders and the chair's side frame. A malleable thermoplastic mask was molded to the subject's face and fastened to the chair behind the subject's head to keep it fixed relative to the rest of the body.

Subjects were tested in two types of experiments (Fig. 1 *A*): *yaw-then-pitch/pitch-then-yaw (yaw-pitch/pitch-yaw) experiment*: yaw rotation through 30° followed by pitch rotation through 30° (*yaw-pitch paradigm*) or pitch rotation through 30° followed by yaw rotation through 30° (*pitch-yaw paradigm*); *yaw-then-roll/roll-then-yaw (yaw-roll/roll-yaw) experiment*: yaw rotation through 30° followed by roll rotation through 20° (*yaw-roll paradigm*) or roll rotation through 20° followed by yaw rotation through 30° (*roll-yaw paradigm*). The yaw motion was performed with the innermost axis and the roll and pitch motion with the middle axis, both moving at constant acceleration/deceleration of ±120°/s^{2} to reach peak velocities of ±60°/s. The yaw-roll/roll-yaw experiment was also performed in supine orientation where the head roll axis coincided with the outermost axis, whose peak acceleration was limited to ±80°/s^{2}. In this case, the peak velocity was only 40°/s.

The time course of a typical experiment is illustrated in Fig. 1*B*. Each trial started with fixation of a small laser spot that appeared in one of the four quadrants for 1 s in otherwise complete darkness. The fixation targets consisted of a small red laser spot, which appeared at an eccentricity of 20° on one of the two diagonals through the fixation point straight ahead, namely “right and up” (45°), “left and up” (135°), “right and down” (−45°), and “left and down” (−135°) (Fig. 2 *D*). The distance between subject and projection screen was 1.55 m in the yaw-roll/roll-yaw and 1.47 m in the yaw-pitch/pitch-yaw experiment. In supine orientation, the distance between subject and screen was 1.50 m. After the fixation time, the target disappeared and the subject was required to fixate the memorized location in space while one of the two rotation sequences started. The rotations were always about cardinal head axes (*x*, *y*, and *z* axes in Fig. 1*A*) and directed such that the imaginary target moved relative to the subject along a rectangular path from one “corner” to the diagonally opposite corner of the oculomotor range. For example, if the subject was fixating a target in the upper left quadrant, the rotation sequences were leftward rotation followed by upward rotation or upward rotation followed by leftward rotation (Fig. 2*D*). In this way, the imaginary target in space never moved out of the oculomotor range. To perform the second rotation about a cardinal head axis, it was necessary in the yaw-roll paradigm to start the yaw rotation from an offset position. As a consequence, in supine, head orientation relative to gravity differed at onset of rotation from that at final head orientation. Moreover, the directions of gravity at the end of the sequence always differed from each other for two complementary sequences of a paradigm. The single axis rotation sequences and the initial and final head orientations are summarized in Table 1, *A* and *B* for the yaw-roll/roll-yaw experiment and in Table 2 for the yaw-pitch/pitch-yaw experiment.

The two motion protocols for the yaw-roll/roll-yaw and yaw-pitch/pitch-yaw experiments, the four target locations (Q1–Q4 in Fig. 2*D*), and the two different start-up orientations for yaw-roll/roll-yaw (upright and supine) and two different offset conditions for the pitch-yaw paradigm in the yaw-pitch/pitch-yaw experiment resulted in a total 28 different protocols, each of which was repeated at least four times. They were randomized and tested in two sessions.

### Control experiments

To check the influence of target eccentricity and motion amplitude in the yaw-pitch/pitch-yaw experiment, we tested in three subjects the target eccentricities of 7, 15, and 20° (= standard eccentricity) and 24° with the respective motion amplitudes of 10, 20, and 30° (= standard amplitude) and 35° in each of the four fixation quadrants. Finally, to check the influence of final head orientation relative to gravity, the pitch-yaw paradigm in the yaw-pitch/pitch-yaw VOR experiment was repeated in three subjects with different initial head orientations as summarized in the Table 2.

### Data analysis

The raw data from each subject's right or left eye, sampled at 833 Hz was first converted into rotation vectors, which represent the horizontal, vertical, and torsional position of the eye in the angle-axis format * E* = (

*E*

_{tor}

*E*

_{ver}

*E*

_{hor}) =

*tan(ρ/2), where ρ describes the angle through which the eye rotated relative to a reference position and*

**e****e**is a unit vector parallel to the axis of the respective ocular rotation (Haustein 1989). Specifically, if the eye moves horizontally, i.e., about the head vertical axis [

**e**= (0 0 1)], this equation implies that

*E*

_{hor}= tan(ϑ/2) where ϑ is the horizontal eccentricity relative to straight ahead (taken as reference position) and

*E*

_{ver}=

*E*

_{tor}= 0. In an analogous way, if the eye moves vertically [

**e**= (0 1 0)], we write

*E*

_{ver}= tan(φ/2), where the angle φ denotes the vertical eccentricity relative straight ahead, and for a pure torsional eye movement about the

*x*axis [

**e**= (1 0 0)], we write

*E*

_{tor}= tan(ψ/2), where the angle ψ is the torsional eccentricity relative to straight ahead. The angles ϑ, φ, and ψ are illustrated in Fig. 2

*D*. From these rotation vector relations, we computed the angular eye velocity as (Hepp 1990) (1) where we use bold letters to design three-component vectors. Note that for an eye-position vector

**E**parallel to its rate of change d

**E**/d

*t*, during a yaw or pitch VOR,

*Eq. 1*simplifies to

**Ω**≈ 2 d

*/d*

**E***t*.

### 3D rotation kinematics of an optimal VOR

Stabilization of the entire retinal image requires that the VOR spins the eye independent of eye position about the same axis as the head but in opposite direction (*optimal VOR strategy*, for details, see appendix). Under this assumption, we can compute the predicted 3D eye positions that keep gaze stable in space in the yaw-pitch/pitch-yaw and in the yaw-roll/roll-yaw experiments for each of the two possible combinations of single-axis rotations in yaw and pitch and yaw and roll, respectively. These calculations show that there is virtually no difference in 2D (i.e., vertical and horizontal) eye end positions for fixation points within a 20° radius relative to straight ahead when comparing the yaw-pitch with the pitch-yaw paradigm (difference <0.2°), whereas in the same oculomotor range eye end positions yield differences of as much as 10.3° in the yaw-roll versus roll-yaw paradigm. However, in both the yaw and pitch VOR, ocular torsion changes as a function of the tertiary gaze position. Thus in the *yaw VOR*, the rotation kinematics predicts that ocular torsion changes as a function of vertical eccentricity (*E*_{ver}) and horizontal eye velocity (d*E*_{hor}/d*t*) as follows (2)

Similarly, during *pitch VOR* ocular torsion changes as a function of horizontal eccentricity (*E*_{hor}) and vertical eye velocity (d*E*_{ver}/d*t*) (3)

In the following, we refer to the ocular torsion (and its time derivative) described in *Eqs. 2* and *3* as *unrestrained kinematic torsional position (and velocity*) to distinguish it from the experimentally measured torsional position (and velocity). The experimentally measured ocular torsion can be a consequence of 3D rotation kinematics (i.e., the result of integrating the differential *Eqs. 2* and *3*), or it can be due to otolith or other contaminating inputs to the motor neurons. Throughout this paper we use the term “kinematic ocular torsion” in the sense of “ocular torsion produced by mathematical integration of *Eqs. 2* and *3*.” If this integration occurs with a gain of unity, the “kinematic” ocular torsion reflects the unrestrained torsional motion component predicted by the kinematics of 3D rotations. Often we are interested more in the change in ocular torsion than in its absolute value. It follows from these equations that the change of kinematic ocular torsion during the VOR should be proportional to the time integral over the vestibularly induced eye velocity times the respective constant eye-position component. In the yaw VOR, we obtain the following ratio of the experimentally measured torsion (denoted by *E*_{tor}*) and the unrestrained kinematic torsion (4) with *t*_{0} = onset, *t*_{1} = offset of the VOR. A similar ratio holds in the pitch VOR (5)

The factors *k*_{Y} and *k*_{P} determine the *torsional position gains* in the yaw and pitch VOR. They depend on the properties of the underlying integrator (see discussion). Note that for *k*_{Y} = *k*_{P} = 1, the experimentally measured torsion equals the torsion predicted by 3D rotation kinematics. Geometrically, these position gains determine the amount of torsion that the eye acquires during the VOR (i.e., its rotation about the *x* axis) depending on gaze eccentricity.

In the yaw VOR, we quantify the deviation of the eye position from the head's rotation axis in the pitch plane (Fig. 2*B*) by the tilt of the line segment from VOR onset to offset, which we characterize by the angle τ_{Y} = tan^{−1}(Δ*E*_{tor}*/Δ*E*_{hor} = −tan^{−1}(*k*_{Y}*E*_{ver}). Using the relation *E*_{ver} = tan(φ/2) with φ denoting vertical gaze eccentricity relative to reference position, we have τ_{Y} ≈ −*k*_{Y}φ/2. In the pitch VOR, we quantify the deviation of the eye position from the rotation axis in the yaw plane (Fig. 2*C*) by the tilt of the line segment from VOR onset to offset, i.e., the angle τ_{P} = tan^{−1}(Δ*E*_{tor}*/Δ*E*_{ver}) = tan^{−1}(*k*_{P}*E*_{hor}). Using *E*_{hor} = tan(ϑ/2)with ϑ denoting horizontal gaze eccentricity, we have τ_{P} ≈ *k*_{P}ϑ/2. Note that if the VOR follows 3D kinematics (described by *Eqs. 2* and *3*), the respective torsional position gains *k*_{Y} and *k*_{P} would be unity, implying a ratio of the angular tilt of the eye-position trajectory to gaze eccentricity of about one-half. In numeric terms, one would expect a change in torsion of as much as 7.6° for eccentric fixations of ≤20°, assuming an optimal vestibular response (i.e., with both unity velocity and torsional position gains) to a triangular head-velocity profile (constant acceleration and deceleration) reaching a peak velocity of 60°/s. In contrast, if the torsional position gain were smaller than unity, e.g., one-half, this ratio would equal approximately one-quarter (with an error <1.1% for gaze eccentricity < 30° due to ignoring the inverse tangent relation).

In the roll VOR, the unrestrained kinematic torsion differs from the experimentally measured torsion in an analogous way to the yaw and pitch VOR (see *Eqs. 2* and *3*). We postulate that (6) where *k*_{R} is the *torsional position gain* in the roll VOR. To justify this relation, we first note that for eccentric gaze positions the kinematically expected torsional eye velocity in the roll VOR can be expressed as a unique function of horizontal and vertical eye position and velocity. Specifically, we have d*E*_{tor}/d*t* = (*E*_{ver}d*E*_{hor}/d*t* − *E*_{hor}d*E*_{ver}/d*t*)/(*E*_{ver}^{2} + *E*_{hor}^{2}), where the right hand side of this equation corresponds to the sum of the two equations (*2* and *3*) divided by the square of the distance of the eye from zero position (for details, see appendix, *Eq. A12*). The crucial point here is that now both the horizontal (*E*_{hor}) and vertical eye position (*E*_{ver}) continuously change in time. Importantly, this relation does not directly depend on head angular velocity. After integrating both sides of *Eq. 6* from VOR onset (*t*_{0}) to end (*t*_{1}), we can define the torsional position gain *k*_{R} as the ratio between the change in the experimentally measured torsion and the unrestrained kinematic torsion that is associated with the horizontal and vertical eye movement (7) See also *Eq. A12* in appendix.

In summary, we measured final torsional eye position at the end of each rotation sequence as indicated in Fig. 1 (vertical line labeled E). To determine the torsional position gains (*Eqs. 4*, *5*, and *7*), we computed the change in torsional eye position by integration of the slow phase velocities from VOR onset to stop (Fig. 1, between vertical lines labeled S and E).

### Tilt of angular eye velocity in the yaw, pitch, and roll VOR

Using the kinematic equations for the rate of change of eye position d**E**/d*t* as a function of eye position **E** and the respective head angular velocity **Ω**_{head} (see *Eqs. A3*, *A7*, and *A11* in appendix), *Eq. 1* yields the following proportionalities (denoted by ∼, see also *Eqs. A13*–*A15* in appendix)

For the yaw VOR with **Ω**_{head} = ω (0, 0, 1) (8)

For the pitch VOR with **Ω**_{head} = ω (0, 1, 0) (9)

And for the roll VOR with **Ω**_{head} = ω (1, 0, 0) (10)

In these equations, ω = ω(t) represents the head angular velocity profile of the rotation about the yaw, pitch, and roll axis, respectively, and the factors *g*_{yaw}, *g*_{pitch}, and *g*_{roll} are the respective velocity gains of the transduction of the physical head angular velocity into a central velocity signal (see *Eqs. A3*, *A7*, and *A11* in the appendix). In contrast to the yaw and pitch VOR, the gain of the roll VOR in *Eq. 10* depends not only on the velocity gain (*g*_{roll}) of the system but also on the torsional position gain (*k*_{R}). We write for the overall roll gain in the roll VOR *G*_{roll} = (Ω_{rollVOR})_{tor}/‖Ω_{head}‖ = *g*_{roll}*k*_{R} that depends on *k*_{R}, whereas in the pitch and yaw VOR, we have *G*_{pitch} = (Ω_{pitchVOR})_{ver}/‖Ω_{head}‖ = *g*_{pitch} and *G*_{yaw} = (Ω_{yawVOR})_{hor}/‖Ω_{head}‖ = *g*_{yaw}, respectively, which do not depend on the torsional position gains *k*_{P} and *k*_{Y} (note that ‖Ω_{head}‖ = |ω|). The proportionality factors for *Eqs. 8*–*10* and other details can be found in the appendix. The crucial point of these equations is the prediction that the torsional position gains *k*_{Y}, *k*_{P}, and *k*_{R} not only limit the torsional oculomotor range but also determine the tilt angles of the angular eye velocity in the pitch and/or yaw plane.

To characterize the angular velocity tilt of the yaw VOR in the pitch plane, we measured the inverse tangent of the ratio of torsional to horizontal angular velocity component, which yields the angle σ_{Y} = tan^{−1}[(Ω_{yaw})_{tor}/(Ω_{yaw})_{hor}] = tan^{−1}[(1 − *k*_{Y})*E*_{ver}]. Expressing vertical gaze eccentricity by *E*_{ver} = tan(φ/2), we can approximately write σ_{Y} ≈ (1 − *k*_{Y})φ/2. Similarly, to characterize the angular velocity tilt of the pitch VOR in the yaw plane, we measured the inverse tangent of the ratio of the torsional to vertical velocity component, yielding σ_{P} = tan^{−1}[(Ω_{pitch})_{tor}/(Ω_{pitch})_{ver}] = tan^{−1}[(*k*_{P} − 1)*E*_{hor}]. Expressing the horizontal gaze eccentricity by *E*_{hor} = tan(ϑ/2), we can write σ_{P} ≈ (*k*_{P} − 1)ϑ/2. For example, if the position gains are about one-half these equations predict *the quarter angle rule* of angular eye-velocity tilt in the yaw and pitch VOR (Misslisch et al. 1994).

We also evaluated the roll VOR in the roll-yaw paradigm to experimentally determine the relations between the torsional position gain (*k*_{R} of *Eq. 7*), the angular eye velocity, and vertical and horizontal gaze direction. The velocity gain of the roll VOR was estimated by *g*_{roll} = *G*_{Roll}/*k*_{R}, where *G*_{roll} is the overall roll VOR gain. Because gaze eccentricity (20°) during fixation of each of the four targets in the four quadrants had a horizontal and vertical component of approximately ±14°, the deviation of the angular eye velocity from the head roll axis theoretically occurs in the diagonal planes bisecting the pitch and the roll plane. To estimate the tilt of roll angular velocity, we fitted straight lines to the experimentally measured torsional component of the roll angular velocity as a function of either the horizontal or the vertical component. From these fits, we computed the tilt angle in the pitch plane, σ_{RP} =tan^{−1}[(Ω_{roll})_{hor}/(Ω_{roll})_{tor}] = tan^{−1}{[(1 − k_{R})/*k*_{R}]*E*_{ver}}, and the tilt angle in the yaw plane, σ_{RY}= tan^{−1}[(Ω_{roll})_{ver}/(Ω_{roll})_{tor}] = tan^{−1}{[(*k*_{R} − 1)/*k*_{R}]*E*_{hor}} (see *Eq.10*). The half angle rule of angular eye velocity in the roll VOR predicts that the torsional position gain *k*_{R} is about one-half (Misslisch et al. 1994). If this is the case, we should find σ_{RY} ≈ −ϑ/2 and σ_{RP} ≈ φ/2 for the respective gaze eccentricities in the horizontal (ϑ) and vertical plane (φ). Easier to test is the prediction of these equations that the angular velocity should tilt opposite to the deviation of gaze direction relative straight ahead.

Finally, to assess the overall influence of torsion, we compared the difference between final eye positions **E**^{a} = (*E*_{tor}^{a}, *E*_{ver}^{a}, *E*_{hor}^{a}) after a yaw-then-roll or yaw-then-pitch head rotation (labeled by a) with the final eye position **E**^{b} = (*E*_{tor}^{b}, *E*_{ver}^{b}, *E*_{hor}^{b}) after a roll-then-yaw or pitch-then-yaw rotation (labeled by “b”), respectively, by calculating the following 2D distances in the yaw-roll/roll-yaw experiment (11) and in the yaw-pitch/pitch-yaw experiment (12)

Similarly, we also calculated the 3D distance (13)

To study its dependency on the order of head rotations, we plotted each component of the eye-position vector **E**^{a} against the respective component of **E**^{b}. Commutative response components must lie on the diagonal of such plots.

### Statistical analyses

Statistical analyses were performed with Matlab (The MathWorks). Repeated-measures two-way ANOVA with factor A order of rotation and factor B fixation quadrants was used to determine the significance of differences in torsion or differences in 2D and 3D final eye positions in the yaw-roll as well as in the yaw-pitch VOR paradigm. If appropriate, variances were computed by considering the propagation of errors of the independent variables (Bevington and Robinson 1992). Linear least-squares fit of the equation *E*_{x}^{β} = *a* + *bE*_{x}^{α} was performed on average eye positions *E*_{x}^{β} (±1 SD), reached after the motion sequence β, plotted versus *E*_{x}^{α} (±1 SD), reached after motion sequence α, for each component “x” (x = tor, ver, hor) of the eye-position vector * E* to check their dependency on the order of head rotations. Because the eye-position data were subject to errors in both

*E*

_{x}

^{α}and

*E*

_{x}

^{β}, we used the method of fitting a straight line to data with errors in both coordinates. More specifically, we minimized the χ

^{2}merit function χ

^{2}(

*a,b*) = (

*E*

_{xi}

^{β}−

*a*−

*bE*

_{xi}

^{α})

^{2}/(

*s*

_{β}

^{2}+

*b*

^{2}

*s*

_{α}

^{2}) where

*s*

_{α}

^{2}is the variance of

*E*

_{x}

^{α}and

*s*

_{β}

^{2}is the variance of

*E*

_{x}

^{β}(Press et al. 2002).

Finally, we used the Bootstrap method to estimate the variability of the parameter *q* in the equation *f*(*k*_{R}) = *q*(1 − *k*_{R})/*k*_{R}, which we fitted to the tilt of the angular velocity in the roll VOR (see *Eq. 10*).

## RESULTS

An example of the eye movements during a pitch followed by a yaw rotation is illustrated in Fig. 1*B*, showing the responses in the time domain. Close inspection of the eye- and head-position traces shows that the horizontal and vertical eye positions appropriately change direction to keep gaze stable on the memorized target location in space. In addition, there is a positive increment in torsion during the pitch-up motion of the head with constant leftward gaze as well as during the leftward yaw motion with constant downward gaze, in accordance with the predictions of *Eqs. 2* and *3* (methods). The torsional end position was typically held constant for ≥1 s after the end of rotation (Fig. 1*B*). The spatial characteristics of these responses are visualized in Fig. 2, *A–C*. The format of this figure is a compromise between a representation of the eye movements as horizontal and vertical gaze movements and a representation as rotations about a head-fixed axis with three components. For computational reasons, 3D eye positions are conveniently represented as rotations in the angle-axis format. In this format, a pure horizontal or a pure vertical eye movement through the angle ϑ and φ, respectively (Fig. 2*D*), is represented as a point moving along the z (horizontal rotation axis) and *y* axes (vertical rotation axis), respectively. Similarly, a pure torsional eye movement through the angle ψ (Fig. 2*D*) is represented as a point moving along the *x* axis (torsional rotation axis). For ease of comparison with the location of the fixation targets (Q1–Q4, Fig. 2*D*), we plotted the horizontal and vertical eye-movement components such that they can be thought of depicting the motion of the tip of the gaze line in space relative to the stationary subject (compare Fig. 2, *A* and *D*). Notice that in fact the subject is moving while gaze remains stationary in space. Although the horizontal and vertical eye positions followed symmetrical trajectories during the yaw-then-pitch and pitch-then-yaw head movement (Fig. 2*A*), the final torsional eye position at the end of the two motion sequences changed orientation by 180° relative to the pitch as well as the yaw plane (Fig. 2, *B* and *C*).

### 3D-rotation kinematics predicts torsional eye position in the yaw-pitch paradigm

To compare the experimentally observed changes in the torsional end position of the eyes with the 3D kinematic predictions, we first computed the predicted torsion at the end of the yaw-pitch and pitch-yaw VOR based on the actual change in horizontal and vertical eye position for each of the fixation targets Q1–Q4 (Fig. 2*D*) as described by the *Eqs. 2* and *3* (methods). Because we were only interested in the absolute change in torsion at the end of the two rotation sequences, we computed the absolute differences (|Δ*E*_{tor}|, *Eqs. 4* and *5*) between the predicted torsion after clockwise (e.g., a leftward followed by an upward rotation; Fig. 2*A*, gray gaze line) and counterclockwise (e.g., an upward followed by a leftward rotation; Fig. 2*A*, black gaze line) head-rotation sequences. We found that the experimentally observed absolute differences in torsion (|Δ*E*_{tor}*|, *Eqs. 4* and *5*) were consistently smaller than the predicted torsional differences for all four target eccentricities (Fig. 3). There was no dependence on the fixation quadrants. The ratio of the experimentally observed and predicted absolute torsional difference (|Δ*E*_{tor}*|/|Δ*E*_{tor}|), averaged across all four quadrants and the four different eccentricities was 0.57 ± 0.18 (*subject TT*), 0.51 ± 0.14 (*subject US*), and 0.52 ± 0.31 (*subject AS*). Using these ratios, we found that on average the predicted torsions, after downscaling by these factors, closely matched the experimentally measured torsions. A linear least-squares fit of the pooled experimental torsions from these three subjects as a function of the respectively downscaled predicted torsions at each eccentricity yielded a straight line with a slope of 0.82 (*r*^{2} = 0.93), supporting the notion of a constant proportion between predicted and experimental ocular torsion as a function of target eccentricity (compare gray and black lower curves in Fig. 3).

Differences in the experimentally observed and theoretically expected ocular torsion, called *unrestricted kinematic* ocular torsion, were also evident in the velocity domain (compare brown and red velocity traces in Fig. 4, *A* and *B*). To quantify these differences, we fitted the torsional eye and gaze velocity by second-order spline functions and computed the area enclosed by each of this functions and the time axis. The ratio of these areas corresponded to the torsional position gain of the yaw and pitch VOR (*k*_{Y}, *Eq. 4*, and *k*_{P}, *Eq. 5*). The torsional position and velocity gains for the yaw and the pitch VOR are summarized in Table 3.

### Noncommutativity of ocular torsions in the yaw-pitch paradigm

We found that the polarity of ocular torsion switched depending on the order of the rotation sequences in the yaw-pitch/pitch-yaw experiment (compare Fig. 2, *B* and *C*), supporting the notion that the VOR is noncommutative in the torsional domain. For any of the four fixation targets, final torsional positions depended significantly on the order of head rotation [factorial ANOVA, targets Q1, Q4: *F*(1,24) = 28, *P* = 0.000; targets Q2, Q3: *F*(1,24) = 38.1, *P* = 0.000; Fig. 5, *A* and *B*]. In contrast to torsion, final horizontal and vertical eye positions generally differed little from each other after yaw-then-pitch and pitch-then-yaw head rotations in upright (Fig. 2*A*). Across all subjects and trials we found an average difference of L_{2D} = 3.6 ± 2.3° (*n* = 78). Despite these differences, neither horizontal nor vertical eye positions depended on the order of head rotation for any of the four fixation targets [factorial ANOVA *F*(1,48) = 0.45, *P* = 0.5 for vertical eye position; *F*(1,48) = 0.0095, *P* = 0.92 for horizontal eye position]. Both vertical (*E*_{ver}) and horizontal (*E*_{hor}) final eye positions lay close to the commutative diagonal in the yaw-then-pitch versus pitch-then-yaw diagrams: denoting final vertical eye position by *E*_{verYP} if measured in the yaw-pitch paradigm and by *E*_{verPY} if measured in the pitch-yaw paradigm, we found that *E*_{verPY} as a function of *E*_{verYP} yielded a straight line fit with minimal offset and close to unity slope: *E*_{verPY} = −0.6° +0.94 *E*_{verYP} (*r*^{2} = 0.99). Similarly, the least squares fit of *E*_{horPY} as a function of *E*_{horYP} yielded a straight line close to the diagonal: *E*_{horPY} = 0.4° + 1.1 *E*_{horYP} (*r*^{2} = 0.98). However, comparison of final 2D and 3D eye positions (upright L_{3D} = 6.8 ± 2.3°, *n* = 78) revealed a significant difference (*t*-test *P* < 0.001), in line with the notion that the overall VOR in the yaw-pitch/pitch-yaw experiment is noncommutative due to the torsional component.

To evaluate the origin of this noncommutativity, we checked to what extent it depended on static otolith input. In the preceding experiments, the ocular torsions reached at the end of the pitch-then-yaw rotation sequences could not be distinguished from a static otolith-induced torsion due to the right or left ear-down component of final head orientations (see Table 2). To determine the contribution due to rotation kinematics, we compared the final ocular torsions in the same final head orientations after a clockwise head-rotation sequence (i.e., rightward–downward, leftward–upward, upward–rightward, downward–leftward rotation sequences; e.g., Q2 to Q1 to Q3 in Fig. 2*D*) and a counterclockwise head-rotation sequence (e.g., leftward-downward, rightward-upward, upward-leftward, downward-rightward rotation sequences; e.g., Q2 to Q4 to Q3 in Fig. 2*D*). In each case, average torsion was negative after a clockwise and positive after a counterclockwise head rotation. This effect was statistically significant [factorial ANOVA with factors direction of torsion and fixation quadrants, *F*(1,48) = 64.7, *P* = 0.000, no interaction: *P* = 0.81]. In particular, the ocular torsion in the same nose-up head orientation after clockwise (i.e., left followed by upward rotation while fixating target Q2 and right followed by downward rotation while fixating target Q3) and counterclockwise rotation (i.e., right followed by upward rotation while fixating target Q1 and left followed by downward rotation while fixating target Q4) significantly differed [univariate ANOVA, *F*(1,24) = 4.26, *P* = 0.05].

To further corroborate the kinematic origin of ocular torsion, we changed the initial start-up position in the pitch-then-yaw rotation sequence in a second series of experiments (3 subjects) from upright to nose-down (for the upper 2 fixation targets Q1, Q2) and to nose-up (for the lower 2 fixation targets Q3, Q4), respectively. In this way, the subjects reached the upright orientation at the end of the two rotations (see Table 2). Again final ocular torsions significantly differed from each other at the end of rotations, depending on whether the same final upright orientation was reached by a clockwise (i.e., an upward and right rotation while fixating target Q1 and a downward and left rotation while fixating target Q4) or a counterclockwise rotation sequence [i.e., an upward and left rotation while fixating target Q2 and a downward and right rotation while fixating target Q3; univariate ANOVA, *F*(1,10) = 14.5, *P* = 0.003].

Averaged across all quadrants and all subjects, we found a difference in magnitude in final ocular torsion of 8.0 ± 3.4° at the target eccentricity of 14° in horizontal and vertical directions (20° diagonal eccentricity) compared with a value of 15.2° = 2 × 7.6° for an optimal VOR response to a yaw-then-pitch or pitch-then-yaw rotation sequence (see appendix). This yields a ratio of *k* = tan(8°/2)/tan(15.2°/2) = 0.52 (±0.22) (see *Eqs. 4* and *5*), which is approximately half of the expected unity ratio of an optimal VOR. We also measured the angle between the two complementary VOR trajectories and found an average angle of 14.4 ± 6.7° compared with a predicted total angle of 2 × 14° for an optimal VOR performance with unity gain. This yields a ratio of *k* = 0.51 ± 0.24, which is again approximately half of the expected unity ratio of an optimal VOR.

The reduced torsional position gain in the yaw and pitch VOR did not covary with a low velocity gain: averaged across all seven subjects, four fixation targets and three to four trials we found a velocity gain of 0.84 ± 0.09 and 0.86 ± 0.13 for the yaw and the pitch VOR, respectively (see Table 3).

### Kinematic versus otolith-dependent noncommutativity in the yaw-roll paradigm

In contrast to the yaw-pitch paradigm, both the final gaze direction and amplitude depended on the order of head rotations in the yaw-roll paradigm, as previously shown by Tweed and colleagues (1999). In this particular case, the vertical rather than the torsional response component does not commute. To investigate whether this noncommutativity depended on otolith inputs, we tested the same sequence of rotations in both upright and supine orientation for each fixation target in the four quadrants of the visual field. In both orientations, the vertical responses did not commute, independent of the four fixation targets [factorial ANOVA, upright: *F*(1,48) = 62.8, *P* = 0.000; supine: *F*(1,48) = 85.1, *P* = 0.000]. In fact, the difference between endpoints (L_{2D} as in *Eq. 11*) averaged across all subjects and fixation targets was statistically indistinguishable for the two orientations (L_{2D}-upright = 8.6 ± 3.0°, L_{2D}-supine = 8.2 ± 2.8°, paired *t*-test, *P* = 0.37, *n* = 81). To compare the noncommutativity in upright and supine, we plotted horizontal and vertical eye positions after roll and yaw rotations (abbreviated as *E*_{horRY} and *E*_{verRY}) against those after yaw and roll rotations (abbreviated as *E*_{horYR} and *E*_{verYR}) for upright and supine position (Fig. 6, *A* and *B*). We found no significant difference in the degree of noncommutativity between upright and supine responses: straight line fits through the horizontal and vertical eye positions, including their variances in both coordinates, yielded statistically indistinguishable parameters: in upright, *E*_{horRY} = *a*_{up} + *b*_{up} *E*_{horYR} with *a*_{up} = 1.0 ± 3.6°, *b*_{up} = 1.2 ± 0.6 (*r*^{2} = 0.98); *a*_{sup} = 0.24 ± 2.6°, *b*_{sup} = 1.2 ± 0.4 (*r*^{2} = 0.99). Similarly, *E*_{verRY} = *a*_{up} + *b*_{up} *E*_{verYR} with *a*_{up} = 7.3 ± 2.3°, *b*_{up} = 0.97 ± 0.36 (*r*^{2} = 0.99); in supine: *a*_{sup} = 7.2 ± 1.9°, *b*_{sup} = 0.97 ± 0.30 (*r*^{2} = 0.99). Both intercepts *a*_{up} and *a*_{sup} are close to the theoretical value of an optimal VOR response. Note that commutative responses must lie on the diagonal, i.e., the line through the origin with unity slope. Assuming a VOR with unity gain in both yaw and roll, the theoretical intercept for a noncommutative response is 8.9°, which is statistically indistinguishable from the experimentally observed values.

Regarding ocular torsion, there was no significant contribution to the overall noncommutativity of 3D eye positions in upright (L_{3D}-upright = 9.0 ± 2.9° vs. L_{2D}-upright = 8.6 ± 3.0°, paired *t*-test, *P* = 0.39, *n* = 78). This was also true when comparing torsion as a function of the order of rotation for each individual target location [factorial ANOVA, *F*(1,48) = 0.05, *P* = 0.82; Fig. 7 *A*]. In line with these observations, the torsional positions reached by yaw-then-roll (abbreviated as *E*_{torYR}) and roll-then-yaw (abbreviated as *E*_{torRY}) were linearly correlated with a slope close to unity, thus corroborating their independence of the order of rotation (Fig. 7*B*, *E*_{torRY} = 0.2° + 0.84 *E*_{torYR}, *r*^{2} = 0.98).

In contrast to the upright orientation, torsional eye positions in supine became depended on the order of yaw and roll rotations (Fig. 7*C*) and thus significantly contributed to the overall difference in final 3D eye positions (L_{3D}-supine = 11.4 ± 2.7° vs. L_{2D}-supine = 8.2 ± 2.8°, *t*-test *P* < 0.001, *n* = 81). Regarding the final torsional eye positions, differences were significant for each of the four fixation targets [factorial ANOVA targets Q1 and Q3, *F*(1,24) = 26.8, *P* = 0.000; targets Q2 and Q4, *F*(1,24) = 9.5, *P* = 0.005]. Next we tested whether this was caused by the different final head orientations or whether it was kinematically induced by the particular sequence of head rotations. In the roll-yaw paradigm, we found that ocular torsion changed from close to zero in supine (0.24 ± 3.6°, *n* = 91) to 5.1 ± 4.5° (*n* = 46) at 30° left ear-down rotation relative to supine and −2.7 ± 3.7° (*n* = 45) at 30° right ear-down rotation relative to supine [factorial ANOVA *F*(1,48) = 5.2, *P* < 0.05] as expected based on the final head orientation (see Table 1*B*). In the yaw-roll paradigm, torsion did insignificantly change from its average initial value of 5.9 ± 4.8° (*n* = 42) in 30° left ear-down rotation relative to supine and of −5.0 ± 4.1° (*n* = 43) in 30° right ear down rotation relative to supine to the final supine orientation [factorial ANOVA *F*(1,48) = 0.55, *P* = 0.46]. Linear correlation of final torsional positions reached by yaw-roll (*E*_{torYR}) versus roll-yaw (*E*_{torRY}) yielded a negative slope indicating a noncommutative dependency on final head rotation (Fig. 7*D*, *E*_{torRY} = 2.8° − 1.2 *E*_{torYR}, *r*^{2} = 0.91).

### Evidence for a reduced torsional velocity-to-position integration in the roll VOR

Similar as in the yaw-pitch paradigm, gaze direction described a counter roll motion during the roll VOR while the subject fixated one of the four eccentric targets (Fig. 2*D*). The key question was whether the experimentally measured torsional eye velocity complied with the unrestrained kinematic torsional velocity that the combined horizontal-vertical eye movement predicted (see *Eqs. 6* and *7*, methods). We found that the predicted torsional velocity not only systematically exceeded the experimentally measured torsional eye velocity but also often overshot the roll head velocity (compare brown and gray velocity traces in Fig. 8, *A* and *B*). In contrast to these overshooting responses, the yaw head motion in the same trials was typically almost perfectly compensated by the yaw VOR with a gain close to unity (compare blue and gray velocity traces in Fig. 8, *A* and *B, bottom*). To further evaluate this effect, we fitted both the unrestricted kinematic and the experimentally observed torsional velocity by a second-order spline function, which perfectly reproduced the response profile and then computed the area enclosed by each of these fitted functions and the time axis. The ratio of these areas corresponded to the torsional position gain (*k*_{R}, see *Eqs. 6* and *7*). The results of evaluations of the torsional position gain, the kinematic roll velocity gain (defined as unrestricted kinematic torsional velocity/head roll velocity) and the roll velocity gain (*G*_{roll}, defined as experimental torsional velocity/ head roll velocity) are summarized in Table 3. The gain of the transduction of the physical angular head velocity in the roll VOR can be estimated by computing the ratio of the roll velocity gain to the torsional position gain (*k*_{R}; see methods, *Eq. 10*). We found on average *g*_{roll} = 0.96 ± 0.29. The large variability is mainly due to the variability of the torsional position gain and to a smaller extent to the variability of the roll velocity gain (see Table 3).

### Tilt of roll angular eye velocity depends on the gain of velocity-to-position integration

To determine the relation between tilt angle of roll angular velocity and the torsional position gain *k*_{R} in the roll VOR, we plotted the ratios of tilt angle and gaze eccentricity from single trials (6 subjects) in the yaw and pitch plane against the estimated torsional position gain at motion onset (Fig. 9). We fitted these data with the function *f*(*k*_{R}) = *q*(1 − *k*_{R})/*k*_{R} [see *Eq. 10*, where (1 − *k*_{R})/*k*_{R} determines the tilt of roll angular velocity as a function of horizontal and vertical eye position]. Accordingly, the parameter *q* describes the unknown ratio of the tilt angle of the roll angular velocity to gaze eccentricity in the pitch and/or the yaw plane. For a VOR with torsional position gain *k*_{R} (0 < *k*_{R} ≤ 1) that fully complies with 3D rotation kinematics, we expected *q* = 1 in the pitch plane and *q* = −1 in the yaw plane (solid black lines in Fig. 9). The particular separation of our data into positive and negative ratios reflects the fact that we had chosen counterclockwise head roll rotations when subjects aimed at fixating the targets Q1 and Q4 and clockwise rotations when they aimed at fixating the targets Q2 and Q3 (Fig. 2*D*). Note that in this analysis, only the roll part in the yaw-roll/roll-yaw experiment was evaluated. Nonlinear least-squares fitting yielded an average *q* = 0.99 ± 0.08 for the tilt angles in the pitch plane and an average *q* = −0.84 ± 0.08 for the tilt in the yaw plane (dashed lines in Fig. 9).

## DISCUSSION

We show that the rotational VORs change ocular torsion as a function of gaze direction and velocity. In the yaw and pitch VOR, this change can best be studied in sequences of head rotations about the respective rotation axis while subjects aim at fixating an eccentric target. The kinematic analysis of such rotation sequences suggests that the VOR modulates ocular torsion proportional to the product of gaze eccentricity and vestibularly induced eye velocity. This modulation changes polarity depending on the order of intervening rotations about the two axes. The stability of the retinal image of the visual surround during such head movements can be characterized by a torsional position gain of the eyes that amounted to ∼0.55 in the yaw and pitch VOR. Values around 0.47 were observed in the pure roll VOR. In the following, we argue that the compensation of the visual consequences of head rotations imposes tight constraints on the basic characteristics of neural eye-velocity and position commands that generate the eye movements, in particular if one takes also the consequences in the roll domain into account. A quantitative analysis of the underlying nonlinear interactions leads to a surprising parallelism between kinematic variables that characterize the head rotation and the form of neural commands necessary to generate the behaviorally observed angular eye-position and -velocity characteristics. In the following discussion, we use the term “noncommutativity” in general in the sense of experimentally verifiable noncommutativity within the limits of the oculomotor range.

### Kinematic and nonkinematic origin of noncommutativity in the VOR

What are the spatial characteristics of the rotational VORs that neural eye-velocity and position commands have to account for to optimally counteract rotational disturbances of the head? It appears that one such characteristics is the dependence of the oculomotor output on the order in which the head rotates about different axes in space (Smith and Crawford 1998; Tweed and Vilis 1987; Tweed et al. 1999). The experimentally observed close correlation between the kinematic variables of head rotation and the angular eye velocity (at the motor output) suggests that the underlying commands implement the proportionality Ω̂_{eye} **∼** dÊ/d*t* + Ê × dÊ/d*t* (to distinguish neural commands from the respective kinematic variables we use the “hat” superscript; see *Eq. 1* in methods). This particular combination of neural velocity and position commands represents the most parsimonious model that can predict all the observed spatial characteristics of the rotational VOR, including the observation of noncommuting vertical (Tweed et al. 1999) but commuting horizontal and torsional response components (Figs. 6*A* and 7, *A* and *B*). Interestingly, the torsional response component in the yaw-roll paradigm can behave in a noncommutative manner if the sequence of rotations results in alternating head roll orientations relative to gravity (Fig. 7, *C* and *D*; Table 1*B*). In this case, it is the otholith input rather than the 3D rotation kinematics that is responsible for it. Both otolith-induced ocular counterroll and what we call in this study kinematic ocular torsion, i.e., ocular torsion as a consequence of changes in horizontal and/or vertical eye position (see *Eqs. 2* and *3* in methods), do eventually work perfectly in register. In contrast, the commutativity of the horizontal and the noncommutativity of the vertical component do not depend on head orientation relative to gravity (Fig. 6, *A* and *B*). This independence on static otolith inputs does not necessarily exclude a possible contribution of dynamic otolith inputs, which have been shown to be little dependent on head orientation (Angelaki 1998; Paige and Tomko 1991; Telford et al. 1997).

In the yaw-pitch paradigm, rotation kinematics predicts that the ocular torsion depends on the order of the intervening yaw and pitch head rotations as indicated by the change of polarity of ocular torsion as a function of eccentric gaze (see *Eqs. 2* and *3* in methods). Our data confirm this prediction and show that it holds true independent of the final head orientation relative to gravity. For example, the comparison of final ocular torsion in rotation sequences that move the head to a final nose-up orientation shows that the observed change in ocular torsion depends on the rotation sense of the sequence. A clockwise yaw-pitch rotation sequence (e.g., a leftward followed by an upward head rotation) results in a clockwise change in gaze and a counterclockwise change in ocular torsion and vice versa. The same is true when the head, starting from a nose-up or nose-down orientation, reaches a final upright orientation at the end of the movement sequence. Thus both clockwise yaw-pitch and clockwise roll (i.e., right ear down rotation from upright) induce a counterclockwise change in ocular torsion, in the former case reflecting the noncommutativity of 3D rotation kinematics and in the latter case due to the otolith input. Furthermore, the change in torsional eye position is linearly related to the actual displacement of the eye in the horizontal and vertical plane during the yaw and pitch VOR, respectively (Fig. 3, *Eqs. 4* and *5* in methods). The proportionality factor is in both VORs ∼0.55. It reflects the gain of the multiplicative interaction of tonic eye-position and -velocity signals and their integration in the yaw and pitch VOR. Little is known about the dynamics and the anatomical site of these computations that ultimately generate the driving signals to the oblique motor neurons. In our experiments, the interaction window is ∼1 s for each individual VOR (Fig. 1*B*). The integration involved in this interaction is bound to low-pass filter the vestibular velocity signal. At the very onset of a head movement, the spatial orientation of the eye angular velocity will depend on the vestibular velocity drive and the momentary tonic eye-position signal that tune the motor neurons. Early in the motion, the VOR should therefore be optimal. We hypothesize that the described interaction signal with low-pass characteristics inhibits the oblique motor neurons through a parallel feed forward loop that manifests itself only with a delay. Because many head movements occur at frequencies >1 Hz, the respective torsional position gains in such a scenario are likely to be larger than the values observed in this study, reflecting responses closer to optimal at higher frequencies. Thurtell et al. (1999) have reported that the eye-position dependence of the VOR does not become apparent until ∼50 ms after the onset of the stimulus. This could reflect the action of a first-order low-pass filtered inhibitory signal with a corner frequency in the order of a few hertz. A similar tendency of an increasing eye-position dependency of the VOR evoked by slower head thrusts (and thus longer integration times) has been reported by Palla et al. (1999).

In the following, we argue that the angular tilt of the eye-velocity axis away from the head-rotation axis in the yaw and pitch VOR is a direct consequence of the limited torsional position gain.

### Ocular torsion and angular eye-velocity tilt

The well-documented observation of the angular eye-velocity tilts in the human and nonhuman primate VOR as a function of eye position (Misslisch and Hess 2000; Misslisch and Tweed 2001; Misslisch et al. 1994) is a direct mathematical consequence of the here-proposed form of kinematic interaction of neural position and velocity signals. Similar nonlinear interactions between eye-position and -velocity signals have earlier been postulated by Tweed (1997). More recently, Glasauer (2007) has extended the Tweed 1997 model to capture the quarter angle rule of the VOR (Misslisch et al. 1994). The limited gain of this interaction is one of the factors that determine the compression of the oculomotor range in the torsional dimension and limit stabilization of the peripheral retinal image in far vision. In an optimal VOR strategy, ocular stability would include the entire retinal image, which requires that the angular velocity of the eye would have to perfectly match in amplitude and direction that of the head. This strategy would imply a torsional position gain of unity (optimal VOR strategy) (Crawford and Vilis 1991; Misslisch and Hess 2000; Misslisch et al. 1994; Robinson 1982). A number of modeling studies of the VOR have attempted to clarify the mathematical nature of the underlying neural computations (Merfeld 1995; Robinson 1982; Schnabolk and Raphan 1994). In a very comprehensive study, Smith and Crawford (1998) have stressed that stabilization of the eye in space requires a 3D multiplicative (tensor) interaction between velocity and position signals, independent of the particular blueprint of the oculomotor plant. Misslisch and Tweed (2001) modeled the response patterns of the angular velocity tilt in the human VOR by the effect of a weak torsional gain. Following up this question, we have asked which aspects of the angular velocity tilt pattern in the VOR are compatible with the rules of 3D kinematics that govern head-in-space-motion (Tweed and Vilis 1987). These rules require that the neural velocity commands to the eye plant should be proportional to both the transduced head angular velocity plus a term proportional to the cross vector product between the head angular velocity and current eye-position signals. The postulated neural velocity command, abbreviated as vector dÊ/d*t*, can formally be written as dÊ/d*t* ∼ Ω̂_{SCC} + Ω̂_{SCC} × Ê, where Ω̂_{SCC} is the transduced head angular velocity and Ê the neural eye-position command (for more details see *Eq. A2* in appendix). Smith and Crawford (1998) implemented a very similar multiplicative interaction between semicircular canal and tonic eye-position signals in their linear plant model. Misslisch and Tweed (2001) also used a multiplicative interaction between these signals to describe the response patterns of the tilt in the human VOR. A fundamental constraint that head rotation imposes on the neural eye-velocity commands is the proportionality Ω̂_{eye} ∼ dÊ/d*t* + Ê × dÊ/d*t* (*Eq. 1* in methods), which ensures a close match between the transduced head and the commanded eye angular velocity. Under this simple condition, the postulated sensorimotor transformation, i.e. dÊ/d*t* ∼ Ω̂_{SCC} + Ω̂_{SCC} × Ê, can explain all the observed facets of the axes tilts, including the noncommutativity of ocular torsion in the yaw-pitch VOR. For example, it follows that the torsional angular velocity component (Ω_{tor}) should depend on vertical eye position as Ω_{tor} = (1 − *k*_{Y})*E*_{ver}Ω_{hor} in the *yaw VOR* and on horizontal eye position as Ω_{tor} = (*k*_{P} − 1)*E*_{hor}Ω_{ver} in the *pitch VOR* (*Eqs. 8* and *9* in methods). The experimentally determined respective *k* values in the yaw and pitch VOR, although quite variable, are compatible with the so called quarter-angle rule. This rule predicts that the eye-velocity axis tilts in the plane formed by the rotation axis and the gaze line by about one-quarter of the angular eccentricity of the gaze line relative to straight ahead. It has been interpreted as a compromise strategy that is halfway between optimal image stabilization and perfect compliance with Listing's law (Migliaccio et al. 2003; Misslisch et al. 1994, 1996; Palla et al. 1999). Listing's law requires that all visually guided eye movements, like fixations, saccades, and smooth pursuit, do not change ocular torsion, i.e., the torsional position must always be zero. Here we show that the tilt of angular velocity is a direct mathematical consequence of a VOR that has evolved in full compliance with the 3D kinematics of head rotations but within a limited torsional oculomotor range.

### Does 3D rotation kinematics fully determine the properties of the roll VOR?

Do the same kinematic principles explain the more complex spatial characteristics of the roll VOR? An interesting feature of the roll VOR is that the angular velocity of the eye tilts in the opposite direction as the gaze line and that the tilt angles might even exceed the half angle ratio (Misslisch and Hess 2000; Misslisch and Tweed 2001; Misslisch et al. 1994). To explain these characteristics, Misslisch and colleagues have proposed that the VOR follows a foveal strategy, i.e., a strategy that stabilizes gaze direction in full compliance with Listing's law (Angelaki et al. 2003; Misslisch et al. 1994). This strategy cannot explain the spatial characteristics of the roll VOR in rhesus monkeys. In these animals, the angular velocity tilt can be accurately described by a hyperbolic function of the form ±(1 − *k*_{R})/*k*_{R}, where *k*_{R} is basically what we describe as torsional position gain (Misslisch and Hess 2000). This gain can range from values as low as 0.2 in vertical semicircular canal plugged animals up to 0.85 in intact animals, yielding tilt ratios varying as widely as from 0.18 to 4. The hyperbolic form of the tilt-gain curve is incompatible with a foveal strategy (Misslisch and Hess 2000). Here we show that it is a direct consequence of 3D rotation kinematics under the same single assumption that also explains the 3D characteristics of the yaw and pitch VOR: it is the torsional position gain *k*_{R} that limits the oculomotor range in the torsional dimension and at the same time causes the experimentally observed characteristic pattern of angular velocity tilts as a function of gaze direction. If one posits that the neural velocity commands to the oculomotor plant depend on eye position as dÊ/d*t* ∼*Ck*_{R}, −Ê_{hor}, Ê_{ver}) (see *Eq. 6* in methods and *Eq. A11* in appendix), where *k*_{R} is the torsional position gain, then the observed tilt pattern follows as a simple consequence (*Eq. 10* in methods). Recall that this particular dependence of eye-velocity command on vertical and horizontal eye-position signals is a consequence of the preceding discussed fundamental constraint on the mathematical form of sensori-motor transformations in the rotational VOR.

On what experimental grounds is the concept of torsional position gain in the roll VOR based? As in the yaw and pitch VOR, it is based on an estimate of the driving forces that control torsional eye position and velocity. The kinematic equations of the relative motion of eccentric gaze with respect to the head during roll rotation show that the torsional velocity associated with the eye movement in the roll plane can be expressed as a unique function of horizontal and vertical eye positions and velocities (see the respective time integral in *Eq. 7* in methods and *Eq. A12* in appendix). We call this the unrestrained *kinematic torsional velocity* (Figs. 4 and 8, brown traces) because it reflects the torsional velocity associated with the experimentally observed roll motion of the gaze line. A key observation is that the experimentally observed torsional eye velocity does not coincide with this theoretically expected torsional velocity but is in fact much smaller. The ratio of these two torsional eye velocities is a measure of the intrinsic torsional position gain in the roll VOR for the following reasons: first, it is independent of head roll velocity and second, it determines the torsional eye position as a multiplicative function of both vertical and horizontal eye positions and velocities alone (see *Eq. 7* in methods, *Eq. A12* in appendix). Interestingly, the unrestrained kinematic torsional velocity often exceeds the head roll velocity (compare brown velocity traces in Fig. 8, *A* and *B*; see kinematic roll velocity gain in Table 3). Because it describes the torsional velocity associated with the observed horizontal and vertical eye movements, we can also define a *kinematic roll gain*, computed as unrestrained kinematic torsional velocity/head roll velocity. Gain values larger than unity indicate that the roll VOR is physiologically not well calibrated in the absence of vision, at least for fixations of eccentric target positions. One reason for this could be that the neural calibration involves two independent variables, namely the torsional position gain and the VOR velocity gain in a multiplicative connection (see *Eq. 10*). Another reason might be that the roll VOR might be more easily influenced by the subjects' mental set than the other two rotational VORs. Because in the same trials the yaw VOR was typically perfectly behaved, it is unlikely that attention deficits play a significant role.

In contrast to the predicted kinematic velocity, the experimental torsional velocity was always much smaller than the head roll velocity. To reduce or eliminate the influence of noise and the disrupting effect of fast phases, we always compared the time integrals of the two kinds of torsional slow phase velocity taken from VOR onset to offset. The resulting average torsional position gain was ∼0.47, which is somewhat lower than the respective values for the yaw and the pitch VOR (see Table 3). One reason for this discrepancy could be that the unrestrained kinematic torsional velocity, which reflects the roll motion of the gaze line about the *x* axis, tended to overshoot the head roll velocity, whereas the analogous compensatory eye movements in the yaw and pitch VOR matched more perfectly the target motion relative to the subject. Our kinematic analysis suggests that the roll VOR gain is the product of both a velocity gain, which we define as the ratio of the transduced to the physically applied head roll velocity, and a torsional position gain (*Eqs. 6* and *10*, and *A11*). After having determined the torsional position gain, we can estimate the velocity gain by dividing the roll VOR gain by the torsional position gain. The results are consistent with those observed in the other two rotational VORs (see Table 3).

To compare the experimental roll VOR data with the expected tilt pattern derived from kinematics, we first computed the tilt of the angular velocity in the yaw and pitch plane as function of horizontal and vertical gaze eccentricity (Fig. 3). Three-dimensional rotation kinematics predicts that the ratio of tilt angles to the respective gaze eccentricities exhibit a hyperbolic dependency on the actual torsional position gain (*k*_{R}, see *Eq. 10*). These predictions are fairly well observed in our data (Fig. 9), supporting the notion that the constraints of 3D rotation kinematics are likely the determinant factor of the particular spatial characteristics of the human roll VOR under the condition of a limited torsional oculomotor range.

### Implications on the neural control of 3D rotational VOR

The discrepancy between the unrestrained kinematic and experimental torsional eye velocity sheds new light on the spatial characteristics underlying vestibular commands that drive the ocular plant in the roll VOR. First, we note that the roll velocity and position signals to the lateral and vertical recti muscles that primarily control horizontal and vertical eye movements must be different from those to the inferior and superior oblique muscles that primarily control the ocular torsion (Simpson and Graf 1981). The experimentally demonstrated torsional noncommutativity is consistent with a neural input pattern to the horizontal and vertical recti muscles that taken together must formally comply with the equation d*E*_{tor}/d*t* = (*E*_{ver}d*E*_{hor}/d*t* − *E*_{hor}d*E*_{ver}/d*t*)/(*E*_{ver}^{2} + *E*_{hor}^{2}) (see *Eq. 7*, methods, and *Eq. A12* in appendix). This pattern changes polarity for clockwise and counterclockwise head rotations, in line with the changing polarity of the respective vestibular head velocity signals for clockwise and counterclockwise head rotations. It implies multiplicative interactions of the position and velocity commands. How the brain parses the angular velocity signals from the semicircular canals into appropriate position and velocity signal such that the roll motion of the gaze line compensate the head roll motion during fixation of an eccentric target is not known. Notice, however, that it cannot be the only input responsible for the observed torsion of the eye in the roll VOR for at least two reasons: first, this signal is bound to break down for gaze close to primary position, where horizontal and vertical eye positions and velocities vanish while the eye still moves in torsional direction. Second, the experimentally measured slow phase torsional velocity at the motor output is only about half as large as the torsional velocity associated with the horizontal and vertical eye movements. This discrepancy of measured versus expected ocular torsion for eccentric gaze directions strongly indicates that the vestibular commands to the ocular motor neurons have to be adjusted such that the resulting torsion of the eye is only a fraction of the torsion produced by the kinematics of the horizontal and vertical eye movements. How the brain solves this dilemma of excessive torsion for eccentric gaze directions in the roll VOR is unknown. One straightforward solution would be to inhibit the vestibular input at the oblique muscle motor neurons by a feed forward signal that is proportional to the kinematic torsion associated with the actually commanded eye movement. As in the yaw or pitch VOR, this signal would have to depend in a highly nonlinear fashion on the horizontal and vertical eye-position and -velocity commands. The resulting ocular torsion would then correspond to the total drive to the oblique muscle motor neurons, modulated by a gain factor. Alternatively, to avoid computationally costly nonlinear feedback signals, it would be possible to send the vestibular roll velocity commands not only to the oblique muscle motor neurons but in parallel also to neurons that collectively regulate the torsional stiffness of the ocular plant with the goal to limit the total ocular torsion produced by the horizontal and vertical eye movements.

A number of authors have claimed that the main functional role of fibromuscular structures in the ocular plant, which seem to play an important role in controlling the action planes of the extraocular muscles (Demer et al. 1995, 2000; Miller 1989), is to simplify oculomotor control (Raphan 1998; Thurtell et al. 2000) such that there is no need for a 3D neural controller at the premotor level as others have proposed (Smith and Crawford 1998; Tweed et al. 1999; for a recent review, see Angelaki and Hess 2004). Although this issue is still controversial regarding visually guided eye movements (Klier and Crawford 1998; Klier et al. 2006; Quaia and Optican 1998; Tweed et al. 1998), there appears to be no way around a 3D premotor neural controller in the rotational VOR (Misslisch and Hess 2000; Misslisch and Tweed 2001; Smith and Crawford 1998). In this scenario, the muscle pulleys might represent a computationally cheap solution to the above-described torsional dilemma, due to their potential capacity of limiting kinematic torsion during the VOR in eccentric gaze directions without simultaneously challenging foveal stability. At a different level, the demonstrated brain's capability of noncommutative processing of sensory inputs emerges also in its capacity of noncommutative updating of perceived self-orientation in 3D space (Glasauer and Brandt 2007; Klier et al. 2007).

## APPENDIX

In the present experiments, rotations in the yaw plane (i.e., about the head-vertical or *z* axis) were combined either with rotations in the roll plane (i.e., about the naso-occipital or *x* axis) or in the pitch plane (i.e., about the interaural or *y* axis). In each of these cases, subjects were tested in both upright and supine position (Fig. 1).

### Eye-position dependence in optimal vestibuloocular responses

In the optimal VOR strategy the subject's gaze line traces a trajectory (Fig. 2) such that the eye and head angular velocity are equal in magnitude but spin in opposite directions. In terms of 3D vectors: **Ω**_{eye} = −**Ω**_{head} (we use bold-faced letters for 3-component vectors). Gaze positions in the optimal VOR strategy can be computed from the following fundamental relation between eye position and angular eye velocity (A1)

Here *Ẽ* represents an eye-position quaternion, that is a four-component vector *Ẽ* = (*E*_{0}, *E*_{tor}, *E*_{ver}, *E*_{hor}), written also as *Ẽ* = (*E*_{0}, * E*) with

*= (*

**E***E*

_{tor},

*E*

_{ver},

*E*

_{hor}), and Ω̃ is the angular eye-velocity quaternion, defined as Ω̃ = (0,

**Ω**), with the angular velocity vector

**Ω**= (Ω

_{tor}, Ω

_{ver}, Ω

_{hor}) (Tweed and Vilis 1987; Westheimer 1957). On the right-hand side of

*Eq. A1*, we have expressed the quaternion product Ω̃

*Ẽ*by 3D-vectors using the more familiar dot (•) and cross vector products (×). If we express the 3D eye-position vector by

*′ = (*

**E***/*

**E***E*

_{0}) = tan(ρ/2)

**e**(Haustein 1989), where

*= (*

**e***E*

_{tor},

*E*

_{ver},

*E*

_{hor})/ √

*E*

_{tor}

^{2}+

*E*

_{ver}

^{2}+

*E*

_{hor}

^{2}represents the axis of rotation (unit vector) and ρ the angle of rotation (relative to a fixed reference position), we can translate

*Eq. A1*into the following equivalent 3D vector relation (in the following we omit the dash in

**E****′**) (see also Hepp 1994) (A2)

This is a coupled system of three nonlinear first-order differential equations for the time evolution of eye position. In the following, we consider the consequences of equation A2 for the yaw and pitch fixed axis rotation VOR.

### Angular eye velocity, vertical and horizontal eye position determine 3D ocular orientation

In the optimal VOR, the angular eye-velocity vector (Ω_{eye}) is thought to result from a neural command that is equal in magnitude but opposite in direction to the physical head angular velocity. In the following, we relax the first requirement of equal magnitude and analyze the more essential requirement of directional parallelism between head and eye angular velocity. For the yaw VOR, we formally write the optimal angular eye velocity as **Ω**_{eye}(*t*) = ω(*t*) (0, 0, 1) = −*g*_{yaw}**Ω**_{head}(*t*), where ω is the head angular velocity profile and *g*_{yaw} is the velocity gain (0 < g_{yaw} ≤ 1). With this relation at hand, we obtain the following set of nonlinear coupled differential equations for eye position from *Eq. A2* [in the following, we write ω for ω(*t*)] (A3)

If the vertical eye position is held constant (*E*_{ver} = const), the second equation of *A3* yields the relation *E*_{tor}(*t*) = −*E*_{ver}*E*_{hor}(*t*). This relation yields after substitution into the first equation and eliminating ω/2 with the help of the third equation of *A3* or alternatively by direct differentiation using the condition d*E*_{ver}/d*t* = 0 (A4)

In general, the equations for eye velocity during yaw VOR write (A5)

Notice the dependence of d*E*_{tor}/d*t* on vertical eye position. The angular relation between changes in torsional and horizontal eye position in the pitch plane is (A6) where we have used the relation * E* = (0,

*E*

_{ver}, 0) = tan(φ/2)(0, 1, 0). Note that the eye-position trajectory will move in the pitch plane away from the yaw axis despite the fact that the angular velocity command is aligned with this axis. Thus for an optimal yaw VOR, we have a ratio τ

_{Y}/φ = −1/2 (inverse half-angle rule for eye position).

Analogous relations hold for the pitch VOR, where the optimal eye angular velocity is **Ω**_{eye} = ω(0, 1, 0) = −*g*_{pitch}**Ω**_{head}(0 < *g*_{pitch} ≤ 1). With this velocity at hand, the set of *Eq. A2* yields (A7)

If horizontal eye position is held constant (*E*_{hor} = const), we find the relation *E*_{tor}(*t*) = *E*_{hor}*E*_{ver}(*t*), which leads to (A8)

In general, the equations for pitch VOR write (A9)

Notice the dependence of d*E*_{tor}/d*t* on horizontal eye position. Accordingly, the eye-position trajectory tilts in the yaw plane away from the pitch axis (A10) resulting again in an inverse half-angle ratio between tilt and horizontal gaze eccentricity ϑ. The same principles hold in the more general case, where the angular eye-velocity command in the VOR reflects a head angular velocity oriented at an oblique angle with a yaw and a pitch component.

Finally, the set of equations (*Eq. A2*) yields for the roll VOR with the optimal eye angular velocity **Ω**_{eye} = ω(1, 0, 0) = −*g*_{roll}**Ω**_{head}(0 < *g*_{roll} ≤ 1) (A11)

The second and third equation of *A11* provide an independent estimate of the kinematically expected torsional eye velocity as a function of vertical and horizontal eye velocity and position for eccentric gaze directions (A12)

This relation is only valid if *E*_{ver} and *E*_{hor} are not simultaneously zero. Because *E*_{tor} = tan(ψ/2) and (d*E*_{tor}/d*t*)/(1 + *E*_{tor}^{2}) = *d*/d*t*(tan^{−1}*E*_{tor}) = 1/2dψ/d*t* on the left-hand side of *A12*, the torsional eye velocity that is expected by 3D kinematics can indeed be estimated from horizontal and vertical gaze direction information alone. For many practical purposes, the factor (1 + *E*_{tor}^{2}) on the left-hand side of *A12* can be approximated by 1 because (d*E*_{tor}/d*t*)/(1 + *E*_{tor}^{2}) ≈ d*E*_{tor}/d*t* with <1% error for angles ψ < 11°.

### Kinematic change in ocular torsion in sequences of yaw-pitch and pitch-yaw VOR

Consider a sequence of rotations with a first rotation about the head *y* axis (pitch) and the second one about the *z* axis (yaw) while the subject is maintaining fixation on a space-fixed target. *Equations A4* and *A8* first predict a change in ocular torsion, which will depend in a noncommutative fashion on the order of rotation. Second, these equations predict that the pitch-then-yaw VOR undoes the torsion generated by the yaw-then-pitch VOR if the directions are reversed while keeping the amplitudes constant. It should be emphasized that although *Eqs. A4* and *A8* predict a change in ocular torsion, the resulting angular eye-velocity vector does not exhibit a torsional component, unless the VOR does not follow an optimal stabilization strategy. This can easily be checked by using the formula for angular eye velocity: **Ω**_{eye} = 2(d* E*/d

*t*+

*×d*

**E***/d*

**E***t*)/(1 +

**E**^{2}) (Hepp 1990). From this equation and

*A5*, we obtain the following angular eye velocity at the motor output (A13)

Similarly from *A9* (A14)

And finally from *A11* (A15)

In these equations, the position gains *k*_{Y}, *k*_{P}, and *k*_{R} are defined as the ratio of the experimentally measured change in torsion (Δ*E*_{tor}*) and the change in unrestrained kinematic torsion (Δ*E*_{tor}; for details, see *Eqs. 4*–*6* in methods). Note that in the yaw and pitch VOR the torsional component of angular velocity becomes zero for *k*_{Y} = *k*_{P} = 1 while there remain still small second-order components along the other nonprincipal response axes. Similarly, in the roll VOR, the response aligns with the *x* axis independent of gaze direction only for *k*_{R} = 1.

### Numerical example

Using the relation Δ*E*_{tor} = tan(Δψ/2) (*Eqs. A4* and *A8*), one can predict a change in torsion of Δψ = ±2tan^{−1}(*E*_{b}Δ*E*_{a}) = ±3.8° for an eccentric fixation *E*_{b} = (1/√2)tan(ρ/2) with ρ = 20° in diagonal direction and an expected change in eye position of Δ*E*_{a}. For both the yaw and the pitch VOR, we obtain the same Δ*E*_{a} = tan(Δϑ/2) = tan(Δφ/2) with Δϑ = Δφ = 30°. The total change in torsion for a yaw-then-pitch or pitch-then-yaw rotation sequence is therefore 2 × Δψ = 7.6°. This example captures the experimental situation of our subjects that were rotated with a triangular velocity profile, reaching peak velocities of ω_{head} = 60°/s after ∼0.5 s assuming a VOR with unity gain (vector equation: Ω_{eye} = *g*_{VOR} Ω_{head}, *g*_{VOR} = −1).

## GRANTS

This work was supported by the Swiss National Science Foundation Grant 31–47 287.96, the Hermann Klaus Foundation, and the Betty and David Koetser Foundation for Brain Research.

## Acknowledgments

We thank B. Disler, E. Buffone, U. Scheifele, and A. Züger for excellent technical assistance.

## Footnotes

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- Copyright © 2008 by the American Physiological Society