Abstract
Hess, Bernhard J. M. and Dora E. Angelaki. Kinematic principles of primate rotational vestibuloocular reflex. II. Gravitydependent modulation of primary eye position. J. Neurophysiol. 78: 2203–2216, 1997. The kinematic constraints of threedimensional eye positions were investigated in rhesus monkeys during passive head and body rotations relative to gravity. We studied fast and slow phase components of the vestibuloocular reflex (VOR) elicited by constantvelocity yaw rotations and sinusoidal oscillations about an earthhorizontal axis. We found that the spatial orientation of both fast and slow phase eye positions could be described locally by a planar surface with torsional variation of <2.0 ± 0.4° (displacement planes) that systematically rotated and/or shifted relative to Listing's plane. In supine/prone positions, displacement planes pitched forward/backward; in left/right eardown positions, displacement planes were parallel shifted along the positive/negative torsional axis. Dynamically changing primary eye positions were computed from displacement planes. Torsional and vertical components of primary eye position modulated as a sinusoidal function of head orientation in space. The torsional component was maximal in eardown positions and approximately zero in supine/prone orientations. The opposite was observed for the vertical component. Modulation of the horizontal component of primary eye position exhibited a more complex dependence. In contrast to the torsional component, which was relatively independent of rotational speed, modulation of the vertical and horizontal components of primary position depended strongly on the speed of head rotation (i.e., on the frequency of oscillation of the gravity vector component): the faster the head rotated relative to gravity, the larger was the modulation. Corresponding results were obtained when a model based on a sinusoidal dependence of instantaneous displacement planes (and primary eye position) on head orientation relative to gravity was fitted to VOR fast phase positions. When VOR fast phase positions were expressed relative to primary eye position estimated from the model fits, they were confined approximately to a single plane with a small torsional standard deviation (∼1.4–2.6°). This reduced torsional variation was in contrast to the large torsional spread (well >10–15°) of fast phase positions when expressed relative to Listing's plane. We conclude that primary eye position depends dynamically on head orientation relative to space rather than being fixed to the head. It defines a gravitydependent coordinate system relative to which the torsional variability of eye positions is minimized even when the head is moved passively and vestibuloocular reflexes are evoked. In this general sense, Listing's law is preserved with respect to an otolithcontrolled reference system that is defined dynamically by gravity.
INTRODUCTION
Recent studies on threedimensional eye movements have emphasized the central role that Listing's law plays in oculomotor control (for a short review see: Hepp 1994). This remarkable principle imposes a strong constraint in the rotatory degrees of freedom of the eyes. When the head is upright and stationary, all accessible eye positions have rotation axes relative to a common reference that are confined to a single headfixed plane (von Helmholtz 1867). Listing's law holds true not only for steady fixations of distant targets but also for smooth pursuit eye movements and for saccade trajectories independently of whether they are fixedaxis rotations or not (Ferman et al. 1987a,b; Haslwanter et al. 1991; Minken et al. 1993; Tweed and Vilis 1990; Tweed et al. 1992; Van Opstal et al. 1991).
Most of the previous work on Listing's law has focused on oculomotor paradigms with the head stationary and upright. Only recently have investigators begun to look at some of the consequences of this principle under the more natural conditions when the head moves in space or during headfree gaze shifts where the vestibuloocular reflex (VOR) stabilizes gaze in space (Glenn and Vilis 1992; Misslisch et al. 1994; Tweed and Vilis 1992; Tweed et al. 1995). It might seem appropriate for gaze stabilization, for example, that the vestibuloocular reflex breaks Listing's law. Closer examination of the VOR, however, has revealed certain limitations to such violations, suggesting a rather complex interplay between gaze stabilization mechanisms and eye movement kinematics. It has been shown, for example, that fast phases of the torsional VOR direct eye positions in the orbit such as to keep the slow phases centered around Listing's plane (Crawford and Vilis 1991). It has been shown further that yaw VOR slow phases could comply to a different degree with the kinematic constraints imposed by Listing's law depending on the particular strategy of image stabilization (Misslisch et al. 1994).
Our present understanding of the role of threedimensional kinematic constraints in the oculomotor system appears to be further complicated by the observation that the orientation of Listing's plane relative to the head depends in a systematic way on static head orientation relative to gravity (Crawford and Vilis 1991; Haslwanter et al. 1992). Little attention has been paid to this result, however, primarily because of the small magnitude of such changes (∼10% of the change in static head position). This result has been taken as evidence for the notion that Listing's law is attributable to factors that optimize the orbital mechanics of the eye muscle plant. There is still much controversy over the nature of these factors, i.e., whether they reside in passive mechanical properties (Demer et al. 1995; Schnabolk and Raphan 1994; Straumann et al. 1995) or whether they are attributable to a neural control strategy (Nakayama 1975; Tweed et al. 1994; van Opstal et al. 1996). Whatever causes for Listing's law have been discussed, primary position of the eyes so far has been considered to remain in essence always headfixed.
There exists an alternative, yet not investigated possibility, however, that primary eye position varies as the head changes its orientation in space. In the preceding paper, we have shown that yaw VOR fast phase velocity axes were not randomly oriented but exhibited a systematic tilt relative to the axis of head rotation. We have proposed that these results emerge as a direct consequence of a change in the oculomotor coordinates and primary eye position as a function of head orientation in space (Hess and Angelaki 1997).
In this paper, we investigate this hypothesis by analyzing the threedimensional organization of eye positions during passive head movements relative to gravity. We show that eye position vectors systematically change their orientation relative to the head. These dynamic orientation changes, as well as the associated changes in fast phase velocity axes described in the preceding paper (Hess and Angelaki 1997), are both consistent with a modulation of primary eye position as a function of dynamic head orientation relative to gravity. Preliminary results of this work have appeared in abstract form (Hess and Angelaki 1995).
METHODS
Animal preparation and eye movement recording
This study presents data obtained from five juvenile rhesus monkeys (Macaca mulatta) that were prepared chronically with a scleral dualsearch coil for threedimensional eye movement recording and head bolts for restraining the head during the experiments. Details of fabrication and implantation of the dualsearch coil have been reported elsewhere (Hess 1990). Threedimensional eye position was measured with a twofield search coil system (Skalar Instruments, Delft). The search coil signals were calibrated as described in Hess et al. (1992). In brief, an in vitro calibration before implantation yielded the coil sensitivities and the angle between the two search coils. The orientation of the dual coil on the eye was determined from the four coil output signals during fixations of three vertically arranged target lights subtending 30° relative to straight ahead. Horizontal, vertical, and torsional eye positions were digitized at a sampling rate of 833 Hz and stored in the computer for offline analysis. Eye positions were expressed as rotation vectors, E = tan (ρ/2) u, where u is a unit vector pointing along the rotation axis of the eye, and ρ is the angle of rotation about u (Haustein 1989). The eye angular velocity vector, Ω, was computed from the eyeposition vector, E, according to the equation (Hepp 1990):Ω = 2(dE/dt + E × dE/dt)/(1 + ‖E‖^{2}).
Listing's plane and primary eye position were determined from spontaneous eye movement data in the light with the head upright and stationary. Rotation vectors were expressed relative to a righthanded coordinate system, referred to as standard headfixed reference system, where the x axis was aligned with primary gaze direction (positive direction is forward) and the y and z axes were lying in Listing's plane (positive directions are leftward and upward, respectively). A positive torsional eye position component (E _{tor}) corresponded to a rotation of the upper pole of the eye toward the right ear, a positive vertical component (E _{ver}) corresponded to a downward rotation, and a positive horizontal component (E _{hor}) corresponded to a leftward rotation. Because Listing's plane was pitched forward in three of the four animals, the standard coordinates were rotated relative to the head roll, pitch, and yaw coordinates by a variable amount (≤6° about the interaural (pitch) axis and <2° about the head vertical (yaw) axis).
Experimental protocols
During the experiments, animals were seated in a primate chair with the head restrained in a position of 15° nosedown relative to the stereotaxic horizontal (defined as “upright” position) to place the lateral semicircular canals approximately earth horizontal. The animals were placed inside the inner frame of a multiaxis turntable with three motordriven gimbaled axes. The effect of dynamic changes in head orientation relative to gravity on fast and slow eye movements was studied during either constantvelocity rotation or sinusoidal oscillations of the animals about their headvertical (yaw) axis, which was oriented in the earthhorizontal plane (90° offvertical). Thus the experimental protocols consisted of: earthhorizontal axis rotations at constant positive or negative speeds of 58, 110, and 184°/s, starting always with the animal in supine position, earthhorizontal axis oscillations at 0.5 Hz (±18°), 0.2 Hz (±45°), and 0.1 Hz (±90°), all of which were centered in supine position.
Data analysis
At the beginning of each experimental session, primary position was determined with the animals upright making spontaneous eye movements while looking around in the normally lit laboratory. Primary eye position was determined by fitting a plane with minimal least squares error to these eye positions. In all VOR protocols, eye positions were expressed relative to upright primary position and rotated into the standard headfixed coordinates x, y, and z such that primary gaze direction aligned with the x coordinate (i.e., Listing's plane corresponded to the plane x = 0). Fast and slow phases of the nystagmus were separated based on a semiautomated, interactive procedure using time and amplitude windows set for the second derivative of the magnitude of the eye velocity vector (see also Hess and Angelaki 1997). The following analysis was performed on these data.
1) Local fit of displacement planes (performed on constantvelocity data only). After eye positions had been expressed in the standard headfixed coordinates, VOR records were divided into 12 sectors (S_{i} , i = 1, . . . , 12) of 30° width, equally spaced throughout each stimulus cycle and shifted by −15° relative to the onset of rotation (supine position) (see also Hess and Angelaki 1997). The sectors for the four cardinal head positions were defined as follows: supine, around 0° (S _{1}: −15–15°); right eardown, around 90° (S _{4}: 75–105°); prone, around 180° (S _{7}: 165–195°); and left eardown, around 270° (S _{10}: 255–285°). The order in which the animal was rotated through these positions was: supine, leftear down, prone, and rightear down for a positive yaw rotation and the other way around for a negative yaw rotation. For each VOR record, five or more cycles were included in this analysis.
After having pooled the data of corresponding sectors, either a single plane (displacement plane) was fitted to the respective eye position vectors in each of the 12 sectors or two separate planes were fitted to both the fast and slow phase components of the respective eye position vectors according to the equation
2) Global fit of a gravitydependent modulation of displacement planes. Rather than separating into discrete sectors, the second set of analyses focused on a global model. It was assumed that there exists always an instantaneous displacement plane of threedimensional (3D) eye positions that changes its orientation dynamically in the head as the head moves in space. Such a timedependent displacement plane is described by (compare with Eq. 1
)
Eq. 4 defines the motion of this plane relative to head coordinates in the case of a passive head movement. The constant parameters a _{0}, b _{0}, and c _{0} account for static changes of the displacement planes in tilted positions compared with upright (c.f. Haslwanter et al. 1992). Parameters a _{1}, b _{1}, and c _{1} describe the linear dependence of the displacement plane orientation, whereas parameters a _{2}, b _{2}, and c _{2} describe secondharmonic components. The motion is described to be a sinusoidal function of angular head position, θ, relative to gravity. In addition, we have included a nonlinear dependency on head position, proportional to the sine of twice the angular head position relative to gravity. As will be shown in results, this second harmonic contribution was only significant for the yaw rotation of the displacement plane, i.e., component b(t).
For constantvelocity rotation data, angular head position changes linearly with time, i.e., θ(t) = ωt = 2π(ν/360°)t, where ν is the rotation velocity in degrees per second. In this simple case, the complete 15parameter model described by Eq. 4 was used. That is, the parameters a _{0}, a _{1}, α_{1}, a _{2}, α_{2}, b _{0}, b _{1}, β_{1}, b _{2}, β_{2}, c _{0}, c _{1}, γ_{1}, c _{2}, and γ_{2} of the cyclic functions a(t), b(t), and c(t) were estimated for a given record of nystagmus by the method of minimal least squares. For oscillatory data, the angular position of the head relative to gravity changes in a sinusoidal fashion, which can be described by θ(t) = −θ_{0} cos (2πft) with the maximal amplitude θ_{0} = 18° (f = 0.5 Hz), 45° (f = 0.2 Hz) or 90° (f = 0.1 Hz). Due to the complexity of this condition, an 11parameter model (whereby a _{2} = 0 and c _{2} = 0) was fit to the oscillatory data. This reduced model was appropriate because the complete 15parameter model showed only a minor contribution of the second harmonic component in a(t) and c(t) when applied to constantvelocity data (see Table 1). For both the 15 and 11parameter model fits, the cyclic modulation of the displacement plane was computed using the relations between the displacement plane vector n(t) and the scalar d(t) with the coefficients a(t), b(t), and c(t), as described in Eq. 2 .
3) Computation of primary eye position in tilted head positions. The relationship between the timedependent displacement plane and Listing's plane in upright position can be described by a unique rotation vector, P, which is given by the equation (for mathematical details, see )
Based on Eq. 5 , primary eye position was evaluated for each of the 12 planar surfaces that were fitted to the displacement planes of eye positions in the 12 sectors into which all VOR records were partitioned and for the timedependent displacement plane fitted by the global model.
4) Transformation of eye positions into “tilted” gravitydependent coordinates. For most of the analysis, eye position, displacement planes, and primary position have been computed relative to the standard headfixed coordinates (x, y, z), as defined above. In addition, selective data sets were also expressed relative to the locally defined tilted coordinates associated with the timevarying (gravitydependent) displacement plane and primary position (Figs. 8 and 9). For this purpose, the eye position vectors were recomputed relative to the estimated gravitydependent primary position and converted into the tilted coordinates, x_{p} , y_{p} , and z_{p} in which primary gaze direction pointed along the (timedependent) x_{p} coordinate. In these coordinates, the corresponding primary eye position coincided with the unique rotation vector with all three components zero similar to the definition of primary position in upright head position relative to the standard headfixed coordinates. The transformation into the tilted gravitydependent coordinates was obtained by a left multiplication of the eye position vectors, E, with the respective inverse primary position, P ^{−1}, i.e., by the equation E′ = P ^{−1} ° E (° denotes multiplication of rotation vector) (for details see and Tweed et al. 1990).
5) Testing Helmholtz's criterion for a Listing's system (performed on constantvelocity data only). To test Helmholtz's criterion of a Listing system, i.e., the geometric relation between eye position and eye velocity of fast phases (see Results and discussion), we estimated the average spatial orientation of fast phase velocity by fitting a straight line in 3D space to the pooled fast phase velocity trajectories in each sector of nystagmus (see also Hess and Angelaki 1997). This procedure ignored possible variations in the velocity profiles of individual fast phases and aimed only at estimating the average direction in space of a given sample of fast phases. The number of fast phases in each sample (i.e., pooled fast phases in each sector) varied typically between 5 and 20. The direction cosines of each fitted 3D line formed a unit vector, k _{(i)}, one for each sector S _{i} (i = 1, . . . , 12). Based on these vectors, the average tilt of fast phase velocity vectors was compared with the respective tilt of the displacement plane of fast phase positions for each sector (Fig. 4).
6) Statistical evaluation of bestfitted planes to eye positions. For the local, sectorwise analyses, we estimated the uncertainty boundaries of the fitted plane parameters, i.e., the unity vector n, the scalar parameter d, and primary position P by a statistical bootstrap method (see e.g., Efron and Tibshirani 1991; Press et al. 1992). For this purpose, the fit procedure for each local displacement plane (see Eq. 1 ) was applied 100 times on random samples drawn with replacement from the observed set of values E ^{(i)} _{x}, E ^{(i)} _{y}, E ^{(i)} _{z}(i = 1,⋅⋅⋅n, n = sample size). The fitted plane parameters n _{x}, n _{y}, n _{z}, and d derived from these 100 bootstrap samples were used to compute means and standard deviations (see also Hess and Angelaki 1997).
In addition, to estimate the “thickness” of the displacement planes and to compare the upright versus the computed time and gravitydependent displacement planes, we calculated: the torsional variability (SD) of all VOR fast phases (expressed in upright standard coordinates) relative to upright Listing's plane; the torsional variability (SD) of VOR fast phases (expressed in upright standard coordinates) relative to the bestfit plane to VOR fast phase eye positions; and, finally, the torsional variability (SD) of fast phase eye positions (expressed in the tilted coordinates x_{p} , y_{p} , z_{p} ) relative to the respective instantaneous displacement plane computed according to the global model presented above.
RESULTS
Local displacement plane description of VOR during constantvelocity rotation
Due to the simplicity of the stimulus, we start by describing the spatial orientation of 3D eye position vectors during constantvelocity yaw rotations. A typical record of horizontal, vertical, and torsional eye positions during such yaw rotations at 184°/s has been illustrated in Fig. 1 of the companion paper (Hess and Angelaki 1997). The corresponding eye position vectors for 30° wide sectors centered around prone, right eardown, supine, and left eardown positions have been illustrated in Fig. 2, separately for VOR slow and fast phases. Qualitative inspection of the eye positions reveals clearly a correlation between the orientation of slow and fast phase axes and head orientation relative to gravity. For example, when the monkey rotated through supine position, slow phase eye positions, pooled from several rotation cycles within a 30° window around supine position, were tilted forward relative to head coordinates (Fig. 2, green lines, side view). The corresponding fast phase position trajectories were also tilted forward by a similar amount. When the monkey rotated through prone position, slow and fast phase positions pooled from 30° windows around prone position were tilted in the opposite direction (Fig. 2, red lines, side view). In contrast, slow and fast phase position trajectories were shifted in positive and negative torsional direction when the monkey moved through left and right eardown positions, respectively (yellow and blue lines in Fig. 2, see side and top views).
ORIENTATION OF DISPLACEMENT PLANES OF FAST AND SLOW PHASE EYE POSITIONS.
For a quantitative analysis of the orientation of eye positions as a function of head orientation relative to gravity, we fitted planes to each of the 12 sectors of slow and fast phase eye positions that were pooled from at least five cycles in each VOR record. In doing so, we presently have assumed that on average both fast and slow phase eye position trajectories can be described adequately in each sector by a planar surface (displacement plane) the orientation of which changed relative to the head as a function of head position relative to gravity (the adequacy of this assumption will be addressed below). The fit of a planar surface to eye positions for each sector provided four quantities describing the orientation of slow and fast phase eye positions: the three components n_{x} , n_{y} , n_{z} of a displacement plane unity vector, n, which was orthogonal to the bestfit plane (Eq. 1 ), and a scalar parameter, d, describing the orthogonal distance of this plane from the coordinate origin. The uncertainty boundaries of the four fitted plane parameters (shown as ±SD in Figs. 3 and 5) were computed by a statistical bootstrap analysis as described in methods.
The spatial orientations of the displacement plane vectors fitted to slow and fast phase trajectories for each 30° sector of a single VOR record (30 cycles of steadystate response, data as in Fig. 2) are plotted in Fig. 3. During one revolution of the head in space, the displacement plane vectors rotated in the pitch plane almost symmetrically around the y axis, from pointing downward when the animal was in or close to supine position to pointing upward when the animal was in or close to prone position (Fig. 3 A, top). Concurrently, there was also a rotation of the displacement plane vectors in the yaw plane (Fig. 3 A, bottom). Due to the systematic pitch and yaw tilts of the displacement planes, the plane vectors fanned out in the frontal plane (Fig. 3 A, middle). These systematic changes in the spatial orientation of the displacement planes can be also described as changes in the components n_{x} , n_{y} , and n_{z} of the unity plane vectors n (Fig. 3 B, top 3 traces). Accordingly, the large tilts of the displacement plane vectors in the pitch plane corresponded to a sinusoidal modulation of the n_{z} component, which reached peak positive and negative values near prone and supine position, respectively. Similarly, the tilts of the displacement plane vectors in the yaw plane corresponded to a modulation of the n_{y} component as a function of head position. In contrast, the modulation of the n_{x} component was small, in agreement with the almost symmetrical clustering of the displacement plane vectors around the x axis. Along with these changes in the orientation of the plane vectors, there was also a systematic sinusoidal shift of the displacement planes. This modulation, described by the parameter d, was zero in supine and prone position and maximal in left and right eardown position (Fig. 3 B, bottom).
SPATIAL ORIENTATION OF FAST PHASE VELOCITY VECTORS.
In the preceding paper, we have shown that the orientation of fast phase velocity axes depends systematically on head orientation relative to gravity. We then have put forward the hypothesis that these systematic gravitydependent changes of fast phase velocity axes could be a direct consequence of a similar systematic change of the oculomotor coordinates in which the VOR operates (Hess and Angelaki 1997). The systematic tilts in the average orientation of fast phase velocities and the change in the displacement plane of eye positions in each sector appear, indeed, to reflect the kinematic restrictions that are characteristic for a Listing's system (for more explanations, see Discussion). As illustrated in Fig. 4, the tilt angle of the displacement planes in the pitch plane (angle α) and the average tilt angle of the fast phase velocity vectors (angle γ) in each sector are correlated for each stimulus velocity. The illustrated plots include average data from four animals during rotation in both directions. This comparison shows that fast phase velocities tilted by an amount equal to twice the angle of tilt of the displacement plane of eye positions. For the remainder of results, we will concentrate on a further analysis of eye position displacement planes as a function of head orientation in space.
GRAVITYDEPENDENT MODULATION OF PRIMARY POSITION.
We have shown so far that the displacement planes of fast phase positions shifted and rotated systematically relative to standard headfixed coordinates as the head changed orientation in space. These effects can be described equivalently by a dynamic change in primary position (see Methods). Primary position has been defined here as a function of the spatial orientation of the displacement plane of eye positions (described by the vector n) as well as the orthogonal shift of the displacement plane (described by the scalar d; the latter defines the “torsional” component of primary position; see Eq. 5 ). As in the upright position, the horizontal and vertical components of primary position determine primary gaze direction, which is, therefore, only a function of the spatial orientation of the displacement plane (vector n).
Separate evaluation of nystagmus slow and fast phases revealed a tight correlation between the displacement planes (and associated primary position) of slow and fast phase eye positions. An example of this correlation in terms of the equivalent description of displacement plane orientation by primary position is shown in Fig. 5 for a vestibular nystagmus elicited during rotation at 184°/s (left; same data as in Figs. 2 and 3) and −184°/s (right). Each of these records comprised 30 cycles of VOR steady state responses. There was no significant difference in primary eye position or displacement plane vector orientation for fast and slow phase data (Fig. 5, ○ vs. •). Bootstrap analysis further indicated a slightly larger scatter for the P _{hor} component of primary position, suggesting a larger variability in the yaw tilts of the planes. Also, primary position for head positions close to prone tended to exhibit larger variations.
Average modulation of primary position during yaw VOR for all animals is illustrated in Fig. 6 for positive and negative rotations at three different speeds. Because there was no significant difference between the spatial orientation of fast and slow phase positions, all eye position vectors were used to compute primary position. As the head changed orientation in space, all three components of primary position changed systematically. Whereas the torsional and vertical components (P _{tor} and P _{ver}) exhibited a simple sinusoidal modulation, the horizontal component (P _{hor}) exhibited a more complex dependence on head orientation (Fig. 6, bottom). The torsional component of primary position was maximal in head orientations close to eardown and nearly zero in supine and prone positions. The opposite was true for the vertical component of primary position: it was maximal positive (gaze down) in supine, maximal negative (gaze up) in prone position, and zero in eardown positions.^{1} Data from all animals were similar and demonstrated the following: 1) the torsional component of primary position depended on instantaneous head orientation but little on the speed of head rotation and 2) the modulation of the vertical as well as the horizontal components of primary position changed as a function of the speed of head rotation: the faster the head moved, the stronger was the modulation of the vertical and horizontal components. Peaktopeak modulation of the vertical component of primary position averaged 70° at 184°/s but only 20° at 58°/s.
Global fit of eye position displacement planes of VOR during constantvelocity rotation
For an independent evaluation of the functional dependence of primary eye position on gravity, we also have used an alternative approach by fitting a sinusoidally moving displacement plane described by Eqs. 3 and 4 to the nystagmus (for details, see Methods). Similar to the sectorbysector analysis, it is hypothesized that all fast phase positions are confined closely to a plane at any instant in time at a given head orientation. Unlike the previously described analysis, which was based on a local fit of a displacement plane for all eye positions in a 30° sector, here we have allowed this plane to vary continuously its orientation relative to the standard headfixed coordinates throughout a head rotation cycle. The instantaneous orientation of the eye position displacement plane was postulated to depend on head orientation in space in a sinusoidal fashion, whereby both first and second harmonic terms were included (Eq. 4 ). A 15parameter model described by Eqs. 3 and 4 therefore was fitted to all fast phase positions collected from typically five or more cycles of nystagmus. In contrast to the local, sectorwise displacement plane analysis, we considered only VOR fast phases for the global fits. An example of such a fit to the VOR data shown in Fig. 2 has been plotted as solid lines in Fig. 3 B. As illustrated by comparing fits of the local, sectorwise model (Fig. 3 B, •) and fits of the global model (Fig. 3 B, ), results were similar with both analyses.
When primary position was computed from the global displacement model fitted to constantvelocity data at 184°/s (according to Eq. 5 ), results for all four animals were similar (Fig. 7, thin lines). The 15 parameters estimated for each run in each of the four animals have been included in Table 1. Examination of the fitted parameters suggests that 1) in all but one case, the second harmonic terms for the x distance and z slope of the fitted displacement planes were negligible (<15% of the fundamental). For the y slope, however, which corresponds to the horizontal component of primary position (P _{hor}), the second harmonic dependence on head position was large. In one animal (lb), the second harmonic term was as large as that of the fundamental. 2) The sinusoidal dependence of the x distance and z slope (i.e., parameters α_{1} and γ_{1}), corresponding to P _{tor} and P _{ver}, respectively, were ∼90° out of phase, as indicated by Figs. 6 and 7. 3) The static terms were generally small, except coefficient c _{0} (z slope of intersection with xz plane), which describes the static tilt of displacement plane when animals are in supine or prone positions.
EYE POSITION TRAJECTORIES EXPRESSED IN TILTED COORDINATES OF THE INSTANTANEOUSLY DEFINED PRIMARY POSITION.
In all analyses presented here, the main and most fundamental assumption we have made is that eye positions are confined locally to a planar surface, even though this plane (displacement plane) changes its orientation relative to headfixed coordinates as a function of the instantaneous orientation of the head relative to gravity. If this hypothesis is true, then eye positions will indeed form a planar surface when examined in the “moving”, gravitydependent coordinate system defined by the instantaneous displacement planes. To show this, we have recomputed the eye position vectors relative to the instantaneously varying primary position as estimated by the global model fits. Expressed in these tilted coordinates (in the following denoted by x_{p} , y_{p} , and z_{p} ),^{2} we are looking at the eye positions from views that are parallel (front view) or perpendicular (side and top views) to the respective time and gravitydependent displacement plane. Figure 8 illustrates such an example for the data set of Figs. 2 and 3. In Fig. 8 B, the fast phase eye positions have been expressed in standard headfixed coordinates. In Fig. 8 C, the fast phase eye positions have been recomputed relative to the sinusoidally “moving” gravitydependent coordinate system. For completeness, the original VOR record (thin lines) (see also Fig. 1 in Hess and Angelaki 1997) with the modulated primary position superimposed (heavy lines) has been also included in Fig. 8 A. By comparing Fig. 8, C and B, it is obvious that the torsional component of fast phase positions of the same VOR record was much reduced in amplitude and centered around the z_{p} y_{p} plane (x_{p} = 0) when expressed in the gravitydependent coordinates. In fact, the torsional range of all fast phase eye positions rarely exceeded a ±2–2.5° range, which is similar to the range scanned by fast phase position vectors during earthvertical axis rotations. The computed thickness of bestfit planes for all VOR fast phases of the whole VOR records, after eye positions have been reexpressed in the moving, gravityrelated coordinates, have been included in the last row of Table 1.
Displacement plane analysis of VOR during sinusoidal oscillations
Due to the simple time dependency of head position relative to gravity, we have concentrated our analysis so far on VOR data elicited by constantvelocity rotations. Similar results were obtained by analyzing VOR responses elicited by sinusoidal head oscillations (both in light and complete darkness). In this case, the angular head position changed as a sinusoidal function, i.e., θ(t) = θ_{0} sin Ωt, rather than as a linear function of time, as in the case of constantvelocity rotation. We have used the global displacement plane model described above to fit sinusoidal data in four rhesus monkeys. Because the second harmonic components of the x distances and z slopes obtained with the 15parameter model for constantvelocity rotation data have been shown to be small, we have fitted a 11parameter model to the sinusoidal data (see Methods).
An example of a VOR record and a superimposed modulation of primary position as determined by the global fit model during sinusoidal oscillations at 0.2 Hz, ±45° has been plotted in Fig. 9. Because of the sinusoidally varying angular head orientation, θ(t) (Eq. 4 ), the fitted planes and the associated primary position varied as a function of head orientation in a strongly nonlinear fashion. The modulation of primary eye position during sinusoidal head oscillations was qualitatively similar as during constantvelocity rotation: in 45° eardown positions (zerocrossing of head velocity signal; Fig. 9 A, bottom), modulation of torsional primary position was maximal whereas modulation of vertical primary position was minimal. Conversely, modulation of vertical primary position peaked in supine position (peak and trough of velocity signal; Fig. 9 A). Because the animal rotated through supine position twice per rotation cycle, vertical primary position modulated at twice the head oscillation frequency. Horizontal primary position exhibited a more complex modulation pattern primarily due to the nonlinear (2nd harmonic) component of the model. Even though the specific data illustrated in Fig. 9 A were collected in complete darkness, similar observations were made during oscillation in the light. In contrast, no consistent change in the torsional and vertical components of primary position were observed during earthvertical axis oscillations.
Similar to VOR responses during constantvelocity rotation (Figs. 2 and 8), all components of eye position were scattered widely when expressed in the standard, headfixed coordinate system (Fig. 9 B). In contrast, when the same eye positions were expressed in the oscillating gravitydependent coordinate system, x_{p} , y_{p} , and z_{p} fast phase axes were more closely confined to a planar surface. In fact, for all three frequencies tested, there was a significant decrease in the standard deviation of torsional eye positions of the whole VOR record when expressed in gravitydependent coordinates compared with the respective standard deviation in headfixed coordinates relative to a planar surface fitted to the VOR data or relative to Listing's plane (as determined in upright head position; Fig. 10). Moreover, this decrease in the standard deviation of torsional eye positions was true for earthhorizontal axis oscillations in both darkness and light. In both cases, the standard deviation decreased to approximately the same values observed during similar oscillations about an earthvertical axis.
DISCUSSION
We have examined the kinematic constraints of eye movements during earthhorizontal axis rotations of the head in the yaw plane. Our results suggest that kinematic constraints similar to those described by Listing's law for the head upright and stationary exist when the head changes dynamically its orientation relative to gravity. These constraints consist in a significant reduction of the rotational degrees of freedom of the eyes analogous to those described by Listing's law yet relative to gravitydependent rather than headfixed coordinates. Our finding implies that there always exists a unique direction in space about which ocular rotations are minimal and that this direction changes as the head rotates relative to gravity. In the following, we first compare our results with previous work and subsequently discuss the possible implications of our findings.
Listing's plane and primary eye position in the head upright and stationary condition
In its original version, Listing's law has been stated for visual fixations of distant targets with the head upright and stationary (von Helmholtz 1867). Under these conditions, the eye may assume only those orientations that can be reached by rotations from a single reference position about axes that are confined to a single plane (usually referred to as the displacement plane) (see Tweed et al. 1990). In other words, the rotational degrees of freedom of the eye are reduced such that there exists a unique direction, defined as primary gaze direction, about which ocular rotations are minimized. Taking the eye position associated with primary gaze direction as reference, the axes of accessible eye rotations form a plane called Listing's plane. It has been demonstrated only relatively recently that saccade trajectories and smooth pursuit eye movements also follow Listing's law with an accuracy of ∼1° (Haslwanter et al. 1991; Minken et al. 1993; Straumann et al. 1996; Tweed and Vilis 1990; Tweed et al. 1992).
Primary eye position and Listing's plane usually have been considered to be headfixed by ignoring the fact that the displacement planes change orientation as a function of static head position relative to gravity (Crawford and Vilis 1991; Haslwanter et al. 1992). This general belief was grounded on the fact that the noted gravitydependent changes, albeit consistent, are usually small and limited to within 10% of static head tilts. Furthermore, it has been assumed tacitly that eye movements are confined only to displacement planes when the head is stationary in space. We have challenged these assumptions by analyzing the geometric constraints of eye positions and velocity under dynamic conditions when the head is passively rotated or oscillated relative to gravity. Even though most of our analysis has been confined to data acquired during constantvelocity yaw rotations, similar results were obtained with sinusoidal oscillations, either in complete darkness or in the light.
Instantaneous eye position vectors lie in displacement planes
The most important aspect of the present analysis is the question of whether eye positions during passive head rotations can be described locally by a planar surface. There are two issues related to this question: First, do eye positions always follow Donders' law if expressed in appropriate coordinates, i.e., are they always confined to a single surface? Second, do eye positions follow Listing's law in any head orientation relative to gravity? It is obvious that Donders' law is violated if one considers eye positions in all possible head orientations. Our results indicate, however, that there exists a set of continuous gravitydependent oculomotor coordinates relative to which all possible eye positions “disentangle” and become arranged in a twodimensional surface (see Figs. 8 C and 9 C). Thus the oculomotor system seems to implement Donders' law as a unique function of head orientation relative to gravity. Moreover, Donders' surfacebecomes flat in these coordinates, i.e., Listing's law is also valid with an accuracy of ∼2° [see Table 1, 2.0 ± (SD) 0.4°].
We have checked the adopted analytic procedure in three different ways: first, to better illustrate the thickness of the traced surfaces, i.e., the torsional variability, eye positions have been expressed in the local primary coordinates (see examples in Figs. 8 and 9). Second, we have used a statistical bootstrap method to estimate the variability of the fitted parameters of the displacement planes and the associated primary eye position (SDs in Fig. 3 B and 5). Third, we have formulated a model that puts forward a mathematical description of the gravitydependent shift and rotation of displacement planes. Based on the model, we have compared primary eye position computed from local plane fits with that obtained from a globally fitted plane.
To further judge the adequacy of our approach, it is important to distinguish between fast and slow phase eye positions. Fast phases of VOR elicited by yaw rotation in upright position follow Listing's law with a standard deviation of ∼1–2° (e.g., Fig. 10). The increase in standard deviation of Listing's plane during VOR compared to visual fixations with a stationary head can be explained in part by the fact that fast phases correct the systematic deviations of slow phases relative to Listing's plane as observed also by Crawford and Vilis (1991). A systematic study of the deviations of yaw VOR slow phases from Listing's plane in humans suggested that slow phase kinematics follows a “half Listing's strategy” (Misslisch et al. 1994). The different kinematic behavior of VOR slow and fast phase positions is correlated with the different functional requirements of the two systems. Although the axis orientation of VOR slow phases should be determined by the imposed head rotation independently of current eye position, axis orientation of fast phases must change as a function of the radial distance from primary position to be in accordance with Listing's law (von Helmholtz 1867).
In our locally fitted displacement plane analysis of constantvelocity VOR, both slow and fast phase eye positions ultimately were analyzed together to estimate the bestfit displacement planes and the modulation of primary position as a function of head orientation (Fig. 6). This approach was adopted for the following reasons. First, a separate analysis of slow and fast phase position trajectories demonstrated that the displacement planes, as well as primary position, estimated from each of them separately were indistinguishable (Fig. 5). A closer inspection of the spatial orientation of eye position revealed that, depending on head position, slow phases exhibited a systematic component oriented across the bestfit plane to fast phase positions. For example, in prone position, slow phase components were oriented parallel to the head rotation axis (approximately z axis, see coordinate system in methods) but fast phase components were not (see Fig. 2, fast and slow phases in prone position, side view). Nevertheless, the bestfit planes to either slow or fast phase position were tilted backward because the fast phases of nystagmus “displace” the eye along such a plane. Therefore, fast and slow phase positions can be described on average by the same plane.^{3} This plane, however, does not necessarily represent the directions of the slow phase displacements. In eardown positions, for example, the bestfit planes to slow and fast phase positions represent at the same time the true displacement planes of the two movement phases (see Fig. 2, fast and slow phase positions in yellow and blue, side view). The local displacement plane analysis (which included both slow and fast phases) was crossexamined by fitting a gravitydependent moving displacement plane to the whole nystagmus record. In this global analysis, we analyzed only the fast phases of nystagmus because of the more complex stimulusinduced behavior of VOR slow phases. Both the local and the global methods yielded closely corresponding results.
Notion of primary eye position extended to dynamic head motion in space
Primary eye position, which is defined on the basis of Listing's law, specifies an orthogonal coordinate system the origin of which coincides with zero ocular rotation (i.e., rotation vector P = 0) and the x axis of which is aligned with the direction of primary gaze. In this coordinate system, the standard Listing's plane of eye positions is centered around the coordinate plane x = 0, implying that eye movements in Listing's plane are constrained to zero torsion with an accuracy of <1° for fixations. The precise orientation of this particular reference frame, for example, with respect to stereotaxic coordinates, depends on head position relative to gravity as previously demonstrated for different static head roll and pitch positions (Haslwanter et al. 1992). Because the existence of a primary position (defined for head upright and stationary) solely depends on the fact that eye positions are confined to a single planar surface, it is natural to extend the notion of primary position not only to different static but also to dynamically changing tilted head positions. This more general notion of primary position makes sense if eye positions are found to be confined to planar surfaces independently of whether the head moves or not. As shown in this study, such local planes, which we will call (instantaneous) displacement planes, can be described indeed. They move relative to the head, whereas the head rotates in space, thereby allowing for a dynamic definition of primary eye position.
Modulation of primary eye position as a function of head orientation in space
According to the notion of a dynamically changing primary position, the modulation of mean torsional and vertical eye position during yaw rotation about an earthhorizontal axis (Fig. 1 in Hess and Angelaki 1997; see also Angelaki and Hess 1996) reflects a systematic modulation of primary eye position as the head changes its orientation in space (e.g., Fig. 6). Although much more variability is present in the modulation of the horizontal component, torsional and vertical primary eye position consistently modulate in phase with head position, independently of the direction and speed of head rotation, as previously also shown for the respective modulations of mean torsional and vertical eye positions (Angelaki and Hess 1996). The modulation of primary eye position appears to shift the plane of desired eye positions in compensatory direction with respect to the orientation of the head relative to gravity. More specifically, the torsional component of primary eye position is maximal around left and right eardown positions. Similarly, the vertical component of primary position is maximal around supine and prone orientations. The same is true during sinusoidal oscillations about an offvertical axis (compare Figs. 8 and 9).
Interestingly, it is the vertical component of primary position that exhibited the largest modulation at the highest frequency (0.5 Hz corresponding to rotation at 180°/s). This modulation decreased with speed, reaching minimum values at 0.16 Hz (corresponding to rotation at 58°/s). Even smaller changes in the vertical component of primary position are observed under static conditions (estimated as double the angle of tilt of the fitted plane of eye positions during static pitch tilts) (e.g., see Haslwanter et al. 1992). This marked frequency dependence of the vertical component of primary eye position suggests that orientation toward a spacefixed vertical reference is of crucial importance only at higher speeds of motion, probably for purposes of proper spatial orientation, which no longer can be maintained by the visual system alone. Because of the foveal organization of the primate visual system, there is no point in controlling the gain of the torsional component as a function of motion frequency.
In addition to these systematic modulations of the torsional and vertical components of primary eye position, a consistent (albeit more variable) modulation of the horizontal component was also observed during yaw rotations, particularly at the highest speeds. That is, as the head changed orientation from eardown to prone/supine positions, primary eye position not only rotated about the x and y axes but also about the z axis. This horizontal modulation appeared not to be a simple sinusoidal function of the oscillating gravity component along the interaural axis. A second harmonic term in the global displacement model gave a reasonable correspondence between results of the local and global displacement plane fits. The significance of this finding is not clear, but it could mean that there is more than one mechanism that determines the horizontal orientation of Listing's plane. One of these mechanisms could be horizontal vergence, which is associated with a disconjugate rotation of the Listing's planes of the two eyes (Mok et al. 1992).
Implications of a dynamically changing primary position for headfree gaze shifts
Our results strongly suggest that kinematic constraints similar to Listing's law are effective during passive motions of the head relative to gravity. This implies that VOR fast phases as well as saccades in general follow Listing's law not only in upright position with the head stationary, but in an analogous way also when the head moves in space. Such a conclusion is further supported by the observation that both primary position and the associated displacement (also called velocity) plane move as a function of gravity in a manner predicted by Listing's law (Fig. 4). The observed relation between gravitydependent changes in primary position and associated displacement plane is analogous to a criterion for Listing's law, first derived by von Helmholtz (1867), which states that when the eye is in a position, E, then the associated eye velocity vectors, Ω, must lie in a plane. For primary position, the associated displacement plane is Listing's plane. For any other position, the associated displacement plane is tilted out of Listing's plane by an angle that corresponds to half its radial distance from primary position. Thus when primary position changes its orientation relative to the head as a function of gravity, fast phase velocity vectors also must change their orientation accordingly. As described in the companion paper (Hess and Angelaki 1997) and as shown in Fig. 4, this was indeed the case.
Eyeinhead positions after large headfree gaze shifts have been reported to violate Listing's law due to an increased twist of the fitted surfaces (Glenn and Vilis 1992). However, Radau et al. (1994) have pointed out that the increased variability of eye positions relative to a planar surface fit (average 1storder SD: 2.6°; 2ndorder SD: 2.2°) could be explained by the simultaneous change in head orientation relative to gravity that was needed to acquire oblique targets. Listing's law therefore still seems to be obeyed, taking into account that the eye position planes change as a function of gravity. Our finding further suggests that primary position might not stay invariant relative to the head during large headfree gaze shifts but could change continuously as a function of head position relative to gravity. Moreover, this modulation of primary position is expected to depart, at any point in time, from the static orientation during a head movement trajectory. Interestingly, Tweed et al. (1995) recently have shown that trajectories of the eye relative to the head during large headfree gaze shifts toward oblique targets were curved strongly and departed also from the static surface. Furthermore, oblique saccades, which involved large changes of head orientation relative to gravity, exhibited consistent looping in their trajectories, even in torsional direction. Although these findings show that eye positions in head during headfree gaze saccades generally break Listing's law, it remains an open question whether such saccades still follow dynamic constraints correlated with the fast change in head orientation relative to gravity. The headfree gaze shifts studied by Tweed, Vilis, and colleagues all involve fast changes not only in yaw but also in head pitch and roll orientation of up to ±30° (Glenn and Vilis 1992; Radau et al. 1994, Tweed et al. 1995). It remains to be shown what role a dynamic modulation of primary position may play during such active headfree gaze shifts.
Acknowledgments
The authors thank K. Hepp for many stimulating discussions.
This work was supported by grants 3132484.91 from the Swiss National Science Foundation, EY10851 from the National Eye Institute, NAGW4377 from the National Aeronautics and Space Administration, F49620 from the Air Force Office of Scientific Research, and 9525 from the Roche Research Foundation.
Footnotes

Address for reprint requests: B.J.M. Hess, Dept. of Neurology, University Hospital Zürich, Frauenklinikstrasse 26, CH8091, Zurich, Switzerland.

1 Because the torsional component was zero in supine (= onset of each rotation cycle) and maximal in eardown positions, it changed polarity with a change in the direction of rotation. In contrast, the vertical component was maximal in supine and zero in eardown positions (i.e., in phase quadrature to torsion), and therefore it did not change polarity with a change in the direction of rotation.

2 In the following, we refer to the components (E _{tor})_{p}, (E _{ver})_{p}, (E _{hor})_{p} of an eye position vector, E, expressed in coordinates x _{p}, y _{p}, and z _{p} also as the “torsional”, “vertical”, and “horizontal” component of E whenever it is clear from the context which coordinate system is used.

3 This observation should not be surprising: the slow phases need on average to follow similar trajectories as the fast phases to limit accumulation of torsion during the head movement.

4 Primary eye position has been computed so far only from displacement planes that run through the origin of the coordinate system (obtained in the condition with the head upright and stationary). If this was not the case, due to measurement noise for example, Tweed et al. (1990) proposed a torsional recalibration of the reference position (which does not affect the reference gaze direction) before computing primary eye position. This procedure could not be applied here because torsion is part of primary position in tilted head orientations.
 Copyright © 1997 the American Physiological Society
APPENDIX
Let us consider two sets of eye position vectors, one set of positions, E, which describe rotations of the eye in Listing's plane expressed in the standard coordinates x, y, z (i.e., E_{x}
= 0) and another set of positions, E′, expressed in the same coordinate system that describes eye rotations in a different plane, tilted relative to Listing's plane by a certain angle (displacement plane). This latter plane of eye positions E′ will in general be displaced relative to the origin of the coordinate system (generalized displacement plane). The geometric relation between these two planes, the Listing's plane of positions E and the generalized displacement plane of positions E′ can be described by a single rotation vector P as follows
The orientation of the plane of eye positions E′ can be determined from the plane equation: E′⋅n = E′_{x}
n
_{x} + E′_{y}
n
_{y} + E′_{z}
n
_{z} = d in terms of the three components n
_{x}, n
_{y}, n
_{z} of a unity vector, n, oriented orthogonal to the plane, and of the orthogonal distance, d, of the plane from the origin (⋅ denotes the vector dot product). This was done experimentally by fitting a plane to the eye positions E′ in each sector (see Methods). Once the orientation of this plane has been determined (i.e., the parameters n and d are known), the components of the unknown rotation vector P can be computed by solving the plane equation: E′⋅n = (P ° E)⋅n = d but now in terms of the components a, b, and c of the vector P (using Eq.
EA1
). In doing so, it suffices to consider the two linearly independent set of vectors E
_{u} = (0, E
_{y}, 0) and E
_{ν} = (0, 0, E
_{z}) and the originE
_{0} = (0, 0, 0). Using these constraints, one obtains three linear equations of the three unknowns a, b, and c that can be written as a matrix equation
The rotation vector P determined from the displacement planes in tilted head positions has the properties of primary position. In this context, primary eye position^{4} can be defined more generally as the unique rotation vector that transforms eye positions that form an arbitrarily oriented plane relative to some (arbitrary) coordinate system into a more natural, canonical coordinate system in which the same plane of eye positions aligns with one of the coordinate planes. It should be noticed that P has nonzero components when expressed in noncanonical (arbitrary) coordinates, whereas its components become all zero when expressed in the associated canonical coordinates. Thus for tilted head positions, these tilted or canonical coordinates play the same role as the standard coordinates defined for upright head position (when the head is stationary).
The transformation of eye positions E′ from the (upright) standard coordinates into their tilted coordinates requires two computational steps (see also Tweed et al. 1990): recomputation of the eye position vectors E′ relative to the new primary position P and conjugation of the recomputed eye positions with primary position. By applying these two operations on the eye positions E′, we have E = P ^{−1} ° (E′ ° P ^{−1}) ° P = P ^{−1} ° E′, which is the inverse of Eq. EA1 obtained by left multiplication with the rotation vector P ^{−1}. Thus left multiplication with the inverse of primary position P ^{−1} (inverse of Eq. EA3 ) converts the general displacement plane of eye positions E′ into tilted coordinates (denoted here by x _{p}, y _{p}, z _{p}) such that it aligns with the coordinate plane x _{p} = 0.