## Abstract

We have examined the spatiotemporal characteristics of postrotatory eye velocity after roll and pitch off-vertical axis rotations (OVAR). Three rhesus monkeys were placed in one of 3 orientations on a 3-dimensional (3D) turntable: upright (90° roll or pitch OVAR), 45° nose-up (45° roll OVAR), and 45° left ear-down (45° pitch OVAR). Subjects were then rotated at ±60°/s around the naso-occipital or interaural axis and stopped after 10 turns, in one of 7 final head orientations, each separated by 30°. We found that postrotatory eye velocity showed horizontal–vertical components after roll OVAR and horizontal–torsional components after pitch OVAR that varied systematically as a function of final head orientation. The quantitative analysis suggests that, in contrast to the analogous yaw OVAR paradigm, a system of up to 3 real, gravity-dependent eigenvectors and eigenvalues determines the spatiotemporal characteristics of the residual eye velocities after roll and pitch OVAR. One of these eigenvectors closely aligned with gravity, whereas the other 2 determined the orientation of the earth horizontal plane. We propose that the spatial characteristics of eye velocity after roll and pitch OVAR follow the physical constraints of stationary orientation in a gravitational field and reflect the brain’s best estimate of head-in-space orientation within an internal representation of 3D space.

## INTRODUCTION

When we move in 3-dimensional (3D) space, the brain uses afferent information from semicircular canals and otolith organs to generate and constantly update an internal representation of current head-in-space orientation with respect to gravity. Most head movements occur in planes that activate both the otolith and semicircular canal organs. Detailed knowledge about the central processing of these signals and their interactions with other self-motion signals is essential for a better understanding of the brain’s programming of motor behavior underlying spatial orientation, navigation, orienting, and postural control.

One fruitful way to explore the interaction of semicircular canal and otolith information involves testing unusual vestibular stimulus configurations. For example, tilting the head after prolonged constant-velocity rotation away from the former spinning axis creates a strong conflict between the canal-borne head rotation and the otolith-borne stationary head orientation signals. The resulting central vestibular signal configuration, i.e., strong rotational velocity cues in the presence of stationary graviceptive information, is physically compatible only with a rotation of the head about an axis parallel to earth vertical. Accordingly, the postrotatory vestibuloocular reflex (VOR) determined in such paradigms shows an eye velocity vector that reorients toward alignment with the direction of gravity (Angelaki and Hess 1994; Merfeld et al. 1993). Another slightly different conflict paradigm consists in evoking a sense of rotation by full-field optic flow stimuli in the head horizontal plane while being tilted relative to gravity. In this case, the velocity vector of the induced optokinetic response reorients toward earth vertical as soon as the dominant optokinetic visual drive is switched off by putting the subject in complete darkness. In contrast to the head-horizontal system, torsional or vertical optic flow stimuli create apparently little conflicting sense of rotation, presumably because they occur more commonly in everyday locomotion in the same plane as a relatively stationary gravity vector (Dai et al. 1991; Gizzi et al. 1994; Raphan and Sturm 1991). In line with this, little or no reorientation of torsional and vertical optokinetic afternystagmus (OKAN) has been observed in static tilt paradigms (Dai et al. 1991).

We have earlier proposed that after yaw head rotations or optokinetic stimulation, the spatial properties of the postrotatory responses can be understood in terms of an active tendency to rotate the head velocity cues toward alignment with the earth vertical. Accordingly, the horizontal postrotatory velocity tends to counter tilt whenever the head is tilted or comes to a halt in a tilted orientation relative to earth vertical (*rotation hypothesis*). To test this hypothesis, we have more recently investigated the spatial properties of horizontal vestibuloocular responses in a static vestibuloocular paradigm, which is comparable to the static optokinetic nystagmus (OKN)/OKAN paradigm. Subjects were rotated in the head yaw plane around off-vertical axes (OVAR) and successively stopped in various positions with respect to the direction of gravity (Jaggi-Schwarz et al. 2000). A detailed quantitative analysis of these 3D postrotatory *yaw OVAR* responses corroborated previous findings using a similar paradigm in humans (Harris and Barnes 1987), monkey (Raphan et al. 1992), and cat (Harris 1987) and showed mathematically that the underlying mechanism can indeed be described by an active rotation of the head velocity signals. This same rotation mechanism, however, cannot readily explain the spatial properties of roll or pitch VOR or OKN responses that have been described so far for the dynamic postrotatory tilt paradigm only. First of all, there is little or no reorientation of torsional or vertical optokinetic afterresponses (Dai et al. 1991) unless when using a dynamic postrotatory tilt paradigm (Hess and Angelaki 1995). Second, the spatial properties of postrotatory roll or pitch VOR or OKN that are observed in the dynamic tilt paradigm appear to result from facilitation of components along the earth vertical and disfacilitation of components that are oriented in the earth horizontal plane. That is, torsional or vertical components of postrotatory velocity perpendicular to gravity are suppressed, suggesting a cosine tuning of these responses with respect to gravity (*projection or cosine tuning hypothesis*; Angelaki and Hess 1994; Hess and Angelaki 1995).

Rather little is known about the spatial characteristics of postrotatory roll or pitch VOR as a function of static head orientations in space, i.e., without applying conflicting postrotatory tilts from supine or ear-down positions toward upright. In contrast to the dynamic tilt paradigm, a spatially stationary orientation of the head rotation axis ensures that during and immediately after stop of rotation the semicircular canal signals and the phasic otolith signals provide congruent information about the axis of ongoing or immediately preceding head rotation. There is thus no physical conflict between the centrally estimated orientation of axis of head rotation in space and the estimated actual head orientation at rotation stop. Because of the opposite directions of the angular velocity information from the semicircular canals and the centrally generated otolith-borne steady-state velocity, both represented in head-fixed coordinates, most of the vestibular head velocity signal will cancel at stop of rotation. Based on the residual eye velocity signals, however, this OVAR paradigm provides the unique opportunity to probe the postrotatory roll and pitch VOR in terms of *1*) which coordinate system is used to represent residual torsional and vertical VOR velocity and *2*) which mechanisms, a rotation or a projection if any, underlies the transformation of postrotatory roll and pitch VOR from head-fixed to space-fixed coordinates (Fig. 1).

The answer to these questions cannot be deduced from dynamic tilt experiments as studies of optokinetic afterresponses show (Dai et al. 1991; Hess and Angelaki 1995). The postrotatory OVAR paradigm, in which otolith and semicircular canal cues interact in different static head orientations, prove to be useful to investigate the coordinate system, in which the central estimation of torsional and vertical head-in-space orientation is represented. It will be shown that, in contrast to the rotation model, the projection model leads to a high correlation between the geometric properties of postrotatory responses and the actual head-in-space orientation.

## METHODS

Experiments were performed on 3 rhesus monkeys (*Macaca mulatta*). Details of animal preparation, measurement, and calibration of 3D eye position were described in Jaggi-Schwarz et al. (2000). All procedures were in accordance with the *National Institutes of Health Guide for the Care and Use of Laboratory Animals* and approved by the Veterinary Office of the Canton of Zurich.

### Experimental setup and protocol

In this study, animals were rotated around the naso-occipital (*x*-axis) or the interaural (*y*-axis; Fig. 2, *A* and *B*) in 2 different experimental configurations with a constant velocity of ±60°/s (positive/negative roll = clockwise/counterclockwise; positive/negative pitch = downward/upward) for 10 cycles (initial acceleration 180°/s^{2}): in one configuration the axis of rotation was oriented 90° and in the other 45° off-vertical. Animals were stopped (decelerations 180°/s^{2}) at one out of 7 end positions in space-fixed coordinates: ±0°, ±30°, ±60°, ±90° (positive, i.e., clockwise roll or downward pitch OVAR; negative, i.e., counterclockwise roll or upward pitch OVAR).

### Data analysis

For data analyses and data presentation, we express the orientation of the animal relative to gravity as equivalent orientation of the gravity vector in a head-fixed coordinate system. Thus a final body orientation of, for instance, −30° roll or 30° pitch expressed relative to space (*x*_{E}–*y*_{E}–*z*_{E} coordinates in Fig. 2, *C* and *D*) corresponded to a rotation of the gravity vector by +30° in roll and −30° in pitch when expressed in the head-fixed coordinate system (*x*–*y*–*z* coordinates in Fig. 2, *C* and *D*).

##### TEMPORAL DOMAIN ANALYSIS.

Details of computation and desaccading of eye angular velocity (**Ω**) as well as the empirical analysis of the spatial orientation of postrotatory VOR velocity have been described previously (Jaggi-Schwarz et al. 2000). Briefly, we first computed postrotatory eye velocity components in the rotation plane, i.e., the plane orthogonal to the rotation axis (see gray discs and dashed lines labeled RP in Fig. 2). After roll OVAR, these orthogonal velocity components represent a trajectory, Ω_{orth} = Ω_{ver} + Ω_{hor}, that lay in the *y*–*z* plane (Fig. 2*C*), whereas following pitch OVAR, the analogous trajectory, Ω_{orth} = Ω_{tor} + Ω_{hor}, lay in the *x*–*z* plane (Fig. 2*D*). Second, we fitted (in a least-squares sense) the sum of 2 exponential functions to each of the 3 components of postrotatory eye velocity as well as to the orthogonal velocity trajectory (Ω_{orth}). Third, we determined the peak of the exponential function fitted to the orthogonal response vector. This peak was used to delimit the final response portion of the postrotatory response, defined as the response from peak velocity to zero or to a nonzero offset value.

##### SPATIAL DOMAIN ANALYSIS.

The remaining analyses were done in the spatial response domain, using the time domain fits of the postrotatory response. By plotting the horizontal, vertical, and torsional components against each other, we analyzed the projection of the 3D velocity trajectories onto the different head planes as follows. In a first empirical approach, we estimated the angle φ̂ that the head-vertical *z*-axis and the orthogonal response trajectory subtended in the rotation plane by measuring the slope of a line fitted to the orthogonal postrotatory trajectory (i.e., the vertical–horizontal trajectory after roll OVAR and the torsional–horizontal trajectory after pitch OVAR; see Fig. 3, *A* and *B*). We call the plane formed by this line and the rotation axis the *resultant response plane* (RRP) (see gray rectangles labeled “RRP” in Fig. 2, *A* and *B* and gray rectangles in Fig. 3). The angle ϑ̂ describes the slope of a straight line fitted to the postrotatory eye velocity trajectory in the resultant response plane relative to the rotation axis. It characterizes how far the final part of the eye velocity vector tilted (in the resultant plane) away from the rotation axis toward alignment with the gravity vector (Fig. 3, *A* and *B*). This angle was difficult to assess empirically because of the generally small magnitude and sluggishness of the peak of the postrotatory signals.

The angular orientation of both, the tangent line and the resultant response plane, approximated the 3D spatial orientation of the final time course of postrotatory eye velocity (see angles ϑ̂ and φ̂ in Fig. 3).

In a parallel theoretical treatment, we fitted the postrotatory response with a system of differential equations that implemented a gravity-dependent 3D projection mechanism (cosine tuning) upstream to a leaky integrator within a feedback system. More specifically, we fitted the following function to the postrotatory VOR responses (1) This function mathematically describes the postrotatory VOR velocity as a sum of a signal coding head velocity (**ω**_{oto}) at stop of rotation (“bias velocity”) and a spatially transformed signal, called inertial velocity (**ω**_{inertial}), based on afferent inputs from the 4 vertical semicircular canals (**ω**_{rlv}: lumped vertical semicircular canal signal generated at stop of rotation) and otolith information about 3D head orientation in space (Fig. 4).

The rationale for this model is in short as follows: As far as the dynamics is concerned *Eq. 1* represents the time domain solution of a 3D generalization of the feedback model proposed by Robinson (1977) to explain the horizontal postrotatory VOR. Translated from the frequency (Laplace) into the time domain this model reads as follows: ω̇+ a(1 − k)ω = ω̇_{rlv} + aω_{rlv}, where “a” is the reciprocal time constant of a leaky integrator and “k” is a feedback constant. In words: head velocity and acceleration signals are driven in a central feedback network by the semicircular canal input that conveys information about head acceleration and velocity in a certain ratio (1:a). Note that the central signals represent the input one-to-one if there is no feedback (k = 0). To extend this model to 3 dimensions we replaced the reciprocal time constant a = 1/T and the feedback constant k with diagonal matrices A (all diagonal elements a_{ii} = a, off-diagonal elements a_{ik} = 0, for i ≠ k) and K (3 diagonal elements: k_{1}, k_{2}, k_{3}, off-diagonal elements zero). To implement the spatial response properties, we introduced a gravity-dependent spatial filter upstream to the integrator in the feedback loop, which modifies the velocity according to its orientation relative to gravity (Fig. 4). The effect of this filter can be described by a cosine tuning of the velocity signals (**ω**) in the feedback loop with respect to gravity (unit vector **ĝ**, expressed in head coordinates), i.e., P**ω** = (**ĝ** · **ω**)**ĝ**, where the dot“ · ” designs the scalar product between **ĝ** and **ω**. Altogether, these changes modify the 1D Robinson equation as follows: **ω̇** + A(1 − KP)**ω** = **ω̇**_{rlv} + A**ω**_{rlv}. The eigenvectors and eigenvalues of the resulting matrix F = A(1 − KP) on the left-hand side of this 1st-order differential equation fully characterize the spatiotemporal characteristics of this feedback system (for details see appendix). Specifically, the eigenvectors determine the response directions of *Eq. 1* as a function of the initial response vector **ω**_{oto} and the centrally estimated direction of gravity **ĝ**, whereas the eigenvalues characterize the dynamics of the response components along each of the 3 eigenvectors (see Table 1 in results). Depending on head orientation the response depends on 2 or 3 distinct eigenvectors (Fig. 5). Alternatively, we also used an analogous form of *Eq. 1* to test the rotation hypothesis. For this, we replaced the cosine tuning in the feedback loop by a rotation operator, i.e., the matrix P in the feedback matrix F by a rotation matrix R [i.e., F = A(1 − KR)]. This system now reflected the spatial properties of the matrix R, which has only one real eigenvector, that is, the axis of rotation, in contrast to the matrix P, which in general has 3 distinct real eigenvectors (for details see appendix).

To further characterize the spatial properties as a function of head orientation we computed the volume of the parallelepiped that is spanned by 3 normalized eigenvectors (i.e., unit vectors **v̂**_{1} = **v**_{1}/‖**v**_{1}‖, **v̂**_{2} = **v**_{2}/‖**v**_{2}‖, and **v̂**_{3} = **v**_{3}/‖**v**_{3}‖), using the equation V = **v̂**_{1} ∧ **v̂**_{2} · **v̂**_{3}, where the wedge (“∧”) designs the cross-product. As shown in the appendix, V depends on the feedback constants and gravity as follows: V = −k_{2}g_{2}g_{1}^{2} ∑_{i} k_{i}g_{i}^{2} (k_{i} ≥ 0). Accordingly, the eigenvectors form a right- (left-) handed triple depending on whether the head rolls right (g_{2} < 0) or left side down (g_{2} > 0). Technically, a value V = 1 or V = −1 means that the vectors are mutually orthogonal, whereas a V = 0 means that the 3 vectors lie in one plane (i.e., they are linearly dependent; Fig. 5).

##### SPATIAL ORIENTATION OF THE EIGENVECTOR v_{3} RELATIVE TO THE HEAD.

The estimated spatial orientation of the head relative to gravity is described by the 3D orientation of the eigenvector **v**_{3} relative to the head. It was obtained by calculating the polar angles ϑ̂ and φ̂ of **v**_{3} as follows: First we calculated the orientation of the eigenvector **v**_{3} relative to the axis of rotation (**e**_{a}: unit vector directed along the *x*-axis for roll OVAR or along the *y*-axis for pitch OVAR) (2) where **v̂**_{3} = **v**_{3}/‖**v**_{3}‖ is the normalized eigenvector **v**_{3} (**v̂**_{3} = unit vector, ‖**v**_{3}‖ = length of **v**_{3}). The angular orientation of the eigenvector **v**_{3} in the rotation plane was obtained by (Fig. 3, *A* and *B*) (3)

##### ESTIMATION OF POSTROTATORY EYE VELOCITY PARAMETERS AS A FUNCTION OF HEAD ORIENTATION.

The solution (*Eq. 1*) of the feedback system (Fig. 4) provides an expression of postrotatory eye velocity as a function of a few relevant physiological parameters. These parameters constitute the angular orientation of eye velocity relative to gravity, the dominant response time constants, and the feedback gains in the horizontal, vertical, and torsional eye movement channel. To obtain a handle on these parameters, we represented the dynamics of the semicircular canals (**ω**_{rlv} in *Eq. 1*) according to the model introduced by Steinhausen (1933), with time constants fixed to 0.003 and 5 s (Wilson and Melvill Jones 1979). To estimate the otolith-driven head velocity signal, we empirically measured in each trial the amplitude of 3D eye velocity at stop of head rotation [**ω**_{oto} = **ω**_{VOR}(t = 0) in *Eq. 1*], which corresponded to the steady-state velocity that was maintained throughout the 10 rotations. Its main component was vertical for pitch OVAR and torsional for roll OVAR. Finally, we analytically solved the integral on the right-hand side of *Eq. 1* by evaluating the matrix exponential e^{tF}, with F = A(1 − KP) being a unique function of final head orientation. According to the head-in-space orientation hypothesis, the spatial orientation of the postrotatory response depends ultimately on estimated head orientation relative to gravity, as described by the 2 polar angles φ̂ and ϑ̂ that characterize the orientation of the eigenvector **v**_{3} in head coordinates (see Fig. 3; see *Eqs. A11* and *A12* in the appendix). Thus the nonlinear least-squares fit of *Eq. 1* to the postrotatory response yielded an estimate of head orientation relative to gravity at stop of rotation (parameters φ̂ and ϑ̂), the time constant T (see *Eq. A4*), the 3 feedback gains k_{1}, k_{2}, and k_{3} (see *Eqs. A4* and *A5* in the appendix), and the amplitude of 3D eye velocity, elicited by activating the vertical semicircular canals at stop of head rotation. A detailed description of *Eq. 1* as a function of these parameters is presented in the appendix.

To illustrate the fitted curves of postrotatory eye velocity in the spatial domain, we used normalized postrotatory curves, **p**_{n}(t), which we computed as follows: If **p**(t) = [p_{tor}(t), p_{ver}(t), p_{hor}(t)]^{T} was the fitted postrotatory curve, the normalized curve was defined as **p**_{n}(t) = **p**(t)/‖**p**_{0}‖, where was the amplitude of the curve **p**(t) at stop of rotation (time t = 0: onset of postrotatory decay).

##### STATISTICS.

Statistical analyses were performed with the Matlab statistics toolbox (The MathWorks, Natick, MA). Repeated-measures Friedman’s ANOVA was used to determine the significance of differences in the time and the feedback constants as a function of the tilt angle and rotation direction. To compute the variances of volume V and the time constant T_{3} we considered the propagation of errors of the 6 independent variables g_{i} (i = 1 to 3) and k_{i} (i = 1 to 3) (Bevington and Robinson 1992).

## RESULTS

Figure 6*A* shows a representative example of perrotatory VOR velocity during the last 2 cycles of 90° roll OVAR at −60°/s (counterclockwise) and the subsequent postrotatory VOR velocity after the stop of rotation (vertical dashed line) in 60° left ear-down position. This corresponds to a tilt of gravity of +60° because we expressed gravity in head-fixed coordinates in a range of ±90° (Fig. 2*A*). Superimposed on the torsional, vertical, and horizontal components of eye velocity (gray data points in *top*, *middle*, and *bottom panels*), we plotted the exponential functions fitted to each component of eye velocity (black lines), starting at stop of rotation. The perrotatory velocity traces exhibit the well-known steady-state torsional bias velocity, in combination with the head-position–dependent modulation of torsional, vertical, and horizontal eye velocity (Angelaki and Hess 1996a, b; Hess and Dieringer 1990). After stop of rotation, the VOR response consisted of a negative torsional (counterclockwise; see methods) eye velocity, which gradually declined to a zero value.

Note that postrotatory eye velocity did not simply decrease along the former (roll) stimulation axis, as indicated by the additional negative vertical (upward) eye velocity, which built up and then slowly declined toward zero, and the positive progressively decreasing horizontal eye velocity. Also postrotatory vertical eye velocity did not decline to zero, an observation that was occasionally found in all animals, for both the 45° and 90° OVAR paradigms.

Figure 6*B*, using the same conventions as in Fig. 6*A*, illustrates an example of reorientation of postrotatory eye velocity after 90° pitch OVAR at 60°/s with a final head orientation of 60° nose down. This corresponds to a tilt of gravity of −60° in head coordinates (Fig. 2*D*). As indicated by the evolution of horizontal and torsional velocity components, postrotatory eye velocity did not decline along the pitch stimulation axis. The emergence of nonprincipal velocity components after stop of head rotation indicates that the postrotatory eye velocity vector reorients such that it aligns with the direction of gravity.

A better way to illustrate this finding, using the same data as shown in Fig. 6, is to plot the postrotatory eye velocity in the spatial domain (Fig. 7).

For this, we used the normalized postrotatory curves, which we computed from the fitted postrotatory curves as described in methods. Inspection of the normalized exponential fits to postrotatory eye velocity after 90° roll OVAR in Fig. 7, *A*–*C* (solid red lines) reveals that from the initial velocity (solid circle) both vertical and horizontal velocity components build up (Fig. 7*B*) and then decrease toward zero (the solid circle specifies the onset of the velocity vector’s trajectory). In other words, postrotatory eye velocity at first reorients and subsequently declines along a line that is in approximate alignment with gravity (compare in Fig. 7, *A* and *B* the dashed lines representing the line fit on eye velocity with the heavy blue line representing the true direction of gravity in head-fixed coordinates). Note that the emergence of nontorsional eye velocity components means that postrotatory eye velocity does not precisely decrease along the roll (*x*-) axis. We quantified the alignment of postrotatory eye velocity within the resultant response plane (plane labeled RRP in Fig. 7*A*) by computing the angle φ̂ between the line fitted to the orthogonal response and the negative *z*-axis (Fig. 7*B*). In this example, φ̂ = 55°, underestimating by 5° the true value of 60°. For a complete characterization of the eye velocity vector’s 3D orientation, we quantified the slope of its decay in the resultant response plane by determining the angle ϑ̂ of the tangent line, fitted to the decreasing part of the postrotatory velocity (Fig. 7*C*). Here, ϑ̂ = 73°, deviating by 17° from the orientation of gravity relative to the rotation axis of 90°.

In other cases the alignment of postrotatory velocity with the direction of gravity was achieved by changing merely the ratio between components rather than by a buildup of nonprincipal velocity components. The postrotatory eye velocity after 90° pitch OVAR illustrated in Fig. 6*B* provides an example of this mechanism. Plots of the respective exponential fits in the spatial domain (Fig. 7, *D*–*F*, solid red lines) show that the alignment with the direction of gravity occurs—after an initial overshooting in vertical direction—by smoothly adjusting the torsional–horizontal (Fig. 7*E*) and the torsional–vertical postrotatory response components (Fig. 7*F*). In this example, the angle φ̂ yielded a value of −59°, closely matching the true head orientation (−60°).

The slope ϑ̂ of the decreasing part of postrotatory orthogonal eye velocity, here the vector sum of horizontal and torsional eye velocity (i.e., **Ω**_{orth} = **Ω**_{tor} + **Ω**_{hor}) was 88°, underestimating the true inclination of the head relative to gravity in the resultant response plane by 2° (see Fig. 7*F*).

### Spatial orientation of postrotatory eye velocity after roll and pitch OVAR

The reorientation of postrotatory eye velocity seen in the examples of Figs. 6 and 7 was observed in all final body orientations independent of the stimulus configuration, i.e., after 90° roll or pitch OVAR (Fig. 8) as well as after 45° roll or pitch OVAR (Fig. 9). More precisely, the diagrams in the *top* and *bottom panels* of Fig. 8, *A* and *B* and Fig. 9, *A* and *B* plot the polar angles ϑ̂ and φ̂ of eye velocity (solid circle) as a function of the head orientation relative to gravity. The open squares in the *bottom panels* represent empirically evaluated φ̂ values, averaged over both rotational directions (clockwise/counterclockwise roll, forward/backward pitch OVAR) and all 3 subjects (means ± 1SD). The solid circles represent the simultaneously determined φ̂ and ϑ̂ angles based on least-squares fits of *Eq. 1* to postrotatory responses.

Because there was no significant difference between clockwise and counterclockwise roll with the exception of ϑ̂ in the 45° roll OVAR paradigm (90° roll OVAR *P*_{φ} > 0.5; *P*_{ϑ} > 0.7; 45° roll OVAR *P*_{φ} > 0.5; *P*_{ϑ} < 0.01) or forward and backward pitch rotations (pitch OVAR paradigms: *P*_{φ} > 0.7; *P*_{ϑ} > 0.2), ϑ̂ and φ̂ angles were averaged over the 2 directions and all 3 subjects (means ± 1SD). Thereby we did not consider the differences in ϑ̂ between clockwise and counterclockwise roll in the 45° roll OVAR paradigm, which was attributed to a response asymmetry in right ear-down position (45° roll OVAR: *P*_{ϑ} > 0.02 after exclusion of right ear-down tilts). Final body orientations of 0°, 90°, and −90° correspond to upright, left ear-down, and right ear-down (roll) or upright, supine, and prone (pitch). The predictions of a perfectly reorienting VOR velocity in the rotation plane are indicated by the dashed diagonal in Figs. 8 and 9. For instance, the estimated φ̂ should ideally be 60° when animals are stopped in a final position such that the gravity vector’s orientation in the pitch or roll plane is 60°. This statement applies to both 90° and 45° roll or pitch OVAR because the φ angle was always measured in the rotation plane. In general, we found a close correlation between the estimated φ̂ values and the direction of gravity in the rotation plane, as supported by the high *r*^{2} values from linear regression analyses: 90° or 45° roll OVAR *r*^{2} = 0.96 or 0.94, 90° or 45° pitch OVAR *r*^{2} = 0.98 or 1.0. Linear regression of the φ̂ angle yielded a slope of 0.75 ± 0.17 and an offset = −0.8 ± 5.4° for 90° roll OVAR compared with 0.86 ± 0.13 and 1.5 ± 8.4° for 90° pitch OVAR. Thus the direction of postrotatory responses after both roll or pitch OVAR rather tightly followed the direction of gravity in the tilt plane. At larger tilt angles there was a tendency to undershoot the veridical head tilt, in particular in the 90° roll OVAR paradigm.

For a complete description of the spatial orientation we also quantified how much gravity determined the trajectory of the postrotatory eye velocity response in a plane orthogonal to the rotation plan. In contrast to the angle φ̂ that described the orientation of eye velocity in the pitch or roll plane, the angle ϑ̂ determined the orientation in the orthogonal plane set up by the estimated direction of eye velocity in the rotation plane (i.e., the line fitted to the decreasing part of orthogonal eye velocity) and the rotation axis (see resultant response plane, Figs. 2, 3, and 7). More specifically, we defined ϑ̂ as the angle subtended by a tangent line to the final portion of the response trajectory and the rotation axis. The average ϑ̂ angle after 90° roll and pitch OVAR was ϑ̂ = 79.7 ± 7.5° and 93.0 ± 24.1°, respectively (in both paradigms the veridical angle is 90°; Figs. 2*A* and 8, *A* and *B*). Whereas the ϑ angle in the roll paradigm was estimated close to veridical, at least for tilt angles between +60° and −60° (Fig. 8*A*) the estimated ϑ̂ values in the 90° pitch paradigm deviated from the true values by an average of 36.9° when the animals were stopped in 30° and 60° nose-up position (Fig. 8*B*). Note also that the estimates ϑ̂ showed much larger SDs in the pitch than in the roll OVAR paradigm. After 45° roll OVAR, ϑ̂ was 110.4 ± 3.2° (the veridical angle ϑ = 135°; Figs. 2*B* and 9*A*) and after 45° pitch OVAR ϑ̂ was 48.4 ± 11.6° (the veridical angle ϑ = 45°; Figs. 2*B* and 9*B*).

### Properties of feedback constants, roll and pitch eigenvectors, and time constants

The feedback strengths in the different eye movement channels (see diagonal matrix K in Fig. 4) ultimately determine, together with the central estimate of head-in-space orientation, **ĝ**, as a function of ϑ̂ and φ̂ (see Fig. 3), the spatial orientation as well as the associated dynamics of the postrotatory response (see *Eqs. A3* and *A4* in the appendix). In fact, the response parallels the estimated head-in-space-orientation only if the ratios of the feedback strengths in the relevant (i.e., nonzero) feedback channels are approximately equal. In the 90° roll OVAR paradigm, the feedback strength of the horizontal (k_{3}) and vertical channel (k_{2}), which control the spatiotemporal characteristics of the orthogonal postrotatory response in the roll plane, did not differ significantly from each other (Friedman’s ANOVA, *P* > 0.5). We therefore pooled these values over the 3 animals and the 2 directions of rotations and compared these averages with the average feedback in the torsional channel k_{1} (Fig. 10*A*). The torsional feedback was significantly smaller than the horizontal or the mean of the horizontal and vertical feedback (Friedman’s ANOVA, *P* < 0.01 for both comparisons). This means that most of the response ultimately decayed in the roll plane as illustrated by the average ϑ̂ angles in Fig. 8*A*. Similarly, one would expect the feedback strength in the vertical channel to be relatively small in the 90° pitch paradigm, such that responses ultimately decayed in the pitch plane. However, this was not observed, consistent with the large variances and the deviations of the average ϑ̂ angles from the pitch plane in Fig. 8*B*. Because there were no significant differences in strength of the three feedback channels in the remaining paradigms (Friedman’s ANOVA, *P* > 0.1), we pooled the values in each of the 90° pitch, 45° roll, and 45° pitch OVAR paradigms over the 3 animals and the 2 directions (Fig. 10, *B*–*D*).

Because the eigenvectors of *Eq. 1* in general change orientation as a function of head-in-space orientation, it is of interest to see whether the orientation of the 3 eigenvectors, which ultimately determine the direction of postrotatory eye velocity, was independent of how final head orientations were reached in different paradigms. As a consequence of *Eq. 1* only the postrotatory responses after 45° roll/pitch OVAR were characterized by 3 linearly independent eigenvectors (see Fig. 5), as indicated by the nonzero volume V of the parallelepiped that these vectors encompassed throughout the tilt range (Fig. 11). Although linearly independent, the eigenvectors were not mutually orthogonal, as demonstrated by the fact that the magnitude of the volume V was less than unity. Furthermore, the sign of V systematically differed in 45° roll responses: In right ear-down/nose-up positions, the orientation of the eigenvectors resulted in a positive V (Fig. 11*A*), whereas V was always negative in left ear-down/nose-up positions. Technically, this observation means that the eigenvectors switched from a right-handed to a left-handed triple when the *y*-component of gravity changed from positive to negative (Fig. 11*A*). In the 45° pitch OVAR paradigm, the orientation of the eigenvectors did not switch, as expected (Fig. 11*B*). Comparison of the orientations of the eigenvectors across the 2 paradigms shows that they were similar in the same final head orientations (led/nu positions in Fig. 11, *A* and *B*).

Associated with each eigenvector is an eigenvalue (= reciprocal time constant) that characterizes the dynamics of the postrotatory response component along this eigenvector. The time constants associated with the first and second eigenvector were equal (see *Eq.*A*5* in the appendix). They were similar for each stimulus configuration and did not differ significantly for clockwise versus counterclockwise and forward versus backward rotation (T_{1}: Friedman’s ANOVA, *P* > 0.25 in 90° OVAR, *P* > 0.05 in 45° OVAR paradigms). They assumed average values between 4 and 6 s (see Table 1). In contrast, the time constants associated with response components along the third eigenvector were on average about 5 times longer and also did not differ significantly for clockwise versus counterclockwise and forward versus backward rotation (T_{3}: Friedman’s ANOVA, *P* > 0.25 in 90° OVAR, *P* > 0.05 in 45° OVAR paradigms).

In addition to the estimated φ and ϑ, the feedback constants, and the time constants, the least-squares fit of *Eq. 1* yielded also estimates of the amplitude (v_{0}) of the semicircular canal–induced response generated at stop of rotation. The averages (±1SD) across the 2 rotation directions and all tilt angles of all 3 animals are summarized for the 90° and 45° roll and pitch OVAR paradigms in Table 1.

We also fitted equation 1 to all postrotatory responses after roll and pitch OVAR in all 3 animals after replacing the projection P in the feedback matrix F with a rotation R (for details see appendix). Although the fitted curves were often indistinguishable from those obtained with the projection model, the corresponding angular head orientation parameters did not correlate well with the final head orientation at stop of rotation. Table 2 summarizes the variances accounted for (VAF) of the estimated φ̂ angles for both the projection and rotation model. It also shows the average estimates of the ϑ̂ angles for the 2 models.

## DISCUSSION

Roll and pitch movements of the head from upright activate both the vertical semicircular canals as well as the otolith organs, which code information about head acceleration, velocity, and position. The brain uses these vestibular and other proprioceptive signals to generate an internal space-centered reference system, which enables orientation of gaze in space independent of self-motion. Our analysis of postrotatory responses after roll and pitch OVAR suggests that the spatial properties of residual 3D eye velocity reflect the brain’s best estimate of head-in-space orientation. The underlying mechanisms can conceptually be thought of as a projection of the vector of residual postrotatory eye velocity onto a central representation of gravity within a feedback system. Alternatively, postrotatory eye velocity could actively rotate toward alignment with an internal estimate of gravity, as described earlier for responses after yaw OVAR (Jaggi-Schwarz et al. 2000). For comparision, simulations of these 2 fundamentally different mechanisms have been illustrated in Fig. 1 for different feedback strengths. In contrast to the yaw OVAR paradigm, we found that after roll and pitch OVAR not only the centrally estimated direction of gravity but also 2 orthogonal directions, which depended on the particular head orientation relative to gravity, determined the geometric properties of postrotatory velocity.

### Residual 3D postrotatory eye velocity reflect 3D head-in-space orientation

In our paradigms the subjects were stopped after several cycles of rotations in complete darkness, well after the rotation cues from the semicircular canals have subsided. In this situation self-orientation in space faces 2 problems, summarized here as the *head-in-space orientation problem*: First, to estimate the final head and body orientation relative to gravity as quickly and accurately as possible and second, based on this estimation, to further estimate whether there is any head translation by comparing the total afferent otolith input with the centrally estimated head orientation. This second step is necessary because the otolith afferents carry information about the sum of gravitational and inertial head accelerations.

After a sudden stop head velocity signals, generated by activation of the vertical semicircular canals and otolith signals, reflecting head-in-space velocity and instantaneous orientation relative to gravity (Angelaki 1992a, b; Hain 1986; Schnabolk and Raphan 1992), are thought to interact within feedback loops with central signals. Immediately after the stop most of the persisting angular eye velocity (**ω**_{oto} in Fig. 4) is cancelled by vertical canal signals (**ω**_{rlv} in Fig. 4), a process that is virtually perfect under visual conditions.

To quantitatively analyze the characteristics of the residual velocity signals when rotation stops in the dark, we adopted the following assumptions: First, we assume that the inertial velocity signals conform to the physical constraints that exist between the rate of change in head orientation, the instantaneous head orientation, and head angular velocity. These constraints can be quantitatively expressed by the vector equation d**a**_{oto}/dt = **ω**_{head} ∧ **a**_{oto} + dα_{oto}/dt **u** (see Goldstein 1980), which captures the spatiotemporal relation between the time derivative of otolith signals (denoted by d**a**_{oto}/dt) that encodes the rate of change in head orientation (i.e., rotation and/or translation), and a rotation of the head relative to gravity (denoted by **w**_{head} ∧ **a**_{oto}), and/or a translation of the head. Note that a pure translation results in a change in acceleration magnitude but not in direction (thus: d**a**_{oto}/dt = dα_{oto}/dt **u**, with **a**_{oto} = α_{oto}**u**, where **u** = unit direction of translation). If the brain’s best estimate is that head-in-space orientation (denoted in Fig. 4 by the estimated orientation of gravity, **ĝ**, in head coordinates) is stationary at stop of rotation (i.e., **ĝ** ≈ **a**_{oto}), it follows from the physical stationary condition for otolith signals that the residual head angular velocity must parallel the estimated head-in-space orientation (i.e., parallel vectors: **ω**_{head} ∼ **ĝ**). An implicit assumption of our approach is that the centrally estimated head orientation relative to gravity does not much change over the time course of postrotatory nystagmus. Recent human psychophysical studies appear to support this notion (Jaggi-Schwarz and Hess 2003; Jaggi-Schwarz et al. 2003), although some earlier studies found considerable variability in estimates of the subjective visual vertical over time, particularly in roll tilt positions beyond 90° (Udo de Haes and Schöne 1970; von Holst and Grisebach 1951). Second, we assume that the mechanisms, which underlie the transformation of residual head velocity signals toward alignment with the estimated head-in-space orientation, are based on cosine tuning, also referred to as projection, because only the vector component along the estimated direction of gravity survives.

Finally, we account for the well-documented central enhancement of the time constants of the postrotatory responses (velocity storage: Cohen et al. 1977; Raphan et al. 1977, 1979) by a leaky integration within a positive feedback loop (Robinson 1977). The stationary hypothesis (vector parallelism, **ω**_{head} ∼ **ĝ**, if the brain estimates d**ĝ**/dt = 0) finds ample support in respective conflict paradigms (Angelaki and Hess 1994; Dai et al. 1991; Gizzi et al. 1994; Harris 1987; Harris and Barnes 1987; Hess and Angelaki 1995; Jaggi-Schwarz et al. 2000; Merfeld et al. 1993, 1999, 2001; Raphan and Sturm 1991; Raphan et al. 1992), where it predicts that postrotatory head velocity signals in the VOR ultimately must align with the brain’s best estimate of direction of gravity. This is so because it represents the only physically possible configuration of a postrotatory head velocity signal and a stationary head-in-space-orientation. That the head must be stationary in these conflict experiments is the brain’s best estimate based on otolith *and* proprioceptive somatosensory cues. Mechanisms underlying the spatial transformations of head angular velocity signals toward alignment with gravity have been suggested in a number of studies (Angelaki and Hess 1994, 1995; Angelaki et al. 1995; Hess and Angelaki 1995; Merfeld 1995; Merfeld and Zupan 2002; Mergner and Glasauer 1999; Raphan and Sturm 1991; Zupan et al. 2002). Specifically, Angelaki and Hess (1994) showed evidence that the spatial transformation of postrotatrory responses show different characteristics in the horizontal versus the torsional and vertical eye movement pathways: Whereas horizontal postrotatory responses after dynamic head tilts in pitch or roll showed characteristics compatible with a rotation, torsional and vertical responses after tilts in the yaw plane showed characteristics of a projection. Recently we have extended this finding by quantitatively showing that the reorientation of angular velocity signals in the yaw OVAR paradigm is compatible with the geometric properties of a rotation (Jaggi-Schwarz et al. 2000). In the following we discuss the experimental evidence for a projection (cosine tuning) mechanism in the torsional and vertical VOR pathways and compare the utility of rotation and projection mechanisms.

### Characteristics of head-in-space orientation estimates after roll or pitch OVAR

Our analysis of the orientation properties of postrotatory responses after roll and pitch OVAR results in reasonably good predictions of the actual head orientation of the monkey in space. In contrast to the yaw OVAR paradigm (Jaggi-Schwarz et al. 2000), head-in-space-orientation could not be predicted from the postrotatory responses after roll and pitch OVAR by implementing a rotation mechanism (rotation matrix R instead of projection P in Fig. 4). As earlier outlined, the 2 transformations have different spatial properties, which entail different dependencies of the postrotatory response profiles on head-in-space orientation. In the projection model, the feedback loop enforces spatial alignment of residual eye velocity with the direction of gravity as the feedback constants tend toward equal strengths in the relevant eye movement channels. In the rotation model, the feedback works differently in that it mainly controls the direction but not as efficiently the magnitude of rotation. The magnitude of rotation depends in a highly nonlinear fashion on the parameters of head orientation in space. Simulations of this model show that the feedback needs to be limited in strength to avoid overshooting responses (see example in Fig. 1*B*). We found that least-squares fits of the rotation model to postrotatory responses yielded angles ϑ̂ and φ̂ that poorly correlated with the final head-in-space orientation in both the roll and pitch OVAR paradigm (see Table 2). This was true even though these fits were often not distinguishable from those of the projection model.

In all OVAR paradigms monkeys estimated head-in-space orientation significantly better in the tilt plane than in a plane perpendicular to it (called “resultant response plane”; see Fig. 2). This is true for the earlier studied yaw OVAR paradigm (Jaggi-Schwarz et al. 2000) as well as for the roll and pitch paradigms. Specifically, we observe negligible differences between the average estimated and true head orientation in the 90° roll or pitch OVAR paradigm if the tilt range is restricted to ±60° around upright or to upright and forward tilted positions, respectively. Nose-up orientations are often under- or overestimated with respect to the resultant plane (ϑ angle) or even systematically underestimated, such as after the animal has been rotated about a 45° nose-up tilted axis (Fig. 9*A*).

To further characterize the otolith-dependent orientation signals, we quantified the mutual orientation of the eigenvectors as a function of head orientation. This analysis demonstrates that, independent of the paradigm, the spatial trajectory of postrotatory responses are attributed to similar orientations of the eigenvectors at the same final head orientation (compare led/nu position in the 45° roll and pitch paradigms, Fig. 11). A comparison of the estimated orientation in the resultant response plane (ϑ angle), however, shows that this orientation parameter is not consistently estimated in the same final head positions in the roll and pitch OVAR paradigm. A possible explanation for this discrepancy is that estimation of the ϑ parameter depends not only on the final head orientation but also on the previous rotation.

### Tilt plane and head position dependent deficits in 3D estimates of head-in-space orientation

What is the origin of the observed head orientation–dependent estimation errors? One possibility is that it reflects a systematic limitation of the projection or cosine-tuning hypothesis. Raphan and Sturm (1991) assumed in their velocity storage model that only the yaw eigenvector reorients toward earth vertical, whereas the roll and pitch eigenvectors remain head fixed. Although these assumptions are perfectly sufficient to explain the reorientation of postrotatory eye velocity in yaw OVAR paradigms they are incompatible with a reorientation of postrotatory responses after pitch and roll OVAR. The projection hypothesis, on the other hand, has been successfully applied in explaining the spatial orientation of torsional and vertical postrotatory eye velocity in dynamic tilt paradigms (Angelaki and Hess 1994; Angelaki et al. 1995). Furthermore it has been used explicitly in some observer-type models to explain the interaction of eye velocity and gravity (Droulez and Darlot 1990; Zupan et al. 2002) but not in others (Merfeld 1995; Merfeld and Zupan 2002; Merfeld et al. 1993). This is not surprising because, to our knowledge, none of these models has explicitly addressed the head-in-space orientation problem after rotations in the roll and pitch plane.

Other reasons that could limit the accuracy of estimating head-in-space orientation in nose-up positions could be rooted in the way of how backward directed otolithic shear forces are encoded and processed. Anatomically it is known that the anterior portion of the utricles is bent upward in primates (Lindenman 1973; Spoendlin 1966), possibly to improve the sensitivity for head orientation cues in a wider range of forward tilted positions. From clinical examinations it is known that static imbalance, such as in one-leg stance, increases when the head is extended, suggesting that the otolith signal processing is less effective in this posture (Brandt 1999).

### Consequences of misjudging head-in-space orientation

Erroneous interpretation of gravitoinertial cues are the source of a number of illusions (for a review see Bos and Bles 2002; Young 2002). In a recent study in humans, Merfeld and colleagues (1999) reported the perception of illusionary tilt in a postrotatory tilt paradigm involving strong conflicting rotational cues. Along these lines, the head-in-space orientation errors in the nose-up positions may suggest that the monkeys tended to misjudge their orientation in a direction perpendicular to the tilt plane and might have experienced an illusionary perception of forward motion at stop of rotation. To our knowledge, there are no quantitative studies on the perception of self-orientation after roll tilt in nose-up orientations.

### Differences in the spatial transformation properties of yaw versus pitch and roll postrotatory OVAR

Why are there fundamental differences in the spatial representation of yaw versus roll or pitch head velocity signals? One reason could be that these differences reflect the geometric constraints on vestibular balance control while performing gaze orientating head and body movements during locomotion and foraging. To maintain balance in a gravitational environment it is advantageous to keep the angular head and body velocity approximately aligned with gravity to comply as much as possible to the dynamic equilibrium condition **ω** ∧ **g** = **0** (i.e., keep head and body angular velocity as much as possible aligned with gravity). Clearly, rotations about roll and/or pitch body axes are much more threatening for balance control than yaw movements because of the nearly orthogonal orientation of these movements relative to gravity. To maintain balance these movements must be limited in amplitude. Along these lines, we propose that the projection mechanism observed in head-in-space orientation signals reflects the signal processing of a central vestibulomotor balance control system. In contrast to roll and pitch, yaw head and body movements are much less threatening because the rotation axis is usually close to earth vertical. To establish a space-fixed relation between self-motion signals and gravity, it is more useful to use a rotation in this case. The rotation is optimally suited to transform velocity signals from head- to space-fixed coordinates because it is symmetric with respect to the longitudinal head and body axis, and therefore also with respect to the usual orientation relative to the earth horizontal plane.

## APPENDIX

### Interaction of otolith and vertical semicircular canal signals

In the present experiments, monkeys were rotated either in roll (about the naso-occipital or *x*-axis) or in pitch (about the interaural or *y*-axis). These rotations were applied with the axis in the earth-horizontal plane (90° OVAR), or tilted nose up or right ear up by 45° relative to the earth-horizontal plane (45° OVAR). Altogether the following 4 paradigms were tested: *1*) 90° roll OVAR (see Fig. 1*A*, ϑ = 0), *2*) 45° roll OVAR (Fig. 1*B*, ϑ = 45°), *3*) 90° pitch OVAR (Fig. 1*C*, ϑ = 0), and *4*) 45° pitch OVAR (Fig. 1*D*, ϑ = 45°). In the following we describe the central interaction of vertical semicircular canal and otolith signals when the rotation stops in different head orientations.

The integration of otolith and semicircular canal-borne head velocity signal can be characterized by a set of linear 1st-order differential equations. Generalizing Robinson’s 1D velocity storage model (Robinson 1977) to 3D we obtain the following set of differential equations in the time domain (A1) The system matrix F is obtained from a diagonal matrix A of 3 equal time constants [A]_{ii} = 1/T (i = 1 … 3) and a diagonal matrix K with 3 different feedback constants [K]_{ii} = k_{i} (i = 1 … 3). The driving inputs to this leaky integration network are the lumped angular velocity signals from the vertical semicircular canals, represented by the vector **ω**_{rlv}. Note that the input **ω**_{rlv} is not modified, that is, **ω** = **ω**_{rlv}, if the feedback is turned off.

##### PROJECTION MODEL.

We have tested the hypothesis that the spatial transformation in this network is realized by a gravity-dependent projection mechanism (cosine tuning). Such a mechanism modifies the velocity signals in the loop according to (A2) where ĝ is the estimated vector of gravity g. Thus the output of the spatial filter P is a vector directed along gravity with magnitude proportional to the cosine of the angle subtended by the input vector and gravity. The effect of this mechanism is to facilitate central integration of vertical canal signals according to their alignment with gravity.

By introducing the projection mechanism (*Eq.*A*2*) into the feedback system (*Eq.*A*1*) we obtain the following system matrix (A3) The eigenvectors and eigenvalues of the matrix F describe the spatiotemporal characteristics of A*1* (A4) (A5) Because of the algebraic multiplicity of the eigenvalue λ_{1}, the eigenvectors (*Eq.*A*4*) must be replaced by generalized eigenvectors to correctly capture the spatial properties of the system in *Eq.*A*1*. Using the Jordan decomposition we find that eigenvector **v**_{1} needs to be replaced by A4′ Depending on the feedback, the eigenvector **v**_{3} tends to align with gravity, whereas the other 2 are always perpendicular to gravity (**v**_{1} · **g** = 0; **v**_{2} · **g** = 0). The time constant of the response component along the eigenvector **v**_{3}, T_{3} = 1/λ_{3} (*Eq.* A*4*) depends on both the feedback in the system and the system’s orientation relative to gravity, whereas the time constants along the other 2 eigenvectors **v**_{1} and **v**_{2} are invariable and equal. As the feedback constants approach unity, the system enhances responses along the eigenvector **v**_{3} by the factor 1/(1 − ∑_{i} k_{i}g_{i}^{2}), whereas responses along orthogonal directions are not affected.

##### ROTATION MODEL.

Along the same lines, we tested the alternative hypothesis that the spatial transformation is a gravity-dependent rotation of postrotatory eye velocity. In this case, the system matrix F (A6) represents a rotation defined by the axis of rotation **a** = [a_{1} a_{2} a_{3}]^{T} (‖**a**‖ = 1) and the angle ϑ of rotation about this axis. The eigenvectors and eigenvalues of matrix F are (if a_{1} and a_{2} are not both equal zero) (A7) (A8) The spatiotemporal properties of this system are characterized by one real eigenvector (**v**_{3} = **a**) and eigenvalue (λ_{3}) and two complex conjugate eigenvectors and eigenvalues (**v**_{1}, λ_{1} and **v**_{2}, λ_{2}). We assume that the rotation axis **a** aligns with the line of intersection of the earth horizontal plane and the plane of head rotation (see Jaggi-Schwarz et al. 2000). Accordingly, we aligned the eigenvector **v**_{3} in the roll OVAR paradigm with the *y*-axis; that is, **v**_{3} = **a** = [0 cos (φ_{a}) sin (φ_{a})]^{T} with φ_{a} = 0 in the upright position. Similarly in the pitch OVAR paradigm we aligned it with the *x*-axis; that is, **a** = [cos (φ_{a}) 0 −sin (φ_{a})]^{T} with φ_{a} = 0 in the upright position. Thus when the subject stops after roll OVAR in 60° left ear-down position, the rotation axis, about which eye velocity is assumed to rotate through an angle ϑ_{a} toward alignment with gravity, would be: **a** = [0 cos (60°) sin (60°)]^{T}. The complete reorientation of postrotatory eye velocity toward gravity is described by the angles φ_{a} and ϑ_{a}.

The solution of *Eq.*A*1* with the gravity-dependent system matrix F (*Eq.*A*3* or *Eq.*A*6*) yields for an initial response vector **ω**_{0} (=**ω**_{oto}) (Kailath 1980) (A9) This solution constitutes a purely *otolith-dependent* and *semicircular canal*–*dependent* component that in turn also depends on gravity by virtue of the gravity-dependent matrix exponential e^{tF} (*Eq.*A*3*). It is easily seen that this matrix exponential has the same eigenvalues and eigenvectors as the matrix F because of the relation (A10) where λ_{i} are the eigenvalues and V is the matrix of eigenvectors of F.

For a parametric evaluation of the matrix elements of F we note that during roll and pitch OVAR, the vector of gravity rotates about the subject as follows (A11) (A12) The angle φ describes the angle of rotation of the vector of gravity about the naso-occipital or interaural axis (positive for leftward or backward rotation), whereas the angle ϑ represents the tilt of the vector of gravity relative to the respective rotation axis (0 ≤ ϑ ≤ 180°). In the 90° roll or pitch OVAR paradigm ϑ = 90°, whereas in the 45° roll or pitch OVAR ϑ = 135° and 45°. In the projection model, *Eqs.*A*11* and A*12* determine the matrix elements of the projection operator P (*Eq.*A*3*) at stop of rotation as functions of φ and ϑ. In the rotation model, the same angles φ and ϑ describe the orientation of the axis **a** (i.e., φ_{a} = φ) and the amount of rotation of postrotatory eye velocity about this axis, away from the head rotation axis (i.e., ϑ_{a} = ϑ). This is so because the vectors **g** and **a** have been defined such that they are always perpendicular to each other.

For the experimental evaluation of *Eq.*A*9* we approximated the postrotatory vertical semicircular canal signals at stop of rotation by **ω**_{rlv}(t) = v_{0}(e−t/T1 − e−t/T0)**ω̂**_{rlv} with T_{1} = 5 s, T_{0} = 0.003 s, and unit vector **ω̂**_{rlv} directed along the naso-occipital and interaural axis for roll and pitch OVAR, respectively. With this information at hand, we analytically solved the time integral on the right-hand side of *Eq.*A*9*. For a detailed quantitative evaluation, we fitted the solution of *Eq.*A*9* with nonlinear least-squares methods to the postrotatory response. The estimated parameters were the tilt angles ϑ and φ (g_{i} in *Eqs.*A*11* and A*12*); the inertial response time constant T (*Eq.*A*5*); the feedback constants k_{1}, k_{2}, and k_{3}; and the sensitivity of the semicircular canal response (v_{0}).

## GRANTS

This work was supported by the Swiss National Science Foundation Grant 31-47 287.96.

## Acknowledgments

We thank B. Disler and E. Buffone for excellent technical assistance.

## Footnotes

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- Copyright © 2005 by the American Physiological Society