Canal-Otolith Interactions After Off-Vertical Axis Rotations I. Spatial Reorientation of Horizontal Vestibuloocular Reflex

Karin Jaggi-Schwarz, Hubert Misslisch, and Bernhard J. M. Hess

Department of Neurology, University of Zurich, CH-8091 Zurich, Switzerland


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Jaggi-Schwarz, Karin, Hubert Misslisch, and Bernhard J. M. Hess. Canal-Otolith Interactions After Off-Vertical Axis Rotations I. Spatial Reorientation of Horizontal Vestibuloocular Reflex. J. Neurophysiol. 83: 1522-1535, 2000. We examined the three-dimensional (3-D) spatial orientation of postrotatory eye velocity after horizontal off-vertical axis rotations by varying the final body orientation with respect to gravity. Three rhesus monkeys were oriented in one of two positions before the onset of rotation: pitched 24° nose-up or 90° nose-up (supine) relative to the earth-horizontal plane and rotated at ±60°/s around the body-longitudinal axis. After 10 turns, the animals were stopped in 1 of 12 final positions separated by 30°. An empirical analysis of the postrotatory responses showed that the resultant response plane remained space-invariant, i.e., accurately represented the actual head tilt plane at rotation stop. The alignment of the response vector with the spatial vertical was less complete. A complementary analysis, based on a 3-D model that implemented the spatial transformation and dynamic interaction of otolith and lateral semicircular canal signals, confirmed the empirical description of the spatial response. In addition, it allowed an estimation of the low-pass filter time constants in central otolith and semicircular canal pathways as well as the weighting ratio between direct and inertially transformed canal signals in the output. Our results support the hypothesis that the central vestibular system represents head velocity in gravity-centered coordinates by sensory integration of otolith and semicircular canal signals.


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

During off-vertical axis rotation the vestibular system detects head velocity and position from semicircular canal and otolith afferent signals. Whereas the semicircular canal afferents encode endolymph velocity relative to the skull (i.e., relative to a head-fixed reference) the otolith organs provide information about absolute head position relative to gravity (i.e., relative to a space-fixed reference). To determine head motion and orientation in space, there must be considerable central processing of these signals based on an internal representation of the outer physical space.

Earlier studies of the vestibuloocular reflex (VOR) during off-vertical axis rotations have shown that a vertical eye velocity component was generated when a subject was stopped in ear-down position after a constant velocity rotation about the yaw axis (cat: Harris 1987; humans: Harris and Barnes 1987; monkey: Raphan et al. 1992). Thus it appeared that postrotatory eye velocity did not decay along the former rotation axis but rather tended to align with gravity. Although it has become clear that this spatial reorientation of the VOR is mediated via the velocity storage network in the brain stem, the underlying computational mechanisms are still a matter of debate. In a recent study on the spatial orientation of the optokinetic nystagmus (OKN) in primates, it has been shown that in the horizontal but not vertical system there is a similar alignment of the optokinetic afterresponse with gravity (Dai et al. 1991; Raphan and Sturm 1991). A reorientation of eye velocity toward gravity can also be found during postrotatory VOR when applying quick changes in head (and body) orientation relative to gravity (Angelaki and Hess 1994; Merfeld et al. 1993). In contrast to OKN findings, this reorientation occurred independently of the head rotation axis (before tilting the head) and the particular tilt plane (Angelaki and Hess 1994). Analyzing the spatial characteristics of the yaw or pitch/roll VOR suggested a rotation or projection of postrotatory eye velocity toward the spatial vertical following the head tilt. It is not known whether the same mechanisms underlie the reorientation of postrotatory VOR in static tilt positions. Off-vertical axis rotation (OVAR) is the vestibular analogue of visual surround rotation about a tilted stationary observer. Because during OVAR the observer is rotated relative to gravity, the postrotatory response results from a complex interaction of otolith and semicircular canal signals. In this paper, we determine the relative contribution of the otolith and canal signals after yaw OVAR by extending an empirical analysis of the VOR spatial characteristics with a parametric analysis based on a three-dimensional (3-D) spatial orientation model. In the companion papers, we shall present the results of a corresponding analysis applied on data collected after pitch and roll OVAR. Preliminary results have been published in abstract form (Jaggi-Schwarz et al. 1999).


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Preparation of animals

Experiments were performed on three juvenile rhesus monkeys (Macaca mulatta; monkeys JU, SU, and RO). The animals were chronically prepared for 3-D eye movement recordings. Using sterile surgical techniques, skull bolts for head restraint and a dual search coil were implanted under intubation anesthesia with O2-N2O supplemented with halothane if required to maintain a constant level of anesthesia (Hess 1990). Animals were trained to fixate small target lights for fluid reward. All procedures were in accordance with the National Institutes of Health Guide for the Care and Use of Laboratory Animals, and the protocol was approved by the Veterinary Office of the Canton of Zürich.

Measurement of 3-D eye position and calibration

3-D eye positions were measured with the magnetic search coil technique using an Eye Position Meter 3000 (Skalar, Delft, The Netherlands). A horizontal and vertical magnetic field (20 kHz; phase and space quadrature) was generated by coils mounted on a cubic frame of 0.3 m side length.

The output of the dual search coil corresponded to the horizontal and vertical angular orientation of two sensitivity vectors: one pointing roughly in the direction of the visual axis and the other about perpendicular to that. The four voltage output signals of the search coil, as well as the head position and velocity signals, were sampled at a rate of 833 Hz (Cambridge Electronics Device 1401 Plus) and stored on the hard disk of a PC for off-line analysis.

3-D eye position was calibrated as described in detail elsewhere (Hess et al. 1992). Briefly, in an in vitro procedure, the magnitude of the two coil sensitivity vectors as well as the angle between them was computed. In an in vivo procedure performed on each experimental day, the monkeys repeatedly fixated three light-emitting diodes placed at straight ahead, 20° up and 20° down. The measured voltages were used in combination with the coil parameters determined in vitro to compute the orientation of the search coil on the eye and offset voltages. Eye positions were expressed as rotation vectors
<B>E</B><IT>=</IT>tan (<IT>&agr;/2</IT>)<B>n</B> (1)
where n is the axis of eye rotation from reference position to the current position, and alpha  is the angle of rotation (Haustein 1989). The direction of n is specified by the right-hand rule. The eye angular velocity vector Omega  was computed as (Hepp 1990)
<B>&OHgr;</B><IT>=2</IT>(d<B>E</B><IT>/d</IT><IT>t</IT><IT>+</IT><B>E</B><IT>×d</IT><B>E</B><IT>/d</IT><IT>t</IT>)<IT>/</IT>(<IT>1+‖</IT><B>E</B><IT>‖<SUP>2</SUP></IT>) (2)
Eye position and eye velocity vectors were expressed relative to a right-handed, orthogonal, head-fixed coordinate system that was defined by the direction of the two magnetic fields relative to the animal in the standard 15° nose-down position (see Experimental setup and protocol). Hereby, the direction of the horizontal magnetic fields determined the direction of the y-axis pointing along the interaural head axis; the direction of the vertical field was inclined backward by 15° relative to the stereotaxic horizontal plane, and the x-axis was determined as the direction perpendicular to the y- and z-axis. Thus Ex, Ey, and Ez (Omega x, Omega y, and Omega z) represent the torsional, vertical, and horizontal component of eye position (velocity). Positive directions of the coordinate axes represent clockwise, downward, and leftward components (as seen from the subject's point of view) of eye position and velocity.

Experimental setup and protocol

The monkeys were seated in a primate chair and secured with shoulder and lap belts. The head was restrained in a 15° nose-down position such that the lateral semicircular canals were approximately earth-horizontal when the monkey was upright. The primate chair was placed inside a vestibular rotator with three motor-driven axes (Acutronic, Bubikon, Switzerland). The accuracy of the position control on each axis was <0.1°. The rotator was surrounded by a light-proof sphere of 0.8 m radius to guarantee complete darkness.

Animals were pitched 24° nose-up (JU and SU; see Fig. 1A) or 90° nose-up (supine; JU, SU, and RO; see Fig. 1B) and rotated around the tilted z-axis with a constant velocity of ±60°/s (left- or rightward) for 10 cycles (initial acceleration 180°/s2). In the following we refer to these two paradigms as "24° yaw OVAR" and "90° yaw OVAR." Then they were stopped with decelerations of 180°/s2 to reach any of 12 predefined end positions in space-fixed coordinates, equally spaced at 30° intervals: 30°, 60°, . . . , 360° (positive, i.e., leftward yaw OVAR) or -330°, -300°, . . . , 0° (negative, i.e., rightward yaw OVAR).



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Fig. 1. Orientation of gravity vector in space- and head-fixed coordinates. A and B: initial body orientations of 24° nose-up [24° yaw off-vertical axis rotation (OVAR)] or 90° nose-up (supine: 90° yaw OVAR); G, direction of gravity vector. C: final head orientation reached after a rightward body rotation through 240°. D: same final body orientation as in C. The final direction of gravity is 60° in the same head-fixed coordinate system in which 3-dimensional eye position and velocity is expressed. Note that the same final end position can be reached by rotating through +x° in positive direction (e.g., +90°) or by rotating through (x°-360°) in negative direction (e.g., -270°).

For data analysis and for representation of results, we describe the orientation of the gravity vector in a head-fixed coordinate system as follows. The rightward rotation of the head by -240° (Fig. 1C) relative to space is equivalent to a rotation of the gravity vector in the opposite direction by +240° relative to the head, corresponding to a final orientation of the gravity vector of 60° (Fig. 1D).

Data analysis

To compute the velocity vector dE/dt, the three eye position components were digitally differentiated with a quadratic polynomial filter using a 15-point forward and backward window (Press et al. 1992; Savitzky and Golay 1964). This filter has a cutoff frequency with respect to smoothing of 29.3 Hz. Eye angular velocity (Omega ) was computed using Eq. 2. To compute slow phase eye velocity, fast phases of vestibular nystagmus were removed based on time and amplitude windows, set for the magnitude of the second derivative of eye velocity (jerk).

In a first approach, the spatial orientation of postrotatory slow phase eye velocity was evaluated in several steps. First, the orthogonal response component Omega orth, i.e., the vectorial sum of the torsional and vertical eye velocity, was computed. The magnitude of this component reflected the deviation of postrotatory eye velocity from the principal response direction (i.e., from a velocity vector parallel to the z-axis). Second, the sum of two exponentials was fitted (in the least-squares sense) to the time course of each of the three components of eye velocity as well as to the orthogonal component, starting at cessation of chair rotation (see Figs. 3 and 4)
<IT>R</IT>(<IT>t</IT>)<IT>=</IT><IT>c</IT><SUB><IT>1</IT></SUB><IT>·</IT>exp (−<IT>t</IT><IT>/</IT><IT>T</IT><SUB><IT>1</IT></SUB>)<IT>+</IT><IT>c</IT><SUB><IT>2</IT></SUB><IT>·</IT>exp (−<IT>t</IT><IT>/</IT><IT>T</IT><SUB><IT>2</IT></SUB>) (3)
Third, the peak of the exponential function fitted to the orthogonal component was determined (see arrows in Fig. 3). Fourth, a line was fitted to the exponential functions in the yaw (xy-), pitch (xz-), and roll (yz-) planes as well as in a plane spanned by the orthogonal component and the z-axis (resultant plane) starting from the peak to the end where the respective eye velocity component had declined to zero or to a nonzero constant value. The slope of these fitted lines described the tilt of the eye velocity vector in the respective planes as illustrated in the 3-D head-fixed coordinate system of Fig. 2. The orthogonal postrotatory VOR response lies in the yaw plane, which is tilted 24 or 90° relative to earth-horizontal for all final body orientations. The angle phi between the positive x-axis and the fitted line in the yaw plane described the tilt of the orthogonal response component as a function of head orientation relative to gravity (see Figs. 5, 7, and 9). Furthermore, the angle theta  between the positive (negative) z-axis and the fitted line in the resultant plane described the tilt away from the principal response direction toward alignment with gravity (see Figs. 6, 7, and 9). We defined the sign of the angle theta  as positive when the orthogonal response was parallel and as negative when it was antiparallel to gravity.



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Fig. 2. Schematic description of the spatial reorientation of postrotatory eye velocity vector (Omega ) in the yaw (tilt angle phi) and resultant plane (tilt angle theta ). The intersection of the yaw plane with the resultant plane describes the estimated direction of gravity (G). For simplicity, we assumed that the orientation of gravity in terms of the estimated angle phi perfectly coincides with the real orientation of gravity in the yaw plane (i.e., phi = phiestimated = phireal). Note that the tilt axis is determined by the intersection of the yaw plane and the earth-horizontal plane (- - -). Its angular orientation relative to head coordinates (x, y, z) is therefore determined by a single parameter in the yaw plane (phi + pi /2).

In a second approach, we fitted the postrotatory responses with an analytic function that was derived from an explicit model of the inertial transformation of vestibular signals (Fig. 10). In this 3-D spatial orientation model we assumed that the postrotatory response following off-vertical axis rotation resulted from linear superposition of three signals: 1) a direct input from the lateral semicircular canal (omega rll, combined right-left lateral semicircular canal input), 2) an otolith-dependent head velocity signal (omega oto), and 3) a spatially transformed (inertial) head velocity signal (omega inertial). Accordingly we fitted the equation
<B>ω</B><SUB><IT>VOR</IT></SUB><IT>=</IT><B>ω</B><SUB><IT>rll</IT></SUB><IT>+</IT><B>ω</B><SUB><IT>oto</IT></SUB><IT>+</IT><IT>k</IT><SUB><IT>g</IT></SUB><B>ω</B><SUB><IT>inertial</IT></SUB> (4)
to the postrotatory responses. Here, the parameter kg described the weighted ratio of the direct and the spatially transformed semicircular canal signal at the motor output site. Each of the three terms on the right-hand side of this equation can be formulated as an analytic function involving a few physiological constants. For example, we used a semicircular canal signal (omega rll) as predicted from the Steinhausen model (Steinhausen 1933) with time constants fixed at 0.003 and 5 s (see Eq. A4 with reciprocal time constants v00 = 1/0.003 s and v11 = 1/5 s). The otolith-dependent head velocity signal (omega oto) was modeled as a single exponential function with amplitude z0 and time constant Toto (see Eq. A7). The amplitude of this signal was the steady-state amplitude of the horizontal response before rotation stop. Finally, an inertial head velocity signal (omega inertial) was computed by spatially transforming and low-pass filtering the lateral semicircular canal signal with a time constant Tstor (see Tstor = 1/a33 in Eq. A6). The dependence on head orientation relative to gravity was described by a linear transformation of gravity in head coordinates using the angles phi and theta  (see Fig. 2). The model was used to estimate the following six physiological parameters by the method of least-square fit of the postrotatory responses: the weighting factor kg, the centrally estimated angles phi and theta  of the head orientation relative to gravity, the time constants Tstor and Toto of the low-pass filtered inertial and otolith-born velocity signals, and the response amplitude of the semicircular canal input at stop of head rotation (v0 in Eq. A4). In the fit procedure, we also took into account a possible small offset in horizontal eye velocity at the end of the postrotatory response phase (see, for example, the offset in horizontal eye velocity at the end of the record in Fig. 3 and w0 in Tables 1 and 2). For a complete derivation of Eq. 4 as a function of these parameters, see APPENDIX.



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Fig. 3. Vestibuloocular reflex (VOR) slow phase velocity before and after stop of constant velocity off-vertical axis rotation (at -60°/s rightward) at final body orientation of 120° left side-down (see Fig. 1, B and C). Horizontal (Omega z), vertical (Omega y), and torsional (Omega x) eye angular velocity of per- and postrotatory VOR plotted vs. time. After the stop (vertical dashed line), horizontal eye velocity reverses direction and decreases slowly, whereas vertical and torsional velocities build up to reach a peak value (arrows) and then gradually decline to zero. H: head position (potentiometer output reset every 360°). Monkey JU.


                              
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Table 1. Fit parameters of spatial orientation model (90° yaw OVAR)


                              
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Table 2. Fit parameters of spatial orientation model (24° yaw OVAR)


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Rotating a monkey around its earth-horizontal longitudinal axis ("barbecue-spit rotation") elicits 3-D vestibuloocular reflexes consisting of a steady-state horizontal nystagmus and a head position-dependent modulation of eye position and velocity (Angelaki and Hess 1996a,b; Hess and Dieringer 1991). This response pattern perseveres as long as the stimulus lasts. Figure 3 shows an example of perrotatory VOR responses during the last two cycles of yaw rotation at -60°/s and the following postrotatory responses after the stop of rotation. Superimposed on the horizontal, vertical, and torsional eye velocity vectors (gray curves on top, middle, and bottom panel) are the exponential functions fitted to each velocity component (solid lines; see Eq. 3 in METHODS).

In this example, the monkey was stopped in a final body orientation of -240° in the space-fixed coordinate system (see Fig. 1C). This corresponded to a rotation of the gravity vector through 240° from an initial position at 180° to a final position at 60° in the head-fixed coordinate system, as illustrated by the head caricature (see also Fig. 1D). The time indicated by the vertical dashed line in Fig. 3 represents the midpoint of the deceleration period (with peak deceleration of 180°/s2). The postrotatory response consisted of a prominent negative horizontal (rightward) slow phase eye velocity which decayed very slowly, whereas a positive vertical velocity (downward) built up to reach a large peak value from which it declined gradually; similarly a torsional positive eye velocity (clockwise) built up, followed by a slow decay to zero. The observation that horizontal eye velocity sometimes decayed to nonzero offset values was found in all animals and for both OVAR conditions (rotations around 90 or 24° tilted axes). This behavior did not systematically depend on the direction of yaw rotation or the static end position.

If the postrotatory response died off along the original stimulation axis, then one would expect to observe only a horizontal eye velocity component. The relatively large vertical and the smaller torsional velocity components reflect a reorientation of the eye rotation axis toward alignment with gravity: initially, just after the stop of rotation, the eye rotation axis is closely aligned with the (former) rotation axis, i.e., parallel to the earth-horizontal body longitudinal axis. In head coordinates, this corresponds to an ocular rotation around the z-axis and a horizontal (rightward) eye velocity component. Over a time course of a few seconds after rotation stop, the eye's rotation axis reorients toward alignment with space-vertical. This corresponds to an ocular rotation around an axis in the yaw plane parallel to gravity, i.e., an axis with a large y-, a smaller x-component, and a vanishing z-component.

The effect of the reorientation mechanism on the spatial orientation of the VOR response can be better understood when plotting the projections of the different response components of eye angular velocity onto the respective (pitch, roll, and yaw) planes. Figure 4 shows the projections of the data starting from the midpoint of deceleration (gray curves), the exponential functions fitted to the data (solid lines), and the lines fitted to these exponential functions (dashed lines) beginning at the peak of the orthogonal response (arrows in Fig. 3). Arrowheads plotted along the exponential function give the direction of buildup and subsequent decay of the fitted postrotatory response. Figure 4A displays torsional velocity plotted versus horizontal velocity in the pitch plane. Stop of rotation after horizontal rightward OVAR elicited a postrotatory rightward horizontal eye velocity (i.e., negative horizontal velocity along the abscissa). While this horizontal velocity component decreased, a torsional component built up. After the torsional component reached its peak value, both horizontal or torsional velocities declined approximately along a straight line to small residual values or zero.



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Fig. 4. Spatial reorientation of horizontal VOR after stop from off-vertical axis rotation at 120° left side-down. A: torsional velocity (Omega x) plotted vs. horizontal velocity (Omega z) as seen from the monkey's right side. B: vertical eye velocity (Omega y) plotted vs. horizontal velocity (Omega z) in a front view. C: torsional velocity plotted against vertical velocity as seen from above the monkey. D: orthogonal velocity (Omega orth, i.e., vectorial sum of torsional and vertical components) plotted vs. horizontal velocity (Omega z). Thick lines denote exponential functions fitted on the data; dashed lines indicate the straight line fit on the exponentials starting from peak until the end of the respective response components. The negative (rightward) horizontal velocity decreases after the stop, whereas the vertical and, to a lesser extent, the torsional velocity increase, reflecting close alignment of the postrotatory velocity vector with the direction of gravity relative to the head. The final orientation of postrotatory eye velocity is described by the angles phi and theta  in the yaw and resultant plane, respectively. Same data as in Fig. 3.

A similar pattern can be seen when plotting the vertical component of eye velocity against the horizontal component (in the roll plane, Fig. 4B): while horizontal velocity decreased, an increasing vertical velocity component was generated, reaching a peak and then declining to zero. The ratio between the (larger) vertical and (smaller) torsional eye velocity buildup reflects the orientation of the orthogonal response component in the yaw plane (Fig. 4C). This orientation can be determined by the angle phi between the line fitted to the declining part of the postrotatory response in the yaw plane and the positive x-axis (see METHODS).

Finally, the orientation of eye velocity in the resultant plane (Fig. 4D) was determined by computing the angle theta  between the z-axis and the fitted line in the resultant plane (see METHODS).

Spatial orientation of the orthogonal response component after rotation about a 90° tilted yaw axis

Postrotatory VOR responses showed precise reorientation toward earth-vertical in two animals (JU and SU) and were somewhat more variable in one animal (RO). This is illustrated in Fig. 5, which shows the tilt angle phi of the orthogonal eye velocity (see METHODS) plotted versus the orientation of gravity in the yaw plane for all final body orientations. On the right side, individual values for each monkey are displayed for both rotation directions (solid line: +60°/s leftward OVAR; dashed line: -60°/s rightward OVAR). The left graph shows tilt angles averaged across all three subjects as well as over positive and negative rotations, with vertical bars denoting one standard deviation. Final body orientations of 90, 0, -90, and ±180° correspond to left ear-down, prone, right ear-down, and supine positions.



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Fig. 5. Tilt of postrotatory eye velocity in the yaw plane (tilt angle phi) shows close alignment with the respective orientation of the gravity vector. Right charts: tilt angles for all subjects (JU, RO, and SU) plotted separately for positive (leftward, ---) and negative (rightward, - - -) OVAR directions. Left graph: average data pooled for both OVAR directions (±SD) and the linear regression line (dotted line). r2 values: mean data, 0.993; JU+, 0.988; JU-, 0.9901; RO+, 0.954; RO-, 0.802; SU+, 0.974; SU-, 0.951. Final body orientations of 90, 0, -90, and ±180° correspond to left ear-down (led), prone, right ear-down (red), and supine positions.

If the orthogonal component of the postrotatory VOR response would perfectly align with gravity, then the tilt angle should be equal to the tilt of the gravity vector relative to the head for each body orientation. This is precisely the pattern seen in the data. For example, when the gravity vector tilted 90° left ear-down, then the tilt angle averaged to ~90°. The close correlation between the tilt angle phi and the direction of the gravity vector was quantified by computing the linear regression, for individual subjects (not shown) and averaged data (dotted line, left graph). The r2 value for the averaged data was 0.993.

Spatial orientation of the resultant response after rotation about a 90° tilted yaw axis

Figure 6 shows the tilt angles (theta ) of the eye velocity vector in the resultant plane as a function of the orientation of gravity in the yaw plane. The individual theta  angles plotted in Fig. 6A reveal that after positive yaw rotations (filled symbols representing the 3 animals) the horizontal component of the postrotatory eye velocity vector predominantly rotates away from the direction of the gravity vector, i.e., toward the zenith (negative values, see METHODS). When the animal is rotated in the negative yaw direction, the postrotatory response rotates mainly toward the direction of the gravity vector (positive values). This pattern was consistently observed in one animal and predominantly in the other two animals. If the horizontal component would rotate exactly into the orthogonal plane, we would expect a tilt angle of ±90°.



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Fig. 6. Tilt angles of postrotatory eye velocity vectors in the resultant plane (theta ) for all final body orientations. A: individual values show that if the rotation is in the positive or negative direction (JU:  or ; RO:  or open circle ; SU: black-triangle or triangle ) the postrotatory response generally moves opposite to or in the direction of the gravity vector (negative or positive angles). B: absolute values averaged across all subjects for both rotational directions (±SD). Mean across all final head orientations yields theta  = 68.6° ( · · · ).

Figure 6B displays absolute tilt angles as a function of final body orientation, averaged over all animals and both rotational directions (mean ± SD). The data cluster around an overall mean tilt angle theta  of 68.6° (dotted line).

Spatial orientation of orthogonal and resultant eye velocity after rotation about a 24° tilted yaw axis

To examine whether the reorientation pattern seen after the barbecue-spit yaw rotations also holds for less extreme cases, we studied the orientation of postrotatory eye velocity in two monkeys after yaw rotations about an axis that was tilted by only 24° from earth-vertical (0.4 G). The orientation of the orthogonal and resultant responses, quantified by the tilt angles phi and theta , are depicted for one animal (JU) in Fig. 7. As can be seen in Fig. 7A, there is a very good correlation between the orientation of declining postrotatory eye velocity in the yaw plane and the orientation of the gravity vector with respect to the head. The regression line ( · · · ) yielded a mean R2 = 0.995 (positive direction: R2 = 0.982; negative direction: R2 = 0.992). Similar to the findings for barbecue spit rotations, the principal component of eye velocity after negative or positive yaw rotations turns in the direction of or opposite to the direction of gravity (data not shown), yielding an average absolute theta  (all final body positions, both directions) of 16.7° ( · · · , Fig. 7B), thus undershooting the spatial vertical by ~29%.



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Fig. 7. Postrotatory eye velocity after 24° yaw OVAR shows the complete reorientation pattern. A: orientation of the velocity vector in the yaw plane (phi) indicates close alignment with gravity. R2 = 0.995 (positive rotation: R2 = 0.982, negative rotation: R2 = 0.992). B: absolute values averaged for both rotational directions (±SD). Mean across all final head orientations yields theta  = 16.7° ( · · · ). Monkey JU.

Identical analysis for the second animal (SU, not shown) yielded correlation coefficients r2 of 0.898, 0.728, and 0.850 (positive/negative yaw rotation and averages) and an average absolute theta  of 29.6°, thus overshooting the spatial vertical on average by ~25%.

Fitting orthogonal and resultant postrotatory yaw VOR responses using the spatial orientation model

The spatial orientation of postrotatory slow phase eye velocity was also determined by fitting the measured response with an analytic function derived from the 3-D spatial orientation model (see METHODS and APPENDIX for details). Examples of VOR responses fitted with this model (solid lines) are plotted on top of the data (gray curves) in Fig. 8. The diagrams show the same data as in Figs. 3 and 4, in the same views as in Fig. 4, A-C. Clearly, the model fits the data well and deviates only little from the exponential function fitting predictions (cf. Fig. 4): while horizontal velocity decreases, torsional and particularly vertical velocity increases until they reach some peak value and then decline to zero or some small residual value.



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Fig. 8. Example of experimental data favorably fitted by the spatial orientation model. Same data and views as in Fig. 4, A-C.

Figure 9 summarizes the performance of the spatial orientation model for this animal and for all final head positions and compares the model predictions with the empirically determined results. The model predicts an accurate reorientation of orthogonal velocity toward earth-vertical (Fig. 9A) along with head orientation invariant reorientation of the resultant response by 73.1° (Fig. 9B).



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Fig. 9. Estimating the head orientation angles (A, phi; B, theta ) by applying the spatial orientation model to postrotatory responses accurately predicts the empirical data. Black curves show averages of values obtained by separately fitting postrotatory responses to positive (R2 = 0.990) and negative barbecue-spit head rotations (R2 = 0.986). The regression line of phi as a function of head tilt yielded R2 = 0.993. The absolute mean theta  = 73.1° (±11.8°). Gray curves denote the empirically determined data averaged for both rotational directions. Animal JU. Empirical data also shown in Figs. 5 and 6.

Tables 1 (data from 90° yaw OVAR) and 2 (data from 24° yaw OVAR) compare the tilt angles predicted from fitting the same responses with the analytic model predictions and empirically with the double exponential functions. Clearly, the estimated tilt angles phi show a closer correspondence between the two methods of analysis than the tilt angle theta  (see Fig. 9, A and B; compare phim with phie and theta m with theta e in Tables 1 and 2). Estimation of the tilt angle theta  is more difficult because it is a nonlinear function of both the magnitude of the horizontal and the orthogonal afterresponse. Because the orthogonal component is the vectorial sum of the torsional and vertical velocity components, it could be biased by a downbeat velocity component that amounted to ~2.5-5°/s or less in our animals. In the model fits we accounted for this vertical velocity bias whenever it could be clearly identified.

The 3-D spatial orientation model provides supplementary information on the time constants of the low-pass filters in the semicircular canal and otolith pathways of the inertial system (int 1 and int 2 in Fig. 10). As a consequence of the model structure, the time constant of the inertial network (Tstor) was not related to the degree of spatial alignment of the inertial response (for further details about the model structure, see DISCUSSION). However, there was a correlation between the length of the otolith and the inertial filter time constants: the shorter the otolith filter time constant (Toto), the shorter the inertial filter time constant (Tstor). Moreover, the two time constants tended to be equal in magnitude, although this was not true when the otolith time constant exhibited only little variation (see Table 1, negative rotation). Finally, the time constants of both the otolith and the inertial filters tended to be shorter for rotations about a 90° tilted axis (barbecue-spit) than for rotation about a 24° tilted axis.



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Fig. 10. Three-dimensional spatial orientation model. The lateral semicircular canal velocity signals (omega rll right-left lateral canal push-pull signal) are transmitted to the final common pathway via a rapid pathway (3-neuron arc) and via a low-pass filter (int 1) upstream to an otolith-dependent spatial transformation filter (Rg). The otolith input signals (goto) provide information about head orientation to the spatial transformation network (Rg). In parallel they also feed into a head-velocity estimation network (Omega g) whose output (omega oto) reaches the final common pathway through a low-pass filter (int 2). Note the (down)scaling of the inertial signal (omega inertial) in the final common pathway (Sigma ) with weighting factor kg. Gray panels highlight an example of processing of lateral semicircular canals signals, the generation of low-pass filtered otolith-born velocity signals and their convergence with the direct semicircular canal signals in the final common pathway. Same signals and parameters as those fitted to experimental data shown in Fig. 11A.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

This work studies the interaction of otolith and lateral semicircular canal signals in postrotatory VOR responses when varying final head position. We find a remarkably precise alignment of the head tilt plane and the plane in which postrotatory eye velocity reorients toward the spatial vertical. The ratio between the tilt of the eye velocity vector and head tilt is more variable. Whereas most of the lateral canal signal is used to cancel the otolith-driven bias velocity, only a relatively small fraction of the canal signal that underwent transformation from a head-centered into a gravity-centered reference frame contributes to the total response. In the following paragraphs we discuss these results in the light of a parametric dynamic model of vestibular processing.

Comparison with previous studies

A number of recent investigations have shown that the vestibular system codes head velocity in the VOR in space- (gravity-) centered rather than in head-centered coordinates (Angelaki and Hess 1994; Dai et al. 1991; Gizzi et al. 1994; Merfeld et al. 1993; Raphan and Sturm 1991). The difference in coding most clearly emerges when the originally coinciding reference frames dissociate. When rotational motion cues from the semicircular canals fade away during prolonged rotation, the otolith organs continue to signal position and, except for pure earth-vertical axis rotations, velocity of the head-in-space motion (Angelaki and Hess 1996a,b; Darlot and Denise 1988; Denise et al. 1988; Guedry 1965; Harris 1987; Haslwanter and Hess 1993; Hess and Dieringer 1990; Raphan et al. 1981; Young and Henn 1975). Postrotatory VOR after off-vertical axis rotations results from a complex interaction of otolith and semicircular canal signals, each of which functions in different reference frames. It is not a priori clear which reference will prevail in the combined afterresponse. Our findings agree with previous studies that have demonstrated that the rotation axis of the afterresponse following a prolonged constant-velocity rotation shifts toward alignment with gravity when the subjects are stopped in ear-down positions (cat: Harris 1987; humans: Harris and Barnes 1987; monkey: Raphan et al. 1992). The present study differs from these investigations, however, by applying a 3-D analysis of postrotatory VOR responses to reveal the underlying computations involved in 3-D spatial reorientation.

Empirical description of spatial reorientation of postrotatory yaw VOR

In our study we applied two methodological approaches to estimate the spatial orientation of postrotatory eye velocity after a constant-velocity rotation. In a first, more descriptive approach, we fitted the sum of two exponentials separately to the horizontal, vertical, and torsional response components as well as to the algebraically determined component orthogonal to the principal (i.e., horizontal) response component. This orthogonal component, i.e., the vectorial sum of the torsional and vertical response component, was by definition located in the yaw plane. Together with the principal response component, it defined the resultant plane, in which postrotatory eye velocity rotated toward the spatial vertical. In all our animals the orientation of this plane was close to parallel to the actual tilt plane, independent of the amount of tilt (Figs. 5 and 7A). This finding proves two important points: 1) rotation of postrotatory eye velocity occurs about a single space-fixed axis, and 2) this axis is always orthogonal to the tilt plane. The internal coding of this plane presumably depends on lateral canal signals that code the axis of head rotation and utricular otolith signals that detect the direction of the projected gravity vector in the utricular plane. The combination of both signals unequivocally determines the orientation of the tilt plane.

The observation of an orthogonal response component that varies with the projection of gravity into the yaw (utricular) plane indicates that the postrotatory velocity vector always rotates toward the spatial vertical. This rotation could be downward, i.e., such that the response vector and gravity end up as parallel vectors (i.e., pointing in the same direction), a pattern usually found after stop of a negative head rotation (solid symbols with negative theta  values in Fig. 6A). Alternatively, the response could also rotate upward, i.e., such that the response vector and gravity end up as antiparallel vectors (i.e., pointing in opposite directions), a pattern generally seen after stop of a positive head rotation (see open symbols with positive theta  values in Fig. 6A). Whereas the spatial reorientation of the postrotatory responses in one animal always followed this rule, i.e., rotation toward (anti-)parallel alignment with gravity after stop of a negative (positive) head rotation, responses in the other two animals occasionally broke this rule. In either case, rotations were always toward the spatial vertical, and absolute tilt angles deviated on average by ~20-30% from the head tilt angle (Figs. 6 and 7B). Whereas most of the orientation responses at 90° tilt were undershooting the spatial vertical, overshooting responses could also be observed. At 24° tilt one animal showed consistently undershooting responses (Fig. 7B), whereas the other animal exhibited both under- and overshooting responses. It remains an open question whether there exists a consistent trend for over- and undershooting responses as a function of the tilt following OVAR as has been reported for the spatial reorientation of optokinetic afterresponses (Dai et al. 1991; Raphan and Cohen 1988).

The results obtained from this first descriptive approach were compared with the predictions obtained by fitting the data with a model that implemented a spatial transformation of the lateral semicircular canal input downstream to a low-pass filter. This low-pass filtered VOR pathway has traditionally been called velocity storage integrator to suggest a supplementary function in gaze stabilization at the low-frequency end of head movements (Raphan et al. 1977, 1979). Recent evidence, however, has proven that this network is part of an inertial vestibular system that encodes head motion in space (Angelaki and Hess 1994, 1995; Raphan and Cohen 1988). In the following paragraph we describe this second analytic approach and its significance in disclosing the mechanisms of spatial orientation.

Underlying canal-otolith interactions: 3-D-spatial orientation model for yaw VOR

To reveal the interaction between semicircular canal and otoliths, we describe postrotatory VOR after horizontal head rotations with a 3-D linear dynamic system, in which the spatial transformation is described by an otolith-dependent rotation (Fig. 10). In this spatial orientation model, the postrotatory response (omega VOR) results from superposition of three different velocity signals: 1) a direct semicircular signal (omega rll) due to activation of the lateral semicircular canals, 2) a velocity signal (omega oto) that has been generated by an otolith-dependent head velocity detection network (Omega g and int 2), and 3) a head-in-space velocity signal (omega inertial) from the inertial transformation network (int 1 and Rg).

The vectorial time signals in the gray panels in Fig. 10 highlight the intermediate steps in lateral semicircular canal signal processing based on the interactions with the static and/or dynamic otolith signals that detect head orientation relative to gravity. In the particular example illustrated (see Fig. 11A) the lateral semicircular canal signals undergo a spatial tranformation such that the vectorial output signals of the network are almost perfectly aligned with the spatial vertical. The animal was stopped from rotating about its longitudinal axis in the earth-horizontal plane when it was close to right ear-down (theta head = 90°, phihead = 240°). As shown in the gray panel next to "omega inertial " in Fig. 10, the spatial transformation network Rg rotates the low-pass filtered head velocity signal in this example such that its z-(horizontal) component decreases, whereas a relatively large y-(vertical) and a smaller x-(torsional) component emerges.



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Fig. 11. Model predictions of horizontal postrotatory VOR as a function of lateral semicircular canal (SCC), otolith and inertially transformed velocity signals after stop of leftward horizontal rotation at 60°/s. A: residual horizontal response (gray curve: data; solid line: model fit) resulting from an almost complete cancellation of the otolith-born velocity signal (dashed line labeled otolith) by the SCC (dashed line labeled SCC). Only a small fraction of the inertially transformed SCC signal (dashed line labeled inertial) contributes to the horizontal response component, whereas the major part aligns with the spatial vertical, i.e., contributes to torsional and vertical velocity (not shown). Fitted parameters: Toto = 5.3 s, Tstor = 5.6 s, kg = 0.12, phi = 254° (240°), theta  = 82.2° (90° yaw OVAR). B: imperfect cancellation of the otolith-born horizontal velocity signal (dashed line labeled otolith) by the SCC signal resulting in an overshooting fast initial rise followed by a slower decay of the horizontal postrotatory response. Note that the inertially transformed SCC signal determines the late time course of the postrotatory response. Fitted parameters: Toto = 7.8 s, Tstor = 15.0 s, kg = 0.15, phi = 259.3° (270°), theta  = 15.2° (24° yaw OVAR). Different time scales in A and B. Animal JU.

How does the inertial transformation network Rg efficiently measure instantaneous head orientation relative to gravity? To see this, it is helpful to notice that any change in head orientation relative to gravity can be understood as a rotation about an earth-horizontal axis. Therefore the transformation Rg can be described by only two parameters: a first parameter, the angular orientation of the tilt axis (phi + pi /2), is defined by the line of intersection of the head yaw plane with the earth-horizontal plane (Fig. 2, dashed line). The orientation of this line could be coded by utricular signals because it is always perpendicular to the projection of gravity onto the yaw plane. Because of the fixed geometric relation between the utricles and the head yaw plane, the transformation from utricular sensory to yaw plane coordinates could be hardwired. The second parameter, the head tilt angle theta , is more difficult to estimate by the vestibular system, presumably requiring both utricular and saccular information. On the other hand, least-squares estimation of this parameter in the framework of our proposed model (Fig. 10) depends on the relative strength and dynamics of the lateral semicircular (omega rll) and the otolith (omega oto) signals, both of which contribute to the spatial characteristics of VOR velocity (omega VOR). Whereas the relatively small scatter found in the estimated phi-angle of final head orientation may be due to the precise spatial tuning of both the canal- and the otolith-born head velocity signals, the larger variability in the dynamics of these signals may underlie the larger scatter in the estimated theta -values.

The head velocity detection network Omega g estimates absolute head angular velocity relative to space based on the characteristic spatiotemporal pattern of otolith activity due to the rotating gravity vector (Angelaki 1992a,b; Hain 1986; Hess 1992; Schnabolk and Raphan 1992). This otolith-born velocity signal (output from Omega g) is low-pass filtered to enhance or sustain VOR eye velocity at low-frequency and constant-velocity head rotations where the canal-born velocity signals are deficient in magnitude or fading away (Angelaki and Hess 1996b). At stop of rotation, the postrotatory VOR (omega VOR) results from convergence of a low-pass filtered output signal from the head velocity detection network Omega g, a directly transmitted and an inertially transformed lateral semicircular canal signal.

An important feature of the model structure is the functional segregation of the spatial orientation and dynamic elements. Whereas the spatial orientation (angles theta  and phi) shows no correlation with the filter time constants (compare theta m* and phim* with Tstor* in Tables 1 and 2), the dynamics of the postrotatory responses basically reflect the interplay of the low-pass frequency canal and otolith-dependent head velocity signals. If the head velocity signals at stop of rotation should efficiently cancel each other (see omega rll and omega oto in Fig. 10), the amplitude and time constants need to be appropriately adjusted. Even though the animals were not reinforced to optimize a certain response behavior, yaw eye velocity was often strongly reduced in amplitude at stop of rotation. An example of this cancellation of the horizontal afterresponse that results from a close match of the amplitude and time constants of the otolith and semicircular canal signals is illustrated in Fig. 11A. If amplitudes and time constants are less well matched, an overshooting horizontal postrotatory response results (see example in Fig. 11B). Note that in this case the late residual horizontal eye velocity consists almost exclusively of the inertial response component.

Our model approach differs from that used by Raphan and Sturm (1991) by the explicit assumption that low-pass filters and spatial transformation networks are functionally segregated elements. This formal difference between the two models becomes important when the head tilts dynamically. In this case our equations yield an additional additive dynamic element in the form of the time derivative of the spatial transformation element Rg (see Angelaki and Hess 1995). Because the same model also predicts the spatial characteristics of dynamic tilt responses (Angelaki and Hess 1994, 1995), it appears that the underlying mechanisms of the spatial reorientation of postrotatory yaw VOR are identical for both static as well as dynamic head tilts. Whether the same is true for the roll and pitch VORs is still a matter of debate (Angelaki and Hess 1994; Dai et al. 1991; Hess and Angelaki 1995; Raphan and Sturm 1991). From a more practical point of view, our explicit formulation of the spatial transformation network allows to derive an analytic solution of the inertial equations as a function of the input signals, filter time constants, convergence ratios, and spatial orientation parameters. This analytic solution allows the extraction of important information about physiologically relevant parameters.

Conclusions

VOR responses following yaw OVAR show a remarkable precision in alignment with the spatial vertical, in particular with respect to the estimated orientation of gravity in the yaw plane (angle phi). A parametric analysis of the afterresponses based on a 3-D spatial orientation model revealed that inertial vestibular signals represent only a relatively small fraction of the total postrotatory response. We propose that these inertial signals are unimportant for gaze stabilization but, instead, reflect a mechanism that realigns the internal coordinates of the vestibulomotor system with gravity after abrupt changes in body motion. Clearly, such a mechanism may be advantageous for visuomotor coordination and posture control during self-motion.


    APPENDIX
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Inertial transformation of lateral semicircular canal signals

It has been shown earlier that the horizontal VOR exhibits different spatial characteristics as compared with the torsional and vertical VOR (Angelaki and Hess 1994, 1995). These characteristics are best modeled by a rotation operator downstream to a leaky integrator as shown in Fig. 10. The integration of the right and left lateral semicircular canal signals that operate in push-pull (omega rll) is described by a first-order linear differential equation (int 1 in Fig. 10)
<B><A><AC>u</AC><AC>˙</AC></A></B><IT>+</IT><IT>A</IT><B>u</B><IT>=ω<SUB>rll</SUB> </IT>with <IT>A</IT><IT>=</IT><FENCE><AR><R><C><IT>a</IT><SUB><IT>11</IT></SUB></C><C><IT>0</IT></C><C><IT>0</IT></C></R><R><C><IT>0</IT></C><C><IT>a</IT><SUB><IT>22</IT></SUB></C><C><IT>0</IT></C></R><R><C><IT>0</IT></C><C><IT>0</IT></C><C><IT>a</IT><SUB><IT>33</IT></SUB></C></R></AR></FENCE> (A1)
where a11, a22, and a33 are the reciprocal integration time constants in the roll, pitch, and yaw plane, respectively. The following spatial transformation (Rg in Fig. 10)
ω=<IT>R</IT><SUB><IT>g</IT></SUB>(<B>e</B><IT>, ϑ</IT>)<B>u</B> with <B>e</B><IT>=</IT><FENCE><AR><R><C>cos (<IT>ϕ</IT>)</C></R><R><C>sin (<IT>ϕ</IT>)</C></R><R><C><IT>0</IT></C></R></AR></FENCE> (A2)
is a rotation through an angle phi about an axis e that always lies in the yaw plane (see Fig. 2). The subscript g indicates that this transformation depends on otolith inputs (see Fig. 10). The matrix elements of the unitary transformation Rg(e,theta ) can be found by evaluating the following equation: Rg(e, theta ) x = (e · x) e - sin(theta )(e and  x) + cos(theta ) [e and  (e and  x) in the 3 basis vectors ( · denotes the scalar product, and  denotes the cross product)]. These basis vectors are the unit vectors defining the orientation of the three orthogonal head coordinate directions (see METHODS). For the yaw VOR, we need only the last column of this matrix (see below) because the input signal is always oriented along the z-axis. With the solution of the homogeneous part of Eq. A1
<B>u</B><SUB><IT>h</IT></SUB>(<IT>t</IT>)<IT>=</IT>exp(<IT>−At</IT>)<B>u</B><SUB><IT>0</IT></SUB><IT>=</IT><FENCE><AR><R><C>exp(−<IT>a</IT><SUB><IT>11</IT></SUB><IT>t</IT>)</C><C><IT>0</IT></C><C><IT>0</IT></C></R><R><C><IT>0</IT></C><C>exp(−<IT>a</IT><SUB><IT>22</IT></SUB><IT>t</IT>)</C><C><IT>0</IT></C></R><R><C><IT>0</IT></C><C><IT>0</IT></C><C>exp(−<IT>a</IT><SUB><IT>33</IT></SUB><IT>t</IT>)</C></R></AR></FENCE><B>u</B><SUB><IT>0</IT></SUB> (A3a)
the full solution of Eq. A1 can be written as follows (e.g., Kailath 1980)
<B>u</B>(<IT>t</IT>)<IT>=</IT>exp(−<IT>At</IT>)<FENCE><B>u</B><SUB><IT>0</IT></SUB><IT>+</IT><LIM><OP>∫</OP><LL><IT>0</IT></LL><UL><IT>t</IT></UL></LIM>  exp(<IT>At</IT>)<IT>ω<SUB>rll</SUB>d</IT><IT>t</IT></FENCE> (A3b)
To evaluate this last equation for the yaw VOR afterresponse, we represent the semicircular canal input at stop of a yaw rotation by a double exponential function (Steinhausen 1933; Wilson and Melvill-Jones 1979)
ω<SUB>rll</SUB>=&ugr;<SUB>0</SUB>[exp(−<IT>&ugr;<SUB>11</SUB></IT><IT>t</IT>)<IT>−</IT>exp(<IT>−&ugr;<SUB>00</SUB></IT><IT>t</IT>)]<B>e</B><SUB><IT>z</IT></SUB> with<B> e</B><SUB><IT>z</IT></SUB><IT>=</IT><FENCE><AR><R><C><IT>0</IT></C></R><R><C><IT>0</IT></C></R><R><C><IT>1</IT></C></R></AR></FENCE> (A4)
Here, the vector ez represents the unit vector normal to the yaw plane, v00 and v11 are the short and long reciprocal time constants of the lateral semicircular canals (0.003 and 5 s for the rhesus monkey), and vo represents the response sensitivity. After evaluating the integral on the right-hand side of Eq. A3b, we obtain with the initial condition vector uo = 0 
<B>u</B>(<IT>t</IT>)<IT>=</IT><FENCE><FR><NU>exp(−<IT>&ugr;<SUB>11</SUB></IT><IT>t</IT>)<IT>−</IT>exp(−<IT>a</IT><SUB><IT>33</IT></SUB><IT>t</IT>)</NU><DE><IT>a</IT><SUB><IT>33</IT></SUB><IT>−&ugr;<SUB>11</SUB></IT></DE></FR><IT>−</IT><FR><NU>exp(−<IT>&ugr;<SUB>00</SUB></IT><IT>t</IT>)<IT>−</IT>exp(−<IT>a<SUB>33</SUB></IT><IT>t</IT>)</NU><DE><IT>a</IT><SUB><IT>33</IT></SUB><IT>−&ugr;<SUB>00</SUB></IT></DE></FR></FENCE><B>e</B><SUB><IT>z</IT></SUB> (A5)
Feeding this signal through the spatial transformation Rg(e, phi) we have
ω<SUB>inertial</SUB>(<IT>t</IT>)<IT>=</IT><IT>R</IT><SUB><IT>g</IT></SUB>(<B>e</B><IT>, ϑ</IT>)<B>u</B>(<IT>t</IT>)<IT>=</IT><FENCE><FR><NU>exp(−<IT>&ugr;<SUB>11</SUB></IT><IT>t</IT>)<IT>−</IT>exp(−<IT>a</IT><SUB><IT>33</IT></SUB><IT>t</IT>)</NU><DE><IT>a</IT><SUB><IT>33</IT></SUB><IT>−&ugr;<SUB>11</SUB></IT></DE></FR><IT>−</IT><FR><NU>exp(−<IT>&ugr;<SUB>00</SUB></IT><IT>t</IT>)<IT>−</IT>exp(−<IT>a</IT><SUB><IT>33</IT></SUB><IT>t</IT>)</NU><DE><IT>a</IT><SUB><IT>33</IT></SUB><IT>−&ugr;<SUB>00</SUB></IT></DE></FR></FENCE> (A6)

× <FENCE><AR><R><C>sin (<IT>ϑ</IT>)sin (<IT>ϕ</IT>)</C></R><R><C>−sin (<IT>ϑ</IT>)cos (<IT>ϕ</IT>)</C></R><R><C>cos (<IT>ϑ</IT>)</C></R></AR></FENCE>
where the angles theta  and phi represent the tilt angle and the orientation of the tilt plane of postrotatory eye velocity, respectively (see Eq. A2 and Fig. 2). At stop of yaw rotation, we have to consider a superposition of three signals at the motor output side, namely the lateral semicircular canal signal (omega rll), the inertial signal (omega inertial), and the otolith-born velocity signal (omega oto) that drove the VOR during the constant-velocity rotation phase. The latter signal is thought to be generated by a central otolith mechanism (Omega g in Fig. 10), the details of which are of no further concern here. At stop of rotation, this signal decays with a certain time constant
ω<SUB>oto</SUB>=<B>z</B><SUB><IT>0</IT></SUB> exp(−<IT>t</IT><IT>/</IT><IT>T</IT><SUB><IT>oto</IT></SUB>)<B>e</B><SUB><IT>z</IT></SUB> (A7)
assuming a low-pass filter with similar dynamics as for the semicircular canal signals ("int 2 " in Fig. 10). The system matrix of this integrator is diagonal like that in Eq. A1. Only the third diagonal element of this matrix comes into play during yaw rotation. It represents the reciprocal integrator time constant of the otolith velocity signal (omega oto) in the yaw plane. At the motor output side, we have finally
ω<SUB>VOR</SUB>=ω<SUB>rll</SUB>+ω<SUB>oto</SUB>+<IT>k</IT><SUB><IT>g</IT></SUB><IT>ω<SUB>inertial</SUB></IT> (A8)
where kg is an unknown coupling constant.


    ACKNOWLEDGMENTS

We thank B. Disler for excellent technical assistance.

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


    FOOTNOTES

Address for reprint requests: B.J.M. Hess, Dept. of Neurology, University of Zurich, Frauenklinikstr. 26, CH-8091 Zurich, Switzerland.

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received 23 August 1999; accepted in final form 15 November 1999.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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