Download Role of Muscle Pulleys in Producing Eye Position

Role of Muscle Pulleys in Producing Eye Position-Dependence in the
Angular Vestibuloocular Reflex: A Model-Based Study
MATTHEW J. THURTELL,1 MIKHAIL KUNIN,2 AND THEODORE RAPHAN2
1
Eye and Ear Research Unit, Department of Neurology, Royal Prince Alfred Hospital, Camperdown, NSW 2050,
Sydney, Australia; and 2Institute of Neural and Intelligent Systems, Department of Computer and Information Science,
Brooklyn College of City University of New York, Brooklyn, New York 11210
Received 8 December 1999; accepted in final form 22 March 2000
Address for reprint requests: T. Raphan, Dept. of Computer and Information
Science, Brooklyn College, 2900 Bedford Ave., Brooklyn, NY 11210 (E-mail:
[email protected]).
www.jn.physiology.org
INTRODUCTION
The angular vestibuloocular reflex (aVOR) has an important
role in minimizing retinal image slip during angular head
rotations. To achieve this minimization, the eye and head
velocity axes must align, and the eye and head velocities must
be opposite in direction but equal in magnitude. That is, the
gain of the aVOR must be 1.0 when viewing a target at far
distance (more than ⬃1.0 m) and when rotating about a headcentered axis. A number of studies have shown that, during
yaw and pitch head rotations, the eye and head velocity axes do
remain approximately aligned and the gain of the aVOR is
approximately equal to 1.0, when the eye is positioned at or
near the center of the ocular motor range at the start of the head
rotation (Aw et al. 1996; Crawford and Vilis 1991). Interestingly, a number of recent studies have revealed that the eye and
head velocity axes do not always align when there is a requirement to fixate a point away from the primary position. In fact,
during low-frequency sinusoidal oscillations in yaw, pitch, and
roll, the eye velocity axis has been observed to systematically
tilt away from the head velocity axis in a manner that is
dependent on eye-in-head position (Frens et al. 1996; Misslisch
et al. 1994). For example, during yaw oscillations the eye
velocity axis tilts back when gaze is directed up and forward
when gaze is directed down. These eye velocity axis tilts result
in mismatches between the roll eye and head velocities, and
hence retinal image slip.
Recently, high acceleration passive (Aw et al. 1996; Halmagyi et al. 1990) and active (Foster et al. 1997) head-on-neck
“impulses” have been utilized to study eye velocity axis tilts in
human subjects fixating targets positioned away from primary
position (Thurtell et al. 1999a). In response to the passive head
impulses, no eye velocity axis tilt was observed for the first
⬃50 ms after the onset of the head rotation, for any eye
position. Palla et al. (1999) have also observed close alignment
of the head and eye velocity axes in the early part of the
response to high velocity passive head impulses. Thurtell et al.
(1999a) found, however, that the axis tilts did become apparent
after the initial 50 ms of the response, and were observed to
occur in the same direction as seen in previous studies. The eye
velocity axis tilts were also observed in the responses to active
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639
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Thurtell, Matthew J., Mikhail Kunin, and Theodore Raphan.
Role of muscle pulleys in producing eye position-dependence in the
angular vestibuloocular reflex: a model-based study. J Neurophysiol
84: 639 – 650, 2000. It is well established that the head and eye
velocity axes do not always align during compensatory vestibular
slow phases. It has been shown that the eye velocity axis systematically tilts away from the head velocity axis in a manner that is
dependent on eye-in-head position. The mechanisms responsible for
producing these axis tilts are unclear. In this model-based study, we
aimed to determine whether muscle pulleys could be involved in
bringing about these phenomena. The model presented incorporates
semicircular canals, central vestibular pathways, and an ocular motor
plant with pulleys. The pulleys were modeled so that they brought
about a rotation of the torque axes of the extraocular muscles that was
a fraction of the angle of eye deviation from primary position. The
degree to which the pulleys rotated the torque axes was altered by
means of a pulley coefficient. Model input was head velocity and
initial eye position data from passive and active yaw head impulses
with fixation at 0°, 20° up and 20° down, obtained from a previous
experiment. The optimal pulley coefficient required to fit the data was
determined by calculating the mean square error between data and
model predictions of torsional eye velocity. For active head impulses,
the optimal pulley coefficient varied considerably between subjects.
The median optimal pulley coefficient was found to be 0.5, the pulley
coefficient required for producing saccades that perfectly obey Listing’s law when using a two-dimensional saccadic pulse signal. The
model predicted the direction of the axis tilts observed in response to
passive head impulses from 50 ms after onset. During passive head
impulses, the median optimal pulley coefficient was found to be 0.21,
when roll gain was fixed at 0.7. The model did not accurately predict
the alignment of the eye and head velocity axes that was observed
early in the response to passive head impulses. We found that this
alignment could be well predicted if the roll gain of the angular
vestibuloocular reflex was modified during the initial period of the
response, while pulley coefficient was maintained at 0.5. Hence a roll
gain modification allows stabilization of the retinal image without
requiring a change in the pulley effect. Our results therefore indicate
that the eye position– dependent velocity axis tilts could arise due to
the effects of the pulleys and that a roll gain modification in the central
vestibular structures may be responsible for countering the pulley
effect.
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M. J. THURTELL, M. KUNIN, AND T. RAPHAN
METHODS
Conventions
The head and eye velocities from the data and model simulations
were expressed relative to a right-handed head-fixed coordinate system. Positive yaw is a leftward rotation, positive pitch is a downward
rotation, and positive roll is a clockwise rotation.
Data collection and analysis
The model simulations presented in this paper were compared
directly with data obtained from a previous study (Thurtell et al.
1999a). The experimental procedures are briefly summarized here.
See Thurtell et al. (1999a) for a complete and detailed description of
the experimental setup and protocols.
Two experiments were conducted: one using high-velocity passive
(manually generated) yaw head rotations, and the other using highvelocity active (self-generated) yaw head rotations. Eight normal
human subjects were tested in each experiment. The head rotation
stimuli typically had an amplitude of 15–25° (with peak velocity
about 250°/s). The velocity vectors corresponding to the head rotation
were approximately aligned with the z-axis of the head-fixed coordinate system (making the rotation predominantly yaw), although the
rotations inevitably had pitch and roll components due to the mechanics of the neck joints and soft tissues. During these experiments,
responses were recorded using the magnetic search coil technique
(Collewijn et al. 1985; Robinson 1963) with subjects maintaining
fixation on different targets (20° up, 0° and 20° down) located on a
tangent screen 94 cm from the front of the cornea. The search coil
signals were digitized at 1 kHz with a 16-bit analog-digital converter.
Rotation vectors corresponding to eye-in-head and head-in-space position, and velocity vectors corresponding to eye-in-head and headin-head angular velocity, were calculated from the search coil data
(Haslwanter 1995; Haustein 1989; Hepp 1990; Tweed et al. 1990).
Analysis was restricted to a 100-ms period, beginning 20 ms before
the onset of head rotation, to exclude the effects of non-VOR systems
such as the cervicoocular reflex and smooth pursuit (Bronstein and
Hood 1986; Carl and Gellman 1987; Tychsen and Lisberger 1986).
In this study, the data and corresponding model predictions will be
presented as time series, and then compared by projecting the predicted eye velocities onto the pitch (xz) and roll (yz) planes. The model
predictions from the 100-ms period will be presented, as for data from
Thurtell et al. (1999a). The predictions for the entire response will be
presented in some cases.
Modeling
The model was implemented using the Microsoft Visual C/C⫹⫹
programming environment. Integration processes in the model were
performed using a regular trapezoidal rule algorithm, with temporal
update every 0.001 s. Because the dominant time constant of the
system is orders of magnitude larger than this value, the integration
technique worked well (Raphan 1998; Yakushin et al. 1998).
Since the model assumed that the reference eye position corresponds with the primary position, we determined primary position
from subjects’ Listing’s plane data and recomputed the rotation vectors and velocity vectors relative to primary position. That is, the data
were rotated so that each subject’s Listing’s plane aligned with the roll
plane of the head-fixed coordinate frame. These computations were
conducted using the methods of Tweed et al. (1990). The new coordinate frame, which is close to the stereotaxic coordinate frame, is
what we will refer to as being the head-fixed coordinate frame.
The analysis of the model predictions was conducted using MATLAB 5.2 running on an IBM-compatible PC under Windows NT. The
figures presented in this paper were generated using Splus running on
a DECstation 5000/240 under Ultrix, and Canvas (Version 6.0) running on an IBM-compatible PC under Windows NT.
Model organization and conceptual basis for study
The model of the aVOR is
comprised of a number of major components that, in concert with one
another, produce an eye rotation response to a head rotation stimulus
(Fig. 1). While detailed descriptions are given elsewhere (Raphan
OVERVIEW OF THE aVOR MODEL.
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head impulses, but they began within 5 ms of the onset of the
head rotation (Thurtell et al. 1999a).
The mechanisms responsible for producing the axis tilts are
unclear. If the eye movements obeyed Listing’s law, a constraint on ocular kinematics (von Helmholtz 1866), the angle of
eye velocity axis tilt would be half that of eye deviation away
from primary position (Tweed and Vilis 1990). However, the
angle of tilt of the velocity axis is less than this (Frens et al.
1996; Misslisch et al. 1994; Solomon et al. 1997). Misslisch et
al. (1994) concluded that the tilts arose due to a compromise
between no axis tilt (perfect image stabilization) and the constraints imposed by Listing’s law. The mechanism for achieving the compromise was thought to be neural in origin. Hence,
in response to identical head rotation stimuli, the signal driving
the eye muscles would alter depending on eye position. Misslisch et al. (1994) also posited that the goal of the compromise
was to minimize retinal image slip on the fovea, while allowing
retinal image slip to occur on the periphery of the retina.
Recently, it has been shown that the paths of the rectus
extraocular muscle bellies are constrained by orbital tissue, the
fibromuscular pulleys (Clark et al. 1998, 1999; Demer et al.
1995, 1997; Miller et al. 1993). If it is assumed that the pulleys
implement a kinematic rotation of the torque axis that is
approximately half the angle of eye deviation away from
primary position, then Listing’s law is obeyed during saccades
when the central drive is confined to the two-dimensional
pitch-yaw plane (Raphan 1997, 1998). If the pulleys produce
such a rotation of the torque axes, the torque generated by the
muscles becomes eye position-dependent. As a result, the eye
velocity axis will tilt as a function of eye position. Therefore it
is no longer clear that eye position-dependence in the aVOR is
neural in origin; it could well arise as a consequence of the eye
plant properties.
The purpose of this study was to investigate the mechanisms
by which the eye position dependence of the aVOR could be
generated during active and passive head rotations, using a
model-based approach. Specifically, we have investigated
whether the eye position-dependence of the aVOR can be
predicted by a model that incorporates a semicircular canal
system (Yakushin et al. 1998), a commutative velocity-position
integrator in the CNS (Schnabolk and Raphan 1994) and a
pulley system in the plant (Raphan 1998). To evaluate the
accuracy of the predictions, we have compared high-resolution
data collected from normal human subjects (Thurtell et al.
1999a) with the predictions of the model. To maximize the
validity of the model predictions, head velocity and initial eye
orientation data were used as input for the model. We have also
sought to determine whether and to what extent additional
“neural processing” is required to predict the patterns observed
in the data. The findings presented in this paper have previously been presented in abstract form (see Thurtell et al.
1999b).
EYE POSITION-DEPENDENCE IN THE ANGULAR VOR
641
1998; Yakushin et al. 1998), a brief description of each of the model
components is included below and in APPENDIXES A and B.
MODEL OF SEMICIRCULAR CANALS AND VESTIBULAR AFFERENTS.
The semicircular canal model was based on the canal model presented
in Yakushin et al. (1998). Since each semicircular canal responds to
angular acceleration about an axis normal to the canal plane, the canal
model must incorporate a kinematic transformation of the head velocity from head-fixed coordinates into canal coordinates. Thus the
head velocity is projected onto each of the canal plane normals, which
form a nonorthogonal basis in humans. In the model, the orientations
of the canals were adjusted to agree with those reported for humans
(Blanks et al. 1975).
The canal model also incorporates a first-order dynamical system,
to produce vestibular afferent firing that is temporally related to the
input angular head acceleration (Yakushin et al. 1998). The dominant
time constant for each canal was set to 4 s (Fernández and Goldberg
1971), as there is now evidence that this may be the time constant in
humans as well as in monkeys (Dai et al. 1999). The equations
describing the canal dynamics, the transformation of the head velocity
signal into canal coordinates, and the transformation of the vestibular
afferent signal back into head-fixed coordinates are given in APPENDIX A.
The vestibular afferent signals activate second-order neurons in the
vestibular nuclei (Waespe and Henn 1977, 1978) (Fig. 1). The sensitivities of the second-order neurons were determined by a gain matrix
(G) relating eye rotation to head rotation along each of the coordinate
axes (roll, pitch, and yaw). The components of the gain matrix (g11,
g22, and g33) were initially adjusted to give gains consistent with those
observed during passive roll, pitch, and yaw head impulses (Aw et al.
1996; Thurtell et al. 1999a). Roll rotations produced an aVOR with
gain approximately 0.7, while yaw and pitch rotations produced an
aVOR with gain approximately 1.0. The gain matrix was represented
with respect to head-fixed coordinates. The effects of velocity storage
were neglected in the model, since the head rotations being considered
in this paper were of short duration (⬍250 ms).
MODEL OF THE CENTRAL aVOR PATHWAYS. The signal from the
vestibular nuclei passes to the ocular motor nuclei via two pathways:
a direct ocular motor pathway (Dp) and one that activates the velocityposition integrator. The velocity-position integrator was implemented
as a commutative vector integrator, characterized by gain matrices (Gp
and Cp) and a system matrix (Hp) that determines the integrator’s
dynamics (see APPENDIX B for further details). The parameters of the
velocity-position integrator were set as in Raphan (1998). The matrices were represented with respect to head-fixed coordinates. The sum
of the outputs from the direct pathway and velocity-position integrator
was used to drive the motoneurons passing to the extraocular muscles.
The direct pathway and velocity-position integrator can also receive
input from a saccadic pulse generator (Raphan 1998). The initial eye
position data were used to produce a two-dimensional saccadic pulse,
to move the eyes into the starting eye position before the onset of the
head rotation. The initial eye position was maintained by the model
for 200 ms before the onset of the head rotation. Because the time
constant of the velocity-position integrator was set as 30 s (Schnabolk
and Raphan 1994), there was negligible decay in the pitch state of the
integrator over the simulation time. The model was not programmed
to produce saccadic eye movements during the head rotation, since
saccadic movements are not normally made in the 100-ms period
extending from 20 ms before the onset of head movement to 80 ms
after onset (Thurtell et al. 1999a).
Torque (m) is generated
by the extraocular muscles as a result of activity in the motoneurons
(mn). Due to the effects of the fibromuscular pulleys, the torque axis
is rotated in a manner depending on eye position (Fig. 2). For
example, when the eye looks up by an angle ⌽, the pulley changes the
pulling direction of the muscle, thereby rotating the torque axis by an
angle ␦. In the model, a muscle matrix M implements the transformation from motoneuron firing to torque. M is a rotation matrix that
incorporates the action of the pulleys in three dimensions. The transformation is given by
MODEL OF THE OCULAR MOTOR PLANT.
m ⫽ Mm n
(1)
The rotation of the torque axis is about the eye orientation axis, and
the angle of rotation is a fraction of the eye rotation angle. The angle
of torque axis rotation is determined by a pulley coefficient, k⌽ (Fig.
2), such that
␦ ⫽ k ⌽⌽
(2)
FIG. 2. The fibromuscular pulleys keep the rectus muscle bellies fixed in
the orbit, regardless of the orientation of the eye. However, the pulling
directions of the extraocular muscles, and hence the torque axis, are altered in
a manner depending on eye position. In the model, the magnitudes of rotation
of the torque (␦) and the eye (⌽) axes are related by a pulley coefficient, k⌽
(figure adapted from Quaia and Optican 1998).
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FIG. 1. Schematic diagram of the 3-dimensional (3-D) model of the vestibuloocular reflex, with saccadic pulse generator. The
matrix M, which transforms motoneuron firing into torque, incorporates the pulley effect by bringing about an eye position–
dependent rotation of the torque axis. See text for a description of the model and explanations for the symbols.
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M. J. THURTELL, M. KUNIN, AND T. RAPHAN
A pulley coefficient of 0.5 constrains the trajectory of a saccadic eye
movement driven by a pitch-yaw command, so that it perfectly obeys
Listing’s law. A pulley coefficient of 0.0, on the other hand, will result
in a large torsional transient and hence a violation of Listing’s law,
since there is no tilt of the torque axis during the saccade (Raphan
1998).
The plant itself was modeled as a second-order dynamical system
governed by the inertia of the eyeball, viscous damping by the fluid
and tissue surrounding the eye, and the elasticity of surrounding
membranes. See APPENDIX B and Raphan (1998) for further details.
OPTIMAL MODEL PARAMETERS. To determine the optimal pulley
coefficient to fit the data in this study, we calculated the pulley
coefficient at which the mean square error (E2) between roll eye
velocities for data [␻x(data)] and model prediction [␻x(pred)] was a
minimum. The pulley coefficient was varied from 0 to 1, in 0.01
increments. The mean square error was computed for each pulley
coefficient value as
1
N
冘
N
共 ␻ x共data兲 关i兴 ⫺ ␻ x共pred兲 关i兴兲 2
(3)
i⫽1
where N is the number of samples in the temporal sequence of
corresponding model and data sets. Since the data were sampled at 1
kHz and the length of the analysis period was 100 ms, N was equal to
100 in this study. The optimal pulley coefficient, which gave the
minimum mean square error, was determined for each subject and eye
velocity trajectory, and over a wide range of roll aVOR gains.
It should be noted that the value of the optimal pulley coefficient
RESULTS
Data-model comparison for active head impulses
Subject 0031m executed active yaw head impulses that had
a large yaw component, and minimal pitch and roll components throughout the head movement (Fig. 3A). The initial
fixation point of the subject did not affect the starting head
position or the head trajectory (Thurtell et al. 1999a). When the
subject was looking 20° up, the yaw and pitch eye velocities
approximately compensated for the yaw and pitch head velocities, respectively (Fig. 3A, heavy and dashed shaded lines).
However, the roll eye velocity was inappropriately clockwise,
as there was little roll head velocity. When the subject was
fixating a central target, all components of eye velocity were
approximately compensatory for the head velocity stimulus.
When the subject was looking 20° down, the yaw and pitch eye
velocities compensated for the yaw and pitch head velocities,
but the roll component was again inappropriate (as in the 20°
FIG. 3. A: time series of head and eye velocity vectors from a normal subject (0031m) during 3 leftward active yaw head
impulses with fixation on targets 20° up, center (0°), and 20° down, respectively. The small arrows indicate the onsets of the head
impulses. B: time series of data head velocity vectors and model predictions of eye velocity vector. With a pulley coefficient (k⌽)
of 0.53, the model predicted the direction and magnitude of all components of eye velocity for each fixation position. The small
saccadic eye movement at the end of the impulse with fixation at 0° was not predicted, since the model was not programmed to
produce saccadic eye movements during the head rotation.
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E2 ⫽
reflects the pulley configuration only for a given roll gain. If the signal
activating the extraocular muscles is a pure pitch-yaw signal, then any
tilting of eye velocity axis in the pitch plane occurs due to the pulley
effect. If there is a roll component to the signal, as during vestibular
slow phases, eye position– dependent tilting of the eye velocity axis in
the pitch plane may occur due to the pulley effect and/or due to an eye
position– dependent alteration in the roll signal activating the muscles.
EYE POSITION-DEPENDENCE IN THE ANGULAR VOR
up condition), on this occasion being counterclockwise. These
patterns are consistent with those observed in data collected
from human subjects during low-frequency yaw oscillations
(Misslisch et al. 1994). A complete description and analysis of
the data for a range of subjects can be found in Thurtell et al.
(1999a).
The head velocity trajectories from the data were used as
input for the model, as were the initial eye position data. The
curve representing mean square error between data and model
torsional eye velocities, as a function of pulley coefficient, was
approximately parabolic for both upward (Fig. 4, – – –) and
downward (Fig. 4, - - - - -) gaze. For the 0° initial fixation
position, the mean square error was not of great magnitude and
varied little over the range of pulley coefficients (Fig. 4, 䡠 䡠 䡠).
The curve representing the mean square error for the average of
all gaze positions had a minimum corresponding to a pulley
coefficient of 0.53 (Fig. 4, 2). Using this mean value, the yaw,
pitch, and roll components of eye velocity were predicted by
the model for all fixation positions (Fig. 3B).
Consistent with the finding that a pulley coefficient of 0.53
closely predicted the time course of the torsional component of
eye velocity (Fig. 5A, middle), the axis tilt was also predicted
at every instant of time during the response (Fig. 5B, middle).
When the pulley coefficient was reduced to 0.00, thereby
removing the effect of the pulleys, the torsional eye velocity
mirrored torsional head velocity (Fig. 5A, bottom), resulting in
approximate head and eye velocity axis alignment independent
of eye position (Fig. 5B, bottom).
FIG. 5. A: time series of torsional head and eye velocity data (top) from subject 0031m executing active head impulses, averaged from
20 ms before to 80 ms after head impulse onset. The small arrow indicates the time of head impulse onset. The model predicts the
magnitude and direction of the torsional eye velocity when the pulley coefficient (k⌽) is optimal (0.53) (middle). However, when the
pulley coefficient is equal to 0, the model predicts loss of eye position-dependence; torsional eye velocity becomes compensatory for
torsional head velocity (bottom). B: the averaged velocity vectors are plotted in the pitch plane for eye velocity data (top) and eye velocity
modeled with pulley coefficients of 0.53 (middle) and 0 (bottom). Note that the roll eye velocity and, hence, velocity axis alignment is
little affected by the change to the pulley coefficient when the fixation position is near the primary position (near 0° in this subject).
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FIG. 4. Averaged mean square error (between data and predicted torsional
eye velocities) plotted as a function of pulley coefficient (k⌽), for subject
0031m executing active head impulses. The curves are approximately parabolic for fixation 20° up and 20° down. The curve for fixation at 0° is flat,
indicating that the pulleys do not alter the roll eye velocity when fixation is
near the primary position. The mean curve is parabolic; the minimum occurs
when the pulley coefficient is 0.53.
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M. J. THURTELL, M. KUNIN, AND T. RAPHAN
We also compared the averaged data from eight subjects
(Fig. 6A), with model predictions (Fig. 6B). The optimal pulley
coefficient value for each subject is given in Table 1. Seven of
the subjects had optimal pulley coefficients ranging from 0.27
to 0.58, with only one of the eight subjects having an optimal
pulley coefficient ⬍0.25. While the mean square error curves
from the subjects demonstrate a single minimum, the gradient
of the curve for most subjects was small over a broad range of
values close to the optimal pulley coefficient, representing a
relative insensitivity in using this method for determining
optimal model parameters. However, introducing weights to
different parameters would bias our results and would not
increase confidence in our findings.
The optimal pulley coefficient across all subjects occurred
when the median of the mean square error over all trials and
subjects was minimal; it was found to be 0.5 (Fig. 7A; Table 1).
With a pulley coefficient of 0.5, the model predicted the
average direction and magnitude of the axis tilts reasonably
well (Fig. 6B), although it was more accurate when the individual pulley coefficients were used for each subject (Fig. 5B,
TABLE
Data-model comparison for passive head impulses
1. Optimal pulley coefficient for active and
passive head impulses
Active Head Impulses
middle). For the 20° up condition, the average eye velocity axis
was noted to be aligned with the head velocity axis initially,
but tilted back for head velocities above about 125°/s (Fig. 6A).
The initial alignment occurred in some subjects, affecting the
average behavior. In other subjects, there was immediate axis
tilt, as predicted by the model (Figs. 5B and 6B). In response to
the center condition, the head and eye velocity axes were noted
to remain fairly well aligned (Fig. 6A); the model accurately
predicted the eye velocity response (Fig. 6B). For the 20° down
condition, there was a significant initial and maintained forward axis tilt (Fig. 6A), as predicted by the model (Fig. 6B).
The axis tilts described above occur predominantly in the
pitch plane, and there is little tilt of the eye velocity axis in the
roll plane (Fig. 6A). In other words, there is good alignment of
head (heavy line) and eye (dashed shaded line) velocity axes in
the roll plane at all points in the trajectory. The model predicted no significant axis tilt in the roll plane for any of the
subjects, agreeing with the patterns observed in the data (Fig.
6B).
Passive Head Impulses
Subject
Optimal k⌽
Subject
Optimal k⌽
0031m
0034m
0043m
0044m
0046m
0047m
0051m
0053m
0.53
0.58
0.39
0.4
0.49
0.15
0.3
0.27
0026m
0030m
0036m
0037m
0041m
0042m
0045m
0055m
0.31
0.24
0.19
0.23
0.31
0.08
0.03
0.27
Median optimal k⌽
0.5
Median optimal k⌽
0.21
As previously shown (Thurtell et al. 1999a), there are eye
position– dependent velocity axis tilts during passive head impulses as during active head impulses. The optimal pulley
coefficient for passive head impulses was computed in all
subjects using a gain matrix equal to that used for active head
impulses; the values of these pulley coefficients are given in
Table 1. Subjects’ optimal pulley coefficients ranged from 0.03
to 0.31. The optimal pulley coefficient was calculated across all
subjects, as for the active head impulses; it was found to be
0.21 (see Fig. 7B; Table 1).
For subject 0036m, the optimal pulley coefficient was found
to be 0.19 for a roll gain of 0.7. Using this value of pulley
coefficient, the model predicted the fundamental features of the
axis tilts (Fig. 8). However, the dynamics of the axis tilts
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FIG. 6. A: mean head and eye velocity data from active head impulses, averaged across 8 subjects, are plotted in the pitch (top)
and roll (bottom) planes. B: for the optimal pulley coefficient (k⌽), 0.5, the model predicts the direction and magnitude of the eye
position– dependent axis tilts.
EYE POSITION-DEPENDENCE IN THE ANGULAR VOR
645
DISCUSSION
Mechanisms for VOR eye position-dependence
FIG. 7. A: averaged mean square error curves from 8 subjects executing
active head impulses, plotted as a function of pulley coefficient (dashed gray
lines). Also plotted is the median curve (black line). The absolute minimum of
the median curve occurred when the pulley coefficient was 0.5. B: averaged
mean square error curves from 8 subjects during passive head impulses, plotted
as a function of pulley coefficient (dashed gray lines). The absolute minimum
of the median curve (black line) occurred when the pulley coefficient was 0.21.
during passive and active head impulses were distinctly different, and the model, with a fixed pulley coefficient and aVOR
roll gain, did not predict the dynamics seen during passive
impulses (Fig. 8, arrows). The axis tilts described for active
head impulses also occurred during passive impulses, but only
after the first 50 ms of the onset of head rotation. During the
first 50 ms of the passive head impulse, there was close
alignment of the head and eye velocity axes (Thurtell et al.
1999a).
In subject 0030m, the head velocity stimulus had a large roll
component. Using this subject’s data, we first considered
whether the close alignment of the axes was due to a dynamic
modification of the pulley effect. Using a roll aVOR gain of 0.7
(Aw et al. 1996) and a pulley coefficient of 0.0, the model
predicted better initial alignment between head and eye velocity axes. But the alignment was not as close as in the data (Fig.
9), suggesting that a change in pulley coefficient alone was
insufficient to bring about the axis alignment. We next exam-
When Misslisch et al. (1994) studied eye position– dependent eye velocity axis tilts in humans using 0.3-Hz sinusoidal
whole-body rotations, they noted that the tilts were consistent
with a compromise between Listing’s law and perfect compensation for head rotation. After studying the predictions of the
SQUINT model of extraocular muscle geometry (Miller and
Robinson 1984), they concluded that the central signals driving
the extraocular muscles must already encode the appropriate
eye velocity axis tilts, and that orbital mechanics contribute
very little to the production of the tilts. They had not, however,
considered the effects of the rectus fibromuscular pulleys
(Clark et al. 1998, 1999; Demer et al. 1995, 1997; Miller et al.
1993) in their study. Preliminary work by Raphan (1997)
indicated that the pulleys might have an important role in
bringing about the tilts. More recently, however, it has been
suggested that even with an ocular motor plant that incorporates pulleys, no integrator model can predict the eye position
dependence of the aVOR, and consequently a neural integrator
that is quaternion-based and noncommutative is required
(Smith and Crawford 1998; Tweed 1997). However, our
model, which incorporates a commutative vector velocityposition integrator, muscle pulleys, and a three-dimensional
semicircular canal system, predicts the axis tilts.
The role of the pulley system was first considered in studies
concerned with modeling eye movement trajectories during
saccades. In those studies, it was shown that if the pulleys are
correctly positioned and the command signal is confined to the
pitch-yaw plane, kinematically accurate saccades can be produced by models that incorporate a commutative vector velocity-position integrator (Quaia and Optican 1998; Raphan 1997,
1998). The value of the pulley coefficient is critical in the
saccadic model, because it determines the magnitude of the eye
velocity axis tilt when the driving signal is in the pitch-yaw
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ined the effects of an alteration in the gain of the roll component of the aVOR (g11 in Fig. 1) in producing the initial axis
alignment. Such a mechanism could maintain the roll gain at
one level during the initial period of the response and then
allow it to revert toward a default value (approximately 0.7)
after the 50 ms had elapsed. We tested this hypothesis by
adjusting the roll gain to fit the data from subject 0030m for the
first 50 ms of the response, while keeping the pulley coefficient
at a constant value of 0.5 (Fig. 9). For the center condition,
model-data differences were minimized during the initial period when the roll gain was equal to 1.0 (Fig. 9). For the 20°
gaze up condition, the roll gain needed to be equal to 0.7 during
this period to maintain axis alignment, while for 20° down, the
best fit to the data occurred when the roll gain was ⬎1.0 (Fig.
9). These variations in the roll gain as a function of initial
eye-in-head position brought about alignment of the head and
eye velocity axes, as observed in the data (Fig. 9), without
requiring a change in pulley coefficient. We found, however,
that there were a number of combinations of pulley coefficient
and roll gain that could produce the initial axis alignment (see
Fig. 10). It is therefore possible that a combination of dynamic
pulley coefficient and roll gain modification is responsible for
producing the initial axis alignment for each subject.
646
M. J. THURTELL, M. KUNIN, AND T. RAPHAN
FIG. 8. Head and eye velocity data from subject
0036m during passive head impulses is plotted in the pitch
plane (top). While the model predicts the general features
of the eye position– dependent axis tilts (bottom), when
the pulley coefficient (k⌽) is optimal for that subject’s
data (0.19), it does not predict the initial alignment of the
eye and head velocity axes (for example, at the arrows).
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FIG. 9. Head and eye velocity data from subject 0030m
during passive head impulses is plotted in the pitch plane
(top). Only the velocity data ⱕ100°/s have been plotted. For
this range, there is almost perfect eye and head velocity axis
alignment for each fixation position. The model was unable
to predict perfect axis alignment when the pulley effect was
eliminated (k⌽ ⫽ 0), although there was no eye position
dependence (middle). However, when the pulley coefficient
was maintained at 0.5, axis alignment could be achieved by
modification of the angular vestibuloocular reflex (aVOR)
roll gain (bottom). The roll gains required to bring about axis
alignment are indicated.
EYE POSITION-DEPENDENCE IN THE ANGULAR VOR
plane and, hence, the degree of adherence to Listing’s law
(Raphan 1998). With a two-dimensional pitch-yaw driving
signal, a pulley coefficient of 0.5 gives perfect adherence to
Listing’s law, since the angle of velocity axis tilt is exactly half
that of eye deviation from primary position. However, for
saccades made between tertiary eye positions, Listing’s law is
not perfectly obeyed, as indicated by the presence of significant
torsional transients (Straumann et al. 1995; Tweed et al. 1994).
The torsional transients occur because the eye velocity axis tilt
is not always half of the angle of eye deviation from primary
position. Indeed, it has been shown that the pulley coefficient
may vary from 0.39 to 0.6 between individual subjects, if the
torsional transients seen in their saccadic data are to be predicted using a model that utilizes a two-dimensional driving
signal (Raphan 1998). Since the pulley system is a feature of
the oculomotor plant, it is reasonable to expect that the pulleys
contribute in the same way during the aVOR as during saccades. Consequently, it would be expected that if the axis of
vestibular stimulation were confined to the pitch-yaw plane,
the velocity axis tilts would primarily arise as a result of the
eye position-dependent changes in the pulling directions of the
muscles (Raphan 1998). The results of this study indicate that
the pulley model accurately predicts axis tilts in accordance
with these expectations. Indeed, the pulley coefficient that gave
optimal performance of the aVOR for active head movements
over all subjects was 0.5, the same as that required for Listing’s
law to be obeyed during saccadic eye movement simulations.
There were, however, considerable inter-subject differences
(see Optimal pulley coefficient calculations).
Differences between saccadic and aVOR kinematics may be
attributed to the fact that the semicircular canals generate roll
control signals, while the saccadic system probably generates
two-dimensional pitch-yaw signals (Raphan 1998). Since we
used actual head impulse data as input to the model, there was,
unavoidably, a roll component to our stimulus. With the gain of
the roll component of the aVOR set to 0.7 during active head
impulses, the model accurately predicted, on average, the vari-
ations in tilt as a function of eye position and head velocity.
Thus the model predicts that the axis tilt patterns may essentially arise as a result of two factors: the pulley effect and the
less-than-unity roll gain. As we have shown, a pulley coefficient of 0.0 gives approximate eye and head velocity axis
alignment and no eye position dependence (Fig. 5B, bottom),
whereas a pulley coefficient of 0.5 would tilt the eye velocity
axis by half the angle of gaze deviation from primary position.
A pulley coefficient ⬍0.5 gives an eye velocity axis tilt that is
less than half the angle of gaze eccentricity. The less-thanoptimal roll gain of the aVOR (which is ⬃0.7 during passive
head impulses) would further diminish the relative tilt of the
eye velocity axis if there is a roll component to the head
movement. Variations in the pulley configuration and roll gain
between individuals, in addition to different stimulus and species characteristics, could account for the variations in the
amount of tilt from subject-to-subject and for the differing
results reported in different studies. Thus for active head impulses, there is no need to postulate an axis tilting mechanism
in the central vestibular structures (Misslisch et al. 1994) or a
complex multiplicative tensor interaction between angular eye
velocity and position with a quaternion integrator (Smith and
Crawford 1998) to bring about this simple result.
Passive-active differences
While the tilt of the eye velocity vector as a function of eye
position was a prominent aspect of the response to active head
impulses, there were striking differences when active and
passive head impulses were compared (Thurtell et al. 1999a).
For active head impulses, the tilts became apparent within the
first 5 ms of the response and were maintained throughout the
trajectory, as predicted by the model. For passive head impulses, there was little tilt of the eye velocity axis during the
first 50 ms (Thurtell et al. 1999a). These findings are consistent
with data showing negligible axis tilt in the averaged eye
velocity vectors during the first 10° of head movement during
passive head impulses (Palla et al. 1999). The axis of eye
velocity was noted to tilt gradually after the initial 50-ms
period (Thurtell et al. 1999a). Thus passive head impulse axis
tilts are not uniform over time but evolve over the duration of
the head trajectory. We found that the initial alignment of the
eye and head velocity axes was inconsistent with model predictions for a fixed pulley coefficient of 0.5 and a fixed aVOR
roll gain of 0.7.
We considered a number of mechanisms that might be
responsible for the temporal evolution of eye velocity axis
tilt. Dynamic changes in pulley effect may arise as a result
of the insertion of the orbital layers of the rectus muscles
into the connective tissue of the pulley structures (Demer
et al. 2000). We found that removing the effect of the
pulleys, by adjusting the pulley coefficient to 0.0 and maintaining the roll gain of the aVOR at 0.7, resulted in closer
initial axis alignment and loss of eye position-dependence.
The model still did not, however, predict the degree of
alignment observed in the data. Thus a dynamic modification in pulley coefficient could not completely account for
the initial axis alignment during passive head impulses. In
addition, such a peripheral mechanism would have to be
specific to the passive aVOR because these dynamic
changes in eye velocity axis tilt were not observed during
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FIG. 10. Mean square error (between data and predicted torsional eye
velocities) plotted as a function of pulley coefficient and aVOR roll gain, for
subject 0030m during passive head impulses with fixation 20° up. The analysis
is restricted to the period from the onset of the head impulse to 50 ms after
onset, when head and eye velocity axis alignment is greatest. The graph
illustrates that instead of there being a unique combination of pulley coefficient
and roll gain that predicts the data, any number of combinations of pulley
coefficient and roll gain can produce the axis alignment.
647
648
M. J. THURTELL, M. KUNIN, AND T. RAPHAN
eration in squirrel monkeys (Paige and Tomko 1991). Furthermore, the latency of compensatory linear vestibuloocular reflex
responses are in the order of 25 ms (Bronstein and Gresty
1988), so the effects we are examining would almost be over
by that time. Therefore it is unlikely that the otoliths significantly contribute to the responses of either active or passive
head impulses in normal human subjects.
The alterations in roll gain may be, in part, related to rapid
feedback from the roll state of the velocity-position integrator
to the secondary vestibular neurons. Such a mechanism would
affect the dynamics of vestibular-evoked eye movements, although its affect on saccades is probably negligible, since the
roll state of the velocity-position integrator is inactive during
saccades in our model (see Raphan 1998). More fundamental
physiological studies are required to determine where such a
roll gain change might occur in the central vestibular structures.
Recent evidence has been presented that neuron responses in
the vestibular nucleus may be attenuated during active head
movements (Cullen and Roy 1999; McCrea et al. 1999) and
may be related to the mechanism that disables a roll gain
modification during active head movements. Cullen and Roy
(1999) found that some, but not all, neuron responses were
attenuated during the gaze stabilization period of an active
head movement. McCrea et al. (1999) suggested that reafferent
neck proprioceptive and efference copy signals could be responsible for the attenuation.
The underlying functional reason for the different dynamic characteristics of the axis tilts in passive and active
head impulses may result from the different goals of the
system during these types of movements. Roll gain modification may not occur during active head impulses because
the goal of the system during normal eye-head refixations is
to shift gaze from one target to another, rather than maintain
fixation on a fixed target. For these refixations, there is
considerable preprogramming but no effort to stabilize the
retinal image initially. Hence there is no reason for the roll
gain of the aVOR to be modified in the initial phase of the
response to active head rotations. In fact, normal aVOR
function is probably suppressed during the initial period of
active eye-head refixations (Laurutis and Robinson 1986),
supporting the idea that the proposed modification of the
initial roll gain of the aVOR during passive head impulses
might be suppressed during active head impulses. Passive
head movements may therefore be unmasking an important
underlying functional component of the pure aVOR.
Optimal pulley coefficient calculations
The optimal pulley coefficient varied considerably between
subjects during the active head impulses, ranging from 0.15 to
0.58. Five of the eight subjects had optimal pulley coefficients
between 0.39 and 0.6, a range that can be used to predict
normal saccade torsional transients (Raphan 1998). Our data
on the range of pulley coefficients are therefore consistent with
the finding that there are large and reproducible inter-individual differences in both the magnitude and direction of torsional
transients in normal subjects (Fig. 4d in Straumann et al. 1995).
Our findings on the range of optimal pulley coefficients are
also consistent with the corresponding range of eye velocity tilt
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active head rotations and have not been observed during the
execution of saccades.
We considered whether the axis tilts were due to a neuralbased preprogrammed mechanism, which was delayed and
therefore responsible for the initial axis alignment. Eye velocity tilting is functionally appropriate for eye positioning mechanisms, such as the saccadic system, which attempt to reduce
torsional transients and maintain the eye orientation axis within
Listing’s plane. Relative tilting of the eye and head velocity
axes is functionally counter to the goal of the aVOR, which is
to compensate for head velocity both in magnitude and direction, thereby bringing about retinal image stabilization. Thus a
preprogrammed mechanism to misalign the eye and head velocity axes would be functionally inappropriate for the aVOR
and unlikely to be responsible for generating the axis tilts.
In contrast to preprogrammed axis tilts, we found that the
axis alignment could be predicted well by modifying the roll
gain of the aVOR during the initial 50-ms period while
maintaining the pulley coefficient at a constant value of 0.5.
It is important to note that an almost perfect match to the
data could be produced using a number of combinations of
pulley coefficient and aVOR roll gain (Fig. 10). Therefore
either a dynamic roll gain modification or dynamic change
in pulley effect, or a combination of both, could be responsible for producing the initial axis alignment during passive
head rotations. The roll gain modification would constitute
a form of eye position– dependent processing in central
vestibular structures, which would help counteract the tilting of the eye velocity axis produced by the pulleys, to bring
the eye and head velocity axes into better alignment. A roll
gain modification may be required in the aVOR and not
during saccades because of their essentially different functional roles. The saccadic system functions to move the eyes
from one fixation point to another. It is, therefore of benefit
if the velocity axis tilts to minimize positional roll transients
(Raphan 1998). The aVOR, on the other hand, functions to
minimize retinal slip. It is, therefore inappropriate for the
eye velocity axis to tilt. The roll gain modification that we
are proposing is entirely different from other proposed
forms of eye position– dependent central processing, because it brings about eye and head velocity axis alignment
and increased retinal image stability, rather than the axis tilt
and decreased retinal image stability that would result from
the other mechanisms (Misslisch et al. 1994).
While it is probable that the pulleys are responsible for
producing the observed axis tilts, as they have been anatomically demonstrated, the site of the proposed roll gain change is
not clear. One possibility is that the otoliths are involved in the
gain modification. During these yaw head rotations, there
would be centripetal acceleration generated at each macula due
to their eccentricity relative to the rotation axis of the head.
During head-on-neck yaw rotations, the head rotates about the
atlantooccipital joint. The distance from this rotation axis to the
maculae is ⬍5 cm (Curthoys et al. 1977). As the velocity
during the initial 50 ms of rotation is ⬍200°/s, the centripetal
acceleration at each macula would be very small (⬍0.05 g). In
addition, the centripetal accelerations at each macula would be
toward the rotation axis. The effect would be similar to accelerating forward and would not be expected to generate strong
compensatory roll eye velocity. Indeed, there has been no
report of compensatory roll eye velocity during fore-aft accel-
EYE POSITION-DEPENDENCE IN THE ANGULAR VOR
APPENDIX A
The model of the three-dimensional semicircular canals (Fig. 1)
was simplified from vector Eqs. 9 –18 given in Yakushin et al. (1998)
冋
cos ␪ a cos ␺ a
⫺cos ␪ p cos ␺ p
cos ␾ 1 sin ␪ 1
sin ␺ a
cos ␺ p
sin ␾ 1 sin ␪ 1
(A1)
where xcup is the cupula state, ␻h is the angular head velocity, and ␻v
is the vestibular afferent signal, referenced to the head-fixed coordinate frame. The matrix Tcan, which transforms head velocity from
head-fixed to canal coordinates, is given by
⫺sin ␪ a cos ␺ a
sin ␪ p sin ␺ p
cos ␪ 1
册
(A2)
where ␪a ⫽ ␪p ⫽ ⫺40°, ␺a ⫽ ␺p ⫽ 135°, ␾l ⫽ 0°, and ␪l ⫽ ⫺30°
(Yakushin et al. 1998). T ⫺1
can transforms the vestibular afferent signal
from canal coordinates back into head-fixed coordinates. The diagonal
matrix Tc is given by
Tc ⫽
冋
4
0
0
0
4
0
0
0
4
册
(A3)
Tc represents the cupula time constant of all canals in the head-fixed
coordinate frame. Since we studied the responses of normal subjects, we
assumed that the time constants for all canals were identical. When the
time constants are altered for specific canals, such as when plugged, more
general equations must be utilized (see Yakushin et al. 1998).
APPENDIX B
The velocity-position integrator (Fig. 1) was implemented in the
model as in Raphan (1998)
d
x p ⫽ H px p ⫹ G pr w
dt
m n ⫽ C px p ⫹ D pr w
(B1)
where xp is the integrator state, rw drives the integrator, and mn is the
motoneuron activity. The matrices were chosen as
H p ⫽ ⫺0.03333/s
C p ⫽ 59.0
冋
冋
1
0
0
1
0
0
0
1
0
0
1
0
0
0
1
0
0
1
册
册
G p ⫽ 0.03333/s
D p ⫽ 0.278
冋
1
0
0
冋
1
0
0
0
1
0
0
1
0
0
0
1
册
0
0
1
册
(B2)
The plant was approximated by a second-order system, given by
d
d
␻ ⫽ ⫺⍀ h ␻ ⫺ ␻ h ⫺ J ⫺1 共B ␻ ⫹ K⌽n̂兲 ⫹ J ⫺1 m
dt
dt
(B3)
where ␻ is the eye velocity relative to the head, ␻h is the head velocity
relative to space, and ⌽n̂ is the orientation of the eye relative to the
head. ⍀h is given by
⍀h ⫽
冋
0
␻ hz
⫺␻ hy
⫺␻ hz
0
␻ hx
␻ hy
⫺␻ hz
0
册
(B4)
where ␻hx, ␻hy, and ␻hz are the components of head velocity relative
to space.
The matrices in the equations were chosen as
J ⫽ 5 ⫻ 10 ⫺7 kg 䡠 m 2
d
x cup ⫽ T c⫺1 共T can ␻ h ⫺ x cup 兲
dt
⫺1
␻ v ⫽ T can
x cup ⫺ ␻ h
T can ⫽
冋
1
0
0
冋
冋
K ⫽ 4.762 ⫻ 10 ⫺4 kg 䡠 m 2 /s 2 /rad
B ⫽ 7.47 ⫻ 10 ⫺5 kg 䡠 m 2 /s/rad
0
1
0
册
0
0
1
1
0
0
1
0
0
0
1
0
0
1
0
0
0
1
0
0
1
册
册
(B5)
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angles that have been reported when making saccades from
secondary to tertiary positions (Table 1 in Palla et al. 1999).
The optimal pulley coefficients calculated for the passive
head impulse stimulus also varied considerably between subjects and were, on average, smaller in magnitude in comparison
with the values obtained for active head impulses. However, as
discussed above (see Model organization and conceptual basis
for study), the values of the optimal pulley coefficient that we
have reported do not necessarily reflect the actual configuration
of the pulleys in the plant. Hence we are unable to determine
the actual pulley configuration in subjects’ orbits from the
calculated optimal pulley coefficient value. The trajectories of
saccadic eye movements would be more useful for determining
actual pulley configuration, as any eye velocity axis tilt that
occurs during these eye movements presumably arises due to
the effect of the pulleys alone (Raphan 1998). We have shown,
however, that in response to passive head impulses the same
eye velocity data can be closely predicted with a number of
different combinations of pulley coefficient and roll aVOR
gain. Indeed, the pulley coefficient need not be changed from
0.5 to achieve the desired effect; the data can be predicted just
by changing roll aVOR gain. Therefore it is possible to have a
fixed pulley configuration in the orbit that would not require
radical readjustment to execute different types of eye movements.
In summary, we have presented a model of the aVOR that
incorporates the kinematics and dynamics of the semicircular
canals, central processing through a modifiable gain matrix, a
vector velocity-position integrator, and an ocular motor plant
that includes muscle pulleys. With muscle pulleys present, the
model predicted the eye position– dependent axis tilts that
occur during both active and passive head impulses. Removal
of the pulley effect resulted in loss of aVOR eye positiondependence and approximate eye and head velocity axis alignment. It was possible to predict the initial alignment of the eye
and head velocity axes, seen in response to passive head
impulses, by introducing an eye position– dependent modification in the roll gain of the aVOR. A change in pulley effect
alone was insufficient to bring about perfect eye and head
velocity axis alignment for those data, but a combination of
changing pulley effect and roll gain change was found to be as
effective as a roll gain change alone. So, the presence of
pulleys takes away the need for the axis tilts to be centrally
programmed, and the roll gain modification serves as a means
of countering the pulley effect, to produce good axis alignment
and retinal image stabilization during high acceleration impulsive head rotations.
649
650
M. J. THURTELL, M. KUNIN, AND T. RAPHAN
A complete derivation of these equations and a description of how eye
orientation is updated based on the Euler-Rodriquez equations can be
found in Raphan (1998).
The authors acknowledge R. Black, Dr. I. Curthoys, Dr. B. Cohen, Dr. G. M.
Halmagyi, and M. Todd for helpful suggestions.
This work was supported by the Royal Prince Alfred Hospital Department
of Neurology Trustees, National Institutes of Health Grants EY-04148 and
DC-02384, and a grant from the National Aeronautics and Space Administration through Cooperative Agreement NCC 9-58 with the National Space
Biomedical Research Institute.
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