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INTRODUCTION |
There is evidence that our
musculature is organized in a way that simplifies motor control. For
instance, several studies (Demer et al. 2000
;
Miller 1989
; Miller et al. 1993
) report
mobile, soft-tissue sheaths or "pulleys" in the orbit that
influence the pulling directions of the extraocular muscles. These
pulleys may simplify the brain's work in implementing Listing's law
(Miller 1989
; Quaia and Optican 1998
;
Raphan 1997
; Tweed 1997
; Tweed et al. 1994a
), which says that during some types of ocular motion, the eye's torsion (its rotation about a forward-backward axis) is zero
(Helmholtz 1867
). Might these same pulleys also play a role in other motor patterns besides Listing's law?
A motor pattern distinct from Listing's law is seen in the human
vestibuloocular reflex (VOR), the reflex that holds the eyeball roughly
steady in space during head motion. When the head turns, the VOR
does not counterrotate the eye about exactly the same axis as the head,
as one might expect. Rather, the eye's axis swings away from the
head's depending on gaze direction. When your head yaws about a
vertical axis, your eye's rotation axis swings up when you look up and
down when you look down, tilting a quarter to a third as far as the
gaze line (Misslisch et al. 1994
, 1996
;
Palla et al. 1999
; Solomon et al. 1997
).
Similarly, when your head pitches about an interaural axis, your eye's
rotation axis swings right when you look right and left when you look
left (Fig. 1), again about a third as far
as the gaze line (Misslisch et al. 1994
,
1996
). During roll head motion, about the
forward-backward axis, the effect is larger and reversed (Fig. 1): your
eye's axis now swings almost as far as the gaze line but in the
opposite direction (Misslisch et al. 1994
,
1996
).

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Fig. 1.
The pattern of eye-velocity vectors in the vestibuloocular reflex
(VOR). Each dot is the tip of an eye-velocity vector sampled during
head oscillation in darkness. During head pitch at 0.6 Hz, eye
velocities (gray dots) swing away from the interaural axis of head
rotation (ordinates), moving left when gaze is 25° left
(A), right when gaze is 25° right (B).
During head roll at 0.3 Hz, eye velocities (black dots) swing away from
the nasooccipital axis of head rotation (abscissas), opposite to gaze:
right when gaze is 25° left (A) and left when gaze is
25° right (B). Subject DT.
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As the VOR's motor pattern breaks Listing's law, it has been assumed
that both patterns cannot be served by the same arrangement of pulleys.
Some authors have therefore suggested that the pulleys might advance
and retract along their muscle paths, adopting one arrangement for
Listing's law and the other for the VOR (Demer et al.
2000
; Thurtell et al. 1999
,
2000
). But the proposed retraction is large (Fig.
2) and has not been observed;
Clark et al. (1997)
and Demer et al.
(2000)
found that the pulleys move as if dragged along with the
turning eyeball (and therefore they move opposite the gaze line in some
dimensions; see APPENDIX), but this motion resembled not at
all the large backward shift that is proposed for switching from
Listing's law to the VOR. Further, we show here that the proposed
retraction would not in fact explain the full pattern of eye axes seen
in the VOR. We show that this pattern can be explained in full given
only the observed actions of the pulleys, if one takes into account the
known neural processing within the VOR, specifically the fact that the
reflex is weak in the torsional dimension (Berthoz et al.
1981
; Collewijn et al. 1985
; Ferman et
al. 1987
; Misslisch and Tweed 2000
;
Robinson 1982
; Seidman and Leigh 1989
;
Tweed et al. 1994b
).

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Fig. 2.
Effects of pulley position on muscle action. A: if a
pulley sits as far behind the eye's center of rotation as the
muscle's insertion lies in front, then the axis of muscle action will
swing with the eye when the eye moves, but only about half as far. This
half-angle property may simplify the implementation of Listing's law
(although exact half-angle behavior requires that the pulleys move
slightly in the orbit when the eye moves). B: if the
pulley is retracted, the muscle axis moves less when the eye moves. To
achieve quarter- or third-angle behavior, the pulley would have to sit
2 or 3 times as far behind the ocular center as the insertion point is
in front. C: if the pulley is pushed forward, the muscle
axis moves more. D: advancing the pulley beyond its
muscle's insertion makes the muscle axis swing opposite the eye's
rotation.
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METHODS |
Six normal human subjects rotated their heads sinusoidally
through about ±10° in yaw, pitch, and roll in time to a computer metronome at three frequencies: 0.3, 0.6, and 1.2. As is usual in daily
life, yaw was about an earth-vertical axis, pitch and roll about
earth-horizontal axes. Subjects were trained, using a head-mounted
laser, to make movements of the desired size and direction. Their head
rotations varied somewhat and were usually larger at the lower
frequencies, reaching ±18° in some subjects, but our
function-fitting analysis described below removed the need for
precisely controlled head motion. Before each trial, subjects fixated a
laser spot 90 cm away. During the subsequent head motion, they tried to
maintain this gaze direction. In dark trials, the laser was
extinguished before the head movements began, and the trial ran in
complete darkness except that the laser flashed for 5 ms once per
second to keep the subjects' eyes from wandering. In light
trials, the laser was constantly visible amid a field of larger white spots.
Rotations of the head and left eye were recorded at 100 Hz using search
coils (Ferman et al. 1987
; Robinson 1963
;
Tweed et al. 1994b
). Eye and head-position quaternions
and angular velocities were computed as described by Tweed et
al. (1990)
. Angular velocities are expressed using the
right-hand rule, i.e., if you point your right thumb in the direction
of the vector, then your fingers curl round in the direction of spin;
for instance a forward-pointing vector represents clockwise spin (from
the subject's viewpoint). Quick phases and saccades were removed
manually and by acceleration criteria chosen individually for each
subject. To quantify the VOR we computed a best-fit quadratic equation
expressing eye velocity as a function of head velocity and eye
position; this form of function is very flexible and accurately
approximates VOR behavior (Misslisch et al. 1994
).
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RESULTS |
The motor pattern shown for one subject in Fig. 1 also held in
general. Figure 3 plots the averaged
behavior of our six subjects during head oscillations at 0.3 Hz in
darkness. During head yaw and head pitch, the eye's rotation axis
swung with the gaze line; during head roll, it swung opposite. This
behavior was entirely consistent, with the complete pattern present in
all six subjects, at all three frequencies, in both darkness and light.
Quantitatively, the eye's axis swung 31% as far as the gaze line
during head pitch; i.e., the angle between the responses for gaze 25°
left and gaze 25° right averaged 15.5°, with angles for individual
subjects ranging from 12.3 to 17.5°. During yaw, the eye's axis
tilted 36% as far as the gaze line. During roll, the axis motion was more than twice as large, 81% as far as the gaze line for left or
right gaze and 93% as far for up or down gaze.

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Fig. 3.
VOR pattern for 0.3-Hz oscillation in darkness, averaged across
subjects. Eye-velocity vectors are drawn as lines through the origin;
in each panel, 3 line styles represent responses evoked with the eye in
3 positions: centered and 25° right and left (A),
centered and 25° up and down (B). A: a
50° leftward change in gaze direction causes a 15.5° leftward swing
of the eye-rotation axis during head pitch and a 46.6° rightward
swing during roll. B: depressing gaze 50° makes the
eye-rotation axis swing 17.8° downward during yaw and 40.7° upward
during roll.
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Axis swing decreased at high frequencies of head rotation. Figure
4 plots the angle of swing for a 50°
change in gaze direction, for the different frequencies of head
rotation and the different visual conditions, averaged over the six
subjects. Two main results are revealed. First, the eye's rotation
axis swung much farther during roll than during pitch and yaw, and in
the opposite direction, at all frequencies. Second, swing tended to
decrease with increasing frequency. Significant differences
(P < 0.05) were found between the following
conditions: pitch at 0.3 versus 1.2 Hz in darkness and light; pitch at
0.3 versus 0.6 Hz in light; yaw at 0.3 versus 1.2 Hz in light; roll
gaze up/down at 0.3 versus 0.6 Hz in darkness and roll gaze right/left
at 0.3 versus 0.6 Hz in light.

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Fig. 4.
Swing angles of eye-rotation axes for different frequencies of head
rotation and different visual conditions, averaged across subjects.
, , , and
, dark trials (top); ,
, , and , light trials
(bottom). and , responses to
head yaw; and , responses to pitch;
and , responses to roll with the gaze
line 25° to the left and right; and
, responses to roll with gaze 25° up and down. The
effect of eye position is much larger in roll than in yaw or pitch and
decreases with increasing frequency. Bars show SE.
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Our computer simulations (see DISCUSSION) showed that
this pattern of axis motion is just what one would expect, given the known anatomy of the pulleys and the known neural processing within the
VOR. Specifically, simulations showed that the axes swing because VOR
responses are weaker in the torsional dimension than in the horizontal
or vertical. This explanation implies a further, surprising prediction:
if the axes swing because torsional gain is weak, then they should
swing farther, the weaker the gain. This prediction is confirmed in
Fig. 5, a plot of swing angles versus
torsional gain. Regression analysis (Sokal and Rohlf
1998
) of the eight data sets (yaw head motion, pitch head
motion, roll while subjects looked up and down, roll while they looked
right and left, all carried out in darkness and in light) showed, in all but one condition (pitch in darkness), a significant
(P < 0.05) inverse relation between swing angle and
torsional gain: as torsional gain decreased, axis swing increased. Also
as predicted (see DISCUSSION), the slope was steeper for
head roll than for yaw or pitch. Other proposed explanations for axis
swing do not predict this relation to torsional gain.

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Fig. 5.
Torsional VOR gain correlates with the eye-position dependence of the
eye's rotation axis. The swing angle of the axis increases
significantly with decreasing torsional gain in both light and darkness
(gray and black symbols) during yaw (A), in light during
pitch (B), and in both light and darkness during roll
with gaze 25° up/down (C) and 25° left/right
(D). Data are for all subjects, at all frequencies.
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DISCUSSION |
Some authors have assumed that pulleys can contribute to the motor
pattern seen in the VOR only if they are strongly retracted along their
muscle paths. But this retraction has not so far been observed and in
any case would not, as we show below, explain in full the observed VOR
responses. We show that VOR kinematics can be explained in full using
only those pulleys properties that have actually been observed, if one
takes into account the neural signals driving the reflex.
No one knows exactly how the pulleys affect muscle action, but the most
popular idea is that they are arranged to implement the half-angle rule
(Tweed et al. 1992
) that is required for Listing's law.
That is, they are placed so that the pulling direction of each muscle
moves when the eye moves, turning in the same direction but only half
as far (Fig. 2A) (Demer et al. 1995
;
Miller and Demer 1997
; Raphan 1997
;
Smith and Crawford 1998
; Thurtell et al.
2000
; Tweed 1997
). During the VOR, however, the
half-angle rule fails: the eye's axis swing significantly less than
half as far as the eye (during yaw and pitch) or significantly farther and in the opposite direction (during roll). To explain the reduced axis motion during yaw and pitch, Demer et al. (2000)
and Thurtell et al. (2000)
suggested that the pulleys
might retract toward the muscle origins during the VOR (Fig.
2B; Thurtell et al. express this retraction as a change in
their variable k
). But simulations using this retracted arrangement do not fit the data during roll: when
the eye looks 25° right, for instance, the eye-rotation axes swing in
the same direction as the eye (Fig.
6B), opposite the swing seen
in the data (Fig. 6A). This reversal error is
large: when the gaze line moves 25° right, the velocity vector,
instead of turning about 25° left as in the data, turns about 8°
right.

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Fig. 6.
Discrepancies between the pulley-retraction hypothesis and VOR
behavior. Both panels plot the vertical vs. the torsional component of
eye velocity during torsional and vertical head oscillations when the
eye looks 25° right (same format as in Fig. 1B).
A: typical behavior of the VOR. B:
predictions of the retraction model. During head pitch the simulated
responses match the data, but during roll the eye-velocity vectors
swing the wrong way: when the gaze line shifts 25° right, the
eye-velocity vector, instead of turning about 25° left as in the
data, turns 8° right.
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The reason is clear from Fig. 2. Retracting the pulleys would reduce
the motion of the eye's axis (Fig. 2B) but would not make
it swing opposite the eye, as seen in the roll VOR. The opposite adjustment, pushing the pulleys forward, would increase the axis motion
(Fig. 2C), but would not reverse it unless the pulley
advanced beyond its muscle's insertion (Fig. 2D). Figure 2
depicts just one muscle, but the principle is the same for the four
muscles that roll and pitch the eye.
One might propose that pulley retraction explains the axis motion
seen during yaw and pitch, whereas some other factor operates during
roll. But as we show next, VOR behavior in all dimensions can be
explained by a single, well-known mechanism
low torsional gain
(Berthoz et al. 1981
; Collewijn et al.
1985
; Ferman et al. 1987
; Misslisch and
Tweed 2000
; Robinson 1982
; Seidman and
Leigh 1989
; Tweed et al. 1994b
)
with no need
for the hypothetical retraction of the pulleys. Of course ocular
mechanics is complex and may have surprises in store, but at the moment
there is no need to assume that the pulleys retract during the VOR. VOR
kinematics are fully explained given the known properties of the
pulleys and the neural drive.
VOR model
Figure 7 simulates a VOR with a weak
torsional gain and pulleys that are set in a Listing configuration and
move slightly with the eye when it turns; the behavior seen by
Clark et al. (1997)
and Demer et al.
(2000)
(see APPENDIX for the model equations). Each
panel of the figure shows the tip of the eye-velocity vector looping
around as the head oscillates. In Fig. 7, A and
B, the torsional gain of the VOR is set to
0.4. When the
head oscillates vertically, at a high frequency of 1.2 Hz, the
eye-velocity loops tilt away from the ordinate (Fig. 7A).
That is, moving the eyes 50° left or right swings the average axis of
eye rotation 14.6° in the same direction. When the head oscillates
torsionally, the same change in eye position swings the eye's axis
50° in the opposite direction; i.e., this model, unlike the one in
Fig. 6B, predicts the entire pattern of axis swing,
including the reversal seen in the torsional cases. At a lower
frequency of head oscillation, 0.3 Hz, the same change in eye position
swings the axes of eye rotation farther: 16.2 and 52.4° (Fig.
7B). So the model may also explain the greater influence of
eye position on the VOR at low frequencies of head motion, although the
mechanism invoked here is not the only one possible (see
below).

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Fig. 7.
VOR kinematics can be attributed to low torsional gain, without
retracted pulleys. In these simulations the oculomotor integrator leaks
torsionally with a time constant of 1 s. Plots show how the tip of
the eye-velocity vector loops around as the head oscillates in pitch
(gray lines) and roll (black lines) when the subject looks 25° right.
A: the torsional gain of the VOR is set to 0.4, and
the head oscillates at 1.2 Hz. Vertical head oscillations evoke
eye-velocity loops tilted 7.3° away from the ordinate; i.e., moving
the eyes 25° right shifts the average axis of eye rotation 7.3°.
The same change in eye position during torsional head oscillation
shifts the eye's axis 25.0°. B: response of the same
system to head oscillations of 0.3 Hz. Now the changes in eye position
cause slightly larger shifts in the axis of eye rotation, because the
leaky integrator magnifies the influence of eye position on the VOR at
low frequencies of head motion. C: torsional gain is set
to zero, flattening the eye-velocity loops and increasing the influence
of eye-position dependence, so that a 25° change in eye position now
causes a 12.5° shift in the response to head pitch, and a 77.5°
shift in the response to head roll. D: torsional gain is
set to 1.0, removing almost all eye-position dependence.
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When torsional gain is reduced to zero in our model, axis swing
is magnified (Fig. 7C); the eye-velocity loops flatten, and the responses to torsional and vertical head motion align. When torsional gain is strengthened to
1.0, the eye-velocity loops hardly
swing at all (Fig. 7D). So in this model, swing depends on
gain. The model therefore predicts the relations seen in Fig. 5 and
also correctly predicts that monkeys, whose torsional gains are
stronger than humans', should show less axis swing (Crawford and Vilis 1991
; Misslisch and Hess 2000
).
Why does low torsional gain reverse the axis tilt in the
torsional case? Given that the pulleys are in their Listing
configuration, any drop in torsional gain pushes the system closer to
Listing's law. So what is the response to torsional head rotation of a
VOR that follows Listing's law perfectly? If for instance the head is
turning counterclockwise (thin gray arrow in Fig.
8) and the eye is looking 25° right,
then what eye velocity, consistent with Listing's law, will best
stabilize the retinal image? To calculate that optimal eye velocity,
first multiply the head velocity by
1 to yield the eye-velocity
vector (thin black arrow in Fig. 8) that would perfectly stabilize the
retinal image (but would break Listing's law). Then project that
vector orthogonally into the eye's velocity plane, which is tilted
12.5° right, in keeping with the half-angle rule. The dashed line in
Fig. 8 depicts this projection and shows that the projected vector
(thick black arrow) tilts leftward away from the vector of perfect
stabilization. That is, the eye's rotation axis tilts opposite the
gaze line (Misslisch et al. 1994
).

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Fig. 8.
Why a low torsional gain reverses the direction of eye rotation
axis tilt in the roll VOR. In this example, the head rotates
counterclockwise at 37.7°/s (gray arrow, peak velocity at 0.6 Hz),
torsional gain is zero, and the eye is looking 25° right (cf. Fig.
7C). The eye velocity that is consistent with Listing's
law and best stabilizes the retinal image must lie in the eye's
velocity plane for that gaze direction, which is tilted 12.5° right.
The eye velocity that stabilizes the entire retina (thin black arrow)
is found by multiplying head velocity by 1. Then the vector in the
eye velocity plane (thick black arrow) is found by projecting the
vector of perfect stabilization orthogonally (dashed line) into the eye
velocity plane. Note that this roll VOR vector tilts leftward, away
from the perfect image stabilization vector, and opposite the gaze
line.
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The frequency effects in Fig. 7, A and B (smaller
axis swings at higher frequencies of head motion) arise because the
model incorporates a torsional leak in the oculomotor integrator
(Seidman et al. 1995
). When the head is stationary, the
torsional eye-position command from the integrator does not hold steady
but drains away to zero with a time constant that may be as low as
1 s. To emphasize the effect, we used this minimal value of 1 s in our simulations (Fig. 7), but if the true time constant is longer,
one may need to invoke other mechanisms to explain the frequency
effects. One possibility is miscalibration: it has been shown that if
the brain's internal estimate of the plant's elasticity were low,
that would explain the small torsional blips that accompany saccades
(Tweed et al. 1994a
); in the VOR, the same
miscalibration would produce smaller axis swings at high frequencies
(this can be simulated by setting
K*11 < K11 in the equations in the
APPENDIX). Another dynamic effect, the weakness or absence
of axis swing during the initial response to an abrupt head jerk
(Thurtell et al. 1999
), may arise by the same
mechanisms, simply because the head motion in this situation is very
high frequency, or it may mean that the neural processes responsible
for axis swing are slightly delayed.
Foveal VOR
Does the swinging-axis pattern serve any purpose? Misslisch
et al. (1994)
showed that it roughly halves the range of
torsional eye positions generated by the VOR; i.e., it reduces
deviations from Listing's law. And what is the advantage of better
obeying Listing's law? Several suggestions have been made (e.g.,
Helmholtz 1867
; Hepp 1990
,
1995
; Hering 1868
; Tweed and Vilis
1990
; Tweed et al. 1992
). Perhaps the most
important factor is that Listing's law keeps the eye near the center
of its motor range, poised for a swift response in any direction.
Of course, when the eye's axis swings away from the head's, eye
motion can no longer perfectly cancel head motion. The visual image is
therefore imperfectly stabilized on the retina. This is an inevitable
geometric consequence of staying close to Listing's law. But it is
theoretically possible to approximate Listing's law while nevertheless
preserving image stability over one small portion of the retina, the
central high-acuity region called the fovea. Theoretically, the brain
has a choice between two extreme strategies: it could eliminate optical
flow entirely over the fovea, or it could minimize the average optical
flow over the whole retina, with no special concern for the fovea. Or
it could do something in between, keeping optical flow over the fovea
somewhat smaller than the average flow elsewhere. All the strategies
along this continuum are equally consistent with Listing's law; they can all be adjusted to yield any desired degree of adherence to the
law, although it can be shown that the strategies that give special
treatment to the fovea require higher eye velocities than the
strategies that treat the whole retina more uniformly. Where does the
actual VOR lie along this continuum? The best test is to measure how
far the eye-velocity vector swings during head roll. If the brain is
trying to eliminate optical flow over the fovea, the vector will swing
100% as far as the gaze line. If the VOR is treating the retina
entirely uniformly, the vector will swing only 50% as far. Our data,
showing ratios between 81 and 93%, support the foveal hypothesis.
Misslisch et al. (1994)
found ratios nearer 50%,
perhaps because they used passive rather than active head motion, but
more likely just because the swing angles vary a lot (Fig. 4).
Presumably these angles need not be controlled with absolute precision.
All that is required is a reasonable balance of several factors
including energy output, ocular torsion, and image stability over
various parts of the retina. In pursuit of this balance, the VOR
restricts its violations of Listing's law while partially preserving
the stability of the foveal image.
This study was supported by the Deutsche Forschungsgemeinschaft
(Sonderforschungsbereich 307/A2). D. Tweed is supported by a Scientist
Award from the Medical Research Council of Canada.
Address for reprint requests: H. Misslisch, Dept. of Neurology,
University of Zurich, Frauenklinikstrasse 26, 8091 Zurich, Switzerland
(E-mail: hubert.misslisch{at}nos.usz.ch).