Department of Physiology, Faculty of Medicine and Health Sciences, Erasmus University Rotterdam, 3000 DR Rotterdam, The Netherlands
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ABSTRACT |
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Collewijn, Han and
Jeroen B. J. Smeets.
Early Components of the Human Vestibulo-Ocular Response to Head
Rotation: Latency and Gain.
J. Neurophysiol. 84: 376-389, 2000.
To characterize vestibulo-ocular
reflex (VOR) properties in the time window in which contributions by
other systems are minimal, eye movements during the first 50-100 ms
after the start of transient angular head accelerations
(~1000°/s2) imposed by a torque helmet were analyzed in
normal human subjects. Orientations of the head and both eyes were
recorded with magnetic search coils (resolution, ~1 min arc; 1000 samples/s). Typically, the first response to a head perturbation was an
anti-compensatory eye movement with zero latency, peak-velocity of
several degrees per second, and peak excursion of several tenths of a
degree. This was interpreted as a passive mechanical response to linear acceleration of the orbital tissues caused by eccentric rotation of the
eye. The response was modeled as a damped oscillation (~13 Hz) of the
orbital contents, approaching a constant eye deviation for a sustained
linear acceleration. The subsequent compensatory eye movements showed
(like the head movements) a linear increase in velocity, which allowed
estimates of latency and gain with linear regressions. After
appropriate accounting for the preceding passive eye movements, average
VOR latency (for pooled eyes, directions, and subjects) was calculated
as 8.6 ms. Paired comparisons between the two eyes revealed that the
latency for the eye contralateral to the direction of head rotation
was, on average, 1.3 ms shorter than for the ipsilateral eye. This
highly significant average inter-ocular difference was attributed to
the additional internuclear abducens neuron in the pathway to the
ipsilateral eye. Average acceleration gain (ratio between slopes of eye
and head velocities) over the first 40-50 ms was ~1.1. Instantaneous
velocity gain, calculated as
Veyet/Vheadtlatency,
showed a gradual build-up converging toward unity (often after a slight overshoot). Instantaneous acceleration gain also converged toward unity
but showed a much steeper build-up and larger oscillations. This
behavior of acceleration and velocity gain could be accounted for by
modeling the eye movements as the sum of the passive response to the
linear acceleration and the active rotational VOR. Due to the latency
and the anticompensatory component, gaze stabilization was never
complete. The influence of visual targets was limited. The initial VOR
was identical with a distant target (continuously visible or
interrupted) and in complete darkness. A near visual target caused VOR
gain to rise to a higher level, but the time after which the difference
between far and near targets emerged varied between individuals.
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INTRODUCTION |
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Passive rotation of the head is accompanied, in
species with mobile eyes, by compensatory eye rotation in the
opposite direction such that gaze direction tends to remain relatively
stable despite head movements. The earliest components of this ocular
response (within 100 ms of the onset of head rotation) are controlled
by the vestibulo-ocular reflex (VOR), which has the following
characteristics: 1) the presence of very short connections
(a three-neuron arc); 2) a common spatial organization
between the sensory organ (the semi-circular canals) and the effector
(the external eye muscles); and 3) sensitivity of the canals
to rotational acceleration as the primary stimulus (see, e.g.,
Highstein 1988). These properties favor an early
stabilization of gaze after sudden disturbances of the orientation of
the head. Indeed, short VOR latencies have been reported: for monkeys,
14.2 ms (Lisberger 1984
), 12 ms (Cullen et al.
1991
), 10 ms (Snyder and King 1992
), and 7.3 ms
(Minor et al. 1999
); for cats, 13 ms (Khater et
al. 1993
); for humans, 6-15 ms (Maas et al.
1989
), 4-13 ms (Johnston and Sharpe 1994
), 7-8
ms (Tabak and Collewijn 1994
), and 10 ms (Crane
and Demer 1998
).
The action of the VOR in generating compensatory gaze-stabilizing eye
movements is complemented by the optokinetic response (OKR), for which
slippage of the retinal image is the primary stimulus. The OKR has a
relatively long delay because the elaboration of visual motion signals
requires considerably more signal processing than the VOR requires. The
shortest latency described for optically driven compensatory eye
movements in humans is 70-80 ms (Gellman et al. 1990).
Any contribution to gaze stabilization in normal humans by
proprioceptive cervico-ocular reflexes appears to be small and
inconsistent (Bronstein and Hood 1986
;
Jürgens and Mergner 1989
).
Thus the best strategy with which to investigate the VOR in a
"pure" form is to measure ocular responses that occur within a
window of ~10-70 ms after the start of a transient, well-defined head movement. While the VOR in this early phase is unlikely to be
affected by visual or propriocepive inflow that is directly derived
from the ongoing head movement, it may still be modulated by factors
that require a modification of the VOR gain, such as the distance of a
visual target, the position of the axis of head rotation, or non-unity
visual magnification factors. The topography of the axes of eye and
head rotation requires an increase in VOR gain as a visual target gets
nearer to the subject. Such a change has been demonstrated repeatedly
(Biguer and Prablanc 1981; Blakemore and Donaghy
1980
; Crane and Demer 1998
; Hine and
Thorn 1987
; Snyder and King 1992
; Snyder
et al. 1992
; Viirre and Demer 1996
;
Viirre et al. 1986
) but its early time course is not
well known.
Research on the human VOR has traditionally used whole-body motion with
low-frequency sinusoidal oscillation or persistent rotation in one
direction. Such long-lasting stimuli often yielded gain-values for the
VOR that were substantially below unity and, moreover, subject to many
extrinsic influences, such as mental frames of reference (for an
overview see Collewijn 1989). Research with transient
stimuli has been sparse, partly because of the technical limitations of
the rotational devices used. Traditional human rotation devices do not
generate accelerations much larger than
100°/s2, but natural head rotations reach
several times this magnitude during walking and running, and can be as
high as 6000-12000°/s2 during vigorous, voluntary head
shaking (Grossman et al. 1988
, 1989
). Such high head
accelerations can thus be considered physiological and apparently
harmless. In some previous experiments, substantial acceleration pulses
of the head alone were achieved. Maas et al. (1989)
were
able to determine the gain and latency of the human VOR by inducing
head accelerations of up to 7100°/s2 by
applying mallet strokes to a yoke that was clenched between the teeth
of the subject. Halmagyi et al. 1990
and Aw et
al. 1996
achieved head accelerations of up to
3000°/s2 in manually applied passive steps in
head orientation in normal subjects and in patients with vestibular
disease. Such transient stimuli proved to be better tests of VOR
performance than traditional motion stimuli but had the disadvantage of
being relatively uncontrolled and variable.
In the last few years, some groups developed more powerful rotational
devices for whole-body rotation of human subjects
[2800°/s2, Crane and Demer
(1998); 284°/s2,
Johnston and Sharpe (1994)
]. In a
different approach, Tabak and Collewijn (1994
, 1995
) and
Tabak et al. (1997a
,b
) introduced a torque-driven helmet
to impose well-controlled transient head accelerations of about
1000°/s2 with great facility. In the present
experiments, the early phase of the normal human VOR in response to
pulses of acceleration was investigated by using this device with
improved recording and analysis procedures, binocular recording, and a
number of different target distances and visibility conditions. In
particular, the latency and early build-up of acceleration gain and
velocity gain were addressed, as well as the occurrence of mechanical
ocular responses in the latency period.
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METHODS |
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Subjects
Healthy subjects without any known vestibular or oculomotor abnormalities were recruited after informed consent. The procedures were approved by the Medical Ethical Committee of the Faculty of Medicine. A few subjects were rejected because they tended to blink in association with the head stimuli, which introduced unmanageable artifacts into their recordings. Ten subjects were retained for analysis; some analyses and comparisons were made in smaller subsets of these subjects.
Motion stimuli
The use of the torque helmet was described previously
(Tabak and Collewijn 1994, 1995
). In the present
experiments, acceleration pulses in the horizontal plane were delivered
by activating the torque motor for 200 ms at maximum power. These
pulses were alternated in the rightward and leftward directions. The
interval between pulses was randomized between 2.5 and 3.5 s
(average, 3 s); one measuring sequence lasted 180 s. Thus
~30 pulses in each direction were delivered in one measuring
sequence. The subjects, while wearing the helmet, were not rigidly
attached to any fixed structure and were relatively free to orient and
move their heads.
Visual stimuli
Seven visual conditions were tested. The first measurement was done in complete darkness. In the other six conditions, a single red light-emitting diode (LED) was presented at two distances, ~220 or ~40 cm, and in three conditions of visibility. The LED was extinguished 50 or 500 ms before the activation of the helmet or was left on throughout the measurement.
Eye movement recording
Movements of both eyes were recorded with the scleral coil
technique (Robinson 1963). Coils embedded in a silicone
annulus (Skalar, Delft, The Netherlands) were inserted in each eye
(Collewijn et al. 1975
). A Remmel EM3 eye-movement
recorder (Remmel Labs, Ashland, MA) was adapted for large, earth-fixed
field coils (pairs of square coils; diameter, 2.5 m; inter-coil
distance, 1.25 m; this provided a "Helmholtz" coil
configuration with sufficient homogeneity). Head movements were
recorded by a third coil, which was mounted to an individually molded
silastic dental-impression bite-board. All coils were pre-calibrated on
an angular rotation device. Gains and offsets of the instrument were
extremely stable. It was verified that calibrations were unaffected by
translations of the coils over a range (up to 20 cm in all directions)
that exceeded any spontaneous head displacements by the subjects. The noise level corresponded to <1 min arc at a recording range of 20°
on each side of the middle position. The resulting signals represented
the orientation of the head in space and the eyes in space (gaze).
Data collection and analysis
Orientations of the eyes and head were sampled at a frequency of
1010 Hz (each channel) with a CED 1401-plus AD-converter with the CED
Spike2 program (Cambridge Electronic Design Ltd., Cambridge, UK) and
stored on disk. The same device was programmed to generate the pulses
that controlled the torque helmet, and marker signals indicating the
timing of these pulses were included in the recordings. In the
subsequent off-line analysis, angular position signals were converted
to angular velocity signals by digital differentiation using five
subsequent samples without time shift (see Collewijn et al.
1995). This routine eliminated much of the noise at the cost of
a mild time-blurring due to smoothing two position samples forward and
backward. Accelerations were calculated by differentiating velocity.
The larger noise inherent to this procedure necessitated the use of
nine subsequent velocity samples with, as a consequence, more time
blurring (smoothing four velocity samples and, therefore, six position
samples forward and backward). After removal of the (occasional) events
that were contaminated by blinks or saccades at critical moments, the
responses in a measurement sequence were averaged for each direction
separately; temporal alignment was achieved by a computer-generated
trigger locked to the electrical command to the helmet. Eye-in-head
movements were computed by subtracting head movements from gaze
movements; vergence and version were computed as the difference between
and the average of the orientations of the two eyes. Statistical
differences were tested with paired t-tests whenever appropriate.
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RESULTS |
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The basic response
The basic result is presented in Fig. 1, which shows head angular velocity and eye angular velocity (the latter is shown inverted for clarity, i.e., head velocity - gaze velocity) as a function of time for a typical subject (MF). These results were obtained with a distant visual target that was extinguished 50 ms prior to the head stimulus. Figure 1A shows 10 subsequent individual head and eye movements superimposed, to show the reproducibility of all main components. Figure 1B shows the average head and eye velocities for the same measurement, which is composed of 23 consecutive head acceleration pulses in the same (rightward) direction. Standard deviations of head and eye velocities (shown as vertical gray bars) were small at all times; the movements were very reproducible within a measurement (which lasted 3 min) with very little variability between impulses. The further analysis of our data is based on such averages of all (uncontaminated) responses (n = 20-30) in a measurement.
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Head acceleration built up over an initial period of ~10 ms to steady values of 1000-1200°/s2 (depending on the subject) that were then maintained for 40-50 ms, during which time velocity rose approximately linearly as a function of time. Later, head velocity tended to saturate smoothly despite the continued force exerted by the helmet (which lasted for 200 ms). This decrease in head acceleration (the exact course of which varied between individuals) may be attributed to the buildup of passive and active mechanical resistance during progressive rotation of the neck. The present analysis is essentially limited to the period with (approximately) constant head acceleration.
Eye velocity approximately mirrored head velocity, showing a similar time course, but separated along the time axis by a delay that was maintained over time. In a majority of the subjects, however, the shape of the early VOR was complicated by the occurrence of an anti-compensatory eye rotation that preceded the compensatory VOR and started at the same time as the head movement. Typically, such early anti-compensatory eye movements reached peak velocities of several °/s, displacements of several hundredths of a degree, and accelerations of several hundred °/s2 (Fig. 1). Their mean peak velocity was 3.26° ± 3.17° (SD) for far targets and 3.60° ± 3.06° (SD) for near targets (pooled results of 6 subjects, 2 eyes, 2 directions, and 2 × 3 visibility conditions; difference not significant in paired t-test). The manifestation of this anti-compensatory component with zero latency relative to the head movement, and its apparent duration commensurate with the probable latency of the active VOR, strongly suggests a passive mechanic origin. We will further analyze its nature in The nature of the anti-compensatory eye movement after proceeding first with calculations of VOR latency and gain.
The occurrence of periods of relatively constant acceleration of head
and eyes allowed a simple analytical procedure for determining the
latency and initial gain of the VOR, as illustrated in Fig. 1B. Linear regressions were fitted to the straight parts of
the velocities of the head, each of the eyes separately, and the two eyes combined (average velocity of the two eyes is called
version).The later parts, in which acceleration declined, as
well as the earliest parts, during which acceleration usually showed a
short buildup and the eyes moved in the anti-compensatory direction,
were not included in the regressions. In general, head and eye
velocities were regressed over a range of 10-50°/s (time span
~15-50 ms after the start of the head movement). These ranges were
individually adjusted whenever visual inspection of the velocity graphs
revealed a different range of the straight parts of the velocity
profiles. The coefficient of determination
(r2) of the linear regressions was
typically ~0.99. Each regression was characterized by its
intersection with the time axis and its slope; the relation between
these parameters for the head- and eye-velocity regressions yield
independent estimates for VOR gain and latency. Gain is estimated as
the ratio between the slopes of the linear regressions of eye and head
velocity; because these slopes represent acceleration, this estimate
reflects the acceleration gain. Latency is the time interval between
the intersections of the linear regressions with the time axis. This
technique for estimating latency is similar to that used by Carl
and Gellman (1987) for the estimation of smooth-pursuit
latencies and by Johnston and Sharpe (1994)
for the VOR.
Acceleration gain
The distribution of acceleration gain values, obtained as described in The basic response, is shown in Fig. 2 for the pooled data of six subjects for whom complete data were collected for two directions and seven conditions. First, it was established that the variations in the visibility of the target during head movement (switched off 50 or 500 ms prior to the stimulus or left on) did not have any systematic effect on the responses in the early period that we analyzed. Accordingly, the results for the three different visibility conditions were pooled for the near and far target. Next, we tested for differences in gain between distant and near target conditions. A paired t-test confirmed that gain was significantly higher (as was theoretically expected, P = 0.04) for near targets than for far targets. Therefore separate histograms were plotted for near and far targets. For far targets, the mean early acceleration was 1.089 (Fig. 2A) whereas for near targets it was 1.124 (Fig. 2C). In darkness, mean gain was similar to that with a distant target (mean, 1.09; Fig. 2E). Ideally, steady-state gain values would be ~1.045 for the far target and ~1.25 for the near target. As will be elucidated in Instantaneous VOR gain, the early gain values do not reflect a steady state and should not be expected to correspond to these ideal values. It can be concluded that in an early period (~15-50 ms after the start of the head movement) VOR gain shows some systematic tendencies: 1) gain is larger for near than for distant targets; 2) all gains (a/a) are systematically larger than unity.
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A further differentiation was a comparison of the gain of the
ipsilateral eye (the eye on the side to which the head rotated) with
the gain of the contralateral eye. A systematic difference could occur
for two possible reasons. First, there could be an intrinsic difference
in the VOR dynamics for nasal and temporal eye movements, like there is
for horizontal "conjugate" saccades in which the abducting eye
usually reaches a higher peak velocity than its fellow eye. This would
allow us to predict a higher gain of the contralateral (abducting) eye
for the VOR. Second, differences in gain could result from the
difference in distance of the two individual eyes to the target as a
function of head position. Inter-ocular differences of this type should
emerge, especially for near targets (Viirre et al.
1986). Paired t-tests comparing gains of the
contralateral and ipsilateral eye were done for the different target
conditions. For far targets, the mean difference in gain (ipsilateral
eye gain
contralateral eye gain = 0.014) was not
significantly different from zero (P = 0.35; see
distribution in Fig. 2B). For near targets, the mean
ipsilateral gain was significantly higher than the contralateral gain
(difference of 0.123; P = 3.7 × 10
5; see Fig.
2D). In darkness, there was again no difference (mean 0.013;
P = 0.55; see Fig. 2F).
To interpret the inter-ocular gain difference for the near targets, the
initial head position has to be known. If the head is rotated to the
right from an initial angular position that is to the left of the
middle, the ipsilateral (right) eye should have a higher initial VOR
gain because it is closer to the target than the left eye
(Viirre et al. 1986). This was actually the case in our
experiments. The alternation of rightward and leftward pulses had the
result that the mean initial positions of the head were ~3° left of
the middle position for rightward pulses that were preceded by leftward
pulses, and vice versa. The fact that the inter-ocular gain difference
virtually disappeared with far targets and darkness supports the
hypothesis that it originates in different eye-target distances and
argues against an intrinsic advantage for nasal or temporal VOR
movements. In particular, there was no evidence for any advantage of
the contralateral (abducting) eye, as occurs in saccades.
VOR latency
Although there is no reason to expect differences in VOR latency
due to visual target conditions, the presence of a minimum of three and
two synapses in the shortest VOR pathways to the medial and lateral
rectus muscles, respectively, suggests a possible shorter latency of
the contralateral than of the ipsilateral eye. A paired
t-test corroborated the absence of a significant difference (P two-tailed = 0.10) between the mean latency for far
targets (10.3 ms) and near targets (10.8 ms). Accordingly, all
conditions (far and near targets and darkness) for the six completely
measured subjects were pooled for a comparison between the latency
estimates for the ipsilateral and the contralateral eyes. The two
distributions, as determined from the intersections of the regression
lines on eye and head velocity with V = 0, are
presented in Fig. 3A. These histograms show that latency was systematically longer for the ipsilateral than for the contralateral eye. For statistical analysis, the two eyes were paired for comparison within a measurement, i.e., for
head pulses to the right the latency of the right eye (ipsilateral) was
compared with the latency of the left eye (contralateral) and vice
versa. Mean values were 11.1 ± 0.2 ms (SE) and 9.8 ± 0.2 ms
(SE); the difference was statistically very significant (two-tailed
paired t-test, P = 5 × 109). The distribution of
the difference (mean value 1.3 ± 0.2 ms, SE) is shown in Fig.
3B.
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The shorter mean latency for the contralateral eye than for the ipsilateral eye was also present in each of the seven stimulus conditions separately. However, it could not be demonstrated at the level of individual measurements, probably as a result of interference by the anti-compensatory component, which showed random variations between eyes and measurements, thus masking the subtle systematic differences in eye latencies. Given the statistical robustness of the effect for the pooled data, a grand average of the VOR of the contra- and ipsilateral eyes (6 subjects, 7 conditions, and 2 directions, normalized for rightward head rotation) is shown in Fig. 4, which shows the consistent delay of the response of the ipsilateral versus the contralateral eye by ~1 ms, which is maintained over time. Figure 4 also prominently shows the early anti-compensatory eye movement in the pooled data.
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Graded effects of anti-compensatory eye movement on latency and gain
The magnitude of the anti-compensatory early eye movement differed
systematically among subjects and, furthermore, randomly between eyes,
directions, and measurements at different times. The latency and gain
measurements from linear regressions as described in VOR latency
and Acceleration gain disregard the anti-compensatory eye velocity attained during the latent period and assume implicitly that the active eye movement starts, like the head movement, from a
velocity of zero. Actually, the active VOR is likely to start before
the anti-compensatory movement has dissipated, i.e., while eye velocity
is negative. In other words, the appropriate reference level for the
start of the VOR is not zero eye velocity but a negative eye velocity.
This would imply that the latencies as estimated in VOR
latency tend to be overestimates; apparent latency is likely to
increase as a function of the magnitude of the anti-compensatory component. If this is the case, then there should be a correlation between the latencies and the magnitude of the anti-compensatory movements in individual measurements. Figure
5A shows the relation between
measured latencies and the absolute values of the maximum anti-compensatory velocity in individual measurements (i.e., averages of 20-30 successive head pulses). Because there was no statistical latency difference between visual conditions, all measurements of the
six complete subjects were again pooled but the data for the
ipsilateral and contralateral eyes were treated separately. Peak
anti-compensatory eye speeds ranged from 0 to ~12°/s; latency increased as a function of the maximum anti-compensatory velocity. Separate linear regressions were done for the ipsilateral and contralateral eyes. Both accounted for about half of the variability (r2 = 0.53 and 0.49, repectively) and
showed a perfectly parallel course [slope = 0.49 ms/(deg × s1)]. They were
therefore separated by a constant time difference corresponding to the
difference in latency between the ipsilateral and contralateral eyes.
The intercepts of the regressions with the latency axis (i.e., for an
anti-compensatory velocity equal to zero) were 8.25 ms for the
contralateral eye and 9.43 ms for the ipsilateral eye. We postulate
that these figures are the best estimates of the true latency of the
active VOR that can be reached in the absence of detailed knowledge of
the passive anti-compensatory component that would allow calculation of
its exact contribution to the apparent latency. The difference between
the intercepts (1.18 ms) agrees well with the estimate reached in Fig.
3B for the mean difference in individual measurements (1.3 ms).
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Figure 5B shows a similar exercise for gain values. Linear regressions were calculated for three subgroups of the data: far target, near target, and darkness (with ipsilateral and contralateral eyes pooled together). For all groups, estimated gain increased as a function of the magnitude of the anti-compensatory response. The intercept of the regressions with the gain-axis (absence of anti-compensatory movement) was 1.02 for the "far" and "dark" conditions whereas it was higher (1.05) for the "near" condition. The difference (0.03) was in good agreement with the difference (0.035) between the mean gains for the populations (cf. Fig. 2, A and C). Again, we postulate that these intercept values are the best estimates for the early VOR gain (averaged over a period of ~15-50 ms after the start of the head movement) whereas the higher apparent values are a side effect of the anti-compensatory eye movement.
The nature of the anti-compensatory eye movement
The appearance of anti-compensatory eye movements simultaneously
with the start of the head rotation, i.e., in the latency period of the
active VOR, strongly suggests that the anti-compensatory movement is
mechanical, not neural, in nature. After carefully considering the
possibility of, but not finding, any plausible errors of measurement,
we assumed that the anti-compensatory movement originates from forces
acting directly on the eye. It cannot be explained, however, as
a reaction purely to the rotation of the head; any
mechanical response of the eye to head rotation would have to be in the
compensatory direction because of inertia of the eye in the
orbit that undergoes a rotational acceleration. The same would apply to
any inertial movement of the coil relative to the eye. It is difficult
to predict the theoretical magnitude and, especially, the dynamics of
an ocular inertial rotational response because the eye is not a rigid
body but a fluid-filled shell. An attempt to calculate the theoretical
mechanical compensatory eye movement during the first 10 ms of an
angular head acceleration of 3000°/s2 yielded a
magnitude of 0.002° (Minor et al. 1999), which is
clearly below the resolution of current recording techniques. However, the mechanical relations are complicated by the fact that the passive
rotation of the eye is eccentric. As illustrated in Fig. 6A, rotation of the head
around its natural axis of rotation near its center (H), as imposed by
the helmet, causes an eccentric rotation of the eye (and orbit) with
radius r. A rotational acceleration (arot,
expressed in radians/s2) around H will induce a
linear acceleration (alin, expressed in
cm/s2) at eccentricity r (expressed in
cm)
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Consider the following mechanical analogue of the orbital tissues in the bony orbit: a glass beaker filled with a semi-solid gel (Fig. 6B; a fluid-filled beaker closed at the open surface with an elastic membrane would be equivalent). When this vessel is tilted to a horizontal position to align gravity with the free surface, or when the vessel is linearly accelerated in a direction parallel to the free surface, pressures in the gel will force the beaker's contents in the direction opposite to the acceleration. The result is a deformation of the surface, with its center rotating in the direction of the acceleration (Fig. 6B). For the head and eye (Fig. 6A), this effect corresponds to an anti-compensatory eye rotation.
To simulate this effect in our experimental conditions, we filled a small glass beaker (50 ml, 3.5 × 6 cm) with a warm 6.5% solution of gelatin and floated an eye coil on the surface. A second coil was glued to the outside of the bottom of the beaker; this mimicked the "head" coil. After the gel solidified and the "eye" coil became embedded in its free surface, the beaker was tilted to a horizontal position in the magnetic field (in two opposite directions) to assess the static steady-state effect of gravity. This effect was in the direction sketched in Fig. 6B and its magnitude was ~2°. The beaker was then mounted horizontally on a rotational device with the free surface at 10 cm eccentricity; this assembly was coupled to the torque helmet. A dummy experiment was then run, with the beaker undergoing dynamic accelerations similar to those applied to the orbits of our subjects, while the angular positions of the beaker ("head") and the free gel surface ("eye") were recorded. After data processing identical to that in the real experiments, results were obtained as shown in Fig. 6C. "Head" velocity accelerated almost uniformly to 60°/s after 70 ms. "Eye" velocity started (with zero latency) in the anti-compensatory direction. Velocity remained negative for ~12 ms but, instead of simply regressing to zero (corresponding to a steady deformation), it showed strong oscillations. This is actually not very surprising because our gelatin analogue represents an elasticity-viscosity-mass system that is unlikely to be critically damped. If acceleration were to be maintained for sufficient time, oscillations would decay and the deformation would reach a steady state, representing an equilibrium between the pressures caused by the acceleration and the elasticity of the deformed material. The primary conclusion at this point is that the early anti-compensatory eye rotation is easily accounted for (at least qualitatively) by a fundamental physical effect.
Isolation and modeling of the passive eye movements
The combined results described in The nature of the
anti-compensatory eye movement suggest that the eye movements
observed in our experiments are the result of two processes:
1) a passive mechanical response to a step in linear
acceleration caused by eccentric rotational acceleration of the orbit;
2) an active, neurally mediated VOR. Presumably, these
processes simply add up to the total eye movement
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To mathematically model the eye movements observed, Eq. 4 was computed over an appropriate time range with a spreadsheet program and the corresponding passive velocity (Vpas) and acceleration (Apas) were obtained by differentiation. Equation 4 models a pure step response, which corresponds to an instantaneous rise of the angular head acceleration to ~1000°/s2. The actual head accelerations did not rise instantaneously; their behavior was well-modeled by an exponential rise to the maximum value with a time constant of 0.01 s. Accordingly, head velocities rose initially smoothly before reaching the period of constant acceleration. These features were implemented in the models of the head movements and the passive eye movements by letting ya rise to its asymptotic value with a time constant of 10 ms. Active VOR eye velocity (Vact) was modeled by Eq. 3 and total eye velocity by Eq. 2.
A typical result is shown in Fig. 7, C (velocities) and
D (passive eye displacement). The parameters were optimized
to match the real results in Fig. 7, A and B
(peak head acceleration = 1150°/s2; = 84 rad/s; ya = 0.05°;
= 0.1 s). The agreement between real data and the model
seems to be quite satisfactory and some features of the data are
clarified. During the latency period (first 8 ms), the eye movement
consists entirely of the passive component. At the end of the latency
period, eye velocity starts to deviate from the passive component;
however, this moment cannot be unambiguously determined in real data.
Subsequently, eye velocity crosses the zero line to become
compensatory. Obviously, this zero-crossing is delayed with respect to
the real start of the active VOR, which causes an increase in the
apparent latencies as measured with simple regression techniques.
Furthermore, the rise in eye velocity is initially steeper than the
rise in head velocity; the acceleration gain is larger than unity. This
is accounted for by the contribution of the passive component. Once the
passive anti-compensatory velocity has reached its maximum and starts
to decrease, its acceleration becomes positive (compensatory) and will
add up to the (approximately unity) acceleration generated by the
active VOR. As a result, the acceleration gain at this time becomes
larger than unity, as consistently observed in our real data.
To quantify these effects, we varied the anti-compensatory
velocity in our model by varying the asymptotic end position
ya. The model was executed using
parameters estimated from real data (Fig. 7) and the apparent gains and
latencies were computed by linear regressions in exactly the same way
as was done initially for the real data. A set of apparent gains and
latencies thus computed from the model ( = 100 rad/s;
= 0.05 s; ya = 0.0-0.1°) is
plotted in the scatter diagrams of Fig. 5, A and
B (crosses). For latency, the model values coincide very
well with the calculated regression lines. The modeled gain values rise
somewhat steeper than the average real data as a function of the
maximum anti-compensatory velocity, but the discrepancy is minor given
the fairly schematic nature of the model.
Instantaneous VOR gain
We have shown that the acceleration gain, calculated from the
slopes of the linear regressions on eye and head velocities as a
function of time and reflecting an average value over a period of ~40
ms after the latency, is larger than unity, even for far targets and in
darkness (Fig. 2). Although this tendency was explained in
Isolation and modeling of the passive eye movements as being the contribution of passive eye movements, a more profound
understanding can be obtained by calculating, instead of this single
gain parameter, the instantaneous gain as a continuous function of
time. This can be done by comparing instantaneous eye and head
velocities (or accelerations) with the appropriate time relations.
Because the VOR has a latency, it is appropriate to calculate
instantaneous velocity gain as the quotient of eye velocity at time
t and head velocity at time t latency
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Figure 8A shows head and eye velocities, as well as instantaneous acceleration and velocity gain, calculated for a latency of 8 ms for a typical subject. (To reduce noise, especially for acceleration gain, Fig. 8A was prepared from a long measurement and includes 165 subsequent rightward head pulses; the target was distant and extinguished 50 ms before the head pulse). Obviously, velocity gain was negative as long as the eye velocity was anti-compensatory; this meaningless part was not plotted. After velocity became compensatory, velocity gain rose to a value of approximately unity. This rise was never instantaneous but took several tens of milliseconds. Acceleration gain was also initially negative but became positive as soon as the anti-compensatory eye velocity had passed its maximum, which occurred, of course, earlier than its crossing to positive values. Thus acceleration gain rose earlier than velocity gain and showed some oscillations while converging slowly toward unity. The first peak of the acceleration curve exceeded unity, which is in agreement with the average acceleration gain values of the early VOR obtained from linear regressions.
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These features are perfectly duplicated by the model described in
Isolation and modeling of the passive eye movements, as shown in Fig. 8B (parameters,
ya = 0.02°; = 84 rad/s;
= 0.1 s). The characteristic features of the gain curves
are entirely determined by the occurrence of the passive, initially
anti-compensatory and later oscillatory eye movements that add up to
the active VOR with a constant gain of 1. The shapes are critically
affected by the correct choice of the latency and by the magnitude of
the passive component. If the latency chosen is too short (or
disregarded, i.e., taken to be zero, as is frequently done in the
literature), gain increases much more slowly because, even with a unity
gain, the latency will cause the eye velocity to remain below the
simultaneous eye velocity as long as the head accelerates. On the other
hand, overestimating the latency for the gain calculation (as occurred in our initial calculations from regressions) results in spurious overshoots. When the passive component is absent, either in the model
or in real data, velocity and acceleration gain are identical and
stable at unity throughout time after latency, when latency is
correctly accounted for as in Eq. 5.
Given the complex interactions of gain, latency, and passive components, even correctly calculated gain values are not a very transparent parameter of VOR performance. The most direct parameter of the effectiveness of the VOR is the residual retinal slip velocity, i.e., the velocity of the eye in space. Such gaze velocities are plotted as additional functions in Fig. 8, A and B. Gaze velocity always had the same sign as head velocity, i.e., the VOR undercompensated. As shown in Fig. 8B (model simulation), a sustained head acceleration will be accompanied by a sustained gaze velocity because of the continued effect of the latency, even though gain is unity. This lag can only be overcome after head acceleration decreases and velocity levels off, as occurred in a real experiment (Fig. 8A; for the saturation effect see Fig. 1). Notice that the earliest gaze movements are even faster than the head movements because of the passive anti-compensatory response.
Effect of target distance on instantaneous gain
The effect of target conditions on the instantaneous VOR gain was studied in four subjects that were free of blinks and saccades during the first 120 ms after the start of head movement at t = 0. This effect could not be adequately studied by comparing eye velocity profiles because head velocity profiles differed among our subjects. Figure 9A shows averaged instantaneous VOR velocity gain for four conditions, total darkness, and a target at a distance of 220 cm that was switched off 50 or 500 ms before the head motion or left on continuously. The two movement directions and eyes were pooled; all gains were computed assuming a VOR latency of 8 ms. The gain profiles (which were aligned at the start of head motion at t = 0) were similar for all four conditions (any differences were not systematic across subjects). Velocity gain rose steeply at first (becoming positive after ~12 ms, which is when eye movement became compensatory) and later gradually, and leveled off ~50 ms after the start of head movement. These results corroborate the fact that, in this early stage, the VOR showed identical responses in the presence of a distant target and in darkness and that short interruptions of the visibility of the target during head movement had no effect on the VOR. The overall average gain for distant targets in the interval 80-100 ms after the start of head movement was 0.998; variability (SD) in this period was ~0.01 for the factor time and ~0.05 for the factor subject. (Perfect compensation at this distance would require a gain of ~1.045, as shown in Fig. 9A.)
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For comparison, responses to head pulses in the presence of a near visual target (distance of ~40 cm) are shown, for the same subjects, in Fig. 9B. Once again, the time course of gain within the analyzed period of ~120 ms was not systematically affected by the visibility of the target (switched off 50 or 500 ms prior to the stimulus or continuously lit). The near target induced, however, a prolonged period of increasing gain as compared with a far target or darkness. After 100 ms, mean gain was ~1.2 and had not quite reached a steady state. (Complete compensation for the near target distance would require a gain of ~1.25.) It was verified that, for all target conditions and all subjects, ocular convergence angles were appropriate for the target distance and were stable throughout the measurements at a specific distance. Thus the time of divergence between the gain for far and near targets was not temporally related to any fast change in convergence.
In Fig. 10, gain curves of these same three subjects and a fourth subject (ST, whose responses were measured with fewer variations in lighting conditions) are pooled in a different way: the average gain curves for far and near targets are shown for each of the subjects separately, and the target conditions (extinguished or not extinguished) are pooled. All subjects showed gain increasing as a function of time with eventually higher values for the near targets. The details, however, differed considerably among the subjects. Two subjects (Fig. 10, C and D) showed an initially steep rise in gain, which was in agreement with the very small anti-compensatory components in these subjects. However, their VOR gain remained initially lower for near targets than it did for far targets, with a crossover occurring after only several tens of milliseconds. A third subject (Fig. 10A) showed a similar slow development of the increase in gain for near targets although gain build-up was slower in general due to a substantial anti-compensatory phase. Only one subject (Fig. 10B) showed a higher gain for the near targets from the very beginning (in combination with a large anti-compensatory component and a slow build-up of gain). Examining the different target conditions separately showed that these individual characteristics were reproducible within a subject. A complete pooling of the results for the four subjects and all of the visibility conditions for near and far targets (excluding the condition "darkness"), which shows the overall trends, is presented in Fig. 9C. The average time courses of velocity gain appeared identical for near and far targets until ~40 ms after the start of head movement at t = 0. After that, the time courses diverged and each of the gain curves gradually approximated the value appropriate for the target distance.
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DISCUSSION |
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The present measurements reveal a number of properties of binocular human VOR that occur during the first tens of milliseconds after a step in head acceleration of ~1000°/s2. The main findings are a short latency (~8-9 ms), a shorter latency for the contralateral eye than for the ipsilateral eye, the frequent occurrence of an anti-compensatory eye movement during the latency period, an initial acceleration gain >1.0 but with a gradual build-up of velocity gain, and a variable latency for an effect of target distance. We will now attempt to explain and relate these findings.
Early anti-compensatory eye movements
Surprisingly, we found that the earliest response to head acceleration was anti-compensatory in most subjects. This component had a latency of zero relative to head movement and increased during the next 7-8 ms, after which eye acceleration in the compensatory direction originated. Given these properties, the anti-compensatory component can only be purely mechanical in origin and can only result from the assembly of eye accelerations (rotational and linear) related to the imposed head rotation. We successfully accounted for and modeled the effect on the basis of elementary physical principles related to the deformation of soft media (the orbital tissues) in a rigid vessel (the bony orbit) under the influence of pressure.
This interpretation of orbital mechanics is supported by previous
studies on the ocular effects of linear accelerations. Steinbach and Lerman (1990) reported the effects of gravity on eye
position in patients that were paralyzed by atracurium in preparation
for a surgical procedure. An effect of gravity (1 g) was
present in 16 of 22 patients; eye rotation (up to 5-10°) was always
away from gravity, suggesting that the center of mass of the
eye is behind its center of rotation. The direction of these effects corresponds to the present findings, in which leftward reactive forces
resulted in a rightward passive eye movement (Fig. 6). Similarly,
Bush and Miles (1996)
noticed that the earliest ocular response of monkeys to a sudden free-fall (corresponding also to a
change in linear acceleration by 1 g) was a
downward eye movement (whereas subsequent compensatory eye
movements were upward). Bush and Miles (1996)
interpreted this as a mechanical effect. A similar anti-compensatory
effect was evident in earlier experiments on the ocular responses of
monkeys from the same laboratory to translation (Schwarz and
Miles 1991
); the authors explicitly discussed this effect and
supported its genuine nature by excluding some potential sources of
artifact. No such anti-compensatory movements were mentioned by
Angelaki and McHenry (1999)
, who performed similar translation experiments in monkeys.
In principle, it should be possible to study the dynamics of the
passive effect of linear accelerations (in pure form or as a component
of eccentric rotation) more directly in an isolated form in subjects
with totally absent vestibular responses. In retrospect,
anti-compensatory eye movements were prominent in a few patients with
bilateral labyrinth defects who were investigated several years ago in
our laboratory with the helmet technique (Fig. 6 in Tabak and
Collewijn 1994; Fig. 4 in Tabak et al. 1997b
). Halmagyi et al. (1990)
detected no mechanical ocular
responses in a human subject with complete bilateral vestibular
neurectomy, but this absence could be due to their use of manual head
rotation, during which acceleration builds up more gradually than with
our helmet. Labyrinthectomy was used by Khater et al.
(1993)
as a control in their experiments on cat VOR, which also
showed a zero-latency mechanical response that was, however,
compensatory in direction. This difference in direction,
compared with the reported results in humans, also emerged from
experiments by Harris et al. (1993)
, who concluded,
because of the effects of gravity, that the effective center of mass of
the cat's eye lies in front of its center of rotation. Our
present interpretation of the results of Harris et al.
(1993)
and Steinbach and Lerman (1990)
is that
effects of linear accelerations on eye orientation are actually not
accounted for by the position of the center of mass of the eye as such, but by the mechanical relations between the soft orbital tissues as a
whole and the surrounding bony orbital structures. These relations are
likely to be different in humans and cats. Minor et al.
(1999)
did not observe passive eye movements in response to
angular accelerations of 3000°/s2 in a monkey
after bilateral labyrinthectomy, but they did not specify the position
of the axis of rotation relative to the eye.
It may be possible to obtain further evidence as to what the mechanism
of passive ocular responses might be by using passive rotation of
normal subjects with varying positions of the rotational axis.
Rotation around an axis centered on an eye should yield minimal
anti-compensatory movement of that eye, whereas rotation around an axis
anterior to the eye should result in inversion of the passive response
to compensatory instead of anti-compensatory movement. Unfortunately,
we were unable to effectively manipulate the axis of head rotation by
varying the axial position of the torque applied to the helmet. The
head tended to rotate around its natural axis no matter which way head
stimulus was applied, and we lacked the facilities for passive
whole-body rotation with comparable accelerations. Such variable axis
conditions were achieved for the vertical human VOR by Viirre
and Demer (1996), who applied impulsive head rotations around a
horizontal axis positioned either through the centers of the eyes or 15 cm posterior to the eyes, and for the horizontal human VOR by
Crane and Demer (1998)
, who varied the vertical axis
position between 20 cm posterior and 10 cm anterior to the eyes.
Neither of these two studies reported any early mechanical eye
responses in the latency period, but neither were they mentioned as a
point of attention. Although we found the anti-compensatory
components to be very conspicuous in our velocity and acceleration
traces, they were quite small (a few minutes of arc) in position
records, which is the form in which they are recorded during
experiments. They could easily be lost or escape attention if
resolution at the min/arc level is slightly compromised during digital
recording or subsequent data processing.
We were able to isolate the passive eye movements in favorable
measurements by subtracting an "ideal" VOR (gain, 1.0; latency, 8 ms) from the total eye movement. This revealed that the passive response to a step in eccentric rotational acceleration shows oscillations of ~12-15 Hz. This suggests a similar natural frequency of oscillation for the orbital tissues. This discovery is also important in interpreting the results of sinusoidal head oscillation in
this frequency range. In previous experiments of this type (Tabak and Collewijn 1994; Tabak et al.
1997a
,b
), we consistently found that after a minimum at 8 Hz
gains increased for oscillation frequencies of 14 and 20 Hz. This trend
was equally present in normal subjects and those with unilateral or
bilateral labyrinth defects. Our present findings suggest that caution
should be used when interpreting VOR measurements in this frequency
range because they are likely to be contaminated by substantial passive contributions.
Latency of the VOR
Reliable estimates of VOR latency require 1) low-noise, high sampling frequency measurements of head and eye rotations (with insensitivity to translations); 2) transient head rotations that are well controlled in timing and magnitude; 3) absence of any spurious mechanical coupling between the stimulus and the response; and 4) suitable analysis techniques. The only recording technique that satisfies these conditions at present is the magnetic search coil technique, with search coils attached to the eye(s) and to a custom molded bite-board (or, in animals, to the bony skull). Such techniques have been applied in a number of primate and human studies (cited in the INTRODUCTION).
The present estimate of mean human VOR latency (~8-9 ms) is
consistent with our first estimate based on the helmet technique (Tabak and Collewijn 1994) and with recent measurements
by Aw et al. (1996)
(7.5 ± 2.9 ms). Furthermore,
the range of the estimated latencies (3-13 ms; see Fig. 7) is
consistent with human data reported by Maas et al.(1989)
(6-15 ms), Johnston and Sharpe (1994)
(4-13 ms), Crane
and Demer (1998)
(7-10 ms), and Minor et al. (1999)
(7.3 ± 1.5 ms).
An intriguing new finding is the statistically robust difference in VOR
latency between the eyes, the contralateral eye being ~1 ms faster
than the ipsilateral eye. This corresponds to a difference of one
synaptic delay between the pathways to the lateral rectus muscle of the
contralateral eye and the medial rectus muscle of the ipsilateral eye,
which is in agreement with the classical description of a disynaptic
pathway (vestibular afferent-medial vestibular nucleus neuron
-contralateral abducens motoneuron) for the abducting eye and a
trisynaptic pathway (vestibular afferent-medial vestibular nucleus
neuron-internuclear neuron in the contralateral abducens
nucleus-ipsilateral medial rectus motoneuron) for the adducting eye
(for a review of these connections see Leigh and Zee
1999). The axons of the abducens internuclear neurons ascend in
the contralateral medial longitudinal fasciculus (MLF). In addition, a
direct pathway from medial vestibular nucleus neurons to ipsilateral
medial rectus motoneurons, which runs through the ascending tract of
Deiters (ATD), was described in the cat (Highstein and Baker
1978
; Reisine and Highstein 1979
) and in the
monkey (McCrea et al. 1987
). The latency difference
found in the present work argues against a strong role of this ATD
pathway in the human VOR. A minor role for the ATD is also suggested by
experimental MLF lesions in monkeys, which cause a VOR with reduced
gain with the adducting eye unable to cross the middle position
(Evinger et al. 1977
).
VOR gain
Gain, the ratio of the magnitudes of eye and head rotation
(expressed in position, velocity, or acceleration), is generally considered to be an adequate measure of VOR performance. For distant targets it should, ideally, be close to unity to eliminate retinal image instability induced by head rotation. But the interpretation of
VOR gain for transient movements is complicated by two factors: latency
and mechanical transients. Confusion between gain and lag time occurs
when velocity gain is calculated as the quotient of simultaneous eye
and head velocities while neglecting the latency. This leads to
spuriously low gain values during head acceleration because even an
actual gain of unity will yield an eye velocity at time t
that matches the head velocity, not at time t, but at time
t latency
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(6) |
The presence of mechanical, initially anti-compensatory, transient eye
responses also affects apparent gain, especially in the very early
stage of the VOR. Although the mechanical response typically has an
amplitude of only a few min/arc, its fast nature and relatively
high-frequency content (12-15 Hz) result in substantial velocities
(several °/s) and high accelerations (several hundreds of
°/s2). The anti-compensatory movement during
the latency of the VOR induces an apparent negative gain. Passive
acceleration reverts to positive as soon as the anti-compensatory
velocity starts to decline. Assuming a pure sinusoidal oscillation at
13 Hz, this would occur after ~19 ms (1/4 cycle); passive velocity
would become positive 19 ms later. This process has a cyclic nature
because the passive movements are underdamped and oscillate at least as long as the period of our analysis (a little over 100 ms). The passive
accelerations and velocities add up to the active VOR, which starts
during the first (negative) passive movement after ~8 ms. As a
consequence, acceleration gain of the VOR will show oscillations around
the genuine active value of ~1.0. The strongest positive effect
should occur in the positive acceleration part of the first cycle of
oscillation, from ~19-58 ms after the start of the head movement.
This is actually what occurs in real data (Fig. 8). This period also
coincides with the period used to estimate gain from linear
regressions. Therefore it is consistent that acceleration gain values
determined in this way are, on average, larger than unity (Fig. 2). The
regressions in the scatter diagram of Fig. 5B strongly
suggest that VOR acceleration gain without a passive contribution is
very close to unity. It should be noted that gain values derived from
regressions are not affected by the magnitude of the latency because
they reflect only the ratio in the slopes of the regressions, which is
independent of time. Velocity gains will also be affected, but weaker.
Assuming that active VOR velocity gain, like acceleration gain, is
constant and near unity for distant targets, the passive contribution
should cause a maximum in the velocity gain at the peak of the second half-cycle of oscillation, i.e., about 55 ms after the start of head
movement. This agrees with the data shown in Figs. 8A,
9A, and 10, A and B. The general
pattern that we found for the time course of VOR gains seems to be
consistent with the observation by Minor et al. (1999)
in the monkey that acceleration gain (measured early in the response)
was higher (mean 1.04) than the velocity gain (mean 0.91) measured
later when velocities had reached a plateau, although Minor's
explanation was different and involved non-linearities.
Effects of target conditions
In the present experiments, visibility of the target, as well as
target distance, were manipulated. The VOR was not affected by the
actual visibility of a target during the transient head movements;
responses were similar whether the target was continuously lit or
extinguished 50 ms or even 500 ms before head rotation (Fig. 9,
A and B). Responses in darkness were identical to
those with a distant target (Fig. 9A). This suggests that
the default gain of the VOR is appropriate for distant targets. A near
target caused an increase in the VOR gain, which is in agreement with the topography of the axes of rotation of eyes and head, as was reported previously (Biguer and Prablanc 1981;
Blakemore and Donaghy 1980
; Crane and Demer
1998
; Hine and Thorn 1987
; Snyder and
King 1992
; Snyder et al. 1992
; Viirre and
Demer 1996
; Viirre et al. 1986
). The time course
of the enhancement of velocity gain by a near target varied
considerably between our subjects (Fig. 10) but, at the average, a
difference in gain was manifest after ~40 ms (Fig. 9C).
Higher acceleration gains for near than for far targets were also
manifest in our gain values obtained from linear regressions (Fig. 2).
The few reports in the literature dealing with this aspect are not
quite congruent. For monkeys, Snyder and King (1992),
using accelerations of 500°/s2, reported a
modulation of the VOR by viewing distance that emerged ~20-30 ms
after the start of head rotation, i.e., ~10 ms after the first
response to head rotation. On this basis, they suggested the existence
of a second, slower channel for the processing of angular head velocity
signals, modified by viewing distance, in addition to a first channel
that relayed only head velocity. For human subjects, Crane and
Demer (1998)
, using accelerations of 2800°/s2, found a higher VOR gain for near than
for far targets throughout the response, without a delay of the
expression of the distance effect such as found by Snyder and
King (1992)
. Crane and Demer (1998)
also found
that a decrease in peak acceleration had nonlinear effects, among which
was an increased latency for the effect of distance. Specifically, for
an acceleration of 1000°/s2 (as used in the
present experiments), they found that gains became larger for near
targets than they did for far targets at ~32 ms after the start of
head movement, as compared with 8 ms for accelerations of
2800°/s2 (their Table 2). Their average value
of 32 ms corresponds reasonably well to the present finding of an
average of ~40 ms (Fig. 9C), but the present data suggest
substantial variation among subjects of the time at which distance
effects emerge.
In agreement with Viirre et al. (1986), we found a
systematic difference between the acceleration gain of the two eyes, in the sense that gain was slightly higher for the eye that was closer to
the target in the starting position of the head. As may be expected,
the effect was statistically significant only for near targets (Fig.
2D), for which the distance of the target to the two eyes
can differ substantially.
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FOOTNOTES |
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Address for reprint requests: H. Collewijn, Dept. of Physiology, H-EE-1540, Faculty of Medicine and Health Sciences, Erasmus University Rotterdam, PO Box 1738, 3000 DR Rotterdam, The Netherlands (E-mail: collewijn{at}fys.fgg.eur.NL).
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 12 November 1999; accepted in final form 6 April 2000.
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REFERENCES |
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