Kinematic Principles of Primate Rotational Vestibulo-Ocular Reflex
II. Gravity-Dependent Modulation of Primary Eye Position
BERNHARD J. M. HESS AND DORA E. ANGELAKI
Department of Neurology, University Hospital Zürich, CH-8091 Zurich, Switzerland; and Department of Surgery
(Otolaryngology), University of Mississippi Medical Center, Jackson Mississippi 39216
Hess, Bernhard J. M. and Dora E. Angelaki. Kinematic principles
of primate rotational vestibulo-ocular reflex. II. Gravity-dependent
modulation of primary eye position. J. Neurophysiol. 78: 2203–
2216, 1997. The kinematic constraints of three-dimensional eye positions were investigated in rhesus monkeys during passive head and
body rotations relative to gravity. We studied fast and slow phase
components of the vestibulo-ocular reflex (VOR) elicited by constant-velocity yaw rotations and sinusoidal oscillations about an earthhorizontal axis. We found that the spatial orientation of both fast and
slow phase eye positions could be described locally by a planar
surface with torsional variation of õ2.0 { 0.47 (displacement planes)
that systematically rotated and/or shifted relative to Listing’s plane.
In supine/prone positions, displacement planes pitched forward/
backward; in left/right ear-down positions, displacement planes were
parallel shifted along the positive/negative torsional axis. Dynamically changing primary eye positions were computed from displacement planes. Torsional and vertical components of primary eye position modulated as a sinusoidal function of head orientation in space.
The torsional component was maximal in ear-down positions and
approximately zero in supine/prone orientations. The opposite was
observed for the vertical component. Modulation of the horizontal
component of primary eye position exhibited a more complex dependence. In contrast to the torsional component, which was relatively
independent of rotational speed, modulation of the vertical and horizontal components of primary position depended strongly on the
speed of head rotation (i.e., on the frequency of oscillation of the
gravity vector component): the faster the head rotated relative to
gravity, the larger was the modulation. Corresponding results were
obtained when a model based on a sinusoidal dependence of instantaneous displacement planes (and primary eye position) on head orientation relative to gravity was fitted to VOR fast phase positions.
When VOR fast phase positions were expressed relative to primary
eye position estimated from the model fits, they were confined approximately to a single plane with a small torsional standard deviation
( Ç1.4–2.67 ). This reduced torsional variation was in contrast to the
large torsional spread (well ú10–157 ) of fast phase positions when
expressed relative to Listing’s plane. We conclude that primary eye
position depends dynamically on head orientation relative to space
rather than being fixed to the head. It defines a gravity-dependent
coordinate system relative to which the torsional variability of eye
positions is minimized even when the head is moved passively and
vestibulo-ocular reflexes are evoked. In this general sense, Listing’s
law is preserved with respect to an otolith-controlled reference system
that is defined dynamically by gravity.
INTRODUCTION
Recent studies on three-dimensional eye movements have
emphasized the central role that Listing’s law plays in oculomotor control (for a short review see: Hepp 1994). This
remarkable principle imposes a strong constraint in the rota-
tory degrees of freedom of the eyes. When the head is upright
and stationary, all accessible eye positions have rotation axes
relative to a common reference that are confined to a single
head-fixed plane (von Helmholtz 1867). Listing’s law holds
true not only for steady fixations of distant targets but also for
smooth pursuit eye movements and for saccade trajectories
independently of whether they are fixed-axis rotations or not
(Ferman et al. 1987a,b; Haslwanter et al. 1991; Minken et
al. 1993; Tweed and Vilis 1990; Tweed et al. 1992; Van
Opstal et al. 1991).
Most of the previous work on Listing’s law has focused
on oculomotor paradigms with the head stationary and upright. Only recently have investigators begun to look at some
of the consequences of this principle under the more natural
conditions when the head moves in space or during headfree gaze shifts where the vestibulo-ocular reflex (VOR)
stabilizes gaze in space (Glenn and Vilis 1992; Misslisch et
al. 1994; Tweed and Vilis 1992; Tweed et al. 1995). It might
seem appropriate for gaze stabilization, for example, that the
vestibulo-ocular reflex breaks Listing’s law. Closer examination of the VOR, however, has revealed certain limitations
to such violations, suggesting a rather complex interplay
between gaze stabilization mechanisms and eye movement
kinematics. It has been shown, for example, that fast phases
of the torsional VOR direct eye positions in the orbit such
as to keep the slow phases centered around Listing’s plane
(Crawford and Vilis 1991). It has been shown further that
yaw VOR slow phases could comply to a different degree
with the kinematic constraints imposed by Listing’s law depending on the particular strategy of image stabilization
(Misslisch et al. 1994).
Our present understanding of the role of three-dimensional
kinematic constraints in the oculomotor system appears to
be further complicated by the observation that the orientation
of Listing’s plane relative to the head depends in a systematic
way on static head orientation relative to gravity (Crawford
and Vilis 1991; Haslwanter et al. 1992). Little attention has
been paid to this result, however, primarily because of the
small magnitude of such changes ( Ç10% of the change in
static head position). This result has been taken as evidence
for the notion that Listing’s law is attributable to factors that
optimize the orbital mechanics of the eye muscle plant. There
is still much controversy over the nature of these factors,
i.e., whether they reside in passive mechanical properties
(Demer et al. 1995; Schnabolk and Raphan 1994; Straumann
et al. 1995) or whether they are attributable to a neural
control strategy (Nakayama 1975; Tweed et al. 1994; van
0022-3077/97 $5.00 Copyright q 1997 The American Physiological Society
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Opstal et al. 1996). Whatever causes for Listing’s law have
been discussed, primary position of the eyes so far has been
considered to remain in essence always head-fixed.
There exists an alternative, yet not investigated possibility,
however, that primary eye position varies as the head
changes its orientation in space. In the preceding paper, we
have shown that yaw VOR fast phase velocity axes were
not randomly oriented but exhibited a systematic tilt relative
to the axis of head rotation. We have proposed that these
results emerge as a direct consequence of a change in the
oculomotor coordinates and primary eye position as a function of head orientation in space (Hess and Angelaki 1997).
In this paper, we investigate this hypothesis by analyzing
the three-dimensional organization of eye positions during
passive head movements relative to gravity. We show that
eye position vectors systematically change their orientation
relative to the head. These dynamic orientation changes, as
well as the associated changes in fast phase velocity axes
described in the preceding paper (Hess and Angelaki 1997),
are both consistent with a modulation of primary eye position
as a function of dynamic head orientation relative to gravity.
Preliminary results of this work have appeared in abstract
form (Hess and Angelaki 1995).
METHODS
Animal preparation and eye movement recording
This study presents data obtained from five juvenile rhesus
monkeys ( Macaca mulatta ) that were prepared chronically with
a scleral dual-search coil for three-dimensional eye movement
recording and head bolts for restraining the head during the experiments. Details of fabrication and implantation of the dualsearch coil have been reported elsewhere ( Hess 1990 ) . Threedimensional eye position was measured with a two-field search
coil system ( Skalar Instruments, Delft ) . The search coil signals
were calibrated as described in Hess et al. ( 1992 ) . In brief, an
in vitro calibration before implantation yielded the coil sensitivities and the angle between the two search coils. The orientation
of the dual coil on the eye was determined from the four coil
output signals during fixations of three vertically arranged target
lights subtending 307 relative to straight ahead. Horizontal, vertical, and torsional eye positions were digitized at a sampling rate
of 833 Hz and stored in the computer for off-line analysis. Eye
positions were expressed as rotation vectors, E Å tan ( r / 2 ) u ,
where u is a unit vector pointing along the rotation axis of the
eye, and r is the angle of rotation about u ( Haustein 1989 ) . The
eye angular velocity vector, V, was computed from the eye
position vector, E, according to the equation ( Hepp 1990 ) :
V Å 2 ( dE / dt / E 1 dE / dt ) / ( 1 / É EÉ2 ) .
Listing’s plane and primary eye position were determined from
spontaneous eye movement data in the light with the head upright
and stationary. Rotation vectors were expressed relative to a righthanded coordinate system, referred to as standard head-fixed reference system, where the x axis was aligned with primary gaze
direction (positive direction is forward) and the y and z axes were
lying in Listing’s plane (positive directions are leftward and upward, respectively). A positive torsional eye position component
(Etor ) corresponded to a rotation of the upper pole of the eye toward
the right ear, a positive vertical component (Ever ) corresponded to
a downward rotation, and a positive horizontal component (Ehor )
corresponded to a leftward rotation. Because Listing’s plane was
pitched forward in three of the four animals, the standard coordinates were rotated relative to the head roll, pitch, and yaw coordi-
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nates by a variable amount ( °67 about the interaural (pitch) axis
and õ27 about the head vertical (yaw) axis).
Experimental protocols
During the experiments, animals were seated in a primate chair
with the head restrained in a position of 157 nose-down relative to
the stereotaxic horizontal (defined as ‘‘upright’’ position) to place
the lateral semicircular canals approximately earth horizontal. The
animals were placed inside the inner frame of a multiaxis turntable
with three motor-driven gimbaled axes. The effect of dynamic
changes in head orientation relative to gravity on fast and slow
eye movements was studied during either constant-velocity rotation
or sinusoidal oscillations of the animals about their head-vertical
(yaw) axis, which was oriented in the earth-horizontal plane (907
off-vertical). Thus the experimental protocols consisted of: earthhorizontal axis rotations at constant positive or negative speeds of
58, 110, and 1847 /s, starting always with the animal in supine
position, earth-horizontal axis oscillations at 0.5 Hz ( {187 ), 0.2
Hz ( {457 ), and 0.1 Hz ( {907 ), all of which were centered in
supine position.
Data analysis
At the beginning of each experimental session, primary position
was determined with the animals upright making spontaneous eye
movements while looking around in the normally lit laboratory.
Primary eye position was determined by fitting a plane with minimal least squares error to these eye positions. In all VOR protocols,
eye positions were expressed relative to upright primary position
and rotated into the standard head-fixed coordinates x, y, and z
such that primary gaze direction aligned with the x coordinate (i.e.,
Listing’s plane corresponded to the plane x Å 0). Fast and slow
phases of the nystagmus were separated based on a semiautomated,
interactive procedure using time and amplitude windows set for
the second derivative of the magnitude of the eye velocity vector
(see also Hess and Angelaki 1997). The following analysis was
performed on these data.
1) Local fit of displacement planes (performed on constantvelocity data only). After eye positions had been expressed in the
standard head-fixed coordinates, VOR records were divided into
12 sectors (Si , i Å 1, . . . , 12) of 307 width, equally spaced throughout each stimulus cycle and shifted by 0157 relative to the onset
of rotation (supine position) (see also Hess and Angelaki 1997).
The sectors for the four cardinal head positions were defined as
follows: supine, around 07 (S1 : 015–157 ); right ear-down, around
907 (S4 : 75–1057 ); prone, around 1807 (S7 : 165–1957 ); and left
ear-down, around 2707 (S10 : 255–2857 ). The order in which the
animal was rotated through these positions was: supine, left-ear
down, prone, and right-ear down for a positive yaw rotation and
the other way around for a negative yaw rotation. For each VOR
record, five or more cycles were included in this analysis.
After having pooled the data of corresponding sectors, either a
single plane (displacement plane) was fitted to the respective eye
position vectors in each of the 12 sectors or two separate planes
were fitted to both the fast and slow phase components of the
respective eye position vectors according to the equation
Ex Å a / bEy / cEz
(1)
where Ex , Ey , and Ez represented the torsional, vertical, and horizontal components of an eye position vector E in the previously
defined standard coordinates x, y, and z. The orientation of each
of these best-fit planes was described by the associated orthogonal
unit vectors, n. The components nx , ny , nz of n are related to the
coefficients a, b, and c of Eq. 1 as follows
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nx Å
q
1
1 / b2 / c2
,
ny Å 0bnx ,
nz Å 0cnx ,
d Å anx
(2)
The scalar parameter d describes the orthogonal distance of this
plane relative to the coordinate origin. The plane orientation vectors, ni , and scalar distances, di (i Å 1, . . . , 12), of each of
the 12 pooled data sectors were used to compute how much the
displacement planes of eye positions obtained from a given VOR
record rotated or translated, respectively, relative to the standard
Listing’s plane of eye positions (i.e., the coordinate plane x Å 0).
2) Global fit of a gravity-dependent modulation of displacement
planes. Rather than separating into discrete sectors, the second set
of analyses focused on a global model. It was assumed that there
exists always an instantaneous displacement plane of three-dimensional (3-D) eye positions that changes its orientation dynamically
in the head as the head moves in space. Such a time-dependent
displacement plane is described by (compare with Eq. 1)
Ex (t) Å a(t) / b(t)Ey (t) / c(t)Ez (t)
(3)
with
a(t) Å a0 / a1 sin [ u(t) / a1 ] / a2 sin [2u(t) / a2 ]
b(t) Å b0 / b1 sin [ u(t) / b1 ] / b2 sin [2u(t) / b2 ]
c(t) Å c0 / c1 sin [ u(t) / g1 ] / c2 sin [2u(t) / g2 ]
(4)
where u(t) is the angular position of the head relative to gravity
as a function of time.
Eq. 4 defines the motion of this plane relative to head coordinates
in the case of a passive head movement. The constant parameters
a0 , b0 , and c0 account for static changes of the displacement planes
in tilted positions compared with upright (c.f. Haslwanter et al.
1992). Parameters a1 , b1 , and c1 describe the linear dependence
of the displacement plane orientation, whereas parameters a2 , b2 ,
and c2 describe second-harmonic components. The motion is described to be a sinusoidal function of angular head position, u,
relative to gravity. In addition, we have included a nonlinear dependency on head position, proportional to the sine of twice the angular
head position relative to gravity. As will be shown in RESULTS,
this second harmonic contribution was only significant for the yaw
rotation of the displacement plane, i.e., component b(t).
For constant-velocity rotation data, angular head position
changes linearly with time, i.e., u(t) Å vt Å 2p( £ /3607 )t, where
£ is the rotation velocity in degrees per second. In this simple case,
the complete 15-parameter model described by Eq. 4 was used.
That is, the parameters a0 , a1 , a1 , a2 , a2 , b0 , b1 , b1 , b2 , b2 , c0 ,
c1 , g1 , c2 , and g2 of the cyclic functions a(t), b(t), and c(t) were
estimated for a given record of nystagmus by the method of minimal least squares. For oscillatory data, the angular position of the
head relative to gravity changes in a sinusoidal fashion, which can
be described by u(t) Å 0 u0 cos (2pft) with the maximal amplitude
u0 Å 187 ( f Å 0.5 Hz), 457 ( f Å 0.2 Hz) or 907 ( f Å 0.1 Hz).
Due to the complexity of this condition, an 11-parameter model
(whereby a2 Å 0 and c2 Å 0) was fit to the oscillatory data. This
reduced model was appropriate because the complete 15-parameter
model showed only a minor contribution of the second harmonic
component in a(t) and c(t) when applied to constant-velocity data
(see Table 1). For both the 15- and 11-parameter model fits, the
cyclic modulation of the displacement plane was computed using
the relations between the displacement plane vector n(t) and the
scalar d(t) with the coefficients a(t), b(t), and c(t), as described
in Eq. 2.
3) Computation of primary eye position in tilted head positions.
The relationship between the time-dependent displacement plane
and Listing’s plane in upright position can be described by a unique
rotation vector, P, which is given by the equation (for mathematical
details, see APPENDI X )
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PÅ
1
nx
S DSD
d
0nz
ny
Å
a
c
0b
2205
(5)
This rotation vector P can be considered as primary position in
a locally tilted coordinate system that is associated to the displacement plane in each head position. To better explain the relationship
between displacement planes and primary position in tilted head
positions, let us consider a simpler and previously dealt with case,
that of static pitch and roll tilts from upright (Haslwanter et al.
1992). For example, if the head pitches nose-down (Fig. 1A), the
displacement plane of eye positions tilts backward by a certain
angle a relative to the standard Listing’s plane (plane x Å 0). In
this case, the new primary position has the coordinates Px Å 0,
Py Å 0nz /nx , and Pz Å 0 corresponding to a gaze direction which
points upward by twice the angle a Å tan 01 ( 0nz /nx ) (Fig. 1A,
right). If the head tilts in the roll plane (Fig. 1B), all eye positions
assume a fixed torsion, resulting in a shift of the displacement
plane along the x axis. Thus the new primary position has the
coordinates Px Å d, Py Å 0, and Pz Å 0 corresponding to a gaze
direction along the x axis with a torsion of twice the angle b Å
tan 01 (d/nx ) (Fig. 1B). In a more general case when head orientation relative to gravity is between pitch and roll, the displacement
plane of eye positions tilts and shifts simultaneously. In this case,
Eq. 5 predicts a more complex change in primary position.
Based on Eq. 5, primary eye position was evaluated for each of
the 12 planar surfaces that were fitted to the displacement planes
of eye positions in the 12 sectors into which all VOR records were
partitioned and for the time-dependent displacement plane fitted
by the global model.
4) Transformation of eye positions into ‘‘tilted’’ gravity-dependent coordinates. For most of the analysis, eye position, displacement planes, and primary position have been computed relative to
the standard head-fixed coordinates (x, y, z), as defined above. In
addition, selective data sets were also expressed relative to the
locally defined tilted coordinates associated with the time-varying
(gravity-dependent) displacement plane and primary position
(Figs. 8 and 9). For this purpose, the eye position vectors were
recomputed relative to the estimated gravity-dependent primary
position and converted into the tilted coordinates, xp , yp , and zp in
which primary gaze direction pointed along the (time-dependent)
xp coordinate. In these coordinates, the corresponding primary eye
position coincided with the unique rotation vector with all three
components zero similar to the definition of primary position in
upright head position relative to the standard head-fixed coordinates. The transformation into the tilted gravity-dependent coordinates was obtained by a left multiplication of the eye position
vectors, E, with the respective inverse primary position, P 01 , i.e.,
by the equation E * Å P 01 7 E ( 7 denotes multiplication of rotation
vector) (for details see APPENDIX and Tweed et al. 1990).
5) Testing Helmholtz’s criterion for a Listing’s system (performed on constant-velocity data only). To test Helmholtz’s criterion of a Listing system, i.e., the geometric relation between eye
position and eye velocity of fast phases (see RESULTS and DISCUSSION ), we estimated the average spatial orientation of fast phase
velocity by fitting a straight line in 3-D space to the pooled fast
phase velocity trajectories in each sector of nystagmus (see also
Hess and Angelaki 1997). This procedure ignored possible variations in the velocity profiles of individual fast phases and aimed
only at estimating the average direction in space of a given sample
of fast phases. The number of fast phases in each sample (i.e.,
pooled fast phases in each sector) varied typically between 5 and
20. The direction cosines of each fitted 3-D line formed a unit
vector, k ( i ) , one for each sector Si (i Å 1, . . . , 12). Based on
these vectors, the average tilt of fast phase velocity vectors was
compared with the respective tilt of the displacement plane of fast
phase positions for each sector (Fig. 4).
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TABLE
B.J.M. HESS AND D. E. ANGELAKI
1.
Global model parameters for constant-velocity data fits
Monkey gg
x shift
Amplitude a1
Amplitude a2
Ratio a2/a1, %
Phase a1
Phase a2
Offset a0
y slope
Amplitude b1
Amplitude b2
Ratio b2/b1, %
Phase b1
Phase b2
Offset b0
z slope
Amplitude c1
Amplitude c2
Ratio c2/c1, %
Phase g1
Phase g2
Offset c0
Thickness, 7
Monkey lb
Monkey ml
Monkey du
/1847/s
01847/s
/1847/s
01847/s
/1847/s
01847/s
/1847/s
01847/s
0.094
0.012
13
9.9
0152.5
0.007
0.097
0.0121
12.5
172.4
25.1
00.022
0.087
0.011
13
21.2
0146.7
00.010
0.091
0.013
14
172.5
012.1
00.0001
0.080
0.011
14
07.9
0156.0
0.013
0.082
0.002
2
189.5
0137.5
00.001
0.091
0.013
14
018.9
119.7
0.033
0.074
0.006
8
151.3
140.5
0.014
0.169
0.048
28.5
25.0
0161.4
0.026
0.149
0.033
22
0147.2
70.0
0.066
0.113
0.080
71
96.8
0117.5
00.096
0.030
0.028
93
0113.1
9.7
0.065
0.094
0.048
51
35.6
101.2
0.025
0.032
0.014
44
107.9
023.2
0.010
0.118
0.007
6
07.9
05.9
0.140
0.072
0.035
49
120.0
0178.5
0.094
0.319
0.103
32
111.9
04.5
00.027
2.0
0.391
0.051
13
73.2
034.0
00.126
1.5
0.320
0.016
5
88.5
25.8
00.091
2.1
0.320
0.027
8
79.2
80.7
00.158
2.2
0.292
0.0415
14
89.5
020.0
00.129
2.2
0.266
0.038
14
93.2
8.4
00.132
1.4
0.260
0.023
9
68.7
147.4
00.041
2.6
0.195
0.010
5
70.1
85.5
00.039
1.9
The coefficients a0, a1, a2, b0, b1, b2, c0, c1, and c2, are unitless numbers. The time-dependent shift and tilt of the associated displacement plane was
obtained by computing the qfunctions a(t), b(t), and c(t) according to Eq. 4. The shift along the x direction, d Å a(t)nx, (nx is the x component of direction
cosine) is obtained as a(t)/ 1 / b(t)2 / c(t)2. Similarly, the tilt relative to the y axis, ny (i.e., y component of the direction
cosine), and the tilt relative
q
to the z-axis, nz, (i.e., z component of direction cosine) are obtained by normalizing b(t) and c(t) with the factor 01/ 1 / b(t)2 / c(t)2 (see Eq. 2). The
associated modulation of primary position P(t) is obtained by the formula P(t) Å (Ptor, Pver, Phor) Å [a(t), c(t), 0b(t)] (see Eq. 5).
6) Statistical evaluation of best-fitted planes to eye positions.
For the local, sector-wise analyses, we estimated the uncertainty
boundaries of the fitted plane parameters, i.e., the unity vector n,
the scalar parameter d, and primary position P by a statistical
bootstrap method (see e.g., Efron and Tibshirani 1991; Press et al.
1992). For this purpose, the fit procedure for each local displacement plane (see Eq. 1) was applied 100 times on random samples
drawn with replacement from the observed set of values E (xi ) ,
E (yi ) , E (zi ) (i Å 1, rrr n, n Å sample size). The fitted plane parameters nx , ny , nz , and d derived from these 100 bootstrap samples
were used to compute means and standard deviations (see also
Hess and Angelaki 1997).
In addition, to estimate the ‘‘thickness’’ of the displacement
planes and to compare the upright versus the computed time- and
gravity-dependent displacement planes, we calculated: the torsional
variability (SD) of all VOR fast phases (expressed in upright
standard coordinates) relative to upright Listing’s plane; the torsional variability (SD) of VOR fast phases (expressed in upright
standard coordinates) relative to the best-fit plane to VOR fast
phase eye positions; and, finally, the torsional variability (SD) of
fast phase eye positions (expressed in the tilted coordinates xp ,
yp , zp ) relative to the respective instantaneous displacement plane
computed according to the global model presented above.
panion paper (Hess and Angelaki 1997). The corresponding
eye position vectors for 307 wide sectors centered around
prone, right ear-down, supine, and left ear-down positions
have been illustrated in Fig. 2, separately for VOR slow
and fast phases. Qualitative inspection of the eye positions
reveals clearly a correlation between the orientation of slow
and fast phase axes and head orientation relative to gravity.
For example, when the monkey rotated through supine position, slow phase eye positions, pooled from several rotation
cycles within a 307 window around supine position, were
tilted forward relative to head coordinates (Fig. 2, green
lines, side view). The corresponding fast phase position trajectories were also tilted forward by a similar amount. When
the monkey rotated through prone position, slow and fast
phase positions pooled from 307 windows around prone position were tilted in the opposite direction (Fig. 2, red lines,
side view). In contrast, slow and fast phase position trajectories were shifted in positive and negative torsional direction
when the monkey moved through left and right ear-down
positions, respectively (yellow and blue lines in Fig. 2, see
side and top views).
RESULTS
ORIENTATION OF DISPLACEMENT PLANES OF FAST AND SLOW
PHASE EYE POSITIONS. For a quantitative analysis of the ori-
Local displacement plane description of VOR during
constant-velocity rotation
Due to the simplicity of the stimulus, we start by describing the spatial orientation of 3-D eye position vectors during
constant-velocity yaw rotations. A typical record of horizontal, vertical, and torsional eye positions during such yaw
rotations at 1847 /s has been illustrated in Fig. 1 of the com-
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entation of eye positions as a function of head orientation
relative to gravity, we fitted planes to each of the 12 sectors
of slow and fast phase eye positions that were pooled from
at least five cycles in each VOR record. In doing so, we
presently have assumed that on average both fast and slow
phase eye position trajectories can be described adequately
in each sector by a planar surface (displacement plane) the
orientation of which changed relative to the head as a func-
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displacement planes can be also described as changes in the
components nx , ny , and nz of the unity plane vectors n (Fig.
3B, top 3 traces). Accordingly, the large tilts of the displacement plane vectors in the pitch plane corresponded to a
sinusoidal modulation of the nz component, which reached
peak positive and negative values near prone and supine
position, respectively. Similarly, the tilts of the displacement
plane vectors in the yaw plane corresponded to a modulation
of the ny component as a function of head position. In contrast, the modulation of the nx component was small, in
agreement with the almost symmetrical clustering of the
displacement plane vectors around the x axis. Along with
these changes in the orientation of the plane vectors, there
was also a systematic sinusoidal shift of the displacement
planes. This modulation, described by the parameter d, was
zero in supine and prone position and maximal in left and
right ear-down position (Fig. 3B, bottom).
FIG . 1. Definition of primary eye position at different head orientations
relative to gravity. A: pitch plane. When the head pitches down, the displacement plane of eye positions rotates upward through an angle a. As a
consequence, the vertical component of primary eye position decreases by
2a. B: roll plane. When the head rolls toward left ear-down, the displacement plane of eye positions shifts along the x axis through an angle b. As
a consequence, the torsional component of primary eye position increases
by 2b. Listing’s plane is assumed to coincide with the frontal plane (y-z
plane) with primary gaze direction, P, pointing along the x axis when the
head is upright.
tion of head position relative to gravity (the adequacy of
this assumption will be addressed below). The fit of a planar
surface to eye positions for each sector provided four quantities describing the orientation of slow and fast phase eye
positions: the three components nx , ny , nz of a displacement
plane unity vector, n, which was orthogonal to the best-fit
plane (Eq. 1), and a scalar parameter, d, describing the
orthogonal distance of this plane from the coordinate origin.
The uncertainty boundaries of the four fitted plane parameters (shown as {SD in Figs. 3 and 5) were computed by a
statistical bootstrap analysis as described in METHODS .
The spatial orientations of the displacement plane vectors
fitted to slow and fast phase trajectories for each 307 sector
of a single VOR record (30 cycles of steady-state response,
data as in Fig. 2) are plotted in Fig. 3. During one revolution
of the head in space, the displacement plane vectors rotated
in the pitch plane almost symmetrically around the y axis,
from pointing downward when the animal was in or close
to supine position to pointing upward when the animal was
in or close to prone position (Fig. 3A, top). Concurrently,
there was also a rotation of the displacement plane vectors
in the yaw plane (Fig. 3A, bottom). Due to the systematic
pitch and yaw tilts of the displacement planes, the plane
vectors fanned out in the frontal plane (Fig. 3A, middle).
These systematic changes in the spatial orientation of the
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FIG . 2. Spatial orientation of vestibulo-ocular reflex (VOR) slow and
fast phase positions as a function of head orientation relative to gravity.
Slow and fast phase eye positions of yaw VOR elicited by constant-velocity
rotation about an earth-horizontal axis (1847 /s) were collected by using
windows of {157 around each of the four cardinal head positions through
which the monkey cycled during the rotation: supine (green lines), left
ear-down (yellow lines), prone (red lines), and right-ear down position
(blue lines). Top to bottom row: side, front, and top view of eye position
vectors plotted in head-fixed x, y, z coordinates (see insets of monkey
heads).
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comparison shows that fast phase velocities tilted by an
amount equal to twice the angle of tilt of the displacement
plane of eye positions. For the remainder of RESULTS, we
will concentrate on a further analysis of eye position displacement planes as a function of head orientation in space.
GRAVITY-DEPENDENT
MODULATION
OF
PRIMARY
POSITION.
We have shown so far that the displacement planes of fast
phase positions shifted and rotated systematically relative to
standard head-fixed coordinates as the head changed orientation in space. These effects can be described equivalently
by a dynamic change in primary position (see METHODS ).
Primary position has been defined here as a function of the
spatial orientation of the displacement plane of eye positions
(described by the vector n) as well as the orthogonal shift
of the displacement plane (described by the scalar d; the
latter defines the ‘‘torsional’’ component of primary position; see Eq. 5). As in the upright position, the horizontal
and vertical components of primary position determine primary gaze direction, which is, therefore, only a function of
the spatial orientation of the displacement plane (vector n).
FIG . 3. Spatial orientation of displacement plane vectors fitted to fast
and slow phase positions of constant-velocity yaw VOR (1847 /s). Same
data record as in Fig. 2. A: side, front, and top views of displacement plane
vectors obtained by fitting a plane to pooled fast and slow phase eye positions in each of 12 consecutive 307 wide sectors into which each rotation
cycle of the data record was partitioned. B: temporal evolution of the
displacement plane vector components (nx , ny , nz ) and the scalar parameter
d (orthogonal shift of displacement plane) as a function of head orientation
relative to gravity. ●, mean values { SD obtained from sector-wise plane
fits;
, obtained by fitting a global model to VOR fast phases; sup,
supine; led, left ear-down; prone; red, right ear-down position.
SPATIAL ORIENTATION OF FAST PHASE VELOCITY VECTORS.
In the preceding paper, we have shown that the orientation
of fast phase velocity axes depends systematically on head
orientation relative to gravity. We then have put forward the
hypothesis that these systematic gravity-dependent changes
of fast phase velocity axes could be a direct consequence of
a similar systematic change of the oculomotor coordinates
in which the VOR operates (Hess and Angelaki 1997). The
systematic tilts in the average orientation of fast phase velocities and the change in the displacement plane of eye positions in each sector appear, indeed, to reflect the kinematic
restrictions that are characteristic for a Listing’s system (for
more explanations, see DISCUSSION ). As illustrated in Fig.
4, the tilt angle of the displacement planes in the pitch plane
(angle a ) and the average tilt angle of the fast phase velocity
vectors (angle g ) in each sector are correlated for each
stimulus velocity. The illustrated plots include average data
from four animals during rotation in both directions. This
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FIG . 4. Comparison between the tilt angle of displacement plane vectors
and fast phase velocity vectors in the pitch plane at 3 different velocities
(184, 110, and 587 /s). Tilt angle of displacement plane vectors (angle a,
● Å 2a ) was defined as the angle between the displacement plane (DP)
vector n for each sector and the x axis (see inset). Tilt angle of the fast
phase velocity vectors ( s, angle g ) was defined as the angle between the
average pooled fast phase velocity vectors, k, in each sector and the z axis.
Mean values { SD (average data from 4 animals pooled for both directions
of rotation).
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Separate evaluation of nystagmus slow and fast phases
revealed a tight correlation between the displacement planes
(and associated primary position) of slow and fast phase
eye positions. An example of this correlation in terms of the
equivalent description of displacement plane orientation by
primary position is shown in Fig. 5 for a vestibular nystagmus elicited during rotation at 1847 /s (left; same data as in
Figs. 2 and 3) and 01847 /s (right). Each of these records
comprised 30 cycles of VOR steady state responses. There
was no significant difference in primary eye position or displacement plane vector orientation for fast and slow phase
data (Fig. 5, s vs. ● ). Bootstrap analysis further indicated
a slightly larger scatter for the Phor component of primary
position, suggesting a larger variability in the yaw tilts of
the planes. Also, primary position for head positions close
to prone tended to exhibit larger variations.
Average modulation of primary position during yaw VOR
for all animals is illustrated in Fig. 6 for positive and negative
rotations at three different speeds. Because there was no
significant difference between the spatial orientation of fast
and slow phase positions, all eye position vectors were used
to compute primary position. As the head changed orientation in space, all three components of primary position
changed systematically. Whereas the torsional and vertical
components (Ptor and Pver ) exhibited a simple sinusoidal
modulation, the horizontal component (Phor ) exhibited a
2209
FIG . 6. Primary eye position as a function of head orientation in the
yaw plane. Primary position computed from plane fits to VOR fast and
slow phases. Mean values (4 animals) computed from responses elicited
by positive (left) and negative (right) yaw rotations at 3 different speeds
(58, 110, and 1847 /s).
more complex dependence on head orientation (Fig. 6, bottom). The torsional component of primary position was maximal in head orientations close to ear-down and nearly zero
in supine and prone positions. The opposite was true for
the vertical component of primary position: it was maximal
positive (gaze down) in supine, maximal negative (gaze up)
in prone position, and zero in ear-down positions.1 Data from
all animals were similar and demonstrated the following: 1)
the torsional component of primary position depended on
instantaneous head orientation but little on the speed of head
rotation and 2) the modulation of the vertical as well as
the horizontal components of primary position changed as
a function of the speed of head rotation: the faster the head
moved, the stronger was the modulation of the vertical and
horizontal components. Peak-to-peak modulation of the vertical component of primary position averaged 707 at 1847 /s
but only 207 at 587 /s.
Global fit of eye position displacement planes of VOR
during constant-velocity rotation
For an independent evaluation of the functional dependence of primary eye position on gravity, we also have used
an alternative approach by fitting a sinusoidally moving displacement plane described by Eqs. 3 and 4 to the nystagmus
(for details, see METHODS ). Similar to the sector-by-sector
analysis, it is hypothesized that all fast phase positions are
confined closely to a plane at any instant in time at a given
FIG .
5. Primary position computed separately from planar surface fits
to slow and fast phase positions ( ● and s, respectively). Mean values {
SD (from bootstrap analysis) computed from VOR responses in 1 animal
to positive (left) and negative (right) yaw rotation at 1847 /s about an earthhorizontal axis.
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1
Because the torsional component was zero in supine ( Å onset of each
rotation cycle) and maximal in ear-down positions, it changed polarity with
a change in the direction of rotation. In contrast, the vertical component
was maximal in supine and zero in ear-down positions (i.e., in phase quadrature to torsion), and therefore it did not change polarity with a change in
the direction of rotation.
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head orientation. Unlike the previously described analysis,
which was based on a local fit of a displacement plane for
all eye positions in a 307 sector, here we have allowed this
plane to vary continuously its orientation relative to the standard head-fixed coordinates throughout a head rotation cycle.
The instantaneous orientation of the eye position displacement plane was postulated to depend on head orientation in
space in a sinusoidal fashion, whereby both first and second
harmonic terms were included (Eq. 4). A 15-parameter
model described by Eqs. 3 and 4 therefore was fitted to all
fast phase positions collected from typically five or more
cycles of nystagmus. In contrast to the local, sector-wise
displacement plane analysis, we considered only VOR fast
phases for the global fits. An example of such a fit to the
VOR data shown in Fig. 2 has been plotted as solid lines in
Fig. 3B. As illustrated by comparing fits of the local, sectorwise model (Fig. 3B, ● ) and fits of the global model (Fig.
), results were similar with both analyses.
3B,
When primary position was computed from the global
displacement model fitted to constant-velocity data at 1847 /
s (according to Eq. 5), results for all four animals were
similar (Fig. 7, thin lines). The 15 parameters estimated for
each run in each of the four animals have been included in
Table 1. Examination of the fitted parameters suggests that
1) in all but one case, the second harmonic terms for the x
distance and z slope of the fitted displacement planes were
negligible ( õ15% of the fundamental). For the y slope,
FIG . 7. Modulation of primary eye position during constant-velocity
rotation in the yaw plane. Primary position was computed by fitting
a sinusoidally moving planar surface to all VOR fast phases obtained
from response to yaw rotations at /1847 /s (left) and 01847 /s (right).
Head position dependent modulation of primary position was modeled
as: Ptor (t) Å a0 / a1 sin [ u (t) / a1 ] / a2 sin [2u(t) / a2 ]; Pver (t) Å
c0 / c1 sin [ u(t) / g1 ] / c2 sin [2u(t) / g2 ]; and Phor (t) Å 0b0 0
b1 sin [ u(t) / b1 ] 0 b2 sin [2u(t) / b2 ]. Coefficients were obtained by
requiring that P(t) Å (Ptor , Pver , Phor ) describes at any instant of time the
best-fitting primary position of fast phase positions in the least squares
sense (see METHODS ). Results from 4 animals (thin lines); mean (thick
line).
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however, which corresponds to the horizontal component of
primary position (Phor ), the second harmonic dependence
on head position was large. In one animal (lb), the second
harmonic term was as large as that of the fundamental. 2)
The sinusoidal dependence of the x distance and z slope
(i.e., parameters a1 and g1 ), corresponding to Ptor and Pver ,
respectively, were Ç907 out of phase, as indicated by Figs.
6 and 7. 3) The static terms were generally small, except
coefficient c0 (z slope of intersection with x-z plane), which
describes the static tilt of displacement plane when animals
are in supine or prone positions.
EYE POSITION TRAJECTORIES EXPRESSED IN TILTED COORDINATES OF THE INSTANTANEOUSLY DEFINED PRIMARY POSITION. In all analyses presented here, the main and most
fundamental assumption we have made is that eye positions are confined locally to a planar surface, even though
this plane ( displacement plane ) changes its orientation relative to head-fixed coordinates as a function of the instantaneous orientation of the head relative to gravity. If this
hypothesis is true, then eye positions will indeed form a
planar surface when examined in the ‘‘moving’’, gravitydependent coordinate system defined by the instantaneous
displacement planes. To show this, we have recomputed
the eye position vectors relative to the instantaneously
varying primary position as estimated by the global model
fits. Expressed in these tilted coordinates ( in the following
denoted by xp , yp , and zp ) , 2 we are looking at the eye
positions from views that are parallel ( front view ) or perpendicular ( side and top views ) to the respective time- and
gravity-dependent displacement plane. Figure 8 illustrates
such an example for the data set of Figs. 2 and 3. In Fig.
8 B, the fast phase eye positions have been expressed in
standard head-fixed coordinates. In Fig. 8C, the fast phase
eye positions have been recomputed relative to the sinusoidally ‘‘moving’’ gravity-dependent coordinate system. For
completeness, the original VOR record ( thin lines ) ( see
also Fig. 1 in Hess and Angelaki 1997 ) with the modulated
primary position superimposed ( heavy lines ) has been also
included in Fig. 8 A. By comparing Fig. 8, C and B, it is
obvious that the torsional component of fast phase positions of the same VOR record was much reduced in amplitude and centered around the zp-yp plane ( xp Å 0 ) when
expressed in the gravity-dependent coordinates. In fact,
the torsional range of all fast phase eye positions rarely
exceeded a {2 – 2.57 range, which is similar to the range
scanned by fast phase position vectors during earth-vertical
axis rotations. The computed thickness of best-fit planes
for all VOR fast phases of the whole VOR records, after
eye positions have been re-expressed in the moving, gravity-related coordinates, have been included in the last row
of Table 1.
Displacement plane analysis of VOR during sinusoidal
oscillations
Due to the simple time dependency of head position relative to gravity, we have concentrated our analysis so far on
2
In the following, we refer to the components ( Etor )p , (Ever )p , (Ehor )p of
an eye position vector, E, expressed in coordinates xp , yp , and zp also as
the ‘‘torsional’’, ‘‘vertical’’, and ‘‘horizontal’’ component of E whenever
it is clear from the context which coordinate system is used.
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FIG . 8. Transformation of VOR fast phase positions from head-fixed into gravity-dependent
tilted coordinates. A: torsional (Etor ), vertical
(Ever ), and horizontal (Ehor ) eye movements elicited by constant-velocity yaw rotations about an
earth-horizontal axis in darkness (1847 /s). Modulation of primary position (heavy lines) superimposed on nystagmic eye movements (thin lines).
Note that torsional fast phases straddle mean torsional primary position. B: side view (top: Ehor
vs. Etor ), top view (middle: Ever vs. Etor ), and front
view (bottom: Ehor vs. Ever ) of spatial orientation
of fast phases in standard head-fixed coordinates.
C: side view [top: (Ehor )p vs. (Etor )p ], top view
[middle panel: (Ever )p vs. (Etor )p ], and front view
[bottom: (Ehor )p vs. (Ever )p ] of spatial orientation
of fast phases in gravity-dependent tilted coordinates. Modulated primary eye position describes
the moving origin of these tilted coordinates at
any instant of time. Listing’s plane is preserved
relative to gravity-dependent tilted coordinates.
Head position (H), potentiometer output reset every 3607.
VOR data elicited by constant-velocity rotations. Similar
results were obtained by analyzing VOR responses elicited
by sinusoidal head oscillations (both in light and complete
darkness). In this case, the angular head position changed
as a sinusoidal function, i.e., u(t) Å u0 sin Vt, rather than
as a linear function of time, as in the case of constantvelocity rotation. We have used the global displacement
plane model described above to fit sinusoidal data in four
rhesus monkeys. Because the second harmonic components
of the x distances and z slopes obtained with the 15-parameter model for constant-velocity rotation data have been
shown to be small, we have fitted a 11-parameter model to
the sinusoidal data (see METHODS ).
An example of a VOR record and a superimposed modulation of primary position as determined by the global fit model
during sinusoidal oscillations at 0.2 Hz, {457 has been plotted in Fig. 9. Because of the sinusoidally varying angular
head orientation, u(t) (Eq. 4), the fitted planes and the associated primary position varied as a function of head orientation in a strongly nonlinear fashion. The modulation of primary eye position during sinusoidal head oscillations was
qualitatively similar as during constant-velocity rotation: in
457 ear-down positions (zero-crossing of head velocity signal; Fig. 9A, bottom), modulation of torsional primary position was maximal whereas modulation of vertical primary
position was minimal. Conversely, modulation of vertical
primary position peaked in supine position (peak and trough
of velocity signal; Fig. 9A). Because the animal rotated
through supine position twice per rotation cycle, vertical
primary position modulated at twice the head oscillation
frequency. Horizontal primary position exhibited a more
complex modulation pattern primarily due to the nonlinear
(2nd harmonic) component of the model. Even though the
specific data illustrated in Fig. 9A were collected in complete
darkness, similar observations were made during oscillation
in the light. In contrast, no consistent change in the torsional
and vertical components of primary position were observed
during earth-vertical axis oscillations.
/ 9k1d$$oc35 J108-7
Similar to VOR responses during constant-velocity rotation (Figs. 2 and 8), all components of eye position were
scattered widely when expressed in the standard, head-fixed
coordinate system (Fig. 9B). In contrast, when the same eye
positions were expressed in the oscillating gravity-dependent
coordinate system, xp , yp , and zp fast phase axes were more
closely confined to a planar surface. In fact, for all three
frequencies tested, there was a significant decrease in the
standard deviation of torsional eye positions of the whole
VOR record when expressed in gravity-dependent coordinates compared with the respective standard deviation in
head-fixed coordinates relative to a planar surface fitted to
the VOR data or relative to Listing’s plane (as determined
in upright head position; Fig. 10). Moreover, this decrease
in the standard deviation of torsional eye positions was true
for earth-horizontal axis oscillations in both darkness and
light. In both cases, the standard deviation decreased to approximately the same values observed during similar oscillations about an earth-vertical axis.
DISCUSSION
We have examined the kinematic constraints of eye movements during earth-horizontal axis rotations of the head in
the yaw plane. Our results suggest that kinematic constraints
similar to those described by Listing’s law for the head
upright and stationary exist when the head changes dynamically its orientation relative to gravity. These constraints
consist in a significant reduction of the rotational degrees of
freedom of the eyes analogous to those described by Listing’s law yet relative to gravity-dependent rather than headfixed coordinates. Our finding implies that there always exists a unique direction in space about which ocular rotations
are minimal and that this direction changes as the head rotates relative to gravity. In the following, we first compare
our results with previous work and subsequently discuss the
possible implications of our findings.
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FIG . 9. Transformation of VOR fast phase
positions from head-fixed into gravity-dependent
tilted coordinates. A: torsional (Etor ), vertical
(Ever ), and horizontal (Ehor ) eye position elicited
by yaw oscillations about an earth-horizontal axis
in darkness (0.2 Hz { 457 ). Modulation of primary position (heavy lines) superimposed on
nystagmic eye movements (thin lines). B: spatial
orientation of fast phases in standard head-fixed
coordinates. C: spatial orientation of fast phases
in tilted coordinates. Modulated primary eye position describes the moving origin of these gravity-dependent tilted coordinates at any instant of
time. Listing’s plane is preserved relative to
gravity-dependent tilted coordinates. C and B
show the same views as in Fig. 8. Head velocity
(Hg ), tachometer output reset every 3607.
Listing’s plane and primary eye position in the head
upright and stationary condition
Instantaneous eye position vectors lie in displacement
planes
In its original version, Listing’s law has been stated for
visual fixations of distant targets with the head upright and
stationary (von Helmholtz 1867). Under these conditions,
the eye may assume only those orientations that can be
reached by rotations from a single reference position about
axes that are confined to a single plane (usually referred to
as the displacement plane) (see Tweed et al. 1990). In other
words, the rotational degrees of freedom of the eye are reduced such that there exists a unique direction, defined as
primary gaze direction, about which ocular rotations are
minimized. Taking the eye position associated with primary
gaze direction as reference, the axes of accessible eye rotations form a plane called Listing’s plane. It has been demonstrated only relatively recently that saccade trajectories and
smooth pursuit eye movements also follow Listing’s law
with an accuracy of Ç17 (Haslwanter et al. 1991; Minken
et al. 1993; Straumann et al. 1996; Tweed and Vilis 1990;
Tweed et al. 1992).
Primary eye position and Listing’s plane usually have
been considered to be head-fixed by ignoring the fact that
the displacement planes change orientation as a function of
static head position relative to gravity (Crawford and Vilis
1991; Haslwanter et al. 1992). This general belief was
grounded on the fact that the noted gravity-dependent
changes, albeit consistent, are usually small and limited to
within 10% of static head tilts. Furthermore, it has been
assumed tacitly that eye movements are confined only to
displacement planes when the head is stationary in space.
We have challenged these assumptions by analyzing the geometric constraints of eye positions and velocity under dynamic conditions when the head is passively rotated or oscillated relative to gravity. Even though most of our analysis
has been confined to data acquired during constant-velocity
yaw rotations, similar results were obtained with sinusoidal
oscillations, either in complete darkness or in the light.
The most important aspect of the present analysis is
the question of whether eye positions during passive head
rotations can be described locally by a planar surface.
There are two issues related to this question: First, do
eye positions always follow Donders’ law if expressed in
appropriate coordinates, i.e., are they always confined to
a single surface? Second, do eye positions follow Listing’s law in any head orientation relative to gravity? It
is obvious that Donders’ law is violated if one considers
eye positions in all possible head orientations. Our results
indicate, however, that there exists a set of continuous
gravity-dependent oculomotor coordinates relative to
which all possible eye positions ‘‘disentangle’’ and become arranged in a two-dimensional surface ( see Figs.
8C and 9C ) . Thus the oculomotor system seems to implement Donders’ law as a unique function of head orientation relative to gravity. Moreover, Donders’ surface
becomes flat in these coordinates, i.e., Listing’s law is
also valid with an accuracy of Ç27 [ see Table 1, 2.0 {
( SD ) 0.47 ] .
We have checked the adopted analytic procedure in three
different ways: first, to better illustrate the thickness of the
traced surfaces, i.e., the torsional variability, eye positions
have been expressed in the local primary coordinates ( see
examples in Figs. 8 and 9 ) . Second, we have used a statistical bootstrap method to estimate the variability of the fitted
parameters of the displacement planes and the associated
primary eye position ( SDs in Fig. 3 B and 5 ) . Third, we
have formulated a model that puts forward a mathematical
description of the gravity-dependent shift and rotation of
displacement planes. Based on the model, we have compared primary eye position computed from local plane fits
with that obtained from a globally fitted plane.
To further judge the adequacy of our approach, it is important to distinguish between fast and slow phase eye posi-
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FIG . 10. Standard deviations (in degrees) of fast phase positions of
sinusoidal yaw VOR from zero torsion in head-fixed and gravity-dependent
tilted coordinates. Top to bottom: standard deviations of VOR elicited by
head oscillations about an earth-horizontal axis (EHA, in light or darkness)
or about an earth-vertical axis (EVA, in darkness) at 0.1 Hz { 907 (top),
0.2 Hz { 457 (middle), and 0.5 Hz { 187 (bottom). Zero torsion was
defined 1) by Listing’s plane fitted to spontaneous eye movements in the
light with the head upright and stationary ( Ω ), 2) statically (i.e., in headfixed coordinates) by fitting a plane to VOR fast phase positions ( … ), and
3) dynamically (i.e., in gravity-dependent tilted coordinates) by fitting a
moving displacement plane to VOR fast phase positions ( h ).
tions. Fast phases of VOR elicited by yaw rotation in upright
position follow Listing’s law with a standard deviation of
Ç1–27 (e.g., Fig. 10). The increase in standard deviation
of Listing’s plane during VOR compared to visual fixations
with a stationary head can be explained in part by the fact
that fast phases correct the systematic deviations of slow
phases relative to Listing’s plane as observed also by Crawford and Vilis (1991). A systematic study of the deviations
of yaw VOR slow phases from Listing’s plane in humans
suggested that slow phase kinematics follows a ‘‘half Listing’s strategy’’ (Misslisch et al. 1994). The different kinematic behavior of VOR slow and fast phase positions is
correlated with the different functional requirements of the
two systems. Although the axis orientation of VOR slow
phases should be determined by the imposed head rotation
independently of current eye position, axis orientation of
fast phases must change as a function of the radial distance
from primary position to be in accordance with Listing’s
law (von Helmholtz 1867).
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2213
In our locally fitted displacement plane analysis of constant-velocity VOR, both slow and fast phase eye positions
ultimately were analyzed together to estimate the best-fit
displacement planes and the modulation of primary position
as a function of head orientation (Fig. 6). This approach
was adopted for the following reasons. First, a separate analysis of slow and fast phase position trajectories demonstrated
that the displacement planes, as well as primary position,
estimated from each of them separately were indistinguishable (Fig. 5). A closer inspection of the spatial orientation
of eye position revealed that, depending on head position,
slow phases exhibited a systematic component oriented
across the best-fit plane to fast phase positions. For example,
in prone position, slow phase components were oriented
parallel to the head rotation axis (approximately z axis, see
coordinate system in METHODS ) but fast phase components
were not (see Fig. 2, fast and slow phases in prone position,
side view). Nevertheless, the best-fit planes to either slow
or fast phase position were tilted backward because the fast
phases of nystagmus ‘‘displace’’ the eye along such a plane.
Therefore, fast and slow phase positions can be described
on average by the same plane.3 This plane, however, does
not necessarily represent the directions of the slow phase
displacements. In ear-down positions, for example, the bestfit planes to slow and fast phase positions represent at the
same time the true displacement planes of the two movement
phases (see Fig. 2, fast and slow phase positions in yellow
and blue, side view). The local displacement plane analysis
(which included both slow and fast phases) was cross-examined by fitting a gravity-dependent moving displacement
plane to the whole nystagmus record. In this global analysis,
we analyzed only the fast phases of nystagmus because of
the more complex stimulus-induced behavior of VOR slow
phases. Both the local and the global methods yielded closely
corresponding results.
Notion of primary eye position extended to dynamic head
motion in space
Primary eye position, which is defined on the basis of
Listing’s law, specifies an orthogonal coordinate system the
origin of which coincides with zero ocular rotation (i.e.,
rotation vector P Å 0) and the x axis of which is aligned
with the direction of primary gaze. In this coordinate system,
the standard Listing’s plane of eye positions is centered
around the coordinate plane x Å 0, implying that eye movements in Listing’s plane are constrained to zero torsion with
an accuracy of õ17 for fixations. The precise orientation of
this particular reference frame, for example, with respect to
stereotaxic coordinates, depends on head position relative to
gravity as previously demonstrated for different static head
roll and pitch positions (Haslwanter et al. 1992). Because
the existence of a primary position (defined for head upright
and stationary) solely depends on the fact that eye positions
are confined to a single planar surface, it is natural to extend
the notion of primary position not only to different static
but also to dynamically changing tilted head positions. This
3
This observation should not be surprising: the slow phases need on
average to follow similar trajectories as the fast phases to limit accumulation
of torsion during the head movement.
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more general notion of primary position makes sense if eye
positions are found to be confined to planar surfaces independently of whether the head moves or not. As shown in this
study, such local planes, which we will call (instantaneous)
displacement planes, can be described indeed. They move
relative to the head, whereas the head rotates in space,
thereby allowing for a dynamic definition of primary eye
position.
Modulation of primary eye position as a function of head
orientation in space
According to the notion of a dynamically changing primary position, the modulation of mean torsional and vertical eye position during yaw rotation about an earth-horizontal axis ( Fig. 1 in Hess and Angelaki 1997; see also
Angelaki and Hess 1996 ) reflects a systematic modulation
of primary eye position as the head changes its orientation
in space ( e.g., Fig. 6 ) . Although much more variability is
present in the modulation of the horizontal component,
torsional and vertical primary eye position consistently
modulate in phase with head position, independently of
the direction and speed of head rotation, as previously also
shown for the respective modulations of mean torsional
and vertical eye positions ( Angelaki and Hess 1996 ) . The
modulation of primary eye position appears to shift the
plane of desired eye positions in compensatory direction
with respect to the orientation of the head relative to gravity. More specifically, the torsional component of primary
eye position is maximal around left and right ear-down
positions. Similarly, the vertical component of primary
position is maximal around supine and prone orientations.
The same is true during sinusoidal oscillations about an
off-vertical axis ( compare Figs. 8 and 9 ) .
Interestingly, it is the vertical component of primary position that exhibited the largest modulation at the highest
frequency ( 0.5 Hz corresponding to rotation at 1807 / s ) .
This modulation decreased with speed, reaching minimum
values at 0.16 Hz ( corresponding to rotation at 587 / s ) .
Even smaller changes in the vertical component of primary
position are observed under static conditions ( estimated as
double the angle of tilt of the fitted plane of eye positions
during static pitch tilts ) ( e.g., see Haslwanter et al. 1992 ) .
This marked frequency dependence of the vertical component of primary eye position suggests that orientation toward a space-fixed vertical reference is of crucial importance only at higher speeds of motion, probably for purposes of proper spatial orientation, which no longer can be
maintained by the visual system alone. Because of the foveal organization of the primate visual system, there is no
point in controlling the gain of the torsional component as
a function of motion frequency.
In addition to these systematic modulations of the torsional and vertical components of primary eye position, a
consistent (albeit more variable) modulation of the horizontal component was also observed during yaw rotations,
particularly at the highest speeds. That is, as the head
changed orientation from ear-down to prone/supine positions, primary eye position not only rotated about the x and
y axes but also about the z axis. This horizontal modulation
appeared not to be a simple sinusoidal function of the oscil-
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lating gravity component along the interaural axis. A second
harmonic term in the global displacement model gave a reasonable correspondence between results of the local and
global displacement plane fits. The significance of this finding is not clear, but it could mean that there is more than
one mechanism that determines the horizontal orientation of
Listing’s plane. One of these mechanisms could be horizontal vergence, which is associated with a disconjugate
rotation of the Listing’s planes of the two eyes (Mok et al.
1992).
Implications of a dynamically changing primary position
for head-free gaze shifts
Our results strongly suggest that kinematic constraints
similar to Listing’s law are effective during passive motions
of the head relative to gravity. This implies that VOR fast
phases as well as saccades in general follow Listing’s law
not only in upright position with the head stationary, but in
an analogous way also when the head moves in space. Such
a conclusion is further supported by the observation that
both primary position and the associated displacement (also
called velocity) plane move as a function of gravity in a
manner predicted by Listing’s law (Fig. 4). The observed
relation between gravity-dependent changes in primary position and associated displacement plane is analogous to a
criterion for Listing’s law, first derived by von Helmholtz
(1867), which states that when the eye is in a position, E,
then the associated eye velocity vectors, V, must lie in a
plane. For primary position, the associated displacement
plane is Listing’s plane. For any other position, the associated displacement plane is tilted out of Listing’s plane by
an angle that corresponds to half its radial distance from
primary position. Thus when primary position changes its
orientation relative to the head as a function of gravity, fast
phase velocity vectors also must change their orientation
accordingly. As described in the companion paper (Hess
and Angelaki 1997) and as shown in Fig. 4, this was indeed
the case.
Eye-in-head positions after large head-free gaze shifts
have been reported to violate Listing’s law due to an increased twist of the fitted surfaces (Glenn and Vilis 1992).
However, Radau et al. (1994) have pointed out that the
increased variability of eye positions relative to a planar
surface fit (average 1st-order SD: 2.67; 2nd-order SD: 2.27 )
could be explained by the simultaneous change in head orientation relative to gravity that was needed to acquire
oblique targets. Listing’s law therefore still seems to be
obeyed, taking into account that the eye position planes
change as a function of gravity. Our finding further suggests
that primary position might not stay invariant relative to the
head during large head-free gaze shifts but could change
continuously as a function of head position relative to gravity. Moreover, this modulation of primary position is expected to depart, at any point in time, from the static orientation during a head movement trajectory. Interestingly,
Tweed et al. (1995) recently have shown that trajectories
of the eye relative to the head during large head-free gaze
shifts toward oblique targets were curved strongly and departed also from the static surface. Furthermore, oblique
saccades, which involved large changes of head orientation
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3-D VOR KINEMATICS II
relative to gravity, exhibited consistent looping in their trajectories, even in torsional direction. Although these findings
show that eye positions in head during head-free gaze saccades generally break Listing’s law, it remains an open question whether such saccades still follow dynamic constraints
correlated with the fast change in head orientation relative
to gravity. The head-free gaze shifts studied by Tweed, Vilis,
and colleagues all involve fast changes not only in yaw but
also in head pitch and roll orientation of up to {307 (Glenn
and Vilis 1992; Radau et al. 1994, Tweed et al. 1995). It
remains to be shown what role a dynamic modulation of
primary position may play during such active head-free gaze
shifts.
APPENDIX
Let us consider two sets of eye position vectors, one set of
positions, E, which describe rotations of the eye in Listing’s plane
expressed in the standard coordinates x, y, z (i.e., Ex Å 0) and
another set of positions, E *, expressed in the same coordinate
system that describes eye rotations in a different plane, tilted relative to Listing’s plane by a certain angle (displacement plane).
This latter plane of eye positions E * will in general be displaced
relative to the origin of the coordinate system (generalized displacement plane). The geometric relation between these two
planes, the Listing’s plane of positions E and the generalized displacement plane of positions E * can be described by a single rotation vector P as follows
1
E* Å P 7 E Å
1 0 bEy 0 cEz
S
a / bEz 0 cEy
b / Ey 0 aEz
c / Ez / aEy
D
with
PÅ
SD
a
b
c
(A1)
where 7 denotes multiplication of rotation vectors (see e.g., Hepp
1990), Ex , Ey , and Ez are the components of the eye position vector
E (in Listing’s plane, i.e., Ex Å 0), and a, b, and c are the unknown
x, y, and z components of the rotation vector P.
The orientation of the plane of eye positions E * can be determined from the plane equation: E *r n Å E x* nx / E y* ny / E z* nz Å
d in terms of the three components nx , ny , nz of a unity vector, n,
oriented orthogonal to the plane, and of the orthogonal distance,
d, of the plane from the origin ( r denotes the vector dot product).
This was done experimentally by fitting a plane to the eye positions
E * in each sector (see METHODS ). Once the orientation of this plane
has been determined (i.e., the parameters n and d are known), the
components of the unknown rotation vector P can be computed by
solving the plane equation: E *r n Å (P 7 E) r n Å d but now in
terms of the components a, b, and c of the vector P (using Eq. A1).
In doing so, it suffices to consider the two linearly independent set
of vectors Eu Å (0, Ey , 0) and E£ Å (0, 0, Ez ) and the origin
E0 Å (0, 0, 0). Using these constraints, one obtains three linear
equations of the three unknowns a, b, and c that can be written as
a matrix equation
S
n1 n2 n3
0n2 n1
d
n3
d 0n1
with the solution vector
PÅ
DS D S D
S D
1
nx
a
b
c
d
0nz
ny
/ 9k1d$$oc35 J108-7
Å
d
0n2
0n3
(A2)
(A3)
2215
which represents the rotation vector P as a function of the components of the plane vector n and the orthogonal distance d.
The rotation vector P determined from the displacement planes
in tilted head positions has the properties of primary position. In
this context, primary eye position 4 can be defined more generally
as the unique rotation vector that transforms eye positions that form
an arbitrarily oriented plane relative to some (arbitrary) coordinate
system into a more natural, canonical coordinate system in which
the same plane of eye positions aligns with one of the coordinate
planes. It should be noticed that P has nonzero components when
expressed in noncanonical (arbitrary) coordinates, whereas its
components become all zero when expressed in the associated
canonical coordinates. Thus for tilted head positions, these tilted
or canonical coordinates play the same role as the standard coordinates defined for upright head position (when the head is stationary).
The transformation of eye positions E * from the (upright) standard coordinates into their tilted coordinates requires two computational steps (see also Tweed et al. 1990): recomputation of the
eye position vectors E * relative to the new primary position P and
conjugation of the recomputed eye positions with primary position.
By applying these two operations on the eye positions E *, we have
E Å P 01 7 (E * 7 P 01 ) 7 P Å P 01 7 E * , which is the inverse of Eq.
A1 obtained by left multiplication with the rotation vector P 01 .
Thus left multiplication with the inverse of primary position P 01
(inverse of Eq. A3) converts the general displacement plane of
eye positions E * into tilted coordinates (denoted here by xp , yp ,
zp ) such that it aligns with the coordinate plane xp Å 0.
The authors thank K. Hepp for many stimulating discussions.
This work was supported by grants 31-32484.91 from the Swiss National
Science Foundation, EY-10851 from the National Eye Institute, NAGW4377 from the National Aeronautics and Space Administration, F49620
from the Air Force Office of Scientific Research, and 95-25 from the Roche
Research Foundation.
Address for reprint requests: B.J.M. Hess, Dept. of Neurology, University
Hospital Zürich, Frauenklinikstrasse 26, CH-8091, Zurich, Switzerland.
Received 4 February 1997; accepted in final form 18 June 1997.
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