Department of Neurology and Center for Sensorimotor Research, Klinikum Grosshadern, Ludwig-Maximilians University, 81377 Munich, Germany
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ABSTRACT |
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Glasauer, S., M. Dieterich, and Th. Brandt. Central Positional Nystagmus Simulated by a Mathematical Ocular Motor Model of Otolith-Dependent Modification of Listing's Plane. J. Neurophysiol. 86: 1546-1554, 2001. To find an explanation of the mechanisms of central positional nystagmus in neurological patients with posterior fossa lesions, we developed a three-dimensional (3-D) mathematical model to simulate head position-dependent changes in eye position control relative to gravity. This required a model implementation of saccadic burst generation, of the neural velocity to eye position integrator, which includes the experimentally demonstrated leakage in the torsional component, and of otolith-dependent neural control of Listing's plane. The validity of the model was first tested by simulating saccadic eye movements in different head positions. Then the model was used to simulate central positional nystagmus in off-vertical head positions. The model simulated lesions of assumed otolith inputs to the burst generator or the neural integrator, both of which resulted in different types of torsional-vertical nystagmus that only occurred during head tilt in roll plane. The model data qualitatively fit clinical observations of central positional nystagmus. Quantitative comparison with patient data were not possible, since no 3-D analyses of eye movements in various head positions have been reported in the literature on patients with positional nystagmus. The present model, prompted by an open clinical question, proposes a new hypothesis about the generation of pathological nystagmus and about neural control of Listing's plane.
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INTRODUCTION |
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When the head is brought into an
off-vertical, lateral, or head-hanging position, there is a change in
graviceptive (otolithic) input. This change is the factor that
precipitates pathological central positional nystagmus. There are
different forms of positional nystagmus with and without vertigo, but
all are due to posterior fossa lesions. They include downbeating
nystagmus in head-hanging position and various types of nystagmus with
torsional, vertical, and horizontal directions. Central positional
nystagmus can be clinically differentiated from peripheral vestibular
positional nystagmus by the presence of a persisting positional
nystagmus (slow-phase velocity >5°/s) without associated vertigo, or
positional nystagmus with pure torsional, downbeat, or upbeat
directions (Baloh 1998; Barber 1984
;
Büttner et al. 1998
; Fisher et al. 1983
; Katsarkas 1981
; Thomsen et al.
1978
). Recently, Bisdorff et al. (2000)
reported
that moderate positional nystagmus in the dark (slow-phase velocity
<10°/s) is a common phenomenon in normal subjects. No current
concept can conclusively explain which ocular motor pathways are
involved or their underlying mechanisms. Involvement of the loop
between the vestibular nuclei and midline archicerebellar structures of
the vermis has been suspected to cause central positional nystagmus
and/or vertigo. Mathematical modeling of the neuronal circuitry that
controls eye movements may be suitable for simulating the occurrence of
central positional nystagmus due to distinct pathway lesions. This
could also help us understand the pathomechanism of positional
nystagmus in patients.
The temporal characteristics of a persisting central positional
nystagmus indicate that otolith function must be involved. A
semicircular canal-induced nystagmus would fade away depending on the
plane-specific central time constants (velocity storage mechanism). Two
aspects of the otolithic influence on eye position are essential to
understand positional nystagmus. First, when the head is tilted
relative to gravity in the frontal roll plane, the torsional component
of eye position changes (ocular counterroll). Second, when the head is
bent forward or backward relative to gravity in the sagittal pitch
plane, the orientation of Listing's plane changes. The latter refers
to ocular torsion that occurs physiologically during lateral gaze. The
quantitative relationship has been shown in monkeys (Haslwanter
et al. 1992) and humans (Haustein 1988
).
Knowledge of vestibular and ocular motor structures and the
quantitative relationship between gravitational input and eye position
allow a mathematical simulation of this biological sensorimotor system.
A three-dimensional mathematical model has been developed
(Glasauer et al. 1998
, 1999
) for the
control of static eye position by utricular input only. Such a static
model is able to simulate pathological eye positions in patients with lesions of the vestibuloocular reflex pathways from the utricles to the
ocular motor nuclei, but it is insufficient for simulating positional
nystagmus. To incorporate the dynamic changes in eye position, a model
is needed that includes a system for generation of rapid eye movements
(saccades) and a velocity-to-position neural integrator operating in
Listing's plane. The latter is required for simulation of static and
dynamic eye positions three-dimensionally (3-D). These two elements
have been demonstrated, quantitatively analyzed, and localized in
animal experiments and studies in humans (Crawford 1994
;
Crawford and Vilis 1992
; Fukushima and Kaneko 1995
). Furthermore, the model has to implement the
otolith-induced changes of Listing's plane.
After reviewing previous work and experimental evidence for the assumptions on which our mathematical model was built, we validated the model by simulating saccades in various head positions. It was then used to test whether distinct lesions of single model elements result in a positional nystagmus similar to that seen in neurological patients. These lesions should cause nystagmus only when the head is brought into an off-vertical position but not when it is in normal upright position.
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BACKGROUND |
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The literature provides a wide spectrum of models of saccade
generation. They range from the classic one-dimensional Robinson model
(Robinson 1975) to detailed modeling of the interplay of different classes of burst neurons (Scudder 1988
) or
fine-tuning of the saccade velocity profile by cerebellar control
(Quaia et al. 1999
). Triggered by the introduction of
3-D eye movement measurement techniques such as scleral search coils,
different models of 3-D saccade generation have been proposed (for
example, Crawford and Guitton 1997
; Quaia and
Optican 1998
; Schnabolk and Raphan 1994
; Tweed 1997a
,b
; Tweed and Vilis 1987
).
Listing's law
One aspect of these models is the implementation of Listing's
law, which was discovered more than a century ago and described by
Helmholtz (1867): "There exists a certain eye position
from which the eye may reach any other position of fixation by a
rotation around an axis perpendicular to the visual axis."
If eye position is represented with respect to this "certain eye
position," called primary position, then Listing's law implies that
eye position (expressed as rotation vectors or quaternions) is
constrained to zero torsion. This constraint has been shown to exist
not only for fixations, but also for smooth pursuit and during
saccades, although to a lesser extent (Straumann et al.
1996
). To achieve such a constraint, recent models of saccade
generation assume that the burst that drives the saccade is coded in
Listing's coordinates, and that the subsequent neural integration that
generates the motor command to hold the eye at the target also operates
in Listing's coordinates. Experimentally, this assumption is supported
by lesion studies in the monkey interstitial nucleus of Cajal
(Crawford 1994
), which have shown that the physiological
neural integrator coordinates are aligned with Listing's plane.
Listing operator
The correct coding of the burst command is assumed to be achieved
by a so-called Listing operator, which transforms the desired direction
of gaze, i.e., the direction of the eye in space, into the desired eye
position that obeys Listing's law, thus restricting torsion to zero.
This occurs prior to the action of the burst generator (Crawford
and Guitton 1997; Quaia and Optican 1998
; Tweed 1997a
,b
; Tweed and Vilis 1990
).
This mechanism would suffice, if Listing's plane were head-fixed under
all conditions, which is not the case. Otolith afferents modify
Listing's plane, although to a small extent. Therefore, if
off-vertical head positions become involved, the Listing operator must
receive otolith input to achieve its task. The saccadic burst is then
generated in the new coordinate system, carrying the necessary amount
of torsion. To model combined eye-head saccades, Tweed
(1997b)
has taken a similar approach of supplying additional
input to the Listing operator.
Holding static ocular torsion
The thus defined operator guarantees correct gaze shifts but is
only sufficient for maintaining the new eye position if the neural
integrator is perfect, i.e., if it holds the eye in any torsional
position without drift. Even though all previous models assumed such a
perfect integration, this is not the case, as Seidman et al.
(1995) demonstrated: the human torsional integrator leaks with
a time constant of about 2 s, while the horizontal integration has
a time constant of 20 s or more (Kramer et al.
1995
). Hence, as will be shown in the following, postsaccadic
torsional drifts would be expected to occur after saccadic eye
movements in off-vertical head positions, which do not actually occur.
For example, when rolling the head to the side, ocular counterroll is
maintained without any drift. Therefore a second mechanism is required:
the torsional zero-position of the integrator has to be adjusted so that the motor command for holding torsion is maintained despite the
leaky integrator. This evidently requires that otolith afferent information is sent either to the neural integrator or to the ocular
motor nuclei. The latter has been excluded experimentally by showing
that ocular counterroll is abolished after completely lesioning the
torsional-vertical neural integrator (Crawford and Vilis
1999
).
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MODEL |
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The model is described in three parts: 1) Listing's
law, 2) the saccadic burst generator, and 3) the
neural velocity-to-position integrator (see Fig.
1). The angular vestibuloocular reflex is of no importance for the following simulations, which are performed only for fixed body positions. All internal model calculations are
implemented using rotation vectors (Haustein 1989).
Lesions of assumed pathways transporting otolith information are
simulated by decreasing the respective inputs to zero (in case of
excitatory input) or by increasing the gain of the input (in case of
inhibitory cerebellar input). A detailed mathematical account of the
model is given in the APPENDIX.
|
Listing's law
Listing's law, i.e., the linear relationship between torsional
and horizontal or vertical eye position, is implemented in two parts.
The baseline tilt of Listing's plane, i.e., the tilt of the internal
neural coordinate system with respect to head coordinates, is
implemented in the eye plant model (Crawford and Guitton
1997; Quaia and Optican 1998
; Raphan
1997
; Tweed et al. 1994
). The otolith-induced
changes obviously require a neural implementation. This neural
implementation is assumed to be located within or prior to the saccadic
burst generator (Listing operator) and in the neural integrator. The
Listing operator also includes the conversion from retinal error to
desired eye position. It is modeled after Crawford and Guitton
(1997)
with the following change: the computation of desired
eye position receives otolith input to achieve a torsional component
within Listing's plane.
Saccadic burst generator
It is assumed that the mechanisms of saccade generation and quick phases of nystagmus are identical. Both are elicited when retinal error exceeds a certain threshold (0.5°) or when the current eye position is too far (1.5°) from Listing's plane. To achieve more realistic eye position time courses, temporal restrictions have been implemented. Retinal error, which is used to trigger a saccadic burst, is delayed by 150 ms to account for visual processing times. An "omnipause" signal generated in this module controls burst duration. An additional time delay within the burst generator prevents saccade generation within a temporal window (100 ms) once the previous saccade or quick phase is finished, as indicated by the omnipause signal. Subsequently, the burst is sent to the neural integrator (indirect pathway) and to the motoneurons (direct pathway).
Neural velocity-to-position integrator
The saccadic burst is integrated to a 3-D eye position signal that
is sent to the motoneurons. The torsional component of the integrator
is assumed to leak with a time constant of 2 s (Seidman et
al. 1995). In head positions other than upright, i.e., when
Listing's plane is shifted (static ocular counterroll) or tilted,
torsion can no longer be generated by the eye plant alone but requires
information from the otoliths about head position relative to gravity.
To be able to hold torsional eye positions despite integrator leakage,
the otolith input modifies the torsional zero position and the feedback
gain matrix of the integrator, which is responsible for the integrator
leakage. Otolith input consists of three parts: 1) additive
superposition of an otolithic torsional offset, 2)
multiplicative modification of torsional zero position, i.e.,
adjustment of the tilt of Listing's plane, depending on horizontal eye
position and head tilt in pitch, and 3) multiplicative
modification of the feedback gain matrix. The first is assumed to be
achieved via direct pathways from the vestibular nuclei via the medial
longitudinal fasciculus (MLF). Lesions of these direct pathways cause
deviations of the static eye position, such as ocular counterroll and
skew deviation. The second and third parts may be caused by cerebellar
input to the neural integrator (see Fig. 1).
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SIMULATIONS |
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Model validation
To show that accurate saccades without postsaccadic drifts or
nystagmus can be simulated by the model, a paradigm devised to examine
patients (Straumann et al. 2000) was used. Eight targets arranged on a square, four at ±20° and four at ±28.3°, had to be
fixated consecutively, starting from a center position. Each fixation
period lasted 2.5 s. The paradigm was simulated for upright and
off-vertical head positions. Figure 2
shows the expected shift and tilt of Listing's plane in response to
head roll and pitch, respectively.
|
To examine the effect of otolith input to the neural integrator, two versions of the model were tested: one included the proposed otolith input to the neural integrator, and one lacked this input. As an example of the performance without any otolith input to the integrator (MLF and proposed cerebellar pathways) but with preserved input to Listing's law operator, Fig. 3 shows simulations for saccades performed with the head in the right-ear-down position. Both model versions generated accurate saccades in the whole range of eye movements. However, the version without otolith input to the neural integrator produced torsional drifts and positional nystagmus as shown in Fig. 3, B and D; it thus fulfills the criteria for central positional nystagmus. Clinically, however, it is unlikely that the entire otolith input to the neural integrator, probably conveyed by separate pathways via the MLF and the cerebellum, would be missing. Note that this is not equivalent to bilateral vestibular failure, since the otolith input to implementation of Listing's law prior to the burst generator is preserved under the tested condition. Figure 3, A and C, shows that the otolith input supplied to the neural integrator effectively ensures gaze holding after saccadic eye movements. Mathematically, this is proven by the fact that the steady-state solution of the neural integrator yields a torsional component that is equivalent to the torsional component of the desired gaze direction fed into the burst generator.
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Simulation of central positional nystagmus
All simulations were performed with gaze straight ahead. Two basic types of pathological otolith input were considered: input to the neural integrator and input prior to the burst generator.
Altered otolith input to the integrator
When the head is bent forward or backward while looking straight ahead, eye positions in the model, expressed in internal coordinates (Listing's coordinates), have no torsional or horizontal components. Due to the assumed baseline tilt of Listing's plane, a vertical eye position command is necessary to maintain gaze straight ahead. Otolith input with the head bent forward or backward has no roll component. Consequently, if only the torsional component of the integrator is assumed to be leaky, bending the head forward or backward does not produce central positional nystagmus. Note, however, that in this situation small deviations from the gaze straight ahead position may evoke positional nystagmus.
During head roll, eye position has a torsional offset (ocular counterroll). Thus modification of otolith inputs to the integrator will cause torsional and/or vertical central positional nystagmus with head roll. In the model we assume two pathways that convey otolith input to the integrator: one via the MLF, mediating static eye position control, and one through the vestibulocerebellum (vermis, fastigial nucleus, etc.), mediating dynamic adjustments. To simulate positional nystagmus, we applied interventions only to the latter. These included complete disruption of inhibitory input, which functionally resulted in an abnormal gain increase of the direct static otolith input through the MLF.
In the model, otolith-dependent inputs to the neural integrator were modified either 1) at the level of the transformation matrix T(u) (see APPENDIX), which is subsequently applied to the direct otolith input supplied via the MLF, 2) for an eye position-dependent normative factor, again applied to the direct otolith input, or by 3) a combination of both (see Fig. 1, lesions A, B, and C). These modifications resulted in different patterns of positional nystagmus with head tilt in roll plane (Fig. 4, A-C). In Fig. 4A a predominantly vertical-torsional nystagmus occurs, in B a torsional nystagmus with a very small horizontal component can be seen, while C produces a predominantly vertical nystagmus. The vertical component of the nystagmus is generated by the erroneous transformation of otolith input by the lesioned transformation matrix. The subdivision of different types of cerebellar input (otolith-dependent feedback modification vs. otolith and eye position-dependent factor) is hypothetical and has to be tested experimentally.
|
The nystagmus frequencies and amplitudes shown in the simulations are determined by the properties of the burst generator (temporal delays, thresholds) and by the velocity of the slow-phase drift, which is induced by the amount of the lesion (gain factor applied in the model, see APPENDIX for an example). The torsional component of nystagmus decays exponentially with the time constant of the integrator (2 s). This, however, is not visible in Fig. 4 due to the short intervals between fast phases. The fast phase of the nystagmus always beats toward the normal (unlesioned), straight-ahead eye position.
Altered otolith input to the implementation of Listing's law prior to the burst generator
Modification of inhibitory input from the cerebellum was simulated by a gain increase of the otolith input to the implementation of Listing's law prior to the burst generator (see Fig. 1, lesion D). Bending the head forward or backward modifies the desired eye position only if gaze is not straight ahead (not shown). In contrast, head roll results in a predominantly torsional positional nystagmus (see Fig. 4D) with pathologically large ocular counterroll and the fast phase beating away from the expected torsional position. Alternatively, a complete lesion of the otolith input to the implementation of Listing's law also causes torsional nystagmus, but leads to a pathologically small counterroll (not shown).
In all cases, changing the head position from right-ear-down to left-ear-down changed the sign of the eye position responses and thus the direction of the quick phases.
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DISCUSSION |
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The development of the described model was prompted by a clinical attempt to explain the mechanisms of central positional nystagmus and to identify the functional consequences of defined lesions within the neuronal circuitry that controls eye position in off-vertical head positions. This required a combination of dynamic and static elements of the oculomotor system to model the head position-dependent modifications of eye position and Listing's plane. These model elements had to be combined with kinematically correct generation of saccades and nystagmus. Simulation of positional nystagmus was based on distinct lesions of model elements. The selection of model lesions was based on the consideration that, to elicit central positional nystagmus, otolith input to the oculomotor system has to be modified without causing spontaneous nystagmus with the head upright. All three assumed lesions resulted in various types of central positional nystagmus.
The construction of the model is based on earlier descriptions in the
literature. However, appropriate 3-D models of saccade generation
(e.g., Crawford and Guitton 1997; Quaia and
Optican 1998
; Tweed 1997b
) do not
include dynamic modifications of Listing's plane by otolith input.
Therefore extensions were necessary to include otolith input and to
account for the leaky torsional integrator. The latter has been
demonstrated experimentally (Seidman et al. 1995
).
Otolith-dependent changes of eye position control
Otolith input is supposed to cause changes in Listing's plane:
head roll induces a shift of Listing's plane, i.e., a superposition of
a torsional component (ocular counterroll), while head pitch causes a
rotation of Listing's plane relative to the head (Haslwanter et
al. 1992; Haustein 1988
). There are several ways
in which otolith input may interact with changes in eye position to
generate Listing's plane. Haustein (1989)
stated that
shifts of Listing's plane can be generated by additive input, while
rotations of Listing's plane require multiplicative input. Such active
modification of Listing's plane by utricular input is necessary at two
levels, as shown above. First of all, the saccadic burst generator must
receive otolith input to generate an accurate burst command.
Furthermore, because the new torsional position has to be held actively
and not only by the mechanical properties of the oculomotor plant, the
neural integrator must receive this input as well. Recent experimental
evidence for neural control of Listing's plane is given by a study on
ocular drifts in patients with cerebellar atrophy. These patients
present with a torsional drift that depends on horizontal gaze
eccentricity (Straumann et al. 2000
).
Concerning the shift of Listing's plane, Crawford and Vilis
(1999) demonstrated that the necessary additive otolith input to ocular torsion is not added directly at the motoneuron level, but
passes through the neural integrator. Bilateral, i.e., complete, inactivation of the vertical-torsional oculomotor integrator, the
interstitial nucleus of Cajal, completely abolishes ocular counterroll
to head tilt in the monkey. The same holds for our model. Further
evidence comes from investigations of the frequency response of ocular
counterroll in humans in response to linear acceleration (Clarke
et al. 1999
), showing the same time constant as for
vestibuloocular reflex-induced changes in torsional eye position
(Seidman et al. 1995
) and galvanic vestibular
stimulation (Schneider et al. 2000
).
A tilt of Listing's plane causes the eyes to adopt torsional
positions, which not only depend on gaze direction but also on head
position, i.e., otolith input. Hence, to make saccades that remain in
Listing's plane, the saccadic burst itself needs to be modified by
otolith input. However, we do not yet know the anatomical site at which
otolith information modifies the eye position commands. Recent models
of saccade generation (Crawford and Guitton 1997;
Tweed 1997a
,b
) propose a so-called Listing's box prior
to the burst generator, which is supposed to implement Listing's law
and to generate a 3-D eye displacement command. This Listing's box is
used in our model for modification of the saccadic burst. To avoid
postsaccadic torsional drifts due to the leaky torsional integrator
(Seidman et al. 1995
), the otolith input has to be sent
to the neural integrator, as proposed above for the shift of Listing's
plane. Therefore we assume that in tilted head positions otolith
information defines the head and eye position-dependent resting
position of the integrator.
Anatomical considerations
The lesion study of the interstitial nucleus of Cajal
(Crawford and Vilis 1999) mentioned above demonstrates
that otolith input generating ocular counterroll must pass through the
neural integrator. The direct otolith pathways via the vestibular
nuclei and the MLF responsible for ocular counterroll have been
previously proposed (Glasauer et al. 1998
,
1999
) on the basis of clinical findings such as
asymmetric dynamic ocular counterroll following MLF lesions
(Averbuch-Heller et al. 1997
). The involvement of the
burst generator in the control of Listing's plane has also been shown
by unilateral lesions of the rostral interstitial nucleus of the MLF in
the monkey (Suzuki et al. 1995
). After this
torsional-vertical burst generator was lesioned, a shift of Listing's
plane was observed, but counterrolling in response to head roll was not
abolished. This is also in accordance with our model. Furthermore, it
has been shown that pathways to the burst generator are indeed involved in the stabilization of Listing's plane (Van Opstal et al.
1996
). Otolith input to the cerebellum, as proposed by our
model, has been demonstrated extensively (see
Büttner-Ennever 1999
for review). Animal studies
have demonstrated pathways not only from the vestibular nuclei via the
inferior olive to the cerebellar vermis (uvula-nodulus), but also
primary otolith projections to the uvula-nodulus via mossy fibers
(Barmack et al. 1993
). Furthermore, single-cell
recordings suggest that the otoliths influence neural activity in the
rostral fastigial nucleus (Siebold et al. 1997
). Cells
in the adjacent caudal fastigial nucleus are known to fire during
saccades (Helmchen et al. 1994
). Afferent
projections from the cerebellum to the neural integrators have been
demonstrated anatomically (see Fukushima et al. 1992
for
review). Their functional significance becomes evident with cerebellar
lesions, which affect the time constant of the integrator: lesion of
the cerebellar flocculus reduces the integrator time constants to only
1.6 s (Zee et al. 1981
). Our model would simulate
such a lesion by increasing the diagonal elements of the feedback
matrix in the neural integrator. Such a functional increase of feedback
gain is similar to the model lesions employed in the present study to
simulate central positional nystagmus. Moreover, clinical findings in
patients with cerebellar atrophy recently showed that the cerebellum is
involved in the adjustment of Listing's plane (Straumann et al.
2000
).
Clinical reports of central positional nystagmus
The pathomechanism of nystagmus proposed in our model does not fit
into the classic hypotheses such as imbalance of the vestibuloocular reflex, integrator failure or deficit, or failure of burst generation. Instead, it results from a mismatch between two internal coordinate systems, namely the burst generator and the neural integrator. In other
words, nystagmus may be generated if the burst generator uses a
Listing's plane that does not coincide with that of the neural
integrator. Such a mismatch could also explain the torsional components
of the more commonly observed nystagmus due to cerebellar atrophy
(Straumann et al. 2000). The predominantly vertical
nystagmus of these patients can be explained by the observed additional vertical integrator leakage, which is known to be caused by cerebellar lesions, and/or by a mechanism similar to that causing the
predominantly vertical nystagmus shown in Fig. 3C
(corresponding to lesion C).
In conclusion, the above-described model enables us to simulate various
patterns of central positional nystagmus, the characteristics of which
are compatible with clinical descriptions. Unfortunately, no systematic
quantitative 3-D eye movement recordings are available from
neurological patients tested in different off-vertical head positions.
Furthermore, the clinical literature is vague about separate and
distinct lesions of the oculomotor and/or vestibular systems that can
be associated with certain types of central positional nystagmus. The
following uni- or bilateral lesions have been reported to occur with
central positional nystagmus: dorsal vermis (Barber 1984; Gregorius et al. 1976
; Sakata et
al. 1991
; Watson et al. 1981
), dorsolateral to
the fourth ventricle including the caudal cerebellar peduncles
(Brandt 1990
), or diffuse cerebellar atrophy (Büttner et al. 1998
). While the simulations shown
produced predominantly torsional and/or vertical positional nystagmus,
significant horizontal components of positional nystagmus can be
observed clinically. Such a predominantly horizontal component is
generated by the model if the effect of the lesion is smaller than
assumed in the present simulations. A vertical positional nystagmus in
supine or prone positions, which is observed even in normal subjects (Bisdorff et al. 2000
), is generated by the model if
gaze straight ahead does not coincide with Listing's horizontal zero
position, as can often be observed.
The ability to simulate failures of a single element or pathway of the oculomotor circuitry allows us to pose clinical questions about syndromes not yet described, the localization of the damage, as well as the mechanisms involved. For central positional nystagmus this means that further testing of patients must include quantitative 3-D analysis of eye position in different head and gaze positions. The model points to potential candidates of structures and pathways, particularly the ocular motor cerebellar pathways, a lesion of which may be causative.
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APPENDIX |
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The model used is based on the spatial saccade model by
Crawford and Guitton (1997) with the exceptions noted
below. A right-handed, head-fixed coordinate system is used
(z-axis pointing upward, x-axis pointing
forward). Eye position, utricular input, and target position were
expressed using rotation vectors (Haslwanter 1995
; Haustein 1989
). A rotation vector is defined as
r = tan (
/2) · n with
being the angle of rotation and
n a unity vector pointing along the axis of
rotation. In terms of rotation vectors, the result of two consecutive
rotations r1 and
r2 is given by
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Retinal error
Retinal error er
is computed from target position in eye coordinates
te by (Haustein
1989)
![]() |
(A1) |
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(A2) |
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(A3) |
Listing's law operator and otolith dependence
In the following it is assumed that the otolith input u acts as a rotation vector. Therefore pure head roll results in u = [ux 0 0]T, while head pitch gives u = [0 uy 0]T. The otolith input is scaled by a diagonal matrix K so that 90° head roll will produce about 5° ocular counterroll (shift of Listing's plane), and 90° pitch forward of the head results in 5° pitch back of Listing's plane.
The desired eye position
rd to be computed by
the Listing's law operator is
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(A4) |
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(A5) |
Burst generator
The burst generator in the model receives two inputs: retinal
error, which mainly triggers the burst, and eye position error re = rd r. The burst trigger logic causes a burst if
retinal error (or alternatively torsional error) exceeds a threshold
and the previous burst is over. Retinal error is subject to a temporal
delay. Additionally, a minimum time delay has to pass between two
successive bursts. As soon as the burst is triggered, eye position
error is fed into the burst generator and is held at a constant level
(sample-hold mechanism) to prevent eye position changes from affecting
the local integrator during the burst. At the same time, the local integrator is reset. This triggers the burst.
Velocity to position integrator
The generated 3-D burst is integrated by the oculomotor velocity
to position integrator to an eye position in Listing's plane. In
contrast to previous models (Crawford and Guitton 1997;
Tweed 1997a
,b
), the neural integrator is modeled to leak
in the torsional component, as suggested by the experiments of
Seidman et al. (1995)
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(A6) |
To avoid torsional postsaccadic drifts in tilted or pitched head
positions, the feedback within the neural integrator has to be adjusted
accordingly
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(A7) |
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(A8) |
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(A9) |
Note that the considerations above (except for Eq. A9) still hold for leakage of the horizontal and vertical components, as observed, for example, in normal subjects in darkness.
Oculomotor plant
The plant is modeled as a so-called linear plant
(Crawford and Guitton 1997; Quaia and Optican
1998
; Raphan 1997
; Tweed et al.
1994
), i.e., the input to the plant is the derivative of eye position (not angular velocity). The direct pathway to the plant is
provided by the output of the burst generator, the indirect input by
the output of the neural integrator. Additionally, the coordinate
transformation from Listing's coordinates to head coordinates is
supposed to take place within the plant. As baseline tilt, an angle of
10° pitch backward of Listing's plane relative to head upright was
assumed for the simulations.
Lesions
The lesion applied to the burst generator (lesion D) is modeled as a gain factor c in Eq. A5 so that the desired torsional component is rdx = c · (uy · rdz + ux). In the upright position, u is zero, and therefore the lesion has no effect.
The lesions applied to the integrator (A, B, and
C) are modeled as constant inputs to parts of Eq. A8. To ensure that only tilted head positions are affected,
Eq. A8 has to be divided into multiple parts
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(A10) |
As an example, lesion A is considered. In Eq. A9,
all occurrences of ux and
uy are replaced by a constant c
resulting in a constant matrix TC. With
u = [ux 0 0]T in
head roll and s = 0 during the slow phase
of the nystagmus, Eq. A10, expressing the slow-phase
velocity, can thus be written as
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ACKNOWLEDGMENTS |
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We are grateful to two anonymous referees for helpful comments and to J. Benson for copy-editing the manuscript.
This work was supported by Fritz Thyssen Stiftung and Deutsche Forschungsgemeinschaft SFB 462, A5.
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FOOTNOTES |
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Address for reprint requests: S. Glasauer, Center for Sensorimotor Research, Dept. of Neurology, Marchioninistrasse 23, 81377 Munich, Germany (E-mail: sglasauer{at}nefo.med.uni-muenchen.de).
Received 31 October 2000; accepted in final form 6 June 2001.
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REFERENCES |
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