Figure: Diagram showing the definition of the vectors used in
the equation of the kinematic model of the vestibulo-ocular
reflex.
The ideal VOR response is a compensatory eye movement which keeps the image fixed on the retina for any head rotations and translations. We therefore derived an equation for the eye rotation velocity by requiring that a target remains stationary on the retina. The velocity of the resulting compensatory eye rotation can be written as (see fig. 1):
where 
is the head rotation velocity sensed by the
semicircular canals, 
is the head translation velocity
sensed by the otoliths, 
, 
is a constant vector
specifying the location of an eye in the head, 
is the
position of either the left or right otolith, 
and 
are the unit vector and amplitude of the gaze vector:
gives the eye position (orientation of the eye relative to the
head), and 
gives the distance from the eye to the
object, and the symbol 
indicates the cross-product between two vectors.
and 
are rotation vectors which
describe the instantaneous angular velocity of the eye and head,
respectively. A rotation vector lies along the instantaneous axis of
rotation; its magnitude indicates the speed of rotation around the
axis, and its direction is given by the right-hand screw rule. A
motion of the head combining rotation (
) and
translation (
) is sensed
as the combination of a rotation velocity 
measured
by the semicircular canals and a translation velocity 
sensed by the otoliths.
The rotation vectors are equal (
),
and the translation velocity vector as measured by the
otoliths is given by:
,
where 
, and 
is
the position vector of the axis of rotation.
The special case where the gaze is horizontal and the rotation
vector is vertical (horizontal head rotation) has been
studied extensively in the literature. We used this special
case in the simulations. In that case 
may be simplify by writing its equation with dot
products. Since 
and 
are then perpendicular (
), the
first term of the following expression in brackets is zero:
The semicircular canals decompose and report acceleration and
velocity of head rotation 
by its components along the
three canals on each side of the head 
: horizontal,
anterior and posterior. The two otolith organs on each side report
the dynamical inertial forces generated during linear motion
(translation) in two perpendicular plane, one vertical and the other
horizontal relative to the head. Here we assume that a translation
velocity signal (
) derived from or reported by the
otolith afferents is available. The otoliths encode as well the head
orientation relative to the gravity vector force, but was not
included in this study.
To complete the correspondence between the equation and a neural
correlate, we need to determine a physiological source for 
and
is assumed to be
given by the output of the velocity-to-position transformation or
so-called ``neural integrator'' which provides eye position
information and which is necessary for the activation of the
motoneuron to sustain the eye in a fixed position. The integrator for
horizontal eye position appears to be located in the nucleus prepositus
hypoglossi in the pons, and the vertical integrator in the midbrain
interstitial nucleus of Cajal. ([Crawford, Cadera and Vilis1991];
[Cannon and Robinson1987]).
We assume that the eye position is given as the
coordinates of the unit vector 
along the 
and
of fig. 1. The eye position depends on the eye
velocity according to
.
For the special case 
, i.e. for horizontal head
rotation, the eye position coordinates are given by:
This is a set of two negatively coupled integrators. The ``neural integrator'' therefore does not integrate the eye velocity directly but a product of eye position and eye velocity. The distance from eye to target
where (
is the vergence angle, and I is
the interocular distance; the angles are measured from a straight
ahead gaze, and take on negative values when the eyes are turned
towards the right. Within the oculomotor system, the vergence angle
and speed are encoded by the mesencephalic reticular formation neurons
([Judge and Cumming1986]; [Mays1984]). The nucleus
reticularis tegmenti pontis with reciprocal connections to the
flocculus, oculomotor vermis, paravermis of the cerebellum also
contains neurons which activity varies linearly with vergence angle
([Gamlin and Clarke1995]).
We conclude that it is possible to perform the computations needed to obtain an ideal VOR with signals known to be available physiologically.
Figure: Anatomical connections considered in the dynamical model.
Only the left side is shown, the right side is identical and connected
to the left side only for the calculation of vergence angle. The
nucleus prepositus hypoglossi and the nucleus reticularis tegmenti
pontis are meant to be representative of a class of nuclei in the
brain stem carrying eye position or vergence signal. All connections
are known to exist except the connection between the prepositus
nucleus to the reticularis nucleus which has not been verified.
Details of the cerebellum are in fig. 3 and of the
vestibular nucleus in fig. 4.
Figure: Contribution of the cerebellum to the dynamical
model. Filtered velocity inputs from the canals and otoliths are
combined with eye position according to equation
(2). These calculations could be performed either
outside the cerebellum in one or multiple brain stem nuclei (as
shown) or possibly inside the cerebellum. The only output is to the
vestibular nucleus. The Laplace notation is used in each boxes to
represent a leaky integrator with a time constant, input derivative
and input gain. The term oe are the coordinates of the vector 
shown in fig. 1. The 
indicates a
multiplication. The term 
multiplies each inputs
individually. The open arrows indicate inhibitory (negative)
connections.
Figure: Contribution of the vestibular nucleus to the dynamical
model. Three pathways in the vestibular nucleus process the canal and
otolith inputs to drive the eye. The first pathway is modulated by
the output of the cerebellum through a FTN (Flocculus Target
Neuron). The second and third pathways report transient
information from the inputs which are combined with eye position in a
manner identical to fig. 3. The location of these
calculations is hypothetical.
