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
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.