Chapter 6 improves on the Simplified Motor Coordinate VOR Model in chapter 5. The Full Model takes into account the precise geometry of the vestibular canals and extraocular muscles in implementing the Kinetic VOR Model. This model makes the most precise predictions for the firing of neurons in the VOR pathway.
As in the previous chapter, the velocity and acceleration equations of the Kinematic VOR Model can be added linearly to form a single equation. This equation describes the transformation from canals and otoliths responses to head motion into eye motor commands needed to stabilize objects on the retina. The Full Motor Coordinate VOR Model implements this equation in the coordinate systems of the canals and extraocular muscles. The model provides an explanation, mainly a posteriori, of the firing discharge of some VOR neurons in the brain stem which discharge to a combination of signals from the canals, otoliths and from eye position. The model shows that a correct VOR response relies upon integrating information from different canals, otoliths and eye position signals. Therefore neurons firing to combinations of these signals are expected from the model.
There are three vestibular semicircular canals on each side of the head. Their orientation in the inner ear defines their directional sensitivity to head rotation ([Blanks, Curthoys and Markham1975]), forming two complete coordinate system in three dimensions. The encoding of head rotations is obtained by taking the projection of the head rotation axis onto the sensitivity vectors of the canals. In this way, the head rotation is represented by two covariant vectors in canal coordinates, one on each side.
Six extraocular muscles, forming three pairs, rotate the eyeball in the ocular cavity. For most eye movements, all six muscles must change their length during eye movements. In addition, the direction in which each muscle pulls on the eyeball depends on current eye position ([Robinson1975]). The six extraocular muscles at each eye form an eye position-dependent coordinate system in three dimensions, which is called here the motor coordinate system. An eye position vector is obtained by summing its components along the six muscle directions; the components form a contravariant vector in motor coordinates.
The goal of this chapter is to solve the computational problem of transforming the covariant vectors of head rotation in six dimensions to the contravariant vectors of eye motor commands, also in six dimensions, required to satisfy the VOR. The dual basis to the canal basis is introduced, as well as a method for estimating the contravariant coordinates of the eye position vector from its covariant coordinates. The responses of model neurons in the vestibular nucleus which combine canal and eye position inputs are plotted. The main result from these model neurons is that one canal and the eye position information from one extraocular muscle should multiply each other. In fact, the responses calculated from the weighted summation of any number of such model neuron could still be homologous to the response of vestibular nucleus neurons. In this case, the combination of canal and eye position information is not a simple multiplication.
Models of VOR transformation have been presented earlier. Robinson described a three-dimensional VOR model using matrices ([Robinson1982]). But as Pellionisz pointed out, this model does not account for the mixing of canal inputs observed experimentally ([Pellionisz1985]). Pellionisz studied the transformation from 3 dimensional canal inputs to 6 dimensional motor coordinate system. This increase in dimensionality and the transformation from canal to motor coordinate system is dealt with through a process he calls sensorimotor covariant embedding combined with the multiplication of a contravariant motor metric tensor. In the Full Motor Coordinate VOR Model, there is no such dimensional increase to tackle since both canal and muscle coordinate system are six dimensional. The problem of calculating the contravariant coordinates of the eye position vector in the overcomplete motor coordinate system was solved by Pellionisz using the Moore-Penrose generalized inverse. Since the generalized inverse did not give results compatible with the muscle innervation patterns ([Robinson1975]), an extension of this method was chosen to calculate the contravariant coordinates.
In contrast to the work of Pellionisz, the present model did not attempt to construct a tensorial theory of the VOR. But as Arbib and Amari pointed out, although Pellionisz uses the terminology, a tensorial theory is never truly developed mathematically ([Arbib and Amari1985]). We agree with the argument of Robinson which refutes the necessity to invoke the existence of a metric tensor in the nervous system ([Robinson1982]). The present model shows that ultimately the transformation by a metric tensor must be combined with other transformations, resulting simply in the weighted sums of products of canal and eye position information. The fact that the weights in the weighted sum can be separated in two parts, one of which includes a metric, is irrelevant to biological concerns, although it remains useful for mathematical manipulations.
There is one major difference with the previous models of VOR transformation. In our model, each muscle defines a coordinate axis along the direction of the muscle, which is different from the moment arm along a virtual axis of rotation associated with each muscle that others have used ([Robinson1982]; [Pellionisz1985]). In this way, the geometrical correspondence between the eye position vector and muscle innervations is more direct and intuitive.