Considerations on Listing's Law and the primary position by means of a matrix description of eye position control

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Abstract
Voluntary gaze control obeys the law of Listing. This law specifies the torsion of the eye in every direction of gaze, when the so-called “primary position” of the eye is known. In the present article, some interesting relations between Listing's Law and the primary position on the one hand, and eye muscle innervation on the other hand are derived. A new vectorial description of eye position is introduced which permits to represent Listing's Law by a very simple equation. It also allows to approximate the transformation between eye muscle innervation and eye position by a 3 x 3-matrix in a large field of gaze and with quantities of innervation which are directly related to motoneuron firing rates. It is shown that these innervation quantities must be linearly coupled to yield Listing's Law and that the parameters of this coupling determine the primary position. Finally, relations between innervations and eye muscle forces are considered, and a hypothesis concerning the co-operation of voluntary gaze control and otolith or vergence inputs is presented.