Modeling control of eye orientation in three dimensions. I. Role of muscle pulleys in determining saccadic trajectory

This study evaluates the effects of muscle axis shifts on the performance of a vector velocity-position integrator in the CNS. Earlier models of the oculomotor plant assumed that the muscle axes remained fixed relative to the head as the eye rotated into secondary and tertiary eye positions. Under this assumption, the vector integrator model generates torsional transients as the eye moves from secondary to tertiary positions of fixation. The torsional transient represents an eye movement response to a spatial mismatch between the torque axes that remain fixed in the head and the displacement plane that changes by half the angle of the change in eye orientation. When muscle axis shifts were incorporated into the model, the torque axes were closer to the displacement plane at each eye orientation throughout the trajectory, and torsional transients were reduced dramatically. Their size and dynamics were close to reported data. It was also shown that when the muscle torque axes were rotated by 50% of the eye rotation, there was no torsional transient and Listing's law was perfectly obeyed. When muscle torque axes rotated >50%, torsional transients reversed direction compared with what occurred for muscle axis shifts of <50%. The model indicates that Listing's law is implemented by the oculomotor plant subject to a two-dimensional command signal that is confined to the pitch-yaw plane, having zero torsion. Saccades that bring the eye to orientations outside Listing's plane could easily be corrected by a roll pulse that resets the roll state of the velocity-position integrator to zero. This would be a simple implementation of the corrective controller suggested by Van Opstal and colleagues. The model further indicates that muscle axis shifts together with the torque orientation relationship for tissue surrounding the eye and Newton's laws of motion form a sufficient plant model to explain saccadic trajectories and periods of fixation when driven by a vector command confined to the pitch-yaw plane. This implies that the velocity-position integrator is probably realized as a subtractive feedback vector integrator and not as a quaternion-based integrator that implements kinematic transformations to orient the eye.