Commutative Saccadic Generator Is Sufficient to Control a 3-D Ocular Plant With Pulleys

Abstract

One-dimensional models of oculomotor control rely on the fact that, when rotations around only one axis are considered, angular velocity is the derivative of orientation. However, when rotations around arbitrary axes [3-dimensional (3-D) rotations] are considered, this property does not hold, because 3-D rotations are noncommutative. The noncommutativity of rotations has prompted a long debate over whether or not the oculomotor system has to account for this property of rotations by employing noncommutative operators. Recently, Raphan presented a model of the ocular plant that incorporates the orbital pulleys discovered, and qualitatively modeled, by Miller and colleagues. Using one simulation, Raphan showed that the pulley model could produce realistic saccades even when the neural controller is commutative. However, no proof was offered that the good behavior of the Raphan-Miller pulley model holds for saccades different from those simulated. We demonstrate mathematically that the Raphan-Miller pulley model always produces movements that have an accurate dynamic behavior. This is possible because, if the pulleys are properly placed, the oculomotor plant (extraocular muscles, orbital pulleys, and eyeball) in a sense appears commutative to the neural controller. We demonstrate this finding by studying the effect that the pulleys have on the different components of the innervation signal provided by the brain to the extraocular muscles. Because the pulleys make the axes of action of the extraocular muscles dependent on eye orientation, the effect of the innervation signals varies correspondingly as a function of eye orientation. In particular, the Pulse of innervation, which in classical models of the saccadic system encoded eye velocity, here encodes a different signal, which is very close to the derivative of eye orientation. In contrast, the Step of innervation always encodes orientation, whether or not the plant contains pulleys. Thus the Step can be produced by simply integrating the Pulse. Particular care will be given to describing how the pulleys can have this differential effect on the Pulse and the Step. We will show that, if orbital pulleys are properly located, the neural control of saccades can be greatly simplified. Furthermore, the neural implementation of Listing's Law is simplified: eye orientation will lie in Listing's Plane as long as the Pulse is generated in that plane. These results also have implications for the surgical treatment of strabismus.

The Sources of Variability in Saccadic Eye Movements

“…We next calculated how this total torque was produced by the muscles. We treated antagonistic muscle pairs as single ideal muscles that can produce positive and negative torques, and whose insertions on the globe are orthogonal to each other (Quaia and Optican, 1998). The net torque generated by each muscle pair can then be found if the pulling directions of all muscle pairs are known.…”

Visuomotor Velocity Transformations for Smooth Pursuit Eye Movements

“…The traditional view of how smooth pursuit eye movements are initiated involves computing a two-dimensional (2D) velocitybased motor plan from the retinal slip information, which is then used to drive the 3D ocular plant (Angelaki and Hess 2004;Dicke and Thier 1999;Ghasia et al 2008;Klier et al 2006;Tweed et al 1992). This view proposes that 3D behavioral constraints such as Listing's law, which allows for 2D control of a 3D plant, are implemented through the extraocular muscle-pulley system (Demer 2004(Demer , 2006(Demer , 2007Quaia and Optican 1998). Whether Listing's law is indeed implemented through the extraocular muscle-pulley system or rather is actively controlled by neurons in the brain is still under debate (Dimitrova et al 2003;McClung et al 2006).…”

Three-Dimensional Kinematics at the Level of the Oculomotor Plant

“…For the past decade, debate has centered around whether such three-dimensional rules are implemented neurally by brainstem circuits (Tweed and Vilis, 1987) or mechanically by the positioning of orbital pulleys (Miller, 1989;Miller et al, 1993;Quaia and Optican, 1998;Demer et al, 2000). In support of the former, Listing's law is obeyed by saccades (Helmholtz, 1867;Tweed and Vilis, 1990), pursuit (Haslwanter et al, 1991;Tweed et al, 1992;Angelaki et al, 2003;Adeyemo and Angelaki, 2005) and vergence (Van Rijn and Van den Berg, 1993) eye movements, but not by others such as the rotational vestibulo-ocular reflex (Crawford and Vilis, 1991;Fetter et al, 1992;Angelaki, 2003;Angelaki et al, 2003).…”