MODELING 3-DIMENSIONAL VELOCITY-TO-POSITION TRANSFORMATION IN OCULOMOTOR CONTROL

1. A considerable amount of attention has been devoted to understandin g the velocity-position transformation that takes place in the control of eye movements in three dimensions. Much of the work has focused on the idea that rotations in three dimensions do not commute and that a ''multiplicative quaternion model'' of velocity-position integration is necessary to explain eye movements in three dimensions. Our study h as indicated that this approach is not consistent with the physiology of the types of signals necessary to rotate the eyes. 2. We developed a three-dimensional dynamical system model for movement of the eye wit hin its surrounding orbital tissue. The main point of the model is tha t the eye muscles generate torque to rotate the eye. When the eye reac hes an orientation such that the restoring torque of the orbital tissu e counterbalances the torque applied by the muscles, a unique equilibr ium point is reached. The trajectory of the eye to reach equilibrium m ay follow any path, depending on the starting eye orientation and eye velocity. However, according to Euler's theorem, the equilibrium reach ed is equivalent to a rotation about a fixed axis through some angle f rom a primary orientation. This represents the shortest path that the eye could take from the primary orientation in reaching equilibrium. C onsequently, it is also the shortest path for returning the eye to the primary orientation. Thus the restoring torque developed by the tissu e surrounding the eye was approximated as proportional to the product of this angle and a unit vector along this axis. The relationship betw een orientation and restoring torque gives a unique torque-orientation relationship. 3. Once the appropriate torque-orientation relationship for eye rotation is established the velocity-position integrator can be modeled as a dynamical system that is a direct extension of the one -dimensional velocity-position integrator. The linear combination of t he integrator state and a direct pathway signal is converted to a torq ue signal that activates the muscles to rotate the eyes. Therefore the output of the integrator is related to a torque signal that positions the eyes. It is not an eye orientation signal. The applied torque sig nal drives the eye to an equilibrium orientation such that the restori ng torque equals the applied torque but in the opposite direction. The eye orientation reached at equilibrium is determined by the unique to rque-orientation relation. Because torque signals are vectors, they co mmute. Thus our model indicates that the signals in the CNS can be tre ated as vectors and that the nonvector orientation properties of the e ye globe are inherent in the dynamical system associated with the glob e and its underlying tissue. 4. Listing's law is explained very simply by our model as being a property of the vector nature of the signals in the CNS driving the eyes, and its implementation is not localized t o any specific locality within the CNS. If the neural vector signal dr iving the eye is confined to Listing's plane, i.e., the pitch-yaw plan e in our model, then eye orientation will obey Listing's law. 5. We pe rformed simulations to show that Listing's law is obeyed by our model for both saccades and smooth pursuit eye movements in the steady state . The simulations also showed that there is commutativity in terms of steady-state eye orientation. We performed simulations that compared t he model output with data of others. Deviations from Listing's law wer e consistent with the physiological findings.