Mathematical Analysis of Control by the Cerebellum of the Oculomotor Neural Integrator

Andrea K. Barriero, Jared C. Bronski, and Tom Anastasio, Beckman Institute, University of Illinois at Urbana-Champaign

In order to control the movement of the eyes, the brain must command the eye muscles to overcome the forces that keep the eyes from moving. Because one of these restraining forces is the elastic force, which is proportional to eye position, the brain must convert sensory signals proportional to velocity into eye position commands. The neural network that accomplished this is called the oculomotor neural integrator. This neural network produces integration through positive feedback and, to keep the integrator away from instability, it is normally leaky, with a time constant of about 20 seconds in humans. In order to ensure that the integrator produces effective eye position commands under changing circumstances, it is regulated by a brain region known as the cerebellum, which takes signals form the integrator, scales them, and feeds them back to the integrator. Through this mechanism the cerebellum is capable of independent adjustment of integrator time constant and gain! . ! This is no mean feat! Simply increasing (or decreasing) integrator positive feedback would increase (or decrease) both the duration (time constant) and amplitude (gain) of its responses. In order to independently adjust integrator time constant and gain the cerebellum employs a dynamic trick. Our mathematical analysis reveals three conditions that are necessary for this trick to be accomplished. First, there must be at least two separate feedback loops onto the integrator through the cerebellum. Second, the coupled cerebellar-integrator system must be asymmetric. Third, the connections from the integrator that form one of the cerebellar feedback loops cannot be an eigenvector of the integrator system by itself. These conditions determine the minimal network configuration capable of independent adjustment of integrator time constant and gain. The analysis also shows that, in order to produce the physiologically realistic long time constant and high gain, the system m! us! t operate near a triple point, where small perturbations can push it into regions of instability, oscillation, or unstable oscillation. The cerebellum is known for its adaptive capability, which under normal circumstances should keep the integrator at its desired operating point, but eye movement abnormalities are common. Congenital eye movement abnormalities are characterized by instability, oscillation, and even unstable oscillation. In familial cases, some family members may have one form and others a different form. Our analysis shows, for the first time, how these different eye movement abnormalities might arise and how they might be related.