Generalization and analysis of the Lisberger-Sejnowski VOR model

Lisberger and Sejnowski (1992) recently proposed a computational model for motor learning in the vestibular-ocular reflex (VOR) system. They showed that the steady-state gain of the system can be modified by changing the ratio of the two time constants along the feedforward and the feedback projections to the Purkinje cell unit in their model VOR network. Here we generalize their model by including two additional time constant variables and two synaptic weight variables, which were set to fixed values in their original model. We derive the stability conditions of the generalized system and thoroughly analyze its steady-state and transient behavior. It is found that the generalized system can display a continuum of behavior with the Lisberger-Sejnowski model and a static model proposed by Miles et al. (1980b) as special cases. Moreover, although mathematically the Lisberger-Sejnowski model requires two precise relationships among its parameters, the model is robust against small perturbations from the physiological point of view. Additional considerations on the gain of smooth pursuit eye movement, which is believed to share the positive feedback loop with the VOR network, suggest that the VOR network should operate in the parameter range favoring the behavior studied by Lisberger and Sejnowski. Under this condition, the steady-state gain of the VOR is found to depend on all four time constants in the network. The time constant of the Purkinje cell unit should be relatively small in order to achieve effective VOR learning through the modifications of the other time constants. Our analysis provides a thorough characterization of the system and could thus be useful for guiding further physiological tests of the model.