The oculomotor integrator is a brainstem neural network that converts velocity signals into the position commands necessary for eye-movement control. The cerebellum can independently adjust the amplitude of eye-movement commands and the temporal characteristics of neural integration, but the percentage of integrator neurons that receive cerebellar input is very small. Adaptive dynamic systems models, configured using the genetic algorithm, show how sparse cerebellar inputs could morph the dynamics of the oculomotor integrator and independently adjust its overall response amplitude and time course. Dynamic morphing involves an interplay of opposites, in which some model Purkinje cells exert positive feedback on the network, while others exert negative feedback. Positive feedback can be increased to prolong the integrator time course at virtually any level of negative feedback. The more these two influences oppose each other, the larger become the response amplitudes of the individual units and of the overall integrator network.