The oculomotor system coordinates different types of eye movements in order to orient the visual axis, including saccade and smooth pursuit,. It was traditionally thought that the premotor pathways for these different eye movements are largely separate. In particular, a group of midline cells in the pons called omnipause neurons were considered to be part of only the saccadic system. Recent experimental findings have shown activity modulation of these brainstem premotor neurons during both kinds of eye movements. In this study, we propose a new computational model of the brainstem circuitry underlying the generation of saccades and smooth pursuit eye movements. Similar models have been developed earlier, but mainly looking at pure saccades. Here, we integrated recent neurophysiological findings on omnipause neuron activity during smooth pursuit. Our computational model can mimic some new experimental findings as the similarity of "eye velocity profile" with "omnipause neuron pattern of activity" in pursuit movement. We showed that pursuit neuron activity is augmented during catch-up saccades; this increment depends on the initial pursuit velocity in catch-up saccade onset. We conclude that saccadic and pursuit components of catch-up saccades are added to each other nonlinearly.