Monkeys fixated a stationary spot during presentation of dot textures that moved in apparent motion defined by the spatial and temporal separations, x and t, between successive flashes of each dot. For each neuron, we assessed speed tuning for smooth motion (t = 2 or 4 ms) at speeds up to 128 °/s and the effect of varying the value of t at speeds of 16 and 32 °/s. Many MST neurons, like MT neurons, were tuned for the speed of smooth motion and showed decreases in firing rate as t increased at a constant speed. A subset of MST neurons, however, showed monotonically increasing firing rates as a function of smooth stimulus speed, and responses to apparent motion that paralleled a previously discovered illusion where estimates of target speed increase with the value of t. Opponent firing rate, defined as the difference between responses for motion in the preferred and opposite directions, peaked at values of t that were consistent with the behavioral illusion. Comparison with a new sample of MT neurons recorded with the same stimuli failed to reveal comparable effects. Attempts to map the population responses in MT and MST onto the behavioral illusion of increased speed succeeded by averaging the opponent response across MST neurons, but only by applying vector averaging to determine the preferred speed of the most active MT neurons. We suggest that a vector averaging computation transforms MT’s place code for target speed into the rate code of some MST neurons.