Perceptual decisions are often modeled as a "race to threshold" where evidence is accumulated up to a set point where the decision occurs. The threshold model is normally applied to decisions made over a long period of time, such as identifying motion or an object in noise. It is unclear how such models apply to rapid perceptual decisions. Previous studies in the macaque monkey measured firing rates in area MT during a rapid coherent motion detection task. I modeled the decision process for this task as "leaky" integration of the MT firing up to a decision threshold. The best fit to animal data was achieved by integrating over 20 Poisson neurons with a time constant of 100 ms and using a variable threshold. The model can account for the performance (correct, miss, and false alarm rates) and reaction times of the animal. This indicates that a similar mechanism may underlie decisions made from weak stimuli presented over a long period of time and strong stimuli presented quickly.