Recently, models in psychology have been shown capable of accounting for the full range of behavioral data from simple two-choice decision tasks: mean reaction times for correct and error responses, accuracy, and the reaction time distributions for correct and error responses. At the same time, recent data from neural recordings have allowed investigation of the neural systems that implement such decisions. In the experiment presented here, neural recordings were obtained from superior colliculus prelude/buildup cells in two monkeys while they performed a two-choice task that has been used in humans for testing psychological models of the decision process. The best-developed psychological model, the diffusion model, and a competing model, the Poisson counter model, were explicitly fit to the behavioral data. The pattern of activity shown in the prelude/buildup cells, including the point at which response choices were discriminated, was matched by the evidence accumulation process predicted from the diffusion model using the parameters from the fits to the behavioral data but not by the Poisson counter model. These results suggest that prelude/buildup cells in the superior colliculus, or cells in circuits in which the superior colliculus cells participate, implement a diffusion decision process or a variant of the diffusion process.