Neurons in cortical area MT respond to visual motion stimuli and drive downstream per- ception of motion and smooth pursuit eye movements. Each neuron responds to parametric changes in stimulus value, and the responses are pooled across a large population of neurons in a population code. The accuracy of such a code can depend subtly on specific features of the response properties of the neurons that make up the code. Trial-to-trial variability, correlated responses, response heterogeneity, and temporal dynamics can all affect how a neural population codes information. This work characterizes the variability of neurons in MT, and describes how the stimulus-dependent structure of this variability can impact the population code. A model of stimulus-dependent variability bridges this result with previous descriptions of variability in MT that lacked stimulus-dependence, and demonstrates how changes in attentional or behavioral state can alter the structure of this variability. This work also characterizes and models the diversity of temporal dynamics in MT neurons, and explores the contribution of temporal heterogeneity to population codes. A generative model of dynamic MT responses provides a foundation for further investigation of the population code.