Computational and behavioral studies suggest that visual motion discrimination is based on quadratic nonlinearities. This raises the question of whether the behavior of motion sensitive neurons early in the visual system is actually quadratic. Theoretical studies show that mechanisms proposed for retinal directional selectivity do not behave quadratically at high stimulus contrast. However, for low contrast stimuli, models for these mechanisms may be grouped into three categories: purely quadratic, quadratic accompanied by a rectification, and models mediated by a high level threshold. We discriminated between these alternatives by analyzing the extracellular responses of ON-OFF directionally selective ganglion cells of the rabbit retina to drifting periodic gratings. The data show that purely-quadratic or high-threshold systems do not account for the behavior of these cells. However, their behavior is consistent with a rectified-quadratic model. © 1990 Springer-Verlag.