The responses of simple cells were recorded from the visual cortex of cats, as a function of the position and contrast of counterphase and drifting grating patterns, to assess whether direction selectivity can be accounted for on the basis of linear summation. The expected responses to a counterphase grating, given a strictly linear model, would be the sum of the responses to the two drifting components. The measured responses were not consistent with the linear prediction. For example, nearly all cells showed two positions where the responses approached zero (i.e. two "null phase positions"); this was true, even for the most direction selective cells. However, the measured responses were consistent with the hypothesis that direction selectivity is a consequence of the linear spatiotemporal receptive-field structure, coupled with the nonlinearities revealed by the contrast-response function: contrast gain control, halfwave rectification, and expansive exponent. When arranged in a particular sequence, each of these linear and nonlinear mechanisms performs a useful function in a general model of simple cells. The linear spatiotemporal receptive field initiates stimulus selectivity (for direction, orientation, spatial frequency, etc.). The expansive response exponent enhances selectivity. The contrast-set gain control maintains selectivity (over a wide range of contrasts, in spite of the limited dynamic response range and steep slope of the contrast-response function). Rectification conserves metabolic energy.