The origin of orientation selectivity in primary visual cortex (V1) is a model problem for understanding cerebral cortical circuitry. A key constraint is that orientation tuning width is invariant under changes in stimulus contrast. We have previously shown that this can arise from the combination of feedforward lateral geniculate nucleus (LGN) input and an orientation-untuned component of feedforward inhibition that dominates excitation. However, these models did not include the large background voltage noise observed in vivo. Here, we include this noise and examine a simple model of cat V1 response. Constraining our simulations to fit physiological data, our single model parameter is the strength of feedforward inhibition relative to LGN excitation. With physiological noise, the contrast invariance of orientation tuning depends little on inhibition level, although very weak or very strong inhibition leads to weak broadening or sharpening, respectively, of tuning with contrast. For any inhibition level, an alternative measure of orientation tuning -- the circular variance -- decreases with contrast as observed experimentally. These results arise primarily because the voltage noise causes large inputs to be much more strongly amplified than small ones in evoking spiking responses, relatively suppressing responses to nonpreferred stimuli. However, inhibition comparable to or stronger than excitation appears necessary to suppress spiking responses to nonpreferred orientations to the extent seen in vivo and to allow the emergence of a tuned mean voltage response. These two response properties provide the strongest constraints on model details. Antiphase inhibition from inhibitory simple cells, and not just untuned inhibition from inhibitory complex cells, appears necessary to fully explain these aspects of cortical orientation tuning.