We consider the role of nonlinear inhibition in physiologically realistic multineuronal models of V1 to predict the dipper functions from contrast discrimination experiments with sinusoidal gratings of different geometries. The dip in dipper functions has been attributed to an expansive transducer function, which itself is attributed to two nonlinear inhibitory mechanisms: contrast normalization and surround suppression. We ran five contrast discrimination experiments, with targets and masks of different sizes and configurations: small Gabor target/small mask, small target/large mask, large target/large mask, small target/in-phase annular mask, and small target/out-of-phase annular mask. Our V1 modeling shows that the results for small Gabor target/small mask, small target/large mask, large target/large mask configurations are easily explained only if the model includes surround suppression. This is compatible with the finding that an in-phase annular mask generates only little threshold elevation while the out-of-phase mask was more effective. Surrounding mask gratings cannot be equated with surround suppression at the receptive-field level. We examine whether normalization and surround suppression occur simultaneously (parallel model) or sequentially (a better reflection of neurophysiology). The Akaike Criterion Difference showed that the sequential model was better than the parallel, but the difference was small. The large target/large mask dipper experiment was not well fit by our models, and we suggest that this may reflect selective attention for its uniquely larger test stimulus. The best-fit model replicates some behaviors of single V1 neurons, such as the decrease in receptive-field size with increasing contrast.