Intracellular studies have revealed the importance of cotuned excitatory and inhibitory inputs to neurons in auditory cortex, but typical spectrotemporal receptive field models of neuronal processing cannot account for this overlapping tuning. Here, we apply a new nonlinear modeling framework to extracellular data recorded from primary auditory cortex (A1) that enables us to explore how the interplay of excitation and inhibition contributes to the processing of complex natural sounds. The resulting description produces more accurate predictions of observed spike trains than the linear spectrotemporal model, and the properties of excitation and inhibition inferred by the model are furthermore consistent with previous intracellular observations. It can also describe several nonlinear properties of A1 that are not captured by linear models, including intensity tuning and selectivity to sound onsets and offsets. These results thus offer a broader picture of the computational role of excitation and inhibition in A1 and support the hypothesis that their interactions play an important role in the processing of natural auditory stimuli.