We propose a model of the primary auditory cortex (A1), in which each iso-frequency column is represented by a recurrent neural network with short-term synaptic depression. Such networks can emit Population Spikes, in which most of the neurons fire synchronously for a short time period. Different columns are interconnected in a way that reflects the tonotopic map in A1, and population spikes can propagate along the map from one column to the next, in a temporally precise manner that depends on the specific input presented to the network. The network, therefore, processes incoming sounds by precise sequences of population spikes that are embedded in a continuous asynchronous activity, with both of these response components carrying information about the inputs and interacting with each other. With these basic characteristics, the model can account for a wide range of experimental findings. We reproduce neuronal frequency tuning curves, whose width depends on the strength of the intracortical inhibitory and excitatory connections. Non-simultaneous two-tone stimuli show forward masking depending on their temporal separation, as well as on the duration of the first stimulus. The model also exhibits non-linear suppressive interactions between sub-threshold tones and broad-band noise inputs, similar to the hypersensitive locking suppression recently demonstrated in auditory cortex. We derive several predictions from the model. In particular, we predict that spontaneous activity in primary auditory cortex gates the temporally locked responses of A1 neurons to auditory stimuli. Spontaneous activity could, therefore, be a mechanism for rapid and reversible modulation of cortical processing.