In this paper we use a continuum model in two spatial dimensions to study the dynamics of the cortex during natural sleep, including explicitly the effects of two key neuromodulators. The model predicts that a number of states could be available to the cortex. We identify two of these with slow-wave sleep and rapid eye movement (REM) sleep, and focus on the transition between the two. Eigenvalue analysis of the linearized model, together with simulations on a two-dimensional grid, show that a number of oscillatory states exist; the occurrence of these is particularly dependent upon the duration in time of the inhibitory postsynaptic potential. These oscillatory states are similar to the cortical slow oscillation and certain types of seizure. Power spectra are evaluated for different parameter sets and compare favorably with experiment. Grid simulations show that transitions between cortical states (e.g., slow-wave to REM) can be seeded at any point in space by random fluctuations in subcortical input.