Perception is strongly affected by the intrinsic state of the brain, which controls the propensity to either maintain a particular perceptual interpretation or switch to another. To understand the mechanisms underlying the spontaneous drive of the brain to explore alternative interpretations of unchanging stimuli, we repeatedly recorded high-density EEG after normal sleep and after sleep deprivation while participants observed a Necker cube image and reported the durations of the alternating representations of their bistable perception. We found that local alpha power around the parieto-occipital sulcus within the first second after the emergence of a perceptual representation predicted the fate of its duration. An experimentally induced increase in alpha power by means of sleep deprivation increased the average duration of individual representations. Taken together, these findings show that high alpha power promotes the stability of a perceptual representation and suppresses switching to the alternative. The observations support the hypothesis that synchronization of alpha oscillations across a wide neuronal network promotes the maintenance and stabilization of its current perceptual representation. Elevated alpha power could also be key to the poorly understood cognitive deficits, that typically accompany sleep deprivation, such as the loss of mental flexibility and lapses of responsiveness.