When a stimulus supports two distinct interpretations, perception alternates in an irregular manner between them. What causes the bistable perceptual switches remains an open question. Most existing models assume that switches arise from a slow fatiguing process, such as adaptation or synaptic depression. We develop a new, attractor-based framework in which alternations are induced by noise and are absent without it. Our model goes beyond previous energy-based conceptualizations of perceptual bistability by constructing a neurally plausible attractor model that is implemented in both firing rate mean-field and spiking cell-based networks. The model accounts for known properties of bistable perceptual phenomena, most notably the increase in alternation rate with stimulation strength observed in binocular rivalry. Furthermore, it makes a novel prediction about the effect of changing stimulus strength on the activity levels of the dominant and suppressed neural populations, a prediction that could be tested with functional MRI or electrophysiological recordings. The neural architecture derived from the energy-based model readily generalizes to several competing populations, providing a natural extension for multistability phenomena.