Investigations into the neural basis of conscious perception span multiple scales and levels of analysis. There is, however, a theoretical and methodological gap between advances made at the microscopic scale in animal models and those made at the macroscopic scale in human cognitive neuroscience that places a fundamental limit on our understanding of the neurobiological basis of consciousness. Here, we use computational modelling to bridge this gap. Specifically, we show that the same mechanism that underlies threshold detection in mice - apical dendrite mediated burst firing in thick-tufted layer V pyramidal neurons - determines perceptual dominance in a thalamocortical model of binocular rivalry - a staple task in the cognitive neuroscience of consciousness. The model conforms to the constraints imposed by decades of previous research into binocular rivalry and generalises to the more sophisticated rivalry tasks studied in humans generating novel, testable, explanations of the role of expectation and attention in rivalry. Our model, therefore, provides an empirically-tractable bridge between cellular-level mechanisms and conscious perception.