Predictive processing (e.g., predictive coding) is a predominant paradigm in cognitive neuroscience. This Primer considers the various levels of commitment neuroscientists have to the neuronal process theories that accompany the principles of predictive processing. Specifically, it reviews and contextualises a recent PLOS Biology study of alpha oscillations and travelling waves. We will see that alpha oscillations emerge naturally under the computational architectures implied by predictive coding-and may tell us something profound about recurrent message passing in brain hierarchies. Specifically, the bidirectional nature of forward and backward waves speaks to opportunities to understand attention and how it nuances bottom-up and top-down influences.