An exploration is given of neural network features now being uncovered in cortical processing which begins to go a little way to help bridge the "Explanatory Gap" between phenomenal consciousness and correlated brain activity. A survey of properties suggested as being possessed by phenomenal consciousness leads to a set of criteria to be required of the correlated neural activity. Various neural styles of processing are reviewed and those fitting the criteria are selected for further analysis. One particular processing style, in which semiautonomous and long-lasting cortical activity "bubbles" are created by input, is selected as being the most appropriate. Further experimental criteria are used to help narrow the possible neural styles involved. This leads to a class of neural models underpinning phenomenal consciousness and to a related set of testable predictions.