Neocortical layer 5 thick-tufted pyramidal cells are prone to exhibiting burst firing on receipt of coincident basal and apical dendritic inputs. These inputs carry different information, with basal inputs coming from feedforward sensory pathways and apical inputs coming from diverse sources that provide context in the cortical hierarchy. We explore the information processing possibilities of this burst firing using computer simulations of a noisy compartmental cell model. Simulated data on stochastic burst firing due to brief, simultaneously injected basal and apical currents allow estimation of burst firing probability for different stimulus current amplitudes. Information-theory-based partial information decomposition (PID) is used to quantify the contributions of the apical and basal input streams to the information in the cell output bursting probability. Four different operating regimes are apparent, depending on the relative strengths of the input streams, with output burst probability carrying more or less information that is uniquely contributed by either the basal or apical input, or shared and synergistic information due to the combined streams. We derive and fit transfer functions for these different regimes that describe burst probability over the different ranges of basal and apical input amplitudes. The operating regimes can be classified into distinct modes of information processing, depending on the contribution of apical input to output bursting: apical cooperation, in which both basal and apical inputs are required to generate a burst; apical amplification, in which basal input alone can generate a burst but the burst probability is modulated by apical input; apical drive, in which apical input alone can produce a burst; and apical integration, in which strong apical or basal inputs alone, as well as their combination, can generate bursting. In particular, PID and the transfer function clarify that the apical amplification mode has the features required for contextually modulated information processing.