The preBötzinger complex (preBötC) of the mammalian brainstem is a heterogeneous neuronal network underlying the inspiration phase of the respiratory rhythm. Through excitatory synapses and a nontrivial network architecture, a synchronous, network-wide bursting rhythm emerges. On the other hand, during synaptic isolation, preBötC neurons display three types of intrinsic dynamics: quiescence, bursting, or tonic activity. This work seeks to shed light on how the network rhythm emerges from the challenging architecture and heterogeneous population. Recent debate surrounding the role of intrinsically bursting neurons in the rhythmogenesis of the preBötC inspires us to evaluate its role in a three-cell network. We found no advantage for intrinsically bursting neurons in forming synchronous network bursting; instead, intrinsically quiescent neurons were identified as a key mechanism. This analysis involved only studying the persistent sodium (NaP) current. Another important current for the preBötC is the calcium-activated nonspecific cationic (CAN) current, which, when combined with a Na/K pump, was previously shown to be capable of producing bursts in coupled tonically active cells.

In the second part of this study, we explore the interactions of the NaP and CAN currents, both currents are ubiquitous in the preBötC. Using geometric singular perturbation theory and bifurcation analysis, we established the mechanisms through which reciprocally coupled pairs of neurons can generate various activity patterns. In particular, we highlighted how the NaP current could enhance the range of the strength of the CAN current for which bursts occur. We also were able to detail a novel bursting pattern seen in data, but not seen in previous models.

With a foundation of understanding heterogeneity in the NaP and CAN currents, we again turned our attention to networks. For the third portion of the dissertation, we examine the effects that heterogeneity in the neuronal dynamics and coupling architecture can impose upon synchronous bursting of the entire network. We again found no significant advantage to including intrinsically bursting neurons in the network, and the best networks were characterized by an increased presence of quiescent neurons. We also described the way the NaP and CAN currents interact on the network scale to promote synchronous bursting.