Dynamical mechanisms and transitions of mixed-mode bursting in a closed-loop respiratory control model

Abnormal respiratory rhythms, a hallmark of many respiratory diseases, largely arise from transitional firing dynamics in the pre-Bötzinger complex (pre-BötC) neurons. Mixed-mode bursting (MMB), characterized by alternating depolarization block (DB) and square-wave (SW) bursts, represents a particularly complex firing pattern of significant research interest. Based on Diekman's closed-loop respiratory control model, this study applies a fast-slow dynamical analysis and the bifurcation theory to investigate how variations in hemoglobin concentration ([Hb]) govern the emergence and disappearance of MMB. Numerical simulations show that [Hb] variations induce clear transitions between MMB and single SW bursting. A dynamical analysis reveals that the intrinsic slow variable $h$ determines the type of individual bursting, while the slow feedback variable $g_{tonic}$, which is regulated by the arterial blood oxygen partial pressure, evolves on a comparable timescale and influences the global trajectories by shifting the critical bifurcation structures. The interaction between these slow variables enables the emergence of MMB. When the slow feedback pathway is fixed such that $g_{tonic}$ remains constant, MMB can no longer be sustained. These results indicate that MMB in closed-loop systems relies on the dynamic coupling of multiple slow variables rather than on the static value of a single parameter, thus providing a dynamical mechanism for complex rhythm generation.