The pre-Bötzinger complex, which is located at a ventrolateral medulla of human and mammal, is considered to be the center for the generation of respiratory rhythms. In a normal state, the respiratory rhythm is uniform and orderly. Otherwise, the respiratory rhythm will change to a pathological state. Therefore, the monitoring of respiratory rhythm is of great significance in monitoring the health. In this paper, according to a two-coupled model of pre-Bötzinger complex with calcium ion current, we investigate the generation and transition mechanism of anti-phase bursting synchronization by using phase-plane analysis, bifurcation and fast-slow decomposition. It is found that the pre-Bötzinger complex model can exhibit mixed bursting when calcium ion concentration is at steady state, which indicates that the oscillation of calcium is not a necessary condition for the generation of mixed bursting. This is quite different from the results obtained in previous studies, indicating that the mixed bursting is caused by the periodic fluctuations of calcium. The methods used in this paper can provide a new idea for investigating the dynamics of mixed bursting, and it can also be applied to the study of other neuronal systems on a multiple time scale.

The dynamics of neuronal firing activity is vital for understanding the pathological respiratory rhythm. Studies on electrophysiology show that the magnetic flow is an essential factor that modulates the firing activities of neurons. By adding the magnetic flow to Butera’s neuron model, we investigate how the electric current and magnetic flow influence neuronal activities under certain parametric restrictions. Using fast-slow decomposition and bifurcation analysis, we show that the variation of external electric current and magnetic flow leads to the change of the bistable structure of the system and hence results in the switch of neuronal firing pattern from one type to another.

<abstract><p>In this paper, the transition from anti-phase spike synchronization to in-phase spike synchronization within mixed bursters is investigated in a two-coupled pre-Bözinger complex (pre-BötC) network. In this two-coupled neuronal network, the communication between two pre-BötC networks is based on electrical and synaptic coupling. The results show that the electrical coupling accelerates in-phase spike synchronization within mixed bursters, but synaptic coupling postpones this kind of synchronization. Synaptic coupling promotes anti-phase spike synchronization when electrical coupling is weak. At the same time, the in-phase spike synchronization within dendritic bursters occurs earlier than that within somatic bursters. Asymmetric periodic somatic bursters appear in the transition state from anti-phase spikes to in-phase spikes. We also use fast/slow decomposition and bifurcation analysis to clarify the dynamic mechanism for the two types of synchronization.</p></abstract>