Synchronization transition from bursting to spiking and bifurcation mechanism of the pre-Botzinger complex

The pre-Botzinger complex is a neuronal network with excitatory coupling, which participates in modulation of respiratory rhythms via the generation of complex firing rhythm patterns and synchronization transitions of rhythm patterns. In the present paper, a mathematical model of single neuron that exhibits complex transition processes from bursting to spiking is selected as a unit, the network model of the pre-Botzinger complex composed of two neurons with excitatory coupling is constructed, multiple synchronous rhythm patterns and complex transition processes of the synchronous rhythm patterns related to the biological experimental observations are simulated, and the corresponding bifurcation mechanism is acquired with the fast-slow variable dissection method. When the initial values of two neurons of the pre-Botzinger complex are the same, with increasing the excitatory coupling strength, the theoretical model of the pre-Botzinger complex shows complete synchronization transition processes from fold/homoclinic bursting, to subHopf/subHopf bursting, and at last to period-1 spiking. When the initial values are different, with the increases of the excitatory coupling intensity, the rhythm transition processes begin from phase synchronization behaviors including fold/homoclinic bursting, fold/fold limit cycle bursting, mixed bursting composed of subHopf/subHopf bursting and fold/fold limit cycle bursting, and subHopf/subHopf bursting in sequence, and to anti-phase synchronous behavior of the period-1 spiking. The complete (in-phase) synchronous period-1 spiking for the same initial values exhibits bifurcation mechanism different from the anti-phase synchronous period-1 spiking for different initial values. The anti-phase synchronous period-1 spiking presents a novel and abnormal example of the synchronization at large excitatory coupling strength, which is different from the traditional viewpoint that large excitatory coupling often induces in-phase synchronous behavior. The results present the synchronization transition process and complex bifurcation mechanism from bursting to period-1 spiking of the pre-Botzinger complex, and the abnormal synchronization example enriches the contents of nonlinear dynamics.