Bifurcation and Spike Adding Transition in Chay–Keizer Model

Electrical bursting is an activity which is universal in excitable cells such as neurons and various endocrine cells, and it encodes rich physiological information. As burst delay identifies that the signal integration has reached the threshold at which it can generate an action potential, the number of spikes in a burst may have essential physiological implications, and the transition of bursting in excitable cells is associated with the bifurcation phenomenon closely. In this paper, we focus on the transition of the spike count per burst of the pancreatic β-cells within a mathematical model and bifurcation phenomenon in the Chay–Keizer model, which is utilized to simulate the pancreatic β-cells. By the fast–slow dynamical bifurcation analysis and the bi-parameter bifurcation analysis, the local dynamics of the Chay–Keizer system around the Bogdanov–Takens bifurcation is illustrated. Then the variety of the number of spikes per burst is discussed by changing the settings of a single parameter and bi-parameter. Moreover, results on the number of spikes within a burst are summarized in ISIs (interspike intervals) sequence diagrams, maximum and minimum, and the number of spikes under bi-parameter value changes.