The central nervous system extensively expresses Ca²⁺-stimulated K⁺ channels, which serve to use Ca²⁺ to control their opening and closing. In this study, we explore the numerical computation of Hopf bifurcation in the Chay model based on the equilibrium point's stability and the center manifold theorem to illustrate the emergence of complicated neuronal bursting induced by variation of the conductance of the Ca²⁺-sensitive K⁺ channel. The results show that the formation and removal of various firing activities in this model are due to two subcritical Hopf bifurcations of equilibrium based on theoretical computation. Furthermore, the computational simulations are shown to support the validity of the conceptual approach. Consequently, the conclusion could be helpful to improve and deepen our understanding of the contribution of the Ca²⁺-sensitive K⁺ channel.

In this work, we focus on a nonlinear dynamical model proposed by Lavrentovich et al. to compute and simulate spontaneous Ca²⁺ oscillations evoked by calcium ion efflux in astrocytes. Selected parameters are chosen, with observation of periodic and chaotic Ca²⁺ oscillations in cytosol. The stability analysis of equilibrium is conducted using the center manifold theorem to investigate the dynamics underlying spontaneous Ca²⁺ oscillations in astrocytes. The results indicate that the Hopf bifurcation represents the dynamical changes in stability of spontaneous Ca²⁺ oscillations. In addition, numerical simulations are performed to further assess the validity of the aforementioned analysis.