Bifurcation analysis on the effect of store-operated and receptor-operated calcium channels for calcium oscillations in astrocytes

Abstract

Experimental evidence has proved that calcium ions ( $$\mathrm {Ca^{2+}}$$ ) play an important role in cellular physiological processes via calcium oscillations. The entry rate of $$\mathrm {Ca^{2+}}$$ into cells through plasma membrane cells is a major modulator of intracellular $$\mathrm {Ca^{2+}}$$ dynamics, including the voltage-gated $$\mathrm {Ca^{2+}}$$ channel, the store-operated $$\mathrm {Ca^{2+}}$$ channel (SOCC) and the receptor-operated $$\mathrm {Ca^{2+}}$$ channel (ROCC). In this paper, we modify an established four-dimensional dynamical model, which contains the SOCC and ROCC, and carry out a bifurcation analysis to study dynamics of the model. In particular, Hopf bifurcation is identified with the maximum flow of the SOCC chosen as the bifurcation parameter, and normal form theory is applied to consider the stability of bifurcating limit cycles. Bifurcation of multiple limit cycles arising from generalized Hopf bifurcation is also discussed, which may yield complex dynamical behaviors. Further, it is shown that the variation of the maximum flows for different calcium channels determines the parameter range for stable oscillations, as well as for the frequency and amplitude of oscillations. The results indicate that Hopf bifurcation is the main source to generate oscillating behaviors, yielding a different bistable phenomenon which involves stable limit cycle and stable equilibrium. Moreover, it is shown that partially blocking the SOCC or the ROCC can change the parameter region of stable calcium oscillations, and the ROCC has more impact than the SOCC on amplitude or frequency of calcium oscillations.