Evaluation of bifurcation phenomena in a modified Shen–Larter model for intracellular $$\hbox {Ca}^{\varvec{2+}}$$ Ca 2 + bursting oscillations

Abstract

The present work describes an evaluation of the bifurcation phenomena in a modified Shen–Larter model based on calcium-induced calcium release and inositol triphosphate $$(\hbox {IP}_{3})$$ crosscoupling for calcium ion $$(\hbox {Ca}^{2+})$$ bursting oscillations. A time delay for negative $$\hbox {Ca}^{2+}$$ feedback on the $$(\hbox {IP}_{3})$$ receptor is added to the original Shen–Larter model, by introducing the proportion of receptors not inactivated by $$\hbox {Ca}^{2+}$$ as a new variable. Compared with the original model, the number of chaotic regions for a stimulation level r is significantly reduced, and regions of $$\hbox {Ca}^{2+}$$ oscillations (particularly bursting) appear to become slightly enlarged. Different topological types of bursting oscillations in this modified model are classified by fast/slow dynamical analysis and codimension-2 bifurcations of fast subsystem, when choosing better of two slow variables the free $$\hbox {Ca}^{2+}$$ concentration in the endoplasmic reticulum and the $$\hbox {IP}_{3}$$ concentration in the cytosol. Furthermore, classification and transition mechanisms of bursting $$\hbox {Ca}^{2+}$$ oscillations could help to understand or detect more distinctive oscillatory behaviors of real cells in response to different levels of stimulation.