The local behaviour of the global attractor can be partially correlated with the bifurcation diagram of the fast subsystem. Figure 3 shows the evolution of the global attractor around the bifurcation curve. The alternation between the silent and the active phase of the bursting process is the result of the global attractor that jumps between a branch of spiralling-type equilibria (active phase) and another branch of node-type equilibria (silent phase).

Figure 3 The evolution of the global attractor around the bifurcation curve.

The cell remains quiescent during the silent phase when the global attractor traverses along the branch of stable nodes. This branch loses its stability via a saddle-node bifurcation where it meets with the branch of saddle equilibria (green dotted line). Subsequently, the global attractor jumps to the branch of stable spiralling equilibria, the only remaining branch of attracting equilibria after the saddle-node bifurcation. This movement initiates the active phase where Ca starts to rise. The global attractor oscillates and traverses with some overshoots and increasingly damped oscillations. At the right end of the branch, there is a family of unstable limit cycles, emanating from a subcritical Hopf  bifurcation. However, the family of limit cycles plays no role in the structure of the global attractor as the global attractor drops down back to the stable nodes and terminates the active phase before it reaches the limit cycles. We provide an explanation for the termination of the active phase by investigating the structure of the stable manifolds of the saddle equilibria.