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Bifurcation Mechanisms of Electrical bursting with Different-Time-Scale Slow Variables

A compositive pancreatic β-cells model is developed by focusing on two important slow negative feedback mechanisms, where three slow processes Css, Cer and a with different time scales can interact to drive fast, medium and slow electrical bursting typically observed in pancreatic β-cells. We found that diverse patterns of electrical bursting with shapes varying from only waves to the coexistence of waves and spikes to only spikes, which are robust to fast, medium and slow oscillation periods. Therefore, dynamic bifurcation mechanisms of these bursting patterns could be surveyed by fast/slow analysis of only fast bursting. Here, the faster slow variable Css is regarded as a bifurcation parameter and Cer has no effect on the fast subsystem, while the slower variable a is nearly constant. In the light of all bifurcations associated with bursting, three topological types of bursting are attained, that is, "fold/fold" point–point hysteresis loop bursting, "fold/Hopf" bursting via the "fold/fold" hysteresis loop and "fold/homoclinic" bursting via the "fold/homoclinic" hysteresis loop.