## Type Ib Bursting

The final type of bursting that we discuss (although there are more) is a subclass of type I. It is important because although the models for this subclass of bursting have a similar underlying bifurcation structure to those for type I, the bursts nevertheless can behave quite differently.

In type Ib bursting (Fig. 6.8), the stable limit cycle surrounds all three fixed points, and the burst cycle is similar to that of type I (which we call type Ia from now on). When V is on the lower branch of the Z-shaped curve, c decreases, as V lies below the c nullcline. As c decreases, the solution crosses the saddle-node bifurcation at the lower knee of the Z-shaped curve and jumps to the branch of stable periodic orbits. Although c does not increase monotonically during each oscillation, the average value of V is

Figure 6.8 Bifurcation diagram of the fast subsystem for a type Ib burster. HB denotes a Hopf bifurcation, HC denotes a homo-clinic bifurcation, and SN denotes a saddle-node bifurcation. The position of the homoclinic bifurcation is denoted by chc. Type Ib is similar to type la, with the major difference that the stable periodic orbit surrounds all three steady states.

Figure 6.8 Bifurcation diagram of the fast subsystem for a type Ib burster. HB denotes a Hopf bifurcation, HC denotes a homo-clinic bifurcation, and SN denotes a saddle-node bifurcation. The position of the homoclinic bifurcation is denoted by chc. Type Ib is similar to type la, with the major difference that the stable periodic orbit surrounds all three steady states.

Figure 6.9 Comparison of type la (A) and type Ib (B) bursting in the Chay-Cook model. Although this model is not discussed in detail in the text, these numerical solutions of Bertram et al. (1995) provide an excellent comparison of the two bursting types. (Bertram et al., 1995, Fig. 3.)

high enough to cause a net increase in c over each cycle. Thus, the solution moves to the right until the branch of periodic orbits disappears at a homoclinic bifurcation, at which time the solution reverts to the lower branch of the Z-shaped curve, completing the burst cycle. In Fig. 6.9 we compare type Ia and Ib bursting patterns. The numerical simulations are from a model not discussed here (Chay and Cook, 1988), but the figure serves as an excellent comparison of the bursting types. In type Ia, the burst pattern is superimposed on a high-voltage baseline, forming a square-wave pattern as seen in pancreatic ^-cells. In type Ib, the minimum of the fast oscillation lies below the lower branch of the Z-shaped curve, and thus the minimum of the fast oscillation lies below the quiescent phase.