Dynamical Systems in Neuroscience:

368 BurstingFigure 9.26: Putative “fold/homoclinic” bursting in a pancreatic β-cell (modified fromKinard et al. 1999).0 mV20 mV1 secFigure 9.27: Putative ”fold/homoclinic” bursting in a cell located in pre-Botzingercomplex of rat brain stem (data kindly shared by Christopher A. Del Negro and JackL. Feldman, Systems Neurobiology Laboratory, Department of Neurobiology, UCLA.)Suppose that the hysteresis loop oscillation of the slow variable u has a small amplitude.That is, the saddle-node bifurcation and the saddle homoclinic orbit bifurcationoccur for nearby values of the parameter u. In this case, the fast subsystem of (9.1)is near co-dimension-2 saddle-node homoclinic orbit bifurcation, depicted in Fig. 9.28and studied in Sect. 6.3.6. The figure shows a two-parameter unfolding of the bifurcation,treating u ∈ R 2 as the parameter. A stable equilibrium (resting state) exists inthe left half-plane, and a stable limit cycle (spiking state) exists in the right half-planeof the figure and in the shaded (bistable) region. “Fold/homoclinic” bursting occurswhen the bifurcation parameter, being a slow variable, oscillates between the restingand spiking states through the shaded region. Due to the bistability, the parametercould be one-dimensional. Other trajectories of the slow parameter correspond to othertypes of bursting.In Ex. 16 we prove that there is a piece-wise continuous change of variables thattransforms any “fold/homoclinic” burster with fast subsystem near such a bifurcationinto the canonical model (see Sect. 8.1.5)˙v = I + v 2 − u ,˙u = −µu ,(9.7)