Dynamical Systems in Neuroscience:

Bursting 365bifurcations of limit cyclessaddle-nodeon invariantcirclesaddlehomoclinicorbitsupercriticalAndronov-Hopffoldlimitcyclesaddle-node(fold)fold/circlefold/homoclinicfold/Hopffold/fold cyclebifurcations of equilibriasaddle-nodeon invariantcirclesupercriticalAndronov-HopfsubcriticalAndronov-Hopfcircle/circleHopf/circlesubHopf/circlecircle/homoclinicHopf/homoclinicsubHopf/homocliniccircle/HopfHopf/HopfsubHopf/Hopfcircle/fold cycleHopf/fold cyclesubHopf/fold cycleFigure 9.23: Classification of planar point-cycle fast-slow bursters based on the codimention-1bifurcations of the resting and spiking states of the fast subsystem.9.3.1 fold/homoclinicWhen the resting state disappears via a saddle-node (fold) bifurcation and the spikinglimit cycle disappears via saddle homoclinic orbit bifurcation, the burster is said to beof the “fold/homoclinic” type depicted in Fig. 9.25. Notice the bistability of restingand spiking states, resulting in hysteresis loop oscillation of the slow subsystem.“Fold/homoclinic” bursting is quite common in neuronal models, for example inthe I Na,p +I K +I K(M) -model considered in this chapter; see Fig. 9.13. It was first characterizedin the context of the insulin-producing pancreatic β-cells in Fig. 9.26, withintracellular concentration of Ca 2+ ions being the slow resonant variable (Chay andKeizer 1983). Neurons located in the pre-Botzinger complex, a region that is associatedwith generating the rhythm for breathing, also exhibit this kind of bursting(Butera et al. 1999), as shown in Fig. 9.27. Intrinsic bursting (IB) and chattering(CH) behavior of the simple model in Sect. 8.2 could be of the “fold/homoclinic” typetoo, provided that the parameter a is sufficiently small. Because of the distinct squarewaveshape of oscillations of the membrane potential in Figures 9.26 and Fig. 9.27,this bursting was called “square-wave” bursting in earlier studies. Since many types ofbursters resemble square waves, referring to a burster by its shape is misleading andshould be avoided.In Fig. 9.25, bottom, we depict a typical configuration of nullclines of the fastsubsystem during “fold/homoclinic” bursting. The resting state of the membrane po-