Dynamical Systems in Neuroscience:

Bursting 3819.3.4 fold/fold cycleWhen the stable equilibrium corresponding to the resting state disappears via saddlenode(fold) bifurcation and the limit cycle attractor corresponding to the spiking statedisappears via fold limit cycle bifurcation, the burster is said to be of the “fold/foldcycle” type, as in Fig. 9.43. This type was first discovered in the Chay-Cook (1988)model of a pancreatic β-cell by Bertram et al. (1995), who referred to it as being TypeIV bursting (the three bursters we considered so far were referred to as being TypeI, II, and III, respectively). Since both bifurcations result in a co-existence of restingand spiking states, the “fold/fold cycle” bursting can occur via a hysteresis loop in a“2+1” system.An interesting geometrical feature of the “fold/fold cycle” bursting is that thereis an unstable limit cycle that appears in the middle of a burst and that participatesin the “fold cycle” bifurcation to terminate the burst. The cycle appears via saddlehomoclinic orbit bifurcation in Fig. 9.43, but other scenarios are possible too. It is agood exercise of one’s geometrical intuition and understanding of the fast-slow burstingmechanisms to come up with alternative scenarios of the “fold/fold cycle” bursting.For example, consider the case of the unstable limit cycle being inside the stable one.9.3.5 fold/HopfWhen the stable equilibrium corresponding to the resting state disappears via saddlenode(fold) bifurcation and the limit cycle attractor corresponding to the spiking stateshrinks to a point via supercritical Andronov-Hopf bifurcation, the burster is said to beof the “fold/Hopf” type; see Fig. 9.44. This type of bursting, called “tapered” in someearlier studies, was found in models of insulin-producing pancreatic β-cells (Smolen etal. 1993, Pernarowski 1994) and in models of certain enzymatic systems (Holden andErneux 1993a,b).As one can see in the figure, the fast subsystem undergoes two bifurcations whileit is in the excited state: One corresponds to the termination of repetitive spikingvia supercritical Andronov-Hopf bifurcation, and the other one corresponds to thetransition from the excited equilibrium to resting equilibrium via saddle-node (fold)bifurcation. The first bifurcation, i.e., bifurcation of a spiking limit cycle attractor,determines the topological type of bursting. The second bifurcation is essential forthe “fold/fold” hysteresis loop, and it only determines the subtype of the “fold/Hopf”bursting. Using ideas described in Ex. 19, one can come up with another subtype of“fold/Hopf” burster having “fold/subHopf” hysteresis loop.9.3.6 fold/circleWhen the stable equilibrium corresponding to the resting state disappears via saddlenode(fold) bifurcation and the limit cycle attractor corresponding to the spiking statedisappears via saddle-node on invariant circle bifurcation, the burster is said to be ofthe “fold/circle” type, as in Fig. 9.45. This type was first discovered in the model of