Dynamical Systems in Neuroscience:

364 Bursting9.3 ClassificationIn Fig. 9.22 we identify two important bifurcations of the fast subsystem that areassociated with bursting activity in the fast-slow burster (9.1):• (resting → spiking) Bifurcation of an equilibrium attractor that results intransition from resting to repetitive spiking.• (spiking → resting) Bifurcation of the limit cycle attractor that results intransition from spiking to resting.The ionic basis of bursting, i.e., the fine electrophysiological details, determine thekind of bifurcations in Fig. 9.22. The bifurcations, in turn, determine the neurocomputationalproperties of fast-slow bursters, discussed in Sect. 9.4.A complete topological classification of bursters based on these two bifurcations isprovided by Izhikevich (2000), who identified 120 different topological types. Here, weconsider only 16 planar point-cycle co-dimension-1 fast-slow bursters. We say that afast-slow burster is planar when its fast subsystem is two-dimensional. We emphasizeplanar bursters because they have a greater chance to be encountered in computersimulations (but not necessarily in nature). We say that a burster is of the pointcycletype when its resting state is a stable equilibrium point and its spiking state is astable limit cycle. All bursters considered so far, including those in Fig. 9.1, are of thepoint-cycle type. Other, less common types, such as cycle-cycle and point-point, areconsidered as exercises.We consider here only bifurcations of co-dimension 1, i.e. those that need onlyone parameter and hence are more likely to be encountered in nature. Having a twodimensionalfast subsystem imposes severe restriction on possible co-dimension-1 bifurcationsof the resting and spiking states. In particular, there are only 4 bifurcations ofequilibria and 4 bifurcations of limit cycles, which we consider in Chap. 6 and summarizein Figures 6.46 and 6.47. Any combination of them results in a distinct topologicaltype of fast-slow bursting, hence there are 4 × 4 = 16 such bursters, summarized inFig. 9.23.We name the bursters according to the type of the bifurcations of the resting andspiking states. To keep the names short, we refer to saddle-node on invariant circlebifurcation as just a “circle” bifurcation because it is the only co-dimension-1 bifurcationon a circle manifold S 1 . We refer to supercritical Andronov-Hopf bifurcationas just “Hopf” bifurcation, to subcritical Andronov-Hopf as “subHopf”, to fold limitcycle bifurcation as “fold cycle”, and to saddle homoclinic orbit bifurcation as “homoclinic”bifurcation. Thus, the bursting pattern exhibited by the I Na,p +I K +I K(M) -modelin Fig. 9.13 is of the “fold/homoclinic” type because the resting state disappears via“fold” bifurcation and the spiking limit cycle attractor disappears via saddle “homoclinic”orbit bifurcation.Similarly to Fig. 9.13, we depict the geometry of the other bursters in Fig. 9.24.This figure gives only examples and it does not exhaust all possibilities. Let us considersome most common bursting types in detail.