Dynamical Systems in Neuroscience:

spiking380 Burstingfoldbifurcationsaddlehomoclinicorbitbifurcationfold limit cyclebifurcationxurestfoldsaddlehomoclinic orbitfold limitcycleFigure 9.43: “Fold/fold cycle” bursting: The resting state disappears via saddle-node(fold) bifurcation and the spiking limit cycle disappears via fold limit cycle bifurcation(modified from Izhikevich 2000).bifurcation of the fast subsystem in (9.1) occur for nearby values of the parameter u.In this case, the fast subsystem is near a co-dimension-2 Bautin bifurcation, whichwas studied in Sect. 6.3.5. Its two-parameter unfolding is depicted in Fig. 9.42, left.“SubHopf/fold cycle” bursting occurs when the bifurcation parameter, being a slowvariable, oscillates between the resting and spiking regions through the shaded region.Due to the bistability, the parameter could be one-dimensional. Other trajectories ofthe slow parameter correspond to other types of bursting shown in Fig. 9.42, right.If the slow variable has an equilibrium near the Bautin bifurcation point, then thefast-slow burster (9.1) can be transformed into the following canonical “2+1” modelby a continuous change of variablesż = (u + iω)z + 2z|z| 2 − z|z| 4 ,˙u = µ(a − |z| 2 ) ,(9.11)where z ∈ C and u ∈ R are the canonical fast and slow variables, respectively, and a, ωand µ ≪ 1 are parameters. In Ex. 14 we show that the model exhibits hysteresis loopperiodic point-cycle bursting behavior depicted in Fig. 9.37 when 0 < a < 1.