Dynamical Systems in Neuroscience:

400 BurstingFigure 9.61: A cycle-cycle bursting:The resting state is not an equilibrium,but a small-amplitude limit cycle attractor."Fold/Fold" Bursting"Fold/SubHopf" BurstingUp-State(Rest)FoldBifurcationUp-State(Rest)FoldBifurcationFoldBifurcationSaddle HomoclinicOrbit BifurcationSubcriticalAndronov-HopfBifurcationxuDown-State(Rest)xuDown-State(Rest)FoldSubcriticalAndronov-HopfBifurcationFoldFoldSaddle HomoclinicOrbit BifurcationFigure 9.62: Two examples of point-point (not fast-slow) bursters (modified fromIzhikevich 2000).19. (Point-point hysteresis loops) Consider (9.1) and suppose that the fast subsystemhas only equilibria for any value of the one-dimensional slow variable u. If there isa co-existence of equilibrium points of the fast subsystem, then (9.1) can exhibitpoint-point hysteresis loop oscillation. Classify all co-dimension-1 point-pointhysteresis loops.20. (Point-point bursting) In Fig. 9.62 we present two geometrical examples of pointpointbursters that have no limit cycle attractors, yet are capable of exhibitingspike-like dynamics in the active phase. Construct a model for each type of pointpointburster in the figure. Use phase portrait snapshots at the bottom of thefigure as hints. What makes such bursting possible?21. (Cycle-cycle bursters) Consider a fast-slow burster (9.1) and suppose that theresting state is not an equilibrium, but a limit cycle attractor, as in Fig. 9.61.