Dynamical Systems in Neuroscience:

Bursting 401bifurcations of limit cyclessaddle-nodeon invariantcirclesaddlehomoclinicorbitsupercriticalAndronov-Hopffoldlimitcyclebifurcations of equilibriasaddle-node(fold)saddle-nodeon invariantcirclesupercriticalAndronov-HopfsubcriticalAndronov-Hopfsaddle-nodehomoclinic orbitv' = I+v2+u 1u 1 ' = -µ 1 u 2u 2 ' = -µ 2 (u 2 -u 1 )v' = I+v2-uu' = -µuBautinz'=(u+iω)z+2z|z| 2 -z|z| 4u'=µ(a-|z| 2 )Figure 9.63: Some canonical models of fast-slow bursters; see Ex. 26.Such a bursting is called cycle-cycle. Classify all co-dimension-1 planar cyclecyclefast-slow bursters. Is bursting in Fig. 9.10 of cycle-cycle type?22. (Minimal models for bursting) Fill in the blank squares in Fig. 9.8.23. Choose a minimal model from Fig. 9.8 and simulate it. Change the parametersto get as many different bursting types as possible.24. [M.S.] Determine the bifurcation diagram of the canonical model for “fold/homoclinic”bursting (9.7).25. [M.S.] Determine the bifurcation diagrams of the canonical models for “circle/circle”bursters (9.9) and (9.10).26. [Ph.D.] Consider fast-slow bursters of the form (9.1) and assume that the fastsubsystem is near a bifurcation of high co-dimension, as in Fig. 9.28 or in Fig. 9.42.Treating the bifurcation point as an organizing center for the fast subsystem(Bertram et al. 1995, Izhikevich 2000, Golubitsky et al. 2001), use unfoldingtheory to derive canonical models for the remaining fast-slow bursters in Fig. 9.63.Do not assume that the slow subsystem has an autonomous oscillation or thatthe fast oscillations have small amplitude.27. [Ph.D.] Classify all possible mechanisms of emergence of bursting oscillationsfrom resting or spiking, as in Fig. 9.19.