Dynamical Systems in Neuroscience:

Bursting 393Saddle-Node Saddle Supercritical FoldBifurcations on Invariant Homoclinic Andronov- LimitCircle Orbit Hopf Cycletriangular square-wave taperedFold Type I Type V Type IVSaddle-Nodeon InvariantCircleparabolicType IISupercriticalAndronov-HopfSubcriticalAndronov-HopfellipticType IIIFigure 9.53: Bifurcation mechanisms and classical nomenclature of the 6 burstersknown in the XX century. Compare with Fig. 9.23 and Fig. 9.24.The complete classification of bursters was provided by Izhikevich (2000), and itwas actually motivated by the paper of Guckenheimer et al. (1997). There is a drasticdifference between Izhikevich’s approach, and that of the scientists mentioned above.They used a bottom-up approach; that is, they considered biophysically plausible conductancebased models describing experimentally observable cellular behavior and thenthey determined the types of bursting these models exhibited. In contrast, Izhikevich(2000) used the top-down approach and considered all possible pairs of co-dimension-1bifurcations of rest and spiking states, which resulted in different types of bursting. Itwas an easy task to provide a conductance-based model exhibiting each bursting type.Thus, many of the bursters are “theoretical” in the sense that they have yet to be seenin experiments.Interestingly, a challenging problem was to suggest a naming scheme for the bursters.The names should be self-explanatory and easy to remember and understand. Thus,the numbering scheme suggested by Bertram et al. (1995) would lead, e.g., to burstersof Type XXVII, Type LXIII, Type LCXVI, etc. We cannot use descriptions such as“elliptic”, “parabolic”, “hyperbolic”, “triangular”, “rectangular”, etc., since they aremisleading. In this book we follow Izhikevich (2000) and name the bursters accordingto the two bifurcations involved, as in Fig. 9.23.Not all bursters can be represented in the fast-slow form with a clear separation