Dynamical Systems in Neuroscience:

394 BurstingspikingrestingFigure 9.54: A hedgehog-like limit cycle attractor results in bursting dynamics even intwo-dimensional systems; see Ex. 1 (modified from Hoppensteadt and Izhikevich 1997).of the time scales. Those that cannot are referred to as hedgehog bursters (Izhikevich2000), since they have a limit cycle (or a more complicated attractor) with some spikyparts corresponding to repetitive spiking and some smooth parts corresponding toquiescence, as in Fig. 9.54. An interesting example of the hedgehog burster is themodel of sensory processing neuron of weakly electric fish (Doiron et al. 2002). Theauthors refer to the model as “ghostburster” because repetitive spiking corresponds toa slow transition of the full system through the ghost of a fold limit cycle attractor.As a dynamical system, the ghostburster is near a co-dimention-2 bifurcation of limitcycle attractor, and it exhibits chaotic dynamics.Betram et al. (1995) noticed that bursting often occurs when the fast subsystemis near a co-dimension-2 bifurcation. Izhikevich (2000) suggested that many simplemodels of bursters could be obtained by considering unfoldings of various degeneratebifurcations of high co-dimension (organizing centers) and treating the unfolding parametersas slow variables rotating around the bifurcation point, as in Fig. 9.28 orFig. 9.42. Considering the Bautin bifurcation, Izhikevich (2001) obtained the canonicalmodel for “elliptic” burster (9.11). Golubitsky et al. (2001) applied this idea toother local bifurcations (spiking with infinitesimal amplitude). Global bifurcations areconsidered in Ex. 26.Izhikevich and Hoppensteadt (2004) extend the classification of bursters to oneandtwo-dimensional mappings, identifying 3 and 20 different classes, respectively. Acollection of chapters “Bursting: The Genesis of Rhythm in the Nervous System” editedby Coombes and Bressloff (2005) provides recent developments in the field of burstingdynamics.Studying bursting dynamics is still one of the hardest problems in applied mathematics.The method of dissection of fast-slow bursters of the form (9.1), pioneeredby Rinzel (1987), is part of the asymptotic theory of singularly perturbed dynamicalsystems (Mishchenko et al. 1994). One would expect the theory to suggest other,quantitative methods of analyses of fast-slow bursters. However, the basic assumptionof the theory is that the fast subsystem has only equilibria, e.g., up- and down-states as