Dynamical Systems in Neuroscience:

Bursting 371Figure 9.31: Putative “circle/circle” bursting pacemaker activity of neuron R 15 inabdominal ganglion of the mollusk Aplysia (modified from Levitan and Levitan 1988).interspike period (sec)21.510.500 5 10 15spike numberinterspike frequency (Hz)2.521.510.500 5 10 15spike numberFigure 9.32: The interspike period in the “circle/circle” bursting in Fig. 9.31 resemblesa parabola, which motivates the name “parabolic bursting” used in earlier studies.9.3.2 circle/circleWhen the equilibrium corresponding to the resting state disappears via a saddle-nodeon invariant circle bifurcation, and the limit cycle attractor corresponding to spikingstate disappears via another saddle-node on invariant circle bifurcation, the burster issaid to be of the “circle/circle” type shown in Fig. 9.30. Since the bifurcation doesnot produce a co-existence of attractors, there is usually no hysteresis loop, and thebursting is of the slow-wave type with at least two slow variables. (An example of“circle/circle” hysteresis loop bursting in a “2+1” system is provided by Izhikevich(2000)).“Circle/circle” bursting is a prominent feature of the R 15 cells in the abdominal ganglionof the mollusk Aplysia, shown in Fig. 9.31 (Plant 1981). It was called “parabolic”bursting in earlier studies because the interspike period depicted in Fig. 9.32 was erroneouslythought to be a parabola. In Sect. 6.1.2 we showed that when a systemundergoes a saddle-node on invariant circle bifurcation, its period scales as 1/ √ λ,where λ is the distance to the bifurcation. Two pieces of this function, put togetheras in Fig. 9.32, do indeed resemble a parabola. But so does the interspike period of a“circle/homoclinic” burster.To transform the I Na,p +I K -model to a “circle/circle” burster, we take the parametersas in Fig. 4.1a so that there is a saddle-node on invariant circle bifurcation whenI = 4.51 (see Sect. 6.1.2). Its nullclines and phase portrait look similar to those inFig. 9.30. Then, we add one amplifying and one resonant current with gating variablesṁ slow = (m ∞,slow (V ) − m slow )/τ Na,slow (V ) (slow I Na,slow ),ṅ slow = (n ∞,slow (V ) − n slow )/τ K(M) (V ) (slow I K(M) ),