Dynamical Systems in Neuroscience:

382 BurstingspikingsupercriticalAndronov-HopfbifurcationfoldbifurcationfoldbifurcationxurestingfoldsupercriticalAndronov-HopffoldFigure 9.44: “Fold/Hopf” bursting: The resting state disappears via saddle-node (fold)bifurcation and the spiking limit cycle shrinks to a point via supercritical Andronov-Hopf bifurcation (modified from Izhikevich 2000).thalamo-cortical relay neuron by Rush and Rinzel (1994), and it was called “triangular”in earlier studies (Wang and Rinzel 1995) because of the shape of the voltage envelope.As one can see in the figure, the fast subsystem can have five equilibria, two ofwhich are stable nodes. This is a consequence of the quintic shape of the V -nullclineof the fast subsystem. While the trajectory is at the lower equilibrium, the V -nullclinemoves up, the equilibrium disappears via fold bifurcation, and the fast subsystemstarts to fire spikes. During this active period, the V -nullcline slowly moves down,and the spiking limit cycle disappears via saddle-node on invariant circle bifurcation.The fast subsystem, however, is at the second stable equilibrium corresponding to adepolarized state. The slow V -nullcline continues to move down, and this equilibriumdisappears via another fold bifurcation, thereby closing the “fold/fold” hysteresis loop.Alternatively, “fold/circle” bursting can be of the slow-wave type depicted in Fig. 9.28having only three equilibria. The slow subsystem needs to be at least two-dimensionalin this case, though.