Dynamical Systems in Neuroscience:

386 Bursting9.4.3 BistabilitySuppose the transition from resting to spiking state occurs via saddle-node bifurcation(off an invariant circle) or subcritical Andronov-Hopf bifurcation of the fast subsystem,as in Fig. 6.46. In these cases, the trajectory jumps to a pre-existing limit cycleattractor corresponding to the spiking state, not shown in the figure. In contrast,saddle-node on invariant circle bifurcation or supercritical Andronov-Hopf bifurcationcreates such a limit cycle attractor. Thus, there must be a co-existence of stable restingand stable spiking states in the former case, but not necessarily in the latter case. Thissimple observation has far reaching consequences described below. In particular, itimplies that all “fold/*” and “subHopf/*” bursters exhibit bistability, at least beforethe onset of a burst, while “circle/*” and “Hopf/*” bursters may not; see Fig. 9.47.Similarly, if the transition from spiking to resting state of the fast subsystem occursvia saddle homoclinic orbit bifurcation or fold limit cycle bifurcation, then there is apre-existing stable equilibrium, and hence a co-existence of attractors. Thus, “*/homoclinic”and “*/fold cycle” bursters also exhibit bistability, at least at the end of aburst, while “*/circle” and “*/Hopf” bursters may not, as we summarize in Fig. 9.47.An obvious consequence of bistability is that an appropriate stimulus can switch thesystem from resting to spiking and back. We illustrate this phenomenon in Fig. 9.48using the I Na,p +I K +I K(M) -model, which exhibits a hysteresis loop “fold/homoclinic”bursting when I = 5. All three simulations in the figure start with the same initialconditions. In Fig. 9.48b we apply a brief pulse of current while the fast subsystem is atthe resting state. This stimulation pushes the membrane potential over the thresholdstate into the attraction domain of the spiking limit cycle of the fast subsystem, therebyevoking a burst.Notice that the evoked burst is one spike shorter than the control one in Fig. 9.48a.This is expected, since the K + M-current did not have enough time to recover fromthe previous burst (not shown in the figure), therefore, there is a residual outwardcurrent that shortens the active phase. From the geometrical point of view, this occursbecause the transition to the spiking manifold in Fig. 9.48b, right, occurs before theslow variable reaches the fold knee, hence the distance to the homoclinic bifurcation isshorter. An interesting observation is that the first spike in the evoked burst actuallycorresponds to the second spike in the control burst in Fig. 9.48a. The earlier thestimulation acts, the sooner the trajectory jumps to the spiking manifold and thefewer spikes the evoked burst has.In Fig. 9.48c we applied a brief pulse of current in the middle of a burst to switchthe system to the resting state. Notice that the quiescent period, i.e., the time periodto the second burst, is shorter than the control one in Fig. 9.48a or in Fig. 9.48b. Thisis also to be expected, since the K + M- current was not fully activated during theinterrupted burst, therefore it does not need that much time to deactivate during theresting period. Geometrically, the short duration of the resting phase is a consequenceof the distance the slow variable needs to travel to get to the fold knee being small.