Dynamical Systems in Neuroscience:

192 Bifurcationsamplitude (max-min), mV8070605040302010fold limit cyclebifurcationstable limit cyclesunstable limit cyclessubcriticalAndronov-Hopfbifurcation040 41 42 43 44 45 46 47 48 49 50injected dc-current, IFigure 6.24: Bifurcation diagram of the I Na,p +I K -model. Parameters as in Fig. 6.16.appear or disappear, but it does not explain how stable periodic spiking behaviorappears. Indeed, let us start with I = 42 in Fig. 6.23 and slowly increase the parameter.The state of the I Na,p +I K -model is at the stable equilibrium. When I passes thebifurcation value, a large-amplitude stable limit cycle corresponding to periodic spikingappears, yet the model is still quiescent, because it is still at the stable equilibrium.Thus, the limit cycle is just a geometrical object in the phase space that corresponds tospiking behavior. However, to actually exhibit spiking, the state of the system must besomehow pushed into the attraction domain of the cycle, say by external stimulation.This issue is related to the computational properties of neurons, and it is discussed indetail in the next chapter.In Fig. 6.24 we depict the bifurcation diagram of the I Na,p +I K -model. For eachvalue of I, we simulate the model forward (t → ∞) to find the stable limit cycle andbackward (t → −∞) to find the unstable limit cycle. Then we plot their amplitudes(maximal voltage minus minimal voltage along the limit cycle) on the (I, V )-plane.One can clearly see that there is a fold limit cycle bifurcation (left) and a subcriticalAndronov-Hopf bifurcation (right). The left part of the bifurcation diagram looksexactly like the one for saddle-node bifurcation, which explains why the fold limitcycle bifurcation is often referred to as fold or saddle-node of periodics.The similarity of the fold limit cycle bifurcation and the saddle-node bifurcationis not a coincidence. Stability of limit cycles can be studied using Floquet theory,Poincaré cross-section maps (Kuznetsov 1995), or by brute force, e.g., by reducingthe model to an appropriate polar coordinate system. When a limit cycle attractorundergoes fold limit cycle bifurcation, its radius undergoes saddle-node bifurcation(this is a hint to Ex. 10).