Dynamical Systems in Neuroscience:

188 BifurcationsBifurcation of a limit cycle attractor amplitude frequencysaddle-node on invariant circle non-zero A √ I−I b → 0supercritical Andronov-Hopf A √ I−I b → 0 non-zerofold limit cycle non-zero non-zerosaddle homoclinic orbit non-zero−Aln |I−I b | →0Figure 6.19: Summary of codimension-1 bifurcations of a limit cycle attractor on aplane. Here, I denotes the amplitude of the injected current, I b is the bifurcationvalue, A is a parameter that depends on the biophysical details.In Fig. 6.19 we summarize how the bifurcations affect a periodic attractor. Saddlenodeon invariant circle and saddle homoclinic orbit bifurcations involve homoclinictrajectories having infinite period (zero frequency). They result in oscillations withdrastically increasing interspike intervals as the system approaches the bifurcation state(see Fig. 6.18).In contrast, supercritical Andronov-Hopf bifurcation results in oscillations withvanishing amplitude, as one can clearly see in Fig. 6.18. If neither the frequency northe amplitude vanishes, then the bifurcation is of the fold limit cycle type. Indeed,the amplitude and the interspike period are constant before the arrow in Fig. 6.18corresponding to the fold limit cycle bifurcation. Damped small-amplitude oscillationafter the arrow occurs because of oscillatory convergence to the equilibrium.We start with a brief review of the saddle-node on invariant circle and the supercriticalAndronov-Hopf bifurcations, which we considered in detail in Sect. 6.1. Thesebifurcations can explain not only transitions from rest to spiking but also transitionsfrom spiking to rest states. Then, we consider fold limit cycle and saddle homoclinicorbit bifurcations.6.2.1 Saddle-node on invariant circleA stable limit cycle can disappear via a saddle-node on invariant circle bifurcationas depicted in Fig. 6.20. The necessary condition for such a bifurcation is that thesteady-state I-V relation is not monotonic.We considered this bifurcation in Sect. 6.1.2 as a bifurcation from an equilibriumto a limit cycle; that is from left to right in Fig. 6.20. Now consider it from rightto left: As a bifurcation parameter changes, e.g., the injected dc-current I decreases,a stable limit cycle (circle in Fig. 6.20, right) disappears because there is a saddle-