Dynamical Systems in Neuroscience:

Bifurcations 189bifurcation parameter changesinvariant circleinvariant circlestable limit cyclenode saddle saddle-nodeFigure 6.20: Saddle-node on invariant circle (SNIC) bifurcation of a limit cycle attractor.bifurcation parameter changesstable equilibriumstable limit cycleunstable equilibriumFigure 6.21: Supercritical Andronov-Hopf bifurcation of a limit cycle attractor.node bifurcation (Fig. 6.20, center) that breaks the cycle and gives birth to a pairof equilibria– stable node and unstable saddle (Fig. 6.20, left). After the bifurcation,the limit cycle becomes an invariant circle consisting of a union of two heteroclinictrajectories.Depending on the direction of change of a bifurcation parameter, the saddle-nodeon invariant circle bifurcation can explain either appearance or disappearance of alimit cycle attractor. In either case, the amplitude of the limit cycle remains relativelyconstant but its period becomes infinite at the bifurcation point because the cyclebecomes a homoclinic trajectory to the saddle-node equilibrium (Fig. 6.20, center). Aswe showed in Sect. 6.1.2 (see Fig. 6.8), the frequency of oscillation scales as √ I − I bwhen the bifurcation parameter approaches the bifurcation value I b .6.2.2 Supercritical Andronov-HopfA stable limit cycle can shrink to a point via supercritical Andronov-Hopf bifurcationin Fig. 6.21, which we considered in Sect. 6.1.3. Indeed, as the bifurcation parameterchanges, e.g., the injected dc-current I in Fig. 6.11 decreases, the amplitude of thelimit cycle attractor vanishes, and the cycle becomes just a stable equilibrium. Aswe showed in Sect. 6.1.3 (see Fig. 6.12), the amplitude scales as √ I − I b when the