Dynamical Systems in Neuroscience:

210 BifurcationsSupercritical Andronov-HopfBifurcationc = 0a < 0BautinBifurcation0c2a - 4c a = 02c = 0a > 0FoldLimit CycleBifurcationSubcritical Andronov-HopfBifurcationaFigure 6.42: Supercritical Bautin bifurcation in (6.12); see also Fig. 9.42, left.a < 0 and subcritical otherwise. Moreover, if a and a 2 have different signs, then (6.12)undergoes fold limit cycle bifurcation whena 2 − 4ca 2 = 0 ,as we illustrate in Fig. 6.42. Thus, both Andronov-Hopf and fold limit cycle bifurcationsoccur simultaneously at the Bautin point a = c = 0. Many two-dimensional neuronalmodels, such as the I Na,p +I K -model with low-threshold K + current, are relatively nearthis bifurcation, which explains why the unstable limit cycle involved in the subcriticalAndronov-Hopf bifurcation is usually born via fold limit cycle bifurcation. There issome evidence that rodent trigeminal interneurons, dorsal root ganglion neurons, andmes V neuron in brainstem are also near this bifurcation; see Sect. 9.3.3.6.3.6 Saddle-node homoclinic orbitLet us compare the saddle-node on invariant circle bifurcation and the saddle homoclinicorbit bifurcation depicted in Fig. 6.43, top. In both cases there is a homoclinicorbit, i.e., a trajectory that originates and terminates at the same equilibrium. However,the equilibria are of different types, and the orbit returns to them along differentdirections. Now suppose a system undergoes both bifurcations simultaneously, as weillustrate in Fig. 6.43, bottom. Such a bifurcation, called saddle-node homoclinic orbit