Dynamical Systems in Neuroscience:

208 Bifurcationswith fast and slow nullclines as in Fig. 6.41a and µ ≪ 1. Suppose that there is a stableequilibrium, as in Fig. 6.41a, for some values of the bifurcation parameter b < 0, anda stable limit cycle, as in Fig. 6.41h, for some other values b > 0. What kind of abifurcation of the equilibrium occurs when b increases from negative to positive, andthe slow nullcline passes the left knee of the fast N-shaped nullcline?The Jacobian matrix at the equilibrium has the form( )fx fL =yµg x µg ySince f x = 0 at the knee (prove this), but f y is typically not, the Jacobian matrixresembles the one for the Bogdanov-Takens bifurcation (6.10) in the limit µ = 0.However, the resemblance is only superficial, since the relaxation oscillator does notsatisfy the non-degeneracy conditions. In particular, second-order partial derivativesof µg(x, y, b) vanish in the limit µ → 0, resulting in σ = 0 and in the disappearance ofthe term u 2 in the topological normal form (6.11).A purely geometrical consideration confirms that the transition from Fig. 6.41a toFig. 6.41h cannot be of the Bogdanov-Takens type, since there is a unique equilibriumand no possibility for a saddle-node bifurcation, which always accompanies theBogdanov-Takens bifurcation. Actually, the equilibrium loses stability via Andronov-Hopf bifurcation that occurs whentr L = f x + µg y = 0 and det L = µ(f x g y − f y g x ) > 0 .The loss of stability typically happens not at the left knee, where f x = 0, but a littlebit to the right of the knee, where f x = −µg y > 0 (because g y < 0 in neuronal models).We already saw this phenomenon in Sect. 4.2.6 when we considered FitzHugh-Nagumomodel.An interesting observation is that the period of damped or sustained oscillationsnear the Andronov-Hopf bifurcation point in Fig. 6.41b is of the order 1/ √ µ, becausethe frequency ω = √ det L ≈ √ µ, whereas the period of large-amplitude relaxationoscillation is of the order 1/µ, because it takes 1/µ units of time to slide up anddown along the branches of the fast nullcline in Fig. 6.41h. Thus, the period of smallsubthreshold oscillations of a neural model may have no relation to the period ofspiking, if the model is a relaxation oscillator.The Andronov-Hopf bifurcation could be supercritical or subcritical, depending onthe functions f and g; see Ex. 14 and Ex. 18. Figure 6.41 depicts the supercriticalcase. In the subcritical case, stable and unstable limit cycles are typically born viafold limit cycle bifurcation, then the unstable limit cycle goes through the shapes asin Fig. 6.41g,f,e,d,c, and b, and then shrinks to a point.CanardsThe distinctive feature of limit cycles in Fig. 6.41c-g is that they follow the unstablebranch (dashed curve) of the fast nullcline before jumping to the left or to the right