Dynamical Systems in Neuroscience:

Bifurcations 207(a) (b) (c) (d)g=0f=0(e) (f) (g) (h)Figure 6.41: Canard (French duck) limit cycles in a relaxation oscillator (hand drawing).Interestingly, the global vector field structure of neuronal models with nullclinesas in Fig. 6.39a results in the birth of a spiking limit cycle attractor via a big saddlehomoclinic orbit bifurcation, so the neuronal model undergoes a cascade of bifurcationsdepicted in Fig. 6.40 as the amplitude of the injected current I increases. The localphase portraits corresponding to I 0 , I 1 , and I 2 are topologically equivalent to the phaseportrait “1” in Fig. 6.38, right. (The equivalence is only local near the left knee; thereis no global equivalence because of the extra equilibrium in Fig. 6.40 and because ofthe big homoclinic or periodic orbit.) As I increases, a stable large-amplitude spikinglimit cycle appears via a big supercritical homoclinic orbit bifurcation at some I 1 . Itcoexists with the stable resting state for all I 1 < I < I 5 . At some point I 2 , thesaddle quantity, i.e., the sum of its eigenvalues, changes from negative to positive(it is zero at the Bogdanov-Takens bifurcation), so another saddle homoclinic orbitbifurcation (at some I 3 ) occurs, which is subcritical, giving birth to an unstable limitcycle. The phase portrait at I 3 is locally topologically equivalent to the one markedSHO in Fig. 6.38. Similarly, the phase portrait at I 4 is locally equivalent to the onemarked “2” in Fig. 6.38. The unstable cycle shrinks to the equilibrium and makes it losestability via a subcritical Andronov-Hopf bifurcation at some I 5 , which corresponds tocase AH in Fig. 6.38. Further increase of I converts the unstable focus into an unstablenode, which approaches the saddle and disappears via the saddle-node bifurcation SN 1in Fig. 6.38 (not shown in Fig. 6.40).6.3.4 Relaxation oscillators and CanardsLet us consider a relaxation oscillatorẋ = f(x, y, b) (fast variable)ẏ = µg(x, y, b) (slow variable)