Dynamical Systems in Neuroscience:

122 Two-Dimensional SystemsK + activation variable, n0.70.60.50.40.30.20.1n-nullcline0.7I=00.6I=12V-nullclineK + activation variable, n0.50.40.30.20.1K + activation variable, n00.70.60.50.40.30.20.10-80 -60 -40 -20 0 20membrane voltage, V (mV)0.50.40.30.20.1-80 -60 -40 -20 0 20membrane voltage, V (mV)0.7I=20 I=400.6K + activation variable, nlimit cycle0-80 -60 -40 -20 0 20membrane voltage, V (mV)0-80 -60 -40 -20 0 20membrane voltage, V (mV)Figure 4.34: Supercritical Andronov-Hopf bifurcation in the I Na,p +I K -model (4.1, 4.2)with low-threshold K + current when I = 12; see also Fig. 4.33.4.3.5 Andronov-Hopf bifurcationIn Fig. 4.33 we repeat the current ramp experiment using the I Na,p +I K -model with lowthresholdK + current. The phase portrait of such a model is simple — it has a uniqueequilibrium, as we illustrate in Fig. 4.34. When I is small, the equilibrium is a stablefocus corresponding to the rest state. When I increases past I = 12, the focus losesstability and gives birth to a small-amplitude limit cycle attractor. The amplitude ofthe limit cycle grows as I increases. We see that increasing I beyond I = 12 results inthe transition from rest to spiking behavior. What kind of a bifurcation occurs there?Recall that stable foci have a pair of complex-conjugate eigenvalues with negativereal part. When I increases, the real part of the eigenvalues also increases until itbecomes zero (at I = 12) and then positive (when I > 12) meaning that the focus isno longer stable. The transition from stable to unstable focus described above is calledAndronov-Hopf bifurcation. It occurs when the eigenvalues become purely imaginary,as it happens when I = 12. We will study Andronov-Hopf bifurcations in detail inChap. 6, where we will show that they can be supercritical or subcritical. The formercorrespond to birth of a small-amplitude limit cycle attractor, as in Fig. 4.34. Thelatter correspond to a death of an unstable limit cycle.