Dynamical Systems in Neuroscience:

Bifurcations 1813210eigenvalues = c(I) + ω(I)ilinearization1510514.66 14.66-1000 10 20 30 0 10 20 30 0 10 20 30injected dc-current, I injected dc-current, I injected dc-current, Inumericalω(I)c(I)membrane voltage, (mV)amplitudenumericaltheoreticalfrequency, 2π/period (rads/ms)2.521.510.5frequencytheoreticalnumericalFigure 6.12: Supercritical Andronov-Hopf bifurcation in the I Na,p +I K -model with lowthresholdK + current (see Fig. 6.11). Dots — numerical simulation of the full model,continuous curves — analytical results using the topological normal form (6.8, 6.9).we conclude that r = 0 is an equilibrium for any value of c(b). Since(c(b)r + ar 3 ) r = c(b) at r = 0,the equilibrium is stable for c(b) < 0 and unstable for c(b) > 0, as we illustrate inFig. 6.13. Indeed, the rest state in the I Na,p +I K -model is stable when I < 14.66 andunstable when I > 14.66.When c(b) > 0, the normal form has a family of stable periodic solutions withamplituder = √ c(b)/|a| and (frequency) = ω(b) + d c(b)/|a| .Hence, the I Na,p +I K -model has a family of periodic attractors withandr = √ 0.03{I − 14.66}/0.0026(frequency) = (2.14 + 0.04{I − 14.66}) − 0.0029 · 0.03{I − 14.66}/0.0026 ,depicted in Fig. 6.12. We see that the topological normal form describes not onlyqualitatively but also quantitatively the full I Na,p +I K -model near the Andronov-Hopfbifurcation.6.1.4 Subcritical Andronov-HopfNeuronal models with monotonic steady-state I-V relations can also exhibit subcriticalAndronov-Hopf bifurcations, as we illustrate in Fig. 6.16 using the I Na,p +I K -modelhaving low-threshold K + current and a steep activation curve for Na + current. The