Dynamical Systems in Neuroscience:

Bifurcations 179r0ϕFigure 6.10: Polar coordinates: r is the amplitude(radius) and ϕ is the phase (angle) of oscillation.The function c(b) in the normal form (6.8,6.9) determines the stability of the equilibriumr = 0 corresponding to a non-oscillatory state. (Stable for c < 0 and unstablefor c > 0, regardless of the value of a). The function ω(b) determines the frequency ofdamped or sustained oscillations around this state. The parameter d describes how thefrequency of oscillation depends on its amplitude. A state-dependent change of timecan remove the term dr 2 from (6.9) (Kuznetsov 1995), so many assume that d = 0 tostart with.Example: The I Na,p +I K -modelLet us use the I Na,p +I K -model with low-threshold K + current in Fig. 6.11 to illustratethe three conditions above. As the magnitude of the injected dc-current I increases,the equilibrium loses stability and gives birth to a stable limit cycle with growingamplitude. Using simulations we find that the bifurcation occurs when I ah = 14.66and (V ah , n ah ) = (−56.5, 0.09). The Jacobian matrix at the equilibrium,(L =1 −3350.0166 −1has a pair of complex conjugate eigenvalues ±2.14i, so the non-hyperbolicity conditionis satisfied. Next, we find numerically (in Fig. 6.12 or analytically in Ex. 9) that theeigenvalues at the equilibrium can be approximated byc(I) + ω(I)i ≈ 0.03{I − 14.66} ± (2.14 + 0.04{I − 14.66})iin a neighborhood of the bifurcation point I = 14.66. Since the slope of c(I) is non-zero,the transversality condition is also satisfied. Using Ex. 17 we find that a = −0.0026 andd = −0.0029, so that the non-degeneracy condition is also satisfied, and the bifurcationis of the supercritical type. The corresponding topological normal form isṙ = 0.03{I − 14.66}r − 0.0026r 3 ,˙ϕ = (2.14 + 0.04{I − 14.66}) − 0.0029r 2 .To analyze the normal form we consider the r-equation and neglect the phase variableϕ. Fromr(c(b) + ar 2 ) = 0),