Dynamical Systems in Neuroscience:
Dynamical Systems in Neuroscience: Dynamical Systems in Neuroscience:
172 Bifurcations0.5v 2V-nullclineK + activation, n0.40.30.20.001v 10-63 -62 -61 -60 -59 -580.10n-nullcline-80 -70 -60 -50 -40 -30 -20 -10 0membrane voltage, V (mV)Figure 6.4: Saddle-node bifurcation in the I Na,p +I K -model (4.1, 4.2) with highthresholdK + current (parameters as in Fig. 4.1a) and I = 4.51.-50membrane voltage, V (mV)-55-60-65-70normal formI Na +I K -model-75-15 -10 -5 0 5 10injected dc-current, IFigure 6.5: Bifurcation diagrams of the topological normal form (6.3) and the I Na,p +I K -model (4.1, 4.2).
Bifurcations 173(a) saddle-node bifurcationlimitcyclenodesaddlesaddle-node(b) saddle-node on invariant circle (SNIC) bifurcationinvariant circlenode saddle saddle-nodeFigure 6.6: Two types of saddle-node bifurcation.depicted in the inset in Fig. 6.4. (It is easy to check that Lv 1 = 0 and Lv 2 = −0.9565v 2 .)The non-degeneracy and transversality conditions yields a = 0.1887 and c = 1, so thatthe topological normal form for the I Na,p +I K -model is˙V = (I − 4.51) + 0.1887(V + 61) 2 , (6.3)which can be solved analytically. The corresponding bifurcation diagrams are depictedin Fig. 6.5. There is no surprise that there is a fairly good match when I is near thebifurcation value.6.1.2 Saddle-node on invariant circleAs its name stands, saddle-node on invariant circle bifurcation (also known as SNICor SNLC bifurcation) is a standard saddle-node bifurcation described above with anadditional caveat: it occurs on an invariant circle, compare Fig. 6.6a and b. Here,the invariant circle consists of two trajectories connecting the node and the saddle,called heteroclinic trajectories. It is called invariant because any solution starting onthe circle remains on the circle. As the saddle and node coalesce, the small trajectoryshrinks and the large heteroclinic trajectory becomes a homoclinic invariant circle, i.e.,originating and terminating at the same point. When the point disappears, the circlebecomes a limit cycle.
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172 Bifurcations0.5v 2V-nullcl<strong>in</strong>eK + activation, n0.40.30.20.001v 10-63 -62 -61 -60 -59 -580.10n-nullcl<strong>in</strong>e-80 -70 -60 -50 -40 -30 -20 -10 0membrane voltage, V (mV)Figure 6.4: Saddle-node bifurcation <strong>in</strong> the I Na,p +I K -model (4.1, 4.2) with highthresholdK + current (parameters as <strong>in</strong> Fig. 4.1a) and I = 4.51.-50membrane voltage, V (mV)-55-60-65-70normal formI Na +I K -model-75-15 -10 -5 0 5 10<strong>in</strong>jected dc-current, IFigure 6.5: Bifurcation diagrams of the topological normal form (6.3) and the I Na,p +I K -model (4.1, 4.2).
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