Dynamical Systems in Neuroscience:

Bifurcations 211homoclinic orbithomoclinic orbithomoclinic orbitsaddle-node on invariant circlebifurcationsaddle homoclinic orbitbifurcationsaddle-node homoclinic orbitbifurcationFigure 6.43: Saddle-node homoclinic orbit bifurcation occurs when a systems undergoesa saddle-node on invariant circle and saddle homoclinic orbit bifurcations simultaneously.bifurcation, has codimension 2, since two strict conditions must be satisfied: First,the equilibrium must be at the saddle-node bifurcation point, i.e., must have eigenvalueλ 1 = 0. Second, the homoclinic trajectory must return to the equilibrium alongthe non-central direction, i.e., along the stable direction corresponding to the negativeeigenvalue λ 2 . Since the saddle-node quantity, λ 1 + λ 2 , is always negative, thisbifurcation always results in the (dis)appearance of a stable limit cycle.In Fig. 6.44 we illustrate the saddle-node homoclinic orbit bifurcation using theI Na,p +I K -model with two bifurcation parameters: the injected dc-current I and the K +time constant τ. The bifurcation occurs at the point (I, τ) = (4.51, 0.17). Notice thatthere are three other codimension-1 bifurcation curves converging to this codimension-2point, as we illustrate in Fig. 6.45. Since the model undergoes a saddle-node bifurcationat I = 4.51 and any τ, the straight vertical line I = 4.51 is the saddle-node bifurcationcurve. The point τ = 0.17 on this line separates two cases: When τ > 0.17, theactivation and deactivation of K + current is sufficiently slow so that the membranepotential V undershoots the equilibrium, resulting in the saddle-node on invariantcircle bifurcation. When τ < 0.17, deactivation of K + current is fast, and V overshootsthe saddle-node equilibrium, resulting in the saddle-node off limit cycle bifurcation.Shaded triangular areas in the figures denote the parameter region correspondingto the bistability of stable equilibrium and a limit cycle attractor (resting and spikingstates). Let us decrease the parameter I and cross such a region from right to left.When I = 4.51, a saddle and a node equilibria appear. Decreasing I further moves thesaddle equilibrium rightward and the limit cycle leftward, until they merge. This occurs