Dynamical Systems in Neuroscience:

116 Two-Dimensional Systemsheteroclinic orbithomoclinicorbitFigure 4.25: A heteroclinic orbit starts andends at different equilibria. A homoclinic orbitstarts and ends at the same equilibrium.trajectory will converge to the rest attractor (left). If the initial condition is preciselyon the stable manifold (point C), the trajectory converges neither to rest nor to spikingstate, but to the saddle equilibrium. Of course, this case is highly unstable and smallperturbations will certainly push the trajectory to one or the other side. The importantmessage in Fig. 4.24 is that a threshold is not a point, i.e., a single voltage value, buta trajectory on the phase plane. (Find an exceptional case when the threshold lookslike a single voltage value. Hint: see Fig. 4.17.)4.3.3 Homoclinic/heteroclinic trajectoriesFig. 4.24 shows that trajectories forming the unstable manifold originate from thesaddle. Where do they go? Similarly, the trajectories forming the stable manifoldterminate at the saddle. Where do they come from? We say that a trajectory isheteroclinic if it originates at one equilibrium and terminates at another equilibrium,as in Fig. 4.25. A trajectory is homoclinic if it originates and terminates at the sameequilibrium. These types of trajectories play an important role in geometrical analysisof dynamical systems.Heteroclinic trajectories connect unstable and stable equilibria, as in Fig. 4.26, andthey are ubiquitous in dynamical systems having two or more equilibrium points. Infact, there are infinitely many heteroclinic trajectories in Fig. 4.26, since all trajectoriesinside the bold loop originate at the unstable focus and terminate at the stable node.(Find the exceptional trajectory that ends elsewhere.)In contrast, homoclinic trajectories are rare. First, a homoclinic trajectory divergesfrom an equilibrium, therefore the equilibrium must be unstable. Next, the trajectorymakes a loop and returns to the same equilibrium, as in Fig. 4.27. It needs to hitthe unstable equilibrium precisely, since a small error would make it deviate from theunstable equilibrium. Though uncommon, homoclinic trajectories indicate that thesystem undergoes a bifurcation — appearance or disappearance of a limit cycle. Thehomoclinic trajectory in Fig. 4.27 indicates that the limit cycle in Fig. 4.23 is aboutto (dis)appear via saddle homoclinic orbit bifurcation. The homoclinic trajectory inFig. 4.28 indicates that a limit cycle is about to (dis)appear via saddle-node on invariantcircle bifurcation. We study these bifurcations in detail in Chap. 6.4.3.4 Saddle-node bifurcationIn Fig. 4.29 we simulate the injection of a ramp current I into the I Na,p +I K -modelhaving high-threshold K + current. Our goal is to understand the transition from the