Dynamical Systems in Neuroscience:

Bifurcations 195stable manifoldoutin?saddleunstable manifoldFigure 6.27: Saddle homoclinic orbit bifurcationoccurs when the stable and unstable submanifolds ofthe saddle make a loop.A useful way to look at the bifurcation is to note that the saddle has one stableand one unstable direction on a phase plane. There are two orbits associated withthese directions, called the stable and unstable submanifolds, depicted in Fig. 6.27.Typically, the submanifolds miss each other, that is, the unstable submanifold goeseither inside or outside the stable one. This could happen for two different values ofthe bifurcation parameter. One can image that as the bifurcation parameter changescontinuously from one value to the other, the submanifolds join at some point and forma single homoclinic trajectory that starts and ends at the saddle.The saddle homoclinic orbit bifurcation is ubiquitous in neuronal models, and it caneasily be observed in the I Na,p +I K -model with fast K + conductance, as we illustratein Fig. 6.28. Let us start with I = 7 (top of Fig. 6.28) and decrease the bifurcationparameter I. First, there is only a stable limit cycle corresponding to periodic spikingactivity. When I decreases, a stable and an unstable equilibrium appear via saddlenodebifurcation (not shown in the figure), but the state of the model is still on thelimit cycle attractor. Further decrease of I moves the saddle equilibrium closer tothe limit cycle (case I = 4 in the figure), until the cycle becomes an infinite periodhomoclinic orbit to the saddle (case I ≈ 3.08), and then disappears (case I = 1). Atthis moment, the state of the system approaches the stable equilibrium, and the tonicspiking stops.Similarly to the fold limit cycle bifurcation, the saddle homoclinic orbit bifurcationexplains how the limit cycle attractor corresponding to periodic spiking behavior appearsand disappears. However, it does not explain the transition to periodic spikingbehavior. Indeed, when I = 4 in Fig. 6.28, the limit cycle attractor exists, yet theneuron may still be quiescent because its state may be at the stable node. The periodicspiking behavior appears only after external perturbations push the state of thesystem into the attraction domain of the limit cycle attractor, or I increases furtherand the stable node disappears via a saddle-node bifurcation.We can use linear theory to estimate the frequency of the limit cycle attractornear saddle homoclinic orbit bifurcation. Because the vector field is small near theequilibrium, the periodic trajectory slowly passes through a small neighborhood of theequilibrium, then quickly makes a rotation and returns to the neighborhood, as weillustrate in Fig. 6.29. Let T 1 denote the time required to make one rotation (dashed