Dynamical Systems in Neuroscience:

114 Two-Dimensional SystemsV-nullcline0.2recovery, wseparatrixw-nullcline0separatrixattraction domain-0.4 -0.2 0 0.2 0.4 0.6 0.8 1membrane voltage, VFigure 4.22: Bistability of two equilibrium attractors (black circles) in the FitzHugh-Nagumo model (4.11,4.12). The shaded area — attraction domain of the right equilibrium.Parameters: I = 0, b = 0.01, a = c = 0.1.which is the attraction domain of the limit cycle attractor, the trajectory approachesthe limit cycle attractor and the neuron fires an infinite train of action potentials.4.3.2 Stable/unstable manifoldsIn contrast to one-dimensional systems, in two-dimensional systems unstable equilibriado not necessarily separate attraction domains. Nevertheless, they play an importantrole in defining the boundary of attraction domains, as in Fig. 4.22 and Fig. 4.23.In both cases the attraction domains are separated by a pair of trajectories, calledseparatrices, which converge to the saddle equilibrium. Such trajectories form thestable manifold of a saddle point. Locally, the manifold is parallel to the eigenvectorcorresponding to the negative (stable) eigenvalue; see Fig. 4.24. Similarly, the unstablemanifold of a saddle is formed by the two trajectories that originate exactly from thesaddle (or approach the saddle if the time is reversed). Locally, the unstable manifoldis parallel to the eigenvector corresponding to the positive (unstable) eigenvalue.The stable manifold of the saddle in Fig. 4.23 plays the role of a threshold, since itseparates rest and spiking states. We illustrate this concept in Fig. 4.24: If the initialstate of the system, denoted as A, is in the shaded area, the trajectory will convergeto the spiking attractor (right) no matter how close the initial condition to the stablemanifold is. In contrast, if the initial condition, denoted as B, is in the white area, the