Dynamical Systems in Neuroscience:

Two-Dimensional Systems 113It is easy to check thatτ = tr L = −a − c and ∆ = det L = ac + b .Using Fig. 4.15 we conclude that the equilibrium is stable when tr L < 0 and det L >0, which corresponds to the shaded region in Fig. 4.21. Both conditions are alwayssatisfied when a > 0, hence the equilibrium in Fig. 4.20a is indeed stable. However,both conditions may also be satisfied for negative a, therefore, the equilibrium inFig. 4.20b may also be stable. Thus, the equilibrium loses stability not at the leftknee, but slightly to the right of it, so that a part of the “unstable branch” of thecubic nullcline is actually stable. The part is small when b and c are small, i.e., when(4.11,4.12) is in a relaxation regime.4.3 Phase PortraitsAn important step in geometrical analysis of dynamical systems is sketching of theirphase portraits. The phase portrait of a two-dimensional system is a partitioning of thephase plane into orbits or trajectories. Instead of depicting all possible trajectories, itusually suffices to depict some representative trajectories. The phase portrait containsall important information about qualitative behavior of the dynamical system, suchas relative location and stability of equilibria, their attraction domains, separatrices,limit cycles, and other special trajectories that are discussed in this section.4.3.1 Bistability and attraction domainsNon-linear two-dimensional systems can have many co-existing attractors. For example,the FitzHugh-Nagumo model (4.11,4.12) with nullclines depicted in Fig. 4.22 hastwo stable equilibria separated by an unstable equilibrium. Such a system is calledbistable (multi-stable when there are more than two attractors). Depending on the initialconditions, the trajectory may approach the left or right equilibrium. The shadedarea denotes the attraction domain of the right equilibrium; that is, the set of allinitial conditions that lead to this equilibrium. Since there are only two attractors,the complementary white area denotes the attraction domain of the other equilibrium.The domains are separated not by equilibria as in one-dimensional case, but by specialtrajectories called separatrices, which we discuss in Sect. 4.3.2.Many neural models are bistable or can be made bistable when the parameters haveappropriate values. Often bistability results from the co-existence of an equilibriumattractor corresponding to the rest state and a limit cycle attractor corresponding tothe repetitive firing state. Fig. 4.23 depicts one of many possible cases. Here we usethe I Na,p +I K -model with a high-threshold fast K + current. The rest state exists due tothe balance of partially activated Na + and leak currents. The repetitive spiking statepersists because the K + current deactivates too fast and cannot bring the membranepotential into the subthreshold voltage range. If the initial state is in the shaded area,