Dynamical Systems in Neuroscience:

Two-Dimensional Systems 105Figure 4.13: Neutrally stable equilibria.diverge from the equilibria.Some trajectories neither converge to norabFigure 4.14: Unstable equilibria.Briefly, an equilibrium is stable if any trajectory starting sufficiently close to the equilibriumremains near it for all t ≥ 0. If, in addition, all such trajectories convergeto the equilibrium as t → ∞, the equilibrium is asymptotically stable, as in Fig. 4.3c.When the convergence rate is exponential or faster, then the equilibrium is said tobe exponentially stable. Notice that stability does not imply asymptotic stability. Forexample, all equilibria in Fig. 4.13 are stable but not asymptotically stable. They areoften referred to as being neutrally stable.An equilibrium is called unstable, if it is not stable. Obviously, if all nearby trajectoriesdiverge from the equilibrium, as in Fig. 4.14a, then it is unstable. This, however,is an exceptional case. For instability it suffices to have at least one trajectory thatdiverges from the equilibrium no matter how close the initial condition to the equilibriumis, as in Fig. 4.14b. Indeed, any trajectory starting in the shaded area (attractiondomain) converges to the equilibrium, but any trajectory starting in the white areadiverges from it regardless of how close the initial point to the equilibrium is.In contrast to the one-dimensional case, the stability of a two-dimensional equilibriumcannot be inferred from the slope of the steady-state I-V curve. For example, theequilibrium around −28 mV in Fig. 4.1a is unstable even though the I-V curve haspositive slope.To determine the stability of an equilibrium, we need to look at the behavior of thetwo-dimensional vector field in a small neighborhood of the equilibrium. Quite oftenvisual inspection of the vector field does not give conclusive information about stability.