Dynamical Systems in Neuroscience:

112 Two-Dimensional SystemsV-nullclinew-nullclinerecovery, w0.2w-nullclineV-nullcline00 0.5 1 0 0.5 1membrane voltage, Vmembrane voltage, VabFigure 4.20: Nullclines in the FitzHugh-Nagumo model (4.11, 4.12). Parameters: I =0, b = 0.01, c = 0.02, a = 0.1 (left) and a = −0.1 (right).?ctr L=-a-c = 0stability tr L < 0det L > 0=det L=ac+b0b0baFigure 4.21: Stability diagram of theequilibrium (0, 0) in the FitzHugh-Nagumo model (4.11,4.12).and they can intersect in one, two, or three points resulting in one, two, or threeequilibria, all of which may be unstable. Below we consider the simple case I = 0, sothat the origin, (0, 0), is an equilibrium. Indeed, the nullclines of the model, depictedin Fig. 4.20, always intersect at (0, 0) in this case. The intersection may occur on theleft (Fig. 4.20a) or middle (Fig. 4.20b) branch of the cubic V -nullcline depending onthe sign of the parameter a. Let us determine how the stability of the equilibrium(0, 0) depends on the parameters a, b, and c.There is a common dogma that the equilibrium in Fig. 4.20a corresponding toa > 0 is always stable, the equilibrium in Fig. 4.20b corresponding to a < 0 is alwaysunstable, and the loss of stability occurs “exactly” at a = 0, i.e., at the bottom of theleft knee. Let us check that this is not necessarily true, at least when c ≠ 0. TheJacobian matrix of the FitzHugh-Nagumo model (4.11,4.12) at the equilibrium (0, 0)has the form( ) −a −1L =.b −c