Dynamical Systems in Neuroscience:

104 Two-Dimensional Systemsy1.510.50Dy=x-x3/3x=0C(a)x(t)2.521.510.50BCBC(b)-0.51-1.5AB-2 -1 0 1 2x-0.5-1AA-1.5-2DD-2.50 100 200 300 400time, tFigure 4.12: Relaxation oscillations in the van der Pol model ẋ = x − x 3 /3 − y, ẏ = µxwith µ = 0.01.4.2 EquilibriaAn important step in the analysis of any dynamical system is to find its equilibria, i.e.,points wheref(x, y) = 0 ,g(x, y) = 0(point (x, y) is an equilibrium).As we mentioned before, equilibria are intersections of nullclines. If the initial point(x 0 , y 0 ) is an equilibrium, then ẋ = 0 and ẏ = 0, and the trajectory stays at equilibrium;that is, x(t) = x 0 and y(t) = y 0 for all t ≥ 0. If the initial point is near the equilibrium,then the trajectory may converge to or diverge from the equilibrium depending on itsstability.From the electrophysiological point of view, any equilibrium of a neuronal modelis the zero crossing of its steady-state I-V relation I ∞ (V ). For example, the I Na,p +I K -model (4.1, 4.2) with high-threshold K + current has an I-V curve with three zeroes(Fig. 4.1a), hence it has three equilibria: around −66 mV, −56 mV, and −28 mV. Incontrast, the same model with low-threshold K + current has a monotonic I-V curvewith only one zero (Fig. 4.1b), hence it has a unique equilibrium, which is around −61mV.4.2.1 StabilityIn Chap. 3, Exercise 18 we provide rigorous definitions of stability of equilibria inone-dimensional systems. The same definitions apply to higher-dimensional systems.