Dynamical Systems in Neuroscience:

62 One-Dimensional Systemsat every point V where F (V ) is negative, the derivative ˙V is negative, and hence thestate variable V decreases. In contrast, at every point where F (V ) is positive, ˙V ispositive, and the state variable V increases; the greater the value of F (V ), the fasterV increases. Thus, the direction of movement of the state variable V , and hence theevolution of the dynamical system, is determined by the sign of the function F (V ).The right-hand side of the I leak -model (3.1) or the I Na,p -model (3.5) in Fig. 3.8 is thesteady-state current-voltage (I-V) relation, I L (V ) or I L (V )+I Na,p (V ) respectively, takenwith the minus sign, see Fig. 3.5. Positive values of the right-hand side F (V ) meannegative I-V, corresponding to a net inward current that depolarizes the membrane.Conversely, negative values mean positive I-V, corresponding to a net outward currentthat hyperpolarizes the membrane.3.2.2 EquilibriaThe next step in the qualitative analysis of any dynamical system is to find its equilibriaor rest points, i.e., the values of the state variable whereF (V ) = 0(V is an equilibrium).At each such point ˙V = 0, the state variable V does not change. In the context ofmembrane potential dynamics, equilibria correspond to the points where the steadystateI-V curve passes zero. At each such point there is a balance of the inward andoutward currents so that the net transmembrane current is zero, and the membranevoltage does not change. (Incidentally, the part l ībra in the Latin word aequil ībriummeans balance).The I K - and I h -models mentioned in Sect. 3.1 can have only one equilibrium becausetheir I-V relations I(V ) are monotonic increasing functions. The correspondingfunctions F (V ) are monotonic decreasing and can have only one zero.In contrast, the I Na,p - and I Kir -models can have many equilibria because their I-Vcurves are not monotonic, and hence there is a possibility for multiple intersectionswith the V -axis. For example, there are three equilibria in Fig. 3.8b corresponding tothe rest state (around −53 mV), threshold state (around −40 mV) and the excitedstate (around 30 mV). Each equilibrium corresponds to the balance of the outwardleak current and partially (rest), moderately (threshold) or fully (excited) activatedpersistent Na + inward current. Throughout this book we denote equilibria as smallopen or filled circles depending on their stability, as in Fig. 3.8.3.2.3 StabilityIf the initial value of the state variable is exactly at equilibrium, then ˙V = 0 and thevariable will stay there forever. If the initial value is near the equilibrium, the statevariable may approach the equilibrium or diverge from it. Both cases are depicted inFig. 3.8. We say that an equilibrium is asymptotically stable if all solutions startingsufficiently near the equilibrium will approach it as t → ∞.