Dynamical Systems in Neuroscience:

36 Electrophysiology of Neurons10.8h sm (V)0.60.40.2V sh (V)0-80 -60 -40 -20 0membrane voltage, V (mV)Figure 2.11: Steady-state activation functionm ∞ (V ) from Fig. 2.10, inactivation functionh ∞ (V ), and values h s from Fig. 2.12. Theiroverlap (shaded region) produces a noticeablepersistent “window” current.In Fig. 2.10 we depict a typical experiment to determine m ∞ (V ) of a persistentcurrent, i.e., a current having no inactivation variable. Initially we hold the membranepotential at a hyperpolarized value V 0 so that all activation gates are closed and I ≈ 0.Then we step-increase V to a greater value V s (s = 1, . . . , 7; see Fig. 2.10a) and holdit there until the current is essentially equal to its asymptotic value, which we denotehere as I s (s stands for “step”; see Fig. 2.10b). Repeating the experiment for variousstepping potentials V s , one can easily determine the corresponding I s , and hence theentire steady-state I-V relation, which we depict in Fig. 2.10c. According to (2.7),I(V ) = ḡm ∞ (V )(V − E), and the steady-state activation curve m ∞ (V ) depicted inFig. 2.10d is just I(V ) divided by the driving force (V − E) and normalized so thatmax m ∞ (V ) = 1. To determine the time constant τ(V ), one needs to analyze theconvergence rates. In Ex. 6 we describe an efficient method to determine m ∞ (V ) andτ(V ).2.2.3 Inactivation of transient currentsThe dynamics of the inactivation variable h can also be described by the first-orderdifferential equationḣ = (h ∞ (V ) − h)/τ(V ) . (2.10)where h ∞ (V ) is the voltage-sensitive steady-state inactivation function depicted inFig. 2.11. In Fig. 2.12 we present a typical voltage-clamp experiment to determineh ∞ (V ) in the presence of activation m ∞ (V ). It relies on the observation that inactivationkinetics is usually slower than activation kinetics. First, we hold the membranepotential at a certain pre-step potential V s for a sufficiently long time so that the activationand inactivation variables are essentially equal to their steady-state values m ∞ (V s )and h ∞ (V s ), respectively, which have yet to be determined. Then we step-increase Vto a sufficiently high value V 0 chosen so that m ∞ (V 0 ) ≈ 1. If activation is much fasterthan inactivation, m approaches 1 after a first few milliseconds while h continues tobe near its asymptotic value h s = h ∞ (V s ), which can be found from the peak valueof the current I s ≈ ḡ · 1 · h s (V s − E). Repeating this experiment for various pre-steppotentials, one can determine the steady-state inactivation curve h ∞ (V ) in Fig. 2.11.