Dynamical Systems in Neuroscience:

Conductance-Based Models 143off) by depolarization and deinactivated (turned on) by hyperpolarization. At restingvoltage this current is usually inactivated (turned off). A sufficient hyperpolarizationof V deinactivates (turns on) the h-current resulting in rebound depolarization. Whiledepolarized, the h-current inactivates (turns off), and the leak current repolarizes themembrane potential toward resting state. Without the persistent Na + current, or someother amplifying current, these oscillations always subside, as the reader was asked toprove in Chap. 4, Ex. 10. However, they may become sustained when I Na,p is involved.To study dynamics of the I Na,p +I h -model we assume that the activation kineticsof the Na + current is instantaneous, and use m = m ∞ (V ) in the voltage equation toobtain a two-dimensional systemleak I Linstantaneous I Na,p IC ˙V{ }} { { }} { { }} h{= I − g L (V −E L ) − g Na m ∞ (V )(V − E Na ) − g h h(V − E h ) ,ḣ = (h ∞ (V ) − h)/τ h (V ) .The nullclines of this systemh = I − g L(V −E L ) − g Na m ∞ (V )(V − E Na )g h (V − E h )(V -nullcline)andh = h ∞ (V )(h-nullcline)have the familiar N and sigmoid shapes depicted in Fig. 5.8. We take the parameters forboth currents from the experimental studies of thalamic relay neurons (see Sect. 2.3.5).This choice results in one intersection of the nullclines in the relevant voltage range,which corresponds to only one equilibrium. This equilibrium is a stable resting statewhen no current is injected, i.e., when I = 0. In Fig. 5.8, top, one can clearly seethat h ≈ 0; that is, the h-current is inactivated (turned off). The rest state is dueto the balance of inward persistent Na + current and the Ohmic leak current. A smallhyperpolarization deactivates the fast Na + current and shifts the balance toward theleak current, which brings V closer to E leak . This, in turn, results in slow deinactivation(turning on) of the h-current, which produces strong inward current and brings themembrane voltage back to the resting state.Negative injected current (case I = −1 in Fig. 5.8) destroys the balance of inward(I Na,p ) and outward (I leak ) currents at rest, and makes the resting state unstable. As aresult, the model exhibits sustained subthreshold oscillations of membrane potential.Indeed, prolonged hyperpolarization turns on strong h-current and produces prolongeddepolarization. Such a depolarization turns off the h-current, and the negative injectedcurrent hyperpolarizes the membrane potential again. As a result, the model exhibitssustained oscillations in the voltage range of −55 mV to −65 mV. The frequency of suchoscillations depends on the parameters of the voltage equation and the time constantτ(V ) of the h-current, and it is near 4 Hz in Fig. 5.8.