Dynamical Systems in Neuroscience:

Conductance-Based Models 145leak currenthyperpolarizationdeinactivationof I Kirdeinactivationof I hinactivationof I hinactivationof I KirdepolarizationFigure 5.9: Mechanism of generation of sustainedvoltage oscillations in the I h +I Kir -model.via a positive feedback loop. Such a regenerative process results in a prolonged hyperpolarizationthat deinactivates (turns on) the slower inward current I h and produces arebound depolarization. This depolarization is enhanced by inactivation (turning off)of the fast I Kir . However, the membrane potential cannot hold long in the depolarizedstate because of the slow inactivation of I h , and the leak current repolarizes themembrane potential. The repolarization is enhanced by the deinactivation of I Kir andbecomes a hyperpolarization again, leading to the oscillations summarized in Fig. 5.9.Since the kinetics of I Kir is practically instantaneous, we can use h Kir = h Kir,∞ (V )in the voltage equation above and consider the two-dimensional systemleak I L instantaneous IC ˙V{ }} {Kir I{ }} { { }} h{= I − g L (V −E L ) − g Kir h Kir,∞ (V )(V − E K ) − g h h(V − E h ) ,ḣ = (h ∞ (V ) − h)/τ h (V ) .One can easily find the nullclines of this systemandh = I − g L(V −E L ) − g Kir h Kir,∞ (V )(V − E K )g h (V − E h )h = h ∞ (V ) (h-nullcline) ,(V -nullcline)which have the familiar form depicted in Fig. 5.10. Most values of the parametersresult in a phase portrait similar to the one depicted in Fig. 5.10, left. The V -nullclineis a monotonic curve that intersects the h-nullcline in one point corresponding to astable resting state. An injected dc-current I shifts the resting state, but does notchange its stability: Voltage perturbations always subside, resulting only in dampedoscillations. There is, however, a narrow region in parameter space (it took the authora few hours to find that region) that produces just the right relationship betweeninactivation curves and conductances so that the V -nullcline becomes N-shaped, andthe subthreshold oscillations become sustained, as in Fig. 5.10, right.5.1.6 I K +I Kir -modelThe last two minimal models consist exclusively of outward K + currents, yet they canexhibit sustained oscillations of membrane voltage. The models defy imagination of