Dynamical Systems in Neuroscience:

142 Conductance-Based Modelshyperpolarizationdeactivationof I Na,pdeinactivationof I hleak currentor injecteddc-currentinactivationof I hactivationof I Na,pdepolarizationFigure 5.7: Mechanism of generation of sustainedvoltage oscillations in the I Na,p +I h -model.a periodic trajectory with a long period (compare the time scales in the bottom insetsin Fig. 5.6a and b).When the Na + inactivation curve h ∞ (V ) has a low threshold, the nullclines haveonly one intersection, hence there is only one equilibrium, as in Fig. 5.6b. WhenI = 0, the equilibrium (filled circle) is stable and all trajectories converge to it. Thereare damped oscillations near the equilibrium, though they can hardly be seen in thefigure. The oscillations occur because the I Na,t current is partially inactivated at rest.An increase of V leads to more inactivation, less inward current, and hence rebounddecrease of V , which in turn leads to partial deinactivation, more inward current, andrebound increase of V . When the applied dc-current I increases, the equilibrium losesstability via Andronov-Hopf bifurcation. When I = 4, the equilibrium is an unstablefocus (white circle in the figure), and there is a stable limit cycle attractor around itcorresponding to periodic spiking.We see that the I Na,t -model exhibits essentially the same dynamic repertoire as theI Na,p +I K -model, even though the models are quite different from the electrophysiologicalpoint of view.5.1.4 I Na,p +I h -modelThe systemI Na,pleak IC ˙V{ }} L{ { }} { { }} {= I − g L (V −E L ) − g Na m(V − E Na ) − g h h(V − E h ) ,ṁ = (m ∞ (V ) − m)/τ m (V ) ,ḣ = (h ∞ (V ) − h)/τ h (V ) ,I his believed to describe the essence of the mechanism of slow subthreshold voltage oscillationsin some cortical, thalamic, and hippocampal neurons, which we summarizein Fig. 5.7. As any other minimal model in this section, it consists of one amplifying(I Na,p ) and one resonant (I h ) current. Both currents may be partially active at restingvoltage. Recall that we treat the h-current as being an inward current that is alwaysactivated (its activation variable m = 1 all the time), but can be inactivated (turned