Dynamical Systems in Neuroscience:

140 Conductance-Based Modelsleak currentresting potentialdeinactivationof I Nainjecteddc-currentinactivationof I Naactivationof I NadepolarizationFigure 5.5: Mechanism of generation of sustainedoscillations in the I Na,t -model.equation and obtainC ˙V = I −leak I L{ }} {g L (V − E L ) −ḣ = (h ∞ (V ) − h)/τ h (V ) .I Na,t , inst. activation{ }} {g Na m 3 ∞(V )h(V − E Na ) ,One can easily find the nullclinesh =I − g L (V −E L )g Na m 3 ∞(V )(V − E Na )(V -nullcline)andh = h ∞ (V ) (h-nullcline) .The V -nullcline looks like a cubic parabola (flipped N shape), and the h-nullcline hasa sigmoid shape. In Fig. 5.6 we depict two typical cases (we invert the h-axis so thatthe vector field is directed counterclockwise and this phase portrait is consistent withthe other phase portraits in this book).When the inactivation curve h ∞ (V ) has a high threshold (i.e., I Na,t is a windowcurrent), there are three intersections of the nullclines, hence three equilibria, as inFig. 5.6a. A stable node (filled circle) corresponds to the rest state, and a nearbysaddle corresponds to the threshold state. Another equilibrium, an unstable focusdenoted by white circle at the top of the figure, determines the shape of the actionpotential since all “spiking” trajectories have to go around it. Because of the highthreshold of inactivation, the I Na,t current is deinactivated at rest. Moreover, smallfluctuations of V do not produce significant changes of the inactivation variable hbecause the h-nullcline is nearly horizontal at rest. Such a system does not performdamped oscillations, and the nonlinear dynamics of V near rest state can be describedby the one-dimensional system (where h = h ∞ (V ))C ˙V = I − g L (V − E L ) − g Na m 3 ∞(V )h ∞ (V )(V − E Na )studied in Chap. 3. When I increases, the stable node and the saddle approach,coalesce, and annihilate each other via saddle-node bifurcation. When I = 0.5, there is