Dynamical Systems in Neuroscience:

Conductance-Based Models 149de-inactivationof I Ahyperpolarizationdeactivationof I Ainjecteddc-currentactivation +inactivationof I Aresting potentialFigure 5.13: Mechanism of generation of sustainedvoltage oscillations in the I A -model.current and an Ohmic leak current. The only difference is that A-current is outward,and as a result, the action potentials are fired downward; see Fig. 5.14 and Fig. 5.15below.The A-current has activation and inactivation variables m and h, respectively, andthe model has the formC ˙V = I −leak I L{ }} {g L (V − E L ) −ṁ = (m ∞ (V ) − m)/τ m (V )ḣ = (h ∞ (V ) − h)/τ h (V ) .I A{ }} {g A mh(V − E K )The mechanism of generation of downward action potentials is summarized in Fig. 5.13.Due to a strong injected dc-current, the rest state is in the depolarizing voltage range,and it corresponds to the balance of the partially activated, partially inactivated A-current, leak outward current, and the injected dc-current. A small hyperpolarizationcan simultaneously deactivate and deinactivate the A-current, i.e., decrease variablem and increase variable h. Depending on the relationship between the activation andinactivation time constants, this may result in an increase of the A-current conductance,which is proportional to the product mh. More outward current produces morehyperpolarization and even more outward current. As a result of this regenerativeprocess, the membrane voltage produces a sudden downstroke. While hyperpolarized,the A-current deactivates (variable m → 0), and the injected dc-current slowly bringsthe membrane potential toward the resting state, resulting in a slow upstroke. Fastdownstroke and a slower upstroke from a depolarized resting state look like an actionpotential pointing downwards.If activation kinetics is much faster than the inactivation kinetics, we can substitutem = m ∞ (V ) into the voltage equation above and reduce the I A -model to atwo-dimensional system, which hopefully would have the right kind of nullclines anda limit cycle attractor. After all, this is what we have done with previous minimalmodels and it always worked. As the reader is asked to prove in Ex. 1, the I A -modelcannot have a limit cycle attractor when the A-current activation kinetics is instantaneous.Oscillations are possible only when the activation and inactivation kinetics