Dynamical Systems in Neuroscience:

Conductance-Based Models 147brings the potential up again. Thus, the upstroke of the action potential is due exclusivelyto the injected dc-current I, while the downstroke is due to the persistentoutward K + current.To perform the geometrical phase plane analysis of the model we take advantage ofthe same observation as before: The kinetics of the amplifying current I Kir is relativelyfast, so that h = h ∞ (V ) can be used in the voltage equation to reduce the threedimensionalsystem above to the two-dimensional systeminstantaneous I Kir IC ˙V{ }} { { }} K{= I − g Kir h ∞ (V )(V − E K ) − g K n(V − E K ) ,ṅ = (n ∞ (V ) − n)/τ n (V ) .It is an easy exercise to find the nullclinesn = I/{g K (V − E K )} − g Kir h ∞ (V )/g K(V -nullcline)andn = n ∞ (V ) (n-nullcline) ,which we depict in Fig. 5.12. There are two interesting cases corresponding to highthreshold(Fig. 5.12a) and low-threshold (Fig. 5.12b) K + current I K .When the I K has low threshold, it is partially activated at resting potential. In thiscase, rest state corresponds to the balance of partially activated I K , partially inactivatedI Kir , and a strong injected dc-current I (without the dc-current the membrane voltagewould converge to E K = −80 mV and stay there forever). A small depolarizationpartially inactivates fast I Kir but leaves slower I K relatively unchanged. This resultsin an imbalance of the inward dc-current I and all outward currents, and the netinward current further depolarizes the membrane voltage. Depending on the size of thedepolarization, the model may generate a subthreshold response or an action potential,as one can see in Fig. 5.12b, top. During the generation of the action potential, thepersistent K + current activates and causes after-hyperpolarization. During the afterhyperpolarization,the persistent K + current deactivates below the resting level. Thislets the injected dc-current I depolarize the membrane potential again, provided thatI is strong enough, as in Fig. 5.12b, bottom.In Fig. 5.12a we leave all parameters unchanged except that we increase the halfvoltageactivation V 1/2 of I K by 15 mV and decrease I to compensate for the deficitof outward current. Now, the resting state corresponds to the balance of the I Kir andI, because the high-threshold persistent K + current is completely deactivated in thisvoltage range. The behavior near the rest state is determined by the interplay betweeninstantaneous I Kir and I, and it was studied in Chap. 3 (see I Kir -model). There are twoequilibria: a stable node corresponding to the resting state, and a saddle correspondingto the threshold state. A sufficiently strong perturbation can push V beyond the saddleequilibrium, as in Fig. 5.12a, top, and can cause the minimal model to fire an action