Dynamical Systems in Neuroscience:

150 Conductance-Based Modelshave comparable time constants or inactivation is much faster than activation.Even though none of the experimentally measured A-currents show fast inactivationand a relatively slower activation, this case is still interesting from the pure theoreticalpoint of view, since it shows how a single K + current can give rise to oscillations.Assuming instantaneous inactivation and using h = h ∞ (V ) in the voltage equation, weobtain a two-dimensional systemC ˙V = I −leak I L{ }} {g L (V − E L ) −ṁ = (m ∞ (V ) − m)/τ m (V ) ,I A , inst. inactivation{ }} {g A m h ∞ (V )(V − E K ) ,whose nullclines can easily be found:m = I − g L(V −E L )g A h ∞ (V )(V − E K )(V -nullcline)andm = m ∞ (V ) (m-nullcline) .Two typical cases are depicted in Fig. 5.14a and b. We start with the simpler case inFig. 5.14b.Figure 5.14b depicts nullclines when the A-current has low activation threshold.There is only one intersection of the nullclines, hence there is only one equilibrium,which is a stable focus when the injected dc-current I is not strong enough (Fig. 5.14b,top). Increasing I makes the equilibrium lose stability via a supercritical Andronov-Hopf bifurcation that gives birth to a small amplitude limit cycle attractor (not shownin the figure). A further increase of I increases the amplitude of oscillations, and e.g.,when I = 10 (middle of Fig. 5.14b), the attractor corresponds to periodic firing of actionpotentials. When I = 10.5, the attractor disappears and the equilibrium becomesstable (via Andronov-Hopf bifurcation) again. The model, however, becomes excitable.A small hyperpolarization does not change significantly the A-current, and the voltagereturns back to rest resulting in a “subthreshold response”. A sufficiently largehyperpolarization deinactivates enough I A to open the K + current and hyperpolarizethe membrane even further. This regenerative process produces the downstroke andbrings V close to E K . During the state of hyperpolarization, the A-current deactivates(m → 0), and the dc-current I brings V back to rest. Notice that the action potentialis directed downward.In Fig. 5.14a we consider the I A -model with exactly the same parameters exceptthat we shift the half-voltage activation V 1/2 of I A by 10 mV so that the A-currenthas higher activation threshold. This does not affect much the behavior of the systemwhen I is small. However, when I ≈ 10.7 the spiking limit cycle attractor undergoesa new kind of bifurcation — saddle-node bifurcation — resulting in the appearance oftwo new equilibria: a stable node and a saddle. If the reader looks at Fig. 5.14a upsidedown,he will notice that this figure resembles figures 5.4a, 5.6a, or 5.12a, with all the