Dynamical Systems in Neuroscience:

56 One-Dimensional SystemsActivation Variable10.90.80.70.60.50.40.30.2m(t)Membrane Voltage (mV)20100-10-20-30-40-50-60V(t)τ(V) < 0.5τ(V) < 0.1τ(V) < 0.01Instantaneous0.1-7000 1 2 3 4 5 6 7 8 9 10Time (ms)-800 1 2 3 4 5 6 7 8 9 10Time (ms)Figure 3.1: Solution of the full system (3.2, 3.3) converges to that of the reducedone-dimensional system (3.4) as τ(V ) → 0reduce the two-dimensional system (3.2, 3.3) to a one-dimensional equationC ˙V = −g L (V − E L ) −instantaneous I fast{ }} {g p ∞ (V ) (V − E) . (3.4)This reduction introduces a small error of the order τ(V ) ≪ 1, as one can see inFig. 3.1.Since the hypothetical current I fast can be either inward (E > E L ) or outward(E < E L ), and the gating process can be either activation (p is m, as in the Hodgkin-Huxley model) or inactivation (p is h), there are four fundamentally different choicesfor I fast (V ), which we summarize in Fig. 3.2 and elaborate on below.GatingactivationinactivationCurrentinward outwardI Na,pI hI KI KirFigure 3.2: Four fundamental examples of voltagegatedcurrents with one gating variable. In thisbook we treat “hyperpolarization-activated” currentsI h and I Kir as being inactivating currents,which are turned off (inactivated via h) by depolarizationand turned on (deinactivated) by hyperpolarization;see discussion in Sect. 2.2.4.3.1.1 I-V relations and dynamicsThe four choices in Fig. 3.2 result in four simple one-dimensional models of the form(3.4):I Na,p -model , I K -model , I h -model , and I Kir -model .These models might seem too simple to biologists, who can easily understand theirbehavior just by looking at the I-V relations of the currents depicted in Fig. 3.3 without