Dynamical Systems in Neuroscience:

134 Conductance-Based Models• If one removes any current or gating variable, the model has only equilibriumattractors for any values of parameters.We refer to such a model as being minimal or irreducible for spiking. Thus, minimalmodels can exhibit periodic activity, even if of small amplitude, but their reductionscannot. According to this definition, any space-clamped conductance-based model iseither a minimal model, or could be reduced to a minimal model or models by removinggating variables. This will be the basis for our classification of electrophysiologicalmechanisms in neurons.For example, the Hodgkin-Huxley model considered in Sect. 2.3 is not minimal forspiking. Recall that the model consists of three currents: leakage I L , transient sodiumI Na,t (gating variables m and h) and persistent potassium I K (gating variable n); seeFig. 5.1. Which of these currents are responsible for excitability and spiking?We can remove the leakage current and the gating variable, h, for the inactivationof the sodium current: The resulting I Na,p +I K -modelC ˙V = I −I K{ }} {g K n 4 (V − E K ) −ṅ = (n ∞ (V ) − n)/τ n (V ) ,ṁ = (m ∞ (V ) − m)/τ m (V ) ,I Na,p{ }} {g Na m 3 (V − E Na ) ,was considered in the previous chapter where we have shown that it could oscillate dueto the interplay between the activation of persistent sodium and potassium currents.Alternatively, we can remove the K + current from the Hodgkin-Huxley model, yet thenew I Na,t -modelC ˙V = I −I Na,t{ }} {g Na m 3 h(V − E Na ) −ṁ = (m ∞ (V ) − m)/τ m (V ) ,ḣ = (h ∞ (V ) − h)/τ h (V ) ,I L{ }} {g L (V − E L ) ,can still oscillate via the interplay between activation and inactivation of the Na +current, as we will see later in this chapter. Both models are minimal, because removalof any other gating variable results in either the I Na,p -, I K -, or I h -models, neither ofwhich can have a limit cycle attractor, as the reader is asked to prove at the end of theprevious chapter.We see that the Hodgkin-Huxley model is not minimal, but it is a combination oftwo minimal models. Minimal models are appealing because they are relatively simple;each individual variable has an established electrophysiological meaning, and its rolein dynamics can be easily identified. As we show below, many minimal models can