Dynamical Systems in Neuroscience:

Conductance-Based Models 139This model is in many respects equivalent to the I Ca +I K -model proposed by Morrisand Lecar (1981) to describe voltage oscillations in the barnacle giant muscle fiber.A reasonable assumption based on experimental observations is that the Na + gatingvariable m(t) is much faster than the voltage variable V (t), so that m approaches theasymptotic value m ∞ (V ) practically instantaneously. In this case we can substitutem = m ∞ (V ) in the voltage equation and reduce the three-dimensional system aboveto a planar systemleak I Linstantaneous I Na,p IC ˙V{ }} { { }} { { }} K{= I − g L (V −E L ) − g Na m ∞ (V ) (V −E Na ) − g K n (V −E K ) (5.1)ṅ = (n ∞ (V ) − n)/τ(V ) , (5.2)which was considered in detail in the previous chapter. In Fig. 5.4 we summarize itsdynamic repertoire. A striking observation is that the other minimal models can havesimilar nullclines and similar dynamic repertoire, even though they consist of quitedifferent ionic currents.5.1.3 I Na,t -modelAn interesting example of a spiking mechanism, implicitly present in practically everybiological neuron, is given by the I Na,t -modelC ˙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 Na,t{ }} {g Na m 3 h(V − E Na ) ,consisting only of an Ohmic leak current and a transient voltage-gated inward Na +current. How could such a model generate action potentials? The upstroke of an actionpotential is generated because of the regenerative process involving the activation gatem. This mechanism is similar to the one in the Hodgkin-Huxley model or in theI Na,p +I K -model: Increase of m results in increase of the inward current, hence moredepolarization and further increase of m until the excited state is achieved. At theexcited state there is a balance of the Na + inward current and the leak outward current.Since there is no I K , the downstroke from the excited state occurs via a differentmechanism: While in the excited state, the Na + current inactivates (turns off) and theOhmic leak current slowly repolarizes the membrane potential toward the leak reversepotential E L , which determines the resting state. While at rest, the Na + currentdeinactivates; i.e., becomes available, and the neuron is ready to generate anotheraction potential. This mechanism is summarized in Fig. 5.5.To study the dynamics of the I Na,t -model, we first reduce it to a planar system.Assuming that activation dynamics is instantaneous, we use m = m ∞ (V ) in the voltage