Dynamical Systems in Neuroscience:

Conductance-Based Models 137resonant gating variablesinactivation ofinward currentactivation ofoutward currentamplifying gating variablesactivation ofinward currentinactivation ofoutward currentI Na,t -modelI Na,p +I h -modelI Kir +I h -modelI Na,p +I K -modelI Kir +I K -modelI A -modelFigure 5.3: Any combination of oneamplifying and one resonant gatingvariables results in a spiking model.variable acts as a band-pass filter; It has no effect on oscillations with a period muchsmaller than its time constant. It damps oscillations having period much larger than itstime constant, because the variable oscillates in phase with the voltage fluctuations; Itamplifies oscillations with a period that is about the same as its time constant becausethe variable lags the voltage fluctuations.Since the amplifying gating variable, say m, has relatively fast kinetics, it can bereplaced by its equilibrium (steady-state) value m ∞ (V ). This allows to reduce thedimension of the minimal models from three (say V , m, n) to two (V and n).Two amplifying and two resonant gating variables produce four different combinations,depicted in Fig. 5.3. However, the number of minimal models is not four, but six.The additional models arise due to the fact that a pair of gating variables may describeactivation/inactivation properties of the same current or of two different currents. Forexample, the activation and inactivation gating variables m and h may describe thedynamics of a transient inward current, such as I Na,t , or the dynamics of a combinationof one persistent inward current, such as I Na,p , and one “hyperpolarization-activated”inward current, such as I h . Hence this pair results in two models, I Na,t - and I Na,p +I h -model. Similarly, the pair of activation and inactivation variables of an outward currentmay describe the dynamics of the same transient current, such as I A , or the dynamicsof two different outward currents, hence the two models, I A - and I Kir +I K -model.Below we present the geometrical analysis of the six minimal voltage-gated modelsshown in Fig. 5.3. Though based on different ionic currents, the models have manysimilarities from the dynamical systems point of view. In particular, all can exhibitsaddle-node and Andronov-Hopf bifurcations. For each model we first provide a worddescription of the mechanism of generation of sustained oscillations, and then usephase plane analysis to provide a geometrical description. The first two, I Na,p +I K -and I Na,t -models are common; they describe the mechanism of generation of actionpotentials or subthreshold oscillations by many cells. The other four models are rare;they even might be classified as weird or bizarre by biologists, since these models reveal