Dynamical Systems in Neuroscience:

136 Conductance-Based Modelsinward(Na, Ca)currentsoutward(K, Cl)amplifyingresonantactivation, mreversepotentialreversepotentialgatinginactivation, hresonantreversepotentialreversepotentialamplifyingFigure 5.2: Gating variables maybe amplifying or resonant dependingon whether they represent activation/inactivationof inward/outwardcurrents (see also Fig. 3.3 andFig. 3.4).can, as we will see below.The amplifying gating variable is the activation variable m for voltage-gated inwardcurrent or inactivation variable h for voltage-gated outward current, as in Fig. 5.2.These variables amplify voltage changes via a positive feedback loop. Indeed, a smalldepolarization increases m and decreases h, which in turn increase inward and decreaseoutward currents and increase depolarization. Similarly, a small hyperpolarizationdecreases m and increases h, resulting in less inward and more outward current, andhence in more hyperpolarization.The resonant gating variable is the inactivation variable h for an inward current oractivation variable n for an outward current. These variables resist voltage changes viaa negative feedback loop. A small depolarization decreases h and increases n, which inturn decrease inward and increase outward currents and produce a net outward currentthat resists the depolarization. Similarly, a small hyperpolarization produces inwardcurrent and possibly rebound depolarization.Currents with amplifying gating variables can result in bistability, and they behaveessentially like the I Na,p -model or I Kir -model considered in Chap. 3. Currents withresonant gating variables have one stable equilibrium with possibly damped oscillation,and they behave essentially like the I K -model or I h model (compare Fig. 5.2 withFig. 3.3). A typical neuronal model consists of at least one amplifying and at least oneresonant gating variable. Amplifying and resonant gating variables for Ca 2+ -sensitivecurrents are discussed at the end of this chapter.To get spikes in a minimal model, we need a fast positive feedback and a slowernegative feedback. Indeed, if an amplifying gating variable has a long time constant,it would act more as a low-pass filter, hardly affecting fast fluctuations, and onlyamplifying slow fluctuations. If a resonant gating variable has a fast time constant,it would act to damp input fluctuations (faster than they could be amplified by theamplifying variable), resulting in stability of the resting state. Instead, the resonant