Dynamical Systems in Neuroscience:

46 Electrophysiology of Neurons1m (V)slopeC base+Campτ(V)0.502kV 1/2VC ampCσ C amp/ebaseV maxVFigure 2.20: Boltzmann (2.11) and Gaussian (2.12) functions and geometrical interpretationsof their parameters.Boltzmann function depicted in Fig. 2.20,m ∞ (V ) =11 + exp {(V 1/2 − V )/k}(2.11)The parameter V 1/2 satisfies m ∞ (V 1/2 ) = 0.5, and k is the slope factor (negative forinactivation curve h ∞ (V )). Smaller values of |k| result in steeper m ∞ (V ).The voltage-sensitive time constant τ(V ) can be approximated by the Gaussianfunctionτ(V ) = C base + C amp exp −(V max − V ) 2σ 2 , (2.12)see Fig. 2.20. The graph of the function is above C base with amplitude C amp . Themaximal value is achieved at V max . The parameter σ measures the characteristic widthof the graph, i.e., τ(V max ± σ) = C base + C amp /e. The Gaussian description is often notadequate, so we substitute it by other functions whenever appropriate.Below is the summary of voltage-gated currents whose kinetics were measured experimentally.The division into persistent and transient is somewhat artificial sincemost “persistent” currents can still inactivate after seconds of prolonged depolarization.Hyperpolarization-activated currents, such as the h-current or K + inwardly rectifyingcurrent, are mathematically equivalent to currents that are always activated, but canbe inactivated by depolarization. To avoid possible confusion we mark these currentsas “opened by hyperpolarization”.