Dynamical Systems in Neuroscience:

34 Electrophysiology of Neuronsm (V)1.00.80.60.40.20.0-80 -40 0 40 80V (mV)τ(V) (ms)1.61.41.21.00.80.60.40.2-120 -80 -40 0 40 80V (mV)Figure 2.9: The activation function m ∞ (V ) and the time constant τ(V ) of the fasttransient K + current in layer 5 neocortical pyramidal neurons (modified from Korngreenand Sakmann 2000).open the channels, and those that inactivate or close them; see Fig. 2.8. Accordingto the tradition initiated in the middle of 20th century by Hodgkin and Huxley, theprobability of an activation gate to be in the open state is denoted by the variable m(sometimes the variable n is used for K + and Cl − channels). The probability of aninactivation gate to be in the open state is denoted by the variable h. The proportionof open channels in a large population isp = m a h b , (2.8)where a is the number of activation gates and b is the number of inactivation gates perchannel. The channels can be partially (0 < m < 1) or completely activated (m = 1);not activated or deactivated (m = 0); inactivated (h = 0); released from inactivationor deinactivated (h = 1). Some channels do not have inactivation gates (b = 0), hencep = m a . Such channels do not inactivate, and they result in persistent currents. Incontrast, channels that do inactivate result in transient currents.Below we describe voltage- and time-dependent kinetics of gates. This descriptionis often referred to as the Hodgkin-Huxley gate model of membrane channels.2.2.2 Activation of persistent currentsThe dynamics of the activation variable m is described by the first-order differentialequationṁ = (m ∞ (V ) − m)/τ(V ) (2.9)where the voltage-sensitive steady-state activation function m ∞ (V ) and the time constantτ(V ) can be measured experimentally: They have sigmoid and unimodal shapes,respectively, as in Fig. 2.9; see also Fig. 2.20. The steady-state activation functionm ∞ (V ) gives the asymptotic value of m when the potential is fixed (voltage-clamp).Smaller values of τ(V ) result in faster dynamics of m.