Dynamical Systems in Neuroscience:

Electrophysiology of Neurons 391h (V)8τ (V) hn (V)m (V)τ (V) n0-40 0 100V (mV)τm(V)0-40 0 100V (mV)Figure 2.13: Steady-state (in)activation functions (left) and voltage-dependent timeconstants (right) in the Hodgkin-Huxley model.These parameters, provided in the original Hodgkin and Huxley paper, correspond tothe membrane potential shifted by approximately 65 mV so that the resting potentialis at V ≈ 0. Hodgkin and Huxley did that for the sake of convenience, but the shifthas led to a lot of confusion over the years. The shifted Nernst equilibrium potentialsareE K = −12 mV , E Na = 120 mV , E L = 10.6 mV;see also Ex. 1. Typical values of maximal conductances areḡ K = 36 mS/cm 2 , ḡ Na = 120 mS/cm 2 , g L = 0.3 mS/cm 2 .C = 1 µF/cm 2 is the membrane capacitance and I = 0 µA/cm 2 is the applied current.The functions α(V ) and β(V ) describe the transition rates between open and closedstates of the channels. We present this notation only for historical reasons. In the restof the book, we use the standard formwhereṅ = (n ∞ (V ) − n)/τ n (V ) ,ṁ = (m ∞ (V ) − m)/τ m (V ) ,ḣ = (h ∞ (V ) − h)/τ h (V ) ,n ∞ = α n /(α n + β n ) , τ n = 1/(α n + β n ) ,m ∞ = α m /(α m + β m ) , τ m = 1/(α m + β m ) ,h ∞ = α h /(α h + β h ) , τ h = 1/(α h + β h )are depicted in Fig. 2.13. These functions can be approximated by the Boltzmann andGaussian functions; see Ex. 4. We also shift the membrane potential back to its truevalue, so that the resting state is near -65 mV.The membrane of the squid giant axon carries only two major currents: transientNa + and persistent K + . Most neurons in the central nervous system have additionalcurrents with diverse activation and inactivation dynamics, which we summarize inSect. 2.3.5. The Hodgkin-Huxley formalism is the most accepted model to describetheir kinetics.