Hodgkin-Huxley Dynamics
The Hodgkin-Huxley approach is named after the Nobel laureates who were the first to develop a mathematical model of the neural action potential. Single channel activity recordings show that channels have a small set of discrete conductance states, with the channel stochastically transitioning between closed states and open states. Short time analysis of a single channel is very difficult to interpret but ensemble averaging clearly reveals smooth kinetics in changes of channel conductance. In the Hodgkin-Huxley formulation, channel conductance is assumed to be controlled by gates which take on values between 0 and unity, representing the portion of the cells in one state. Since cells have hundreds of ion channels if not more, this approximation holds well. Current flow produced by ion species :math:X passing through a channel is then described by
\( \begin{equation} I_{\rm S} = \overline{g}_S \prod_n \eta_n ( V_{\rm m} - E_{\rm S} ) \nonumber \end{equation} \)
where \(\overline{g}_{\rm S}\) is the maximum conductance of the channel and \(\eta_{\rm n}\) is a gating variable. Often the gating variable is assumed to follow first order dynamics so
\( \begin{align} \frac{d\eta}{dt} & = \alpha(V_{\rm m})(1-\eta) - \beta(V_{\rm m})\eta \nonumber \\ & = \frac{\eta_{\infty}(V_{\rm m}) - \eta}{\tau_{\eta}(V_{\rm m})} \nonumber \end{align} \)
where \(\alpha\) and \(\beta\) are rates which can be cast into an equivalent form of a steady state value (\(\eta_{\infty}\)) and a rate of change (\(\tau_\eta\)). The advantage of this latter formulation is that mathematically, the update of the gating variables can be performed by a numercial integration method, the Rush-Larsen technique, which is guaranteed to unconditionally keep the gating variable bounded within the range [0, 1] while allowing a large time step.