Interdependence of sodium and potassium gating variables in the Hodgkin-Huxley model

We explore the relationship between sodium (Na$^+$) and potassium (K$^+$) gating variables in the 4-dimensional (4D) Hodgkin-Huxley (HH) electrophysiology model and reduce its complexity by deriving new 3D and 2D models that retain the original model's dynamic properties. The new 3D and 2D models are based on the relationship $h \simeq c - n$ between the gating variables $h$ and $n$ of the 4D HH model, where $c$ is a constant, which suggests an interdependence between the dynamics of Na$^+$ and K$^+$ transmembrane voltage-gated channels. The presence of Na$^+$/K$^+$-ATPase pump may explain this interdependence. We derive the corresponding cable equations for the three HH-type models and demonstrate that the action potential propagates along the axon at with the speed $v(R, C_m) = \alpha / (C_m R^{\beta}) = \gamma D^{\beta}$, where $\alpha > 0$, $0 < \beta < 1$, and $\gamma$ are constants independent of the local stimulus intensity, $D$ is the diffusion coefficient of the electric signal along the axon, $C_m$ is the axon transmembrane capacitance, and $R$ is the axon conducting resistivity.