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

Abstract: We explore the relationship between sodium (Na$^+$) and potassium (K$^+$) gating variables in the 4-dimensional (4D) Hodgkin-Huxley (HH) electrophysiology model, and reducing its complexity by deriving new 3D and 2D models that maintain the dynamic properties of the original model. The new 3D and 2D models are grounded in the relationship $h \simeq c(I) - n$ between the gating variables $h$ and $n$ of the 4D HH model, where $c(I)$ depends of the input external stimulus, indicating an interdependence between the dynamics of Na$^+$ and K$^+$ transmembrane voltage-gated channels. The presence of Na$^+$/K$^+$-ATPase pumps along the axon may explain this interdependence. We derive the corresponding cable equations for the two new HH-type models and demonstrate that the action potential propagates along the axon at a speed given by $v(R, C_m) = \alpha / (C_m R^{\beta}):= \gamma D^{\beta}$, 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.