  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. We derive the corresponding cable
equations for the three HH-type models and demonstrate that the action
potential propagates along the axon at a speed described 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. The
width $w$ of the action potential spikes is inversely related to the
resistivity of the axon, with $w = \alpha_2 / R^{\beta_2}$, where $\alpha_2 >
0$ and $\beta_2 > 0$.
