Reduced Hodgkin-Huxley models based on the correlation between sodium and potassium gating variables

Abstract

We analyse the interaction between the sodium (Na${}^{+}$) and potassium (K${}^{+}$) gating variables in the 4D Hodgkin-Huxley axonal electrophysiological model. We develop simplified 3D and 2D models that capture the core dynamical behaviour of the original. These models rely on the empirical relationship $h\simeq c(I)-n$ between the gating variables $h$ and $n$, with $c(I)$ depending on stimulus intensity and reflecting an effective correlation between Na${}^{+}$ and K${}^{+}$ gating. Additionally, we derive the relevant cable equations for these reduced models and demonstrate that the action-potential propagation speed is well approximated by $v(R,C_m)=a/(C_m{R}^b)$, where $a>0$, $0<b<1$, and the parameters show weak dependence on stimulus strength within the studied range; here $C_m$ is membrane capacitance, and $R$ is the axon's internal resistivity.