Simulation of the basic Wilson model

Starting from the Hodgkin-Huxley model for action potentials, Wilson reduced it for mammalian neocortical neurons to a two-dimensional dynamical system in the following way. First, he adopted the simplification of the Krinsky-Kokoz-Rinzel model for gate functions: $h=1-n$ and $m=m_{\inf}(V)$. Because neocortical neurons show no inactivation of sodium channels, one can even set $h = 1$ and $n = 0$. Wilson also combined the sodium channel and the leackage channel into a new single sodium channel. The recovery of the membrane potential can then be described by a dynamic modulation fuctions $R$. The system of equations becomes:

$$\begin{aligned} {C_m} \cdot \frac{{dV}}{{dt}} &= - {g_{\rm{K}}} \cdot R \cdot (V - {E_{\rm{K}}}) - {g_{{\rm{Na}}}}(V) \cdot (V - {E_{{\rm{Na}}}}) + {I_{{\rm{stim}}}} \\ \tau _{\rm{R}} \cdot \frac{{dR}}{{dt}} &= {R_{\infty}}(V) - R\\ \end{aligned}$$ where $V$ is the membrane potential, $E_X$ is the Nernst potential for a given ion $X$, and $$\begin{aligned} {g_{{\rm{Na}}}}(V) &= 17.8 + 0.476\,\,V + 33.8 \cdot {10^{ - 4}}\,{V^2}\\ {R_\infty }(V) &= 1.24 + 0.037\,V + 3.2\, \cdot {10^{ - 4}}\,{V^2}\end{aligned}$$

In order to avoid numerical difficulties, the basic Wilson model is rescaled.
Let $U = V/100$ then the equations become:

$$\begin{aligned} C_m \cdot \frac{dU}{dt} &= - g_{\rm{K}} \cdot R \cdot (U - E_{\rm{K}}/100) - g_{\rm{K}} \cdot (U - E_{\rm{Na}}/100) + I_{\rm{stim}}/100 \\ \tau _{\rm{R}} \cdot \frac{dR}{dt} &= {R_{\infty}}(U) - R\\ g_{\rm{Na}}(U) &= 17.8 + 47.6\,U + 33.8\,{U^2}\\{R_{\infty}}(U) &= 1.24 + 3.7\,U + 3.2\,{U^2}\end{aligned}$$