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 simplication of the Krinsky-Kokoz-Rinzel model for gate functions: \(h=1-n\) and \(m=m_{\infty}(V)\). 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 fuction \(R\). The system of differential equations becomes: \[\begin{aligned} {C_m} \cdot \frac{{\dd V}}{{\dd t}} &= - {g_{\rm{K}}} \cdot R \cdot (V - {E_{\rm{K}}}) - {g_{{\rm{Na}}}}(V) \cdot (V - {E_{{\rm{Na}}}}) + {I_{{\rm{stim}}}} \\[0.25cm] \tau _{\rm{R}} \cdot \frac{{\dd R}}{{\dd t}} &= {R_{\infty}}(V) - R\\ \end{aligned}\] where \(V\) is the membrane potential, \(E_X\) is the Nernst potential for a given ion \(X\), and \[\begin{aligned} {g_{{\rm{Na}}}}(V) &= 17.8 + 0.476\,\,V + 33.8 \cdot {10^{ - 4}}\,{V^2}\\[0.25cm] {R_\infty }(V) &= 1.24 + 0.037\,V + 3.2\, \cdot {10^{ - 4}}\,{V^2}\end{aligned}\]

In order to avoid numerical difficulties, the basic Wilson model is rescaled.
Let \(U = V/100\) then the equations become: \[\begin{aligned} C_m \cdot \frac{\dd U}{\dd t} &= - g_{\rm{K}} \cdot R \cdot (U - E_{\rm{K}}/100) - g_{\rm{K}} \cdot (U - E_{\rm{Na}}/100) + I_{\rm{stim}}/100\\[0.25cm] \tau _{\rm{R}} \cdot \frac{\dd R}{\dd t} &= {R_{\infty}}(U) - R\\[0.25cm] g_{\rm{Na}}(U) &= 17.8 + 47.6\,U + 33.8\,{U^2}\\[0.25cm] {R_{\infty}}(U) &= 1.24 + 3.7\,U + 3.2\,{U^2}\end{aligned}\]