Simulation of the Krinsky-Kokoz-Rinzel model

The system of four differential equations of the Hodgkin-Huxley model can be reduced to a two-dimensional dynamical system in the following way.

Examining the behaviour of the state variables in the Hodgkin-Huxley model, one may notice that the sum of the gate functions \(n\) and \(h\) is nearly constant and that the gate function \(m\) changes relatively rapidly because its time scale is small compared to the time scales of \(n\) and \(h\) in the relevant voltage range. In mathematical formules this means that \(h = c - n\), for some constant \(c\), and that \(m = m_{\inf}(V)\). Thus we have the following systen of equations: \[\begin{aligned}C_m\frac{\dd V_m}{\dd t} &= - {x_{\rm{K}}}\overline {{g_{\rm{K}}}} {n^4}({V_m} - {E_{\rm{K}}}) - {x_{{\rm{Na}}}}\overline {{g_{{\rm{Na}}}}} {m^3}h({V_m} - {E_{{\rm{Na}}}}) - \overline {{g_{\rm{L}}}} ({V_m} - {E_{\rm{L}}}) + I\\[0.25cm] \frac{{\dd n}}{{\dd t}} &= {\alpha _n}({V_m})(1 - n) - {\beta _n}({V_m})n \\[0.25cm] h &= c - n \\[0.25cm] m &= {m_\infty }({V_m}) = \frac{{{\alpha _m}({V_m})}}{{{\alpha _m}({V_m}) + {\beta _m}({V_m})}}\end{aligned}\] where \(V_m\) is the membrane potential, \(E_X\) is the Nernst potential for a given ion \(X\) or the leakage potential (for \(X=L\)), and the gate function \(n\), satisfies an initial value problems of exponentially restricted growth equations with voltage-dependent coefficients.

We have also introduced blocking of potassium and sodium channels by adding to the model parameters \(x_\mathrm{K}\) and \(x_\mathrm{Na}\), defined as the fractions of active ion channels (in terms of percentages \(b_\mathrm{K}\) and \(b_\mathrm{Na}\) of blocked ion channels): \[x_\mathrm{K}=1-\frac{b_\mathrm{K}}{100}\qquad\text{and}\qquad x_\mathrm{Na}=1-\frac{b_\mathrm{Na}}{100}\] Blocking of ion channels often is a result of neurotoxines: Tetrodoxin (TTX), isolated from the Japanese pufferfish fugu, blocks sodium channels, and Tetraethylammonium (TEA) blocks potassium channels.