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.