Bioelectricity: Simulation of several single neuron models
Hodgkin-Huxley model
Simulation of the Hodgkin Huxley Model
The system of four differential equations in the Hodgkin-Huxley model is
\[\begin{aligned} C_m\frac{dV_m}{dt} &= -\overline{g_\mathrm{K}}n^4(V_m-E_\mathrm{K})-\overline{g_\mathrm{Na}}m^3h(V_m-E_\mathrm{Na})-\overline{g_\mathrm{L}}(V_m-E_\mathrm{L})+I\\
\frac{dn}{dt}&=\alpha_n(V_m)(1-n)-\beta_n(V_m)n\\
\frac{dm}{dt}&=\alpha_m(V_m)(1-m)-\beta_m(V_m)m \\
\frac{dh}{dt}&=\alpha_h(V_m)(1-h)-\beta_h(V_m)h \\
\end{aligned}\]
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 functions \(n\), \(m\), \(h\) satisfy the following initial value problems of exponentially restricted growth equations with voltage-dependent coefficients:
\[\begin{aligned}
\alpha_n(v) &= \frac{0.01\bigl(10-(v-V_r)\bigr)}{\exp\left(\frac{10-(v-V_r)}{10}\right)-1}\\
\beta_n(v)&=0.125\exp\left(\frac{-(v-V_r)}{80}\right)\\
\alpha_m(v)&= \frac{0.01\bigl(25-(v-V_r)\bigr)}{\exp\left(\frac{25-(v-V_r)}{10}\right)-1}\\
\beta_m(v)&=4\exp\left(\frac{-(v-V_r)}{18}\right)\\
\alpha_h(v)&=0.07\exp\left(\frac{-(v-V_r)}{20}\right)\\
\beta_h(v) &= \frac{1}{\exp\left(\frac{30-(v-V_r)}{10}\right)+1}\\
n(0) &= 0.32\\
m(0)&=0.06\\
h(0) &= 0.6\\
\end{aligned}\] We have also introduced blocking of potassium and sodium channels by adding to the original Hoddgkin-Huxley 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.