Hodgkin Huxley model (part 1)

Before starting to program our neuron model, let us talk about the H-H (hodgkin Huxley) model of neuron.

 

Electrical signals are propagated from nerve cell to nerve cell (neuron) via electro-chemical mechanisms. Hodgkin and Huxley experimented on squids and discovered how the signals are produced within the neuron. H.-H. model was published in Jour. of Physiology (1952). Hodgkin and Huxley were awarded 1963 Nobel Prize.

The neuron axon membrane has a potential difference,
\(V=V_{in}-V_{out}\)

 

When the axon is excited, V spikes because sodium Na+ and potassium K+ ions flow through the membrane. An equivalent electrical model proposed by Hodgkin and Huxley is as follows:

This circuit is governed by the differential equations:
$$C\frac{dV}{dt}=-g_{Na}m^3 h(V-V_{Na} )-g_Kn^4(V-V_K )-g_l(V-V_l)+I_{ext} $$$$\frac{dm}{dt}=\alpha_m(v)(1-m)-\beta_m(v)m$$$$\frac{dn}{dt}=\alpha_n(v)(1-n)-\beta_n(v)n$$$$\frac{dh}{dt}=\alpha_h(v)(1-h)-\beta_h(v)h$$

Alpha and beta are also functions of V which are determined by:
$$\alpha_n=\frac{0.01(10-V)}{\exp^{\left(1-\frac{V}{10}\right)}-1}$$ $$\beta_n = 0.125\exp^{-\left(\frac{V}{80}\right)}$$ $$\alpha_m=\frac{0.1(25-V)}{\exp^{\left(2.5-\frac{V}{10}\right)}-1}$$ $$\beta_m = 4\exp^{-\left(\frac{V}{18}\right)}$$ $$\alpha_h = 0.07\exp^{-\left(\frac{V}{20}\right)}$$$$\beta_h=\frac{1}{\exp^{\left(3-\frac{V}{10}\right)}+1}$$

At \(t=\infty\) \(\frac{dn}{dt}=0\), therefore we can calculate:$$n_\infty=\frac{\alpha_n(-V)}{\alpha_n(-V)+\beta_n(-V)}$$

\(m_\infty\) and \(h_\infty\) have similar formulas.
Now, based on these equations we will make our program. So let's get down to some real coding :D. We will make a neuron spiking model first and then write a program to analyse the phase space of the parameters by plotting some graphs (Beware its a 4D phase space).
Please follow my second article of the series where I'll discuss the programs.