Friday, October 15, 2010

The Hodgkin and Huxley Equations

In the biological maths course I took last semester, we looked at the Hodgkin and Huxley model for action potential. I hoped that this chapter would mention it, but it didn’t go into many of the details. So I thought I’d share a bit of what I learnt here.

They modelled the rate of change of the potential as a function of the current injected into a cell minus the change in voltage caused by the change in conductance of each ion (Ex is the equilibrium Nernst potential of the ion x).

They had to find a relation to model the conductances of each ion as function of membrane potenial. They chose the equations below. The variables n and m measure (respectively) the probability of the potassium or sodium voltage gated ion channels being open and thus 0 <= n <=1. The term for the sodium conductance has an extra variable, h, which measures the probability of the sodium gated ion channel being inactivated. As section 12.3.2 on page 540 of Nelson points out, it’s ok to use a probabilistic model for the activation of ion channels, as there are many of them in a single cell.

A model must then be found for these probabilities. Each of the variables n, m and h were assumed to have first order kinetics, where there was one rate constant governing the opening of the channel, and one rate constant governing the closing. These rates are α and β in the equations below, and are assumed to be the same for each variable m, n and h. It is these variables, α and β, which are membrane potential dependant.

This model might seem complicated, and it is, but it produces quite realistic results. One of our assignments for this course was to solve these equations numerically. This is an example of the action potentials produced by this model.

I think this chapter would have been better off discussing this model rather than the model in section 12.2.

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