The Goldman Equation

This chapter derived some interesting models for the way action potentials are generated. But I noticed they let out an equation which I thought was quite important when considering the membrane potential. This equation is the Goldman-Hodgkin-Katz equation. This equation is effectively just the Nernst equation, but for multiple ionic species. But there is one important difference. The Nernst equation uses relative concentrations on each side of the cell membrane to determine the potential, whereas the Goldman equation goes a step further, and includes a permeability term for each ion.

Px represents the permeability of ionic species x, F is Faraday’s constant, and R and T have their usual meaning, the gas constant and temperature.

The permeability term allows this equation to remain valid during the entire action potential, unlike the Ohmic hypothesis used in Nelson. To convince yourself this equation will still give an accurate measurement of the membrane potential during an action potential, consider what happens to V as the permeability of sodium (PNa) increases. [Na+out]>[Na+in], so the numerator in the natural log increases, increasing the membrane potential V.

A realistic ratio of the permeabilities of these ions in the resting state is PNa:PK:PCl = 1 : 0.03 : 0.1. Using these pemeabilities and the values in table 11.1 on page 477 of Nelson, I calculated a resting potential of -63.8mV, which is very close to the measured value of 60mV. The ratio of permeabilities for the excited state is PNa:PK:PCl = 1 : 15 : 0.1, which yields a membrane potential of +43.7mV, which is again close to the accepted value of the peak potential of an action potential.

If you don’t want to repeat these calculations again, but would like to play around with values, I suggest going to this website: http://www.nernstgoldman.physiology.arizona.edu/. It allows you to vary the permeabilities of the ions and observe the effect on the membrane potential using the Goldman equation.