In any science class, math is used to describe phenomena quantitatively. In neuroscience, we use math to describe the conductance of ions and the permeability of the neuronal membrane to ions.
There are two equations, The Nernst Equation and The Goldman-Katz Equation that describe the conductance and permeability of neurons’ protein channels that allow ions to flow, creating current and thus, an Action Potential.
I barely understand the concept myself, which is why I’m writing this article to start to acquaint myself with the ideas that lead it around.
The Nernst Equation
The Nernst Equation describes the permeability of the cell membrane to a particular ion, for example, Na+. Ions want to go to a more positive place, whether inside the cell or outside the cell.
The equation above is a simplified representation of the Nernst Equation. This equation is built for the human body at a temperature of 37’C.
z is the valence of the ionic species. For example, z is +1 for Na+, +1 for K+, +2 for Ca2+, −1 for Cl−, etc. Note that z is unitless.
The equilibrium potenial for an ion is the voltage difference needed to balance that ion’s concentraLon gradient.
Try thinking about the Nernst Equation like this:
Out | In | All in mV | |
1 | 1 | 0mV | |
10 | 1 | +61.54mV | |
1 | 10 | -61.54mV |
Any conclusions you build? I’ll give you a minute.
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The closer to 0 the Eion is, the less driving force acts on the ion to move either in or out.
The more negative an Eion is, the more likely it will move into the ion.
The less negative an Eion is, the more likely it will move out of a neuron.
The Goldman Equation
The Goldman Equation is the mathematical relationship between the permeabilities of all ions with channels in the neuronal membrane.
If two or more ions contribute to the membrane potential, the Nernst potential no longer yields the Vm. In this case, use the Goldman-Hodgkin-Katz (GHK) equation to calculate the Vm.