# Understanding the derivation of the Nernst equation

I am trying to understand how the Nernst equation can be derived and am mostly referring to the explanation given in the book Theoretical Neuroscience by Dayan and Abbott.

Given we have a concentration gradient and consider only Potassium, the overall approach to derive the Nernst equation is to find the equilibrium state where the flow of ions out of the cell and the flow of ions into the cell due to diffusion are the same. For the flow of ions we find the following two expressions: $$F_{out \rightarrow in} = \gamma [K^+]_{out} \\ F_{in \rightarrow out} = \gamma [K^+]_{in} \cdot exp({\frac{q V}{k_b T}})$$ In these two formulas $$\gamma$$ is some complex factor that we don't know that e.g. encodes the number of channels but it does not matter since it will disappear when we equate both equations. $$[K^+]_{out}$$ is the concentration of potassium ions outside the cell and $$[K^+]_{in}$$ is the potassium concentration inside the cell. $$V$$ is the difference in electrical potential between the inside and the outside and $$T$$ is the temperature.

The reasoning given by Abbott and Dayan is that the flow of ions from inside the cell to the outside needs to overcome the energy barrier of the electrical gradient. Only ions that have sufficient thermal energy can do so, this is why the Boltzmann factor appears. The flow of ions from the outside to the inside does not need to overcome such a gradient. It is therefore proportional to the concentration of the outside.

The last part is where I struggle. Don't we need to account for the fact that positively charged ions are pulled into the cell due to the potential difference. Am I wrong in claiming that there is an electrical field that will increase the number of ions traveling into the cell?

• This is not how I would approach thinking about the Nernst equation; check the derivation on Wikipedia I find it very straightforward: en.wikipedia.org/wiki/Nernst_equation – Bryan Krause Feb 5 at 15:46
• Unfortunately, I am not really familiar with electrochemistry and cannot follow the employed formalism on Wikipedia. Do you think the more physical approach presented in the question is not rigorous? – smonsays Feb 6 at 16:06
• I think it's probably perfectly rigorous (though I suspect there is a hidden simplification in there), just not a direction I've approached it from before and I don't have Dayan and Abbott in front of me, though I trust them and typically recommend their book as a starting point for those interested in theoretical neuroscience. See also columbia.edu/cu/biology/courses/w3004/Nernstequationderiv.pdf or onlinelibrary.wiley.com/doi/pdf/10.1002/9781118687864.app1 – Bryan Krause Feb 6 at 16:15