\(\textbf{Neurophysiology}\) \[\textit{The Hodgkin-Huxley Model}\] \[\textit{Part 1 - Nernst Equation}\] \[V_{eq}=-\frac{RT}{zF}ln\frac{[C_{in}]}{[C_{out}]}\]
The Hodgkin-Huxley model for neuronal activity is one of the most well studied. It is also the inspiration for the perceptron and neural network models found in machine learning algorithms of today. More than that however, it was one of our first insights into a physiological mechanism, which to this day, is still in the process of being unraveled.
In explaining the fundamentals of this model, we will walk along a winding road full of neat and often clever mathematical descriptions which will inevitably build upon each other until we arrive at our destination. The first stop, per the title, is the Nernst Equation.
The \(\textit{membrane potential}\) of a cell, is the potential difference between the inside and outside of a cell across its membrane.
The \(\textit{resting potential}\) is the potential across the cellular membrane when the cell is at rest, with the inside being more electronegative. The typical value for a neuron is \(-70mV\).
The \(\textit{equiibrium potential}\) across a membrane for a particular ion is the potential at which the electrical and diffusive movement are equal and opposite, such that there is no net movement across the membrane for that ion. The following relation must therefore be satisfied
\[J_{diff} + J_{drift} = 0\]
\[J_{diff} \propto \frac{\partial [C]}{\partial x}\]
\[J_{drift} \propto z[C]\frac{\partial V}{\partial x}\]
Where \([C]\) is the concentration, and \(V\) is the potential of the given ion at the point \(x\) across the membrane.
It is found by the Einstein relation [3] that, adjusting the coefficients of these terms and summating them, we arrive at the Nernst-Planck equation [4]; an expression for the \(\textit{current flux}\), \(I\)
\[I=-uzRT\frac{\partial [C]}{\partial x}-uz^2F[C]\frac{\partial V}{\partial x}\]
At \(I=0\), we can evaluate the equilibrium potential for ions passing through an open channel by solving the differential equation
\[-uzRT\frac{\partial [C]}{\partial x}-uz^2F[C]\frac{\partial V}{\partial x} =0\]
\[\frac{\partial V}{\partial x} = -\frac{RT}{Fz}\frac{1}{[C]}\frac{\partial [C]}{\partial x}\]
Summating over the domain (from \(x=0\) to \(x=l\), where \(l\) is the thickness of the membrane), which is from the inside to the outside of the cell, we arrive at
\[\int_{out}^{in}VdV=-\frac{RT}{Fz}\int_{out}^{in}\frac{1}{[C]}d[C]\]
\[V_{eq}=-\frac{RT}{Fz}ln\frac{[C_{in}]}{[C_{out}]}\]
This equation provides the equilibrium potential given a concentration gradient of ions across an open channel.
This equation assumes that the ions do not interact, and that the medium of the flux is aqueous. Typically, cell membranes behave in a more complex way, however, deriving this equation is a good start for a simplified model.
