We want to explore here the basic properties of the extra-cellular potential generated by a uniform active---that is, able to propagate an action potential (or spike)---cable or axon. We are going to use the conductance model of Hodgkin and Huxlek (1952) together with the cable model making the "full" H & H model.
Remember that H & H did not solve their full model in their opus magnum, remember also that the mechanical calculator they had at this time was far less powerfull that any of the smartphones everyone has nowadays in his/her pocket. They used a "trick" looking at the propagation of a waveform without deformation at a constant speed $\theta$, that is, a spike or action potential. In this way the spatial derivatives of the membrane potential can be expressed as time derivatives (as we will see bellow) and the partial differential equation (PDE) of the full model can be replaced by ordinary differential equations (ODE).
So we are going to start by deriving the expression of the extra-cellular potential generated by a "cable like neurite"---a neurite with a large length to radius ratio---that can approximated by a line source. We will follow a classical development that is very clearly explained in the book of Plonsey and Barr (2007) Bioelectricity. A Quantitative Approach, published by Springer. This development will lead us to an equation relating the extra-cellular potential to the integral of the weighted second partial derivative of the membrane potential with respect to space, $\partial^2 V_m(x,t) / \partial x^2$. The H & H model will give us actual values for this derivative but that will require a numerical solution. We will then explore the effect of the axonal diameter on the extra-cellular potential.
The electrostatic potential $\Phi_e$ [mV] generated by a constant point source of intensity $I_0$ [mA] is given by: \begin{align} Φ_e = \frac{1}{4 \pi \sigma_e} \frac{I_0}{r} ,\end{align} where $\sigma_e$ [S/cm] is the conductivity of the extracellular medium assumed homogeneous and $r$ [cm] is the distance between the source and the electrode (Plonsey and Barr, 2007, Bioelectricity: A Quantitative Approach, p. 29).
For an extended source with a large length to diameter ratio (a cable) that can be approximated by a line source; the generalization of the previous equation for a continuous line source along the x axis (between $x_{min}$ and $x_{max}$) of the 3D Euclidean space equipped with Cartesian coordinates when the electrode is located at $(X,Y,Z)$ is: \begin{align} Φ_e(X,Y,Z) = \frac{1}{4 \pi \sigmae} \int{x{min}}^{x{max}} \frac{i_m(x)}{r(x)} dx ,\end{align} where: \begin{align} r(x) ≐ \sqrt{(x-X)^2+Y^2+Z^2} ,\end{align} and $i_m(x)$ [mA/cm] is the current density at position $x$ [cm] along the cable.
We get an expression for $i_m(x)$ by considering a small piece of cable of radius $a$ [cm] and of length $\Delta x$ [cm] (Plonsey and Barr, 2007).
If the intracellular potential at position $x$ is written $\Phi_i(x)$, then Ohm's law---the current equals the potential drop multiplied by the conductance---implies that the axial current $I_i(x)$ [mA] is given by ($\sigma_i$ [S/cm] is the intracellular conductivity):
\begin{align} \label{eq:stat2} I_i(x) &= -\pi a^2 \sigma_i \frac{\Phi_i(x+\Delta x) - \Phi_i(x)}{\Delta x} \nonumber \\ &\xrightarrow[\Delta x \to 0]{ } -\pi a^2 \sigma_i \frac{d \Phi_i(x)}{dx} \, . \end{align}Then the charge conservation implies that the membrane current density $i_m(x)$ (positive for an outgoing current) is given by:
\begin{align} \label{eq:stat3} I_i(x+\Delta x) - I_i(x) &= -i_m(x)\, \Delta{}x \nonumber \\ \frac{d I_i(x)}{dx} &= -i_m(x). \end{align}Combining equation \eqref{eq:stat2} and equation \eqref{eq:stat3} we get:
\begin{align} \label{eq:stat4} i_m(x) &= \pi a^2 \sigma_i \frac{d^2 \Phi_i(x)}{d x^2}. \end{align}Now, writing the membrane potential $V_m = \Phi_i - \Phi_e$ we have:
\begin{align} \label{eq:stat5} i_m(x) &= \pi a^2 \sigma_i \frac{d^2 V_m(x)}{dx^2} \,. \end{align}This allows us to rewrite equation \eqref{eq:stat1} as:
\begin{align} \label{eq:stat6} \Phi_e(X,Y,Z) = \frac{a^2 \sigma_i}{4 \sigma_e} \int_{x_{min}}^{x_{max}} \frac{1}{\sqrt{(x-X)^2+Y^2+Z^2}} \frac{d^2 V_m(x)}{dx^2} dx \,. \end{align}The quasi-static approximation (Plonsey, 1967, The bulletin of mathematical biophysics 29:657-664; Nicholson and Freeman, 1975, Journal of Neurophysiology 38: 356-368)---that amounts to considering the extracellular medium as purely resistive--- leads to a more general, time dependent, version of equation \eqref{eq:stat6}:
\begin{align} \label{eq:stat7} \Phi_e(X,Y,Z,t) = \frac{a^2 \sigma_i}{4 \sigma_e} \int_{x_{min}}^{x_{max}} \frac{1}{\sqrt{(x-X)^2+Y^2+Z^2}} \frac{\partial^2 V_m(x,t)}{\partial x^2} dx \,. \end{align}Notice that the derivation of equations \eqref{eq:stat6} and \eqref{eq:stat7} does not assume anything about the origin of the membrane potential non-uniformity.
If the membrane potential and its derivatives are null at the boundaries of the integration domain, then two rounds of integration by part give (with $X=0$ and $h = \sqrt{Y^2+Z^2}$):
\begin{align} \label{eq:statPart} \Phi_e(h) = \frac{a^2 \sigma_i}{4 \sigma_e} \int_{x_{min}}^{x_{max}} \left(\frac{3 u^2}{(u^2+h^2)^{5/2}} - \frac{1}{(u^2+h^2)^{3/2}}\right) V_m(u) du \, . \end{align}
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