Deriving partial differential equation based on Ohm's Law I am doing a project the Fitzhugh-Nagumo equations from a chapter (Chapter 9.5, The Transmission of Nerve Impulses) in Wave Motion by J. Billingham and A.C. King. I have taken basic physics and am familiar with Ohm's law but physics is not my specialty. I'm wondering if someone could help me interpret the components of this derivation of a PDE. (It is not one of the FH-N equations, but an intermediate step in deriving them.) 
The model is as follows: 


*

*Long, thin, cylindrical fiber of radius $a$

*Denote current density flowing by $i(x,t)$

*Resistivity $r$

*Potential at point $x$ and time $t$ denoted by $v(x,t)$

*Slice of the fiber is of thickness $\delta x$
The goal is to measure the change in potential over an infinitesimally thin slice of the fiber. The physical equation obtained from this model is
$$ v(x+\delta x, t) - v(x,t) = -i(x,t)\pi a^2 \delta x \,r$$
Upon taking the limit as $\delta x \xrightarrow{} 0$ we get that $$\frac{\partial v}{\partial x} = -\pi a^2 r i.$$
So one question I have is about the sign of $i(x,t)$, or the fact that it's negative. I'm having trouble understanding this and I'm wondering if someone could give me an intuitive explanation of what this implies. 
Another thing is that I'd like someone to confirm my understanding of the terms involved in the PDE derivation. I know that the $v(x,t)$ is the potential at one end and $v(x + \delta x,t)$ is the potential at the other end of the fiber slice, which has thickness $\delta x$. The $\pi a^2$ is the area of the fiber and when multiplied by $\delta x$ this gives a "volume" of the fiber slice. When further multiplied by the current density this is effectively an amount of current. My confusion is that the text uses the term "resistivity" rather than resistance. I have that $R = \rho L/A$ (and here $\rho = r$), but I don’t see this expression clearly laid out. Specifically, there is no clear "division by area" anywhere. They do not explain how they got the RHS. Is it possible that they meant to say "resistance"? Or else, could someone rederive the final equation (in the limit) for me with steps laid out? Perhaps I am misunderstanding the whole interpretation of the derivation. 
 A: Voltage is electric potential energy - $qV = U$. Remember things flow from high potential energy to low potential energy $$ \text{Force} = - \frac{dU}{dx} = -q \frac{dV}{dx} $$. 
$i(x,t)$ is a measure of the current in the $x$ direction. Current is flowing from high to low potential energy, thus if $i(x,t) >0$, current is flowing towards positive $x$, and thus the potential energy must be decreasing with increasing $x$, or $dv/dx < 0 $. So $i$ and $dv/dx$ have opposite signs.
As for you're second question... I've seen elsewhere resistivity defined as "resistance per unit length" - though your definition is valid too. This would fit with charge density usually being "charge per area" to make your equation: 
$$ \Delta V = \left [  i(x,t) \times \pi a^2 \right] \times \left [ r \times \Delta x \right ]  =  \left [  \frac{\text{Current}}{\text{area}}  \times \text{area} \right] \times \left [ \frac{\text{resistance}}{\text{length}} \times \text{length} \right ]\\
= \text{Current} \times \text{Resistance}$$ 
