## 2 Materials and Methods  
We model the stimulating coil as a straight length of copper wire of radius \(a\) and length 2L, surrounded by a tissue of radius \(b\) (we assume \(b>>a\)). The copper has conductivity \(\sigma_{i}\), and the surrounding tissue has conductivity \(\sigma_{e}\). The resistance per unit length of the wire is then \(r_{i}=1/\left(\sigma_{i}\pi a^{2}\right)\) and that of the surrounding space is \(r_{e}=1/\left(\sigma_{e}\pi b^{2}\right)\). The two are coupled capacitively. The wire insulation has a capacitance per unit area \(C=\frac{\kappa c_{e}}{d}\), determined by the dielectric constant of the insulator \(\kappa\) and its thickness \(d\). We assume the resistance of the insulation is infinite. This model for the wire is essentially the same as the cable model for a nerve axon, except that we take the \"membrane\" conductance to be zero. Figure 1 illustrates the model. Our analysis follows the traditional derivation of the cable equation [23]. The position along the cable is denoted by \(x\), with \(x=0\) at the center and \(x=\pm L\) at the ends. Time is denoted as \(t\). To derive an equation governing the voltages in the wire, \(V_{i}\), and in the surrounding tissue, \(V_{e}\), we consider a section of the wire of length \(\Delta x\). The wire current, \(I_{i}\), entering this section from the left is \(I_{i}(x)\) and the current leaving this section on the right is \(I_{i}(x+\Delta x)\). The current exiting the wire through the capacitance is \(C\left(V_{i}-V_{e}\right)/\partial t\) times the wire's surface area \(2\pi a\Delta x\). Taking the limit as \(\Delta x\) goes to zero, this relationship becomes \(-\frac{\partial I_{i}}{\partial x}=2\pi aC\frac{\partial(V_{i}-V_{e})}{dt}\). In the copper, the current and voltage are related by [PERSON]'s law, \(I_{i}=-\frac{1}{\tau_{i}}\frac{\partial V_{i}}{\Delta x}\), so this equation becomes \(\frac{1}{\tau_{i}}\frac{\partial^{2}V_{i}}{\partial x^{2}}=2\pi aC\frac{ \partial(V_{i}-V_{e})}{dt}\). A similar relationship can be derived for the tissue space, but a minus sign is introduced because capacitive current  
Figure 1: A model of a wire coupled to the surrounding tissue by a capacitance. The current in the wire is \(I_{i}\), the wire resistance per unit length is \(r_{i}\), the current in the tissue is \(I_{e}\), the tissue resistance per unit length is \(r_{e}\), the capacitance per unit area of the insulation is \(C\), and the current applied to the wire at its ends is \(I\). exiting the wire is entering the tissue. We can rearrange these expressions and define a diffusion constant as \(D=\frac{1}{2\pi\alpha C(r_{i}+r_{e})}\), resulting in the pair of differential equations:  
\[\frac{\partial}{\partial t}(V_{i}-V_{e})=D\left(\frac{r_{i}+r_{e}}{r_{i}} \right)\frac{\partial^{2}V_{i}}{\partial x^{2}}\ \ \text{and}\ \frac{\partial}{\partial t}(V_{i}-V_{e})=-D\left(\frac{r_{i}+r_{e}}{r_{e}} \right)\frac{\partial^{2}V_{e}}{\partial x^{2}}, \tag{1}\]  
Next, we introduce two new voltages--the voltage across the insulation, \(V\), and a weighted average voltage, \(\psi\)--defined by  
\[V=V_{i}-V_{e}\ \ \ \text{and}\ \ \ \psi=V_{i}+\frac{r_{i}}{r_{e}}V_{e}, \tag{2}\]  
which can be inverted to give \(V_{i}\) and \(V_{e}\) in terms of \(V\) and \(\psi\)  
\[V_{i}=\frac{r_{e}}{r_{i}+r_{e}}\left(\psi+\frac{r_{i}}{r_{e}}V\right)\ \ \text{and}\ \ \ V_{e}=\frac{r_{e}}{r_{i}+r_{e}}(\psi-V). \tag{3}\]  
We can use Equations (1) and (2) to show that \(V(x,t)\), the voltage across the insulation, obeys the diffusion equation, and \(\psi(x,t)\), a weighted average of the wire and tissue voltages, obeys a one-dimensional version of Laplace's equation:  
\[\frac{\partial V}{\partial t}=D\frac{\partial^{2}V}{\partial x^{2}}\ \ \ \text{and}\ \frac{\partial^{2}\psi}{\partial x^{2}}=0. \tag{4}\]  
Our introduction of the diffusion constant earlier was motivated by its appearance in the diffusion equation for the voltage difference across the wire capacitance.