## 2 Materials and Methods
We assume a sinusoidal stimulus current (the system is driven by a current source, not a voltage source), \(I(t)=\ I_{o}\sin{(\ \omega t)}\), where \(\omega=2\pi t\) and \(f\) is the frequency. The diffusion time can be defined as \(L^{2}/D\) and is the same as the RC time constant of the wire: the resistance \((r_{i}+r_{e})L\) times the capacitance C\(2\pi aL\). For \(t<0\), the initial condition is \(V=\psi=0\). At the ends \(x=\pm L\), all the current is in the wire and none is in the tissue, implying  
\[\frac{\partial V_{e}}{\partial x}=0\ \ \text{and}\ \ \ \frac{\partial V_{i}}{ \partial x}=-I_{o}r_{i}\sin(\omega t). \tag{5}\]  
In terms of \(V\) and \(\psi\), these boundary conditions become  
\[\frac{\partial V}{\partial x}=-r_{i}I_{o}\sin(\omega t)\ \ \ \text{and}\ \frac{ \partial\psi}{\partial x}=-r_{i}I_{o}\sin(\omega t). \tag{6}\]  
The equation for \(\psi(x,t)\) in Equation (4) can be solved analytically:  
\[\psi(x,t)=-r_{i}I_{o}\sin(\omega t)x. \tag{7}\]  
The diffusion equation for \(V(x,t)\) in Equation (4) must be solved numerically [24]. The derivatives are approximated using finite differences with space step \(\Delta x\) and time step \(\Delta t\)  
\[V(x,t+\Delta t)=V(x,t)+D\frac{\Delta t}{\Delta x^{2}}[V(x+\Delta x,t)+V(x- \Delta x,t)-2V(x,t)]. \tag{8}\]  
The calculation must satisfy the stability criterion  
\[\frac{4D\Delta t}{\Delta x^{2}}\leq 1. \tag{9}\]  
The parameters we used in the calculation are given in Table 1, selected to approximately match those used experimentally by [PERSON] et al. [18]. They imply \(D=1.8\) m\({}^{2}\)/s, \(r_{i}=\)850 \(\Omega\)/m, and \(r_{e}=320\) M\(\Omega\)/m. Clearly \(r_{i}<<r_{e}\), as would be expected. The DCresistance of the coil is 1.7 \(\Omega\). The capacitance per unit area is \(C~{}=0.00011~{}\mathrm{F/m^{2}}\). This is about a hundred times smaller than the capacitance of the cell membrane, 0.01 \(\mathrm{F/m^{2}}\) or 1 \(\mathrm{\SIUnitSymbolMicro F/cm^{2}}\), because the insulation thickness, 300 nm, is nearly a hundred times greater than the thickness of a cell membrane. The diffusion time is 0.56 \(\mathrm{\SIUnitSymbolMicro s}\). In the calculation, we used \(\Delta x=0.1~{}\mathrm{mm}\) and \(\Delta t=0.001~{}\mathrm{\SIUnitSymbolMicro s}\), implying that the quantity on the left side of Equation (9) is 0.72 and that the calculation is stable. Using a MATLAB (version R2022b) program, we calculated the electric field generated by the model shown in \[\frac{\partial}{\partial t}(V_{i}-V_{e})=D\left(\frac{r_{i}+r_{e}}{r_{i}} \right)\frac{\partial^{2}V_{i}}{\partial x^{2}}\ \ \text{and}\ \frac{\partial}{\partial t}(V_{i}-V_{e})=-D\left(\frac{r_{i}+r_{e}}{r_{e}} \right)\frac{\partial^{2}V_{e}}{\partial x^{2}}, \tag{1}\]0 The current in the wire was 1 mA with a frequency of 3 kHz. The voltage in the wire \(V_{i}\) and the tissue \(V_{e}\) varied as a function of time and space, and we expressed this variation using the functions \(\psi\) and \(V\) in terms of \(V_{i}\) and \(V_{e}\). After determining the values of \(\psi\) and \(V\) using the parameters outlined in Table 1, we could calculate the voltage along the wire \(V_{i}\) and the voltage in the tissue \(V_{e}\). Finally, we calculated the electric field \(E\) in the tissue by taking the gradient of \(V_{e}\). The MATLAB program used to perform the calculation is given in Supplementary Materials.