## 2 Materials and Methods
### Active Nerve Axon
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(g_{Na}\) & Maximum sodium conductance per unit area & 120 mS cm\({}^{-2}\) \\ \(g_{K}\) & Maximum potassium conductance per unit area & 36 mS cm\({}^{-2}\) \\ \(g_{l}\) & Leakage conductance per unit area & 0.3 mS cm\({}^{-2}\) \\ \(E_{Na}\) & Sodium reversal potential & 50 mV \\ \(E_{K}\) & Potassium reversal potential & \(-77\) mV \\ \(E_{l}\) & Leakage reversal potential & \(-54.387\) mV \\ \(C_{m}\) & Membrane capacitance per unit area & 1 \(\mu\)F cm\({}^{-2}\) \\ \(R_{i}\) & Intracellular resistivity of the axoplasm & 0.0354 k\(\Omega\) cm \\ \(a\) & Radius of the axon & 0.0238 cm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the Hodgkin\(-\)Huxley model [23]. Furthermore, we assume the normalized spatial distribution of the induced electric field is given by  
\[\frac{b^{2}}{b^{2}+x^{2}}\, \tag{16}\]  
where \(b\) indicates the spatial extent of the electric field in centimeters and varies with the coil size and distance from the neuron, and \(x\) is the position along the axon. The electric field is largest at \(x=0\) and decays over a distance \(b\) (Figure 3a). The spatial distribution of the activating function is proportional to the spatial derivative of the function in Equation (16). It is zero at \(x=0\), is positive to the left of \(x=0\), and is negative to the right (Figure 3b)  
\[\frac{-2b^{2}x}{\left(b^{2}+x^{2}\right)^{2}}. \tag{17}\]  
Figure 3: **(a)** The electric field \(E_{x}\), and **(b)** the spatial derivative of the electric field, \(dE_{x}/dx\), as functions of time \(t\) and space \(x\), for \(Q=1\) mV/cm and \(b=1\) cm. The electric field is large at the origin (\(x=0\)) and decays to half its value by \(x=\pm b\). In time, the electric field follows the time course of a biphasic pulse, like \(dl/dt\) in Figure 2b. The gradient of the electric field is zero at the origin and large on either side of the origin, positive in one direction, and negative in the other. Figure 2: **(a)** The coil current \(I(t)\), and **(b)** its time derivative, normalized so that \(dl/dt\) at \(t=0\) is one. The coil current is monophasic, with a fast rise and slow decay. The induced electric field is proportional to \(dl/dt\), which is biphasic with a strong positive phase lasting about \(0.1\) ms followed by a slower, weaker negative phase. Thus, the expression for the activating function is  
\[dE_{x}/dx=Q\left[\left(1-\frac{t}{\tau_{coil}}\right)\mathrm{e}^{-\frac{t}{\tau_{coil }}}\right]\left[\frac{-2b^{2}x}{\left(b^{2}+x^{2}\right)^{2}}\right], \tag{18}\]  
where \(Q\) is the peak electric field at time \(t=0\) and position \(x=0\). Calculating the threshold corresponds to varying \(Q\). We express our results in terms of the peak value of \(dE_{x}/dx\) required for excitation. We started by taking the expression for \(dE_{x}/dx\) as a function of \(x\) and then calculated its derivative and set it equal to zero to determine the value of \(x\) corresponding to the peak value of \(dE_{x}/dx\), which we found to be \(x_{peak}=0.577\)\(b\). By substituting this value into the equation for \(dE_{x}/dx\), we obtained \((dE_{x}/dx)_{peak}=0.65\)\(Q/b\). We observed that the value of \(b\) influences the peak of the electric field gradient; as \(b\) decreases, similar to the distance between the nerve and microcoil, the spatial gradient of the electric field increases. The value of \(Q/b\) decreases significantly as the distance between the nerve and the coil increases.