## 4 Discussion  
When analyzing magnetic stimulation with a microcoil, many researchers assume that excitation will occur if the electric field gradient, \(dE_{x}\)/\(dx\), reaches a threshold value. For instance, both [PERSON] et al. [9] and [PERSON] et al. [16] claim that excitation will occur if \(dE_{x}\)/\(dx\) reaches 11,000 V/m\({}^{2}\) (1100 mV/cm\({}^{2}\)) and justify this value by citing [PERSON] et al. [7] who used a coil with a diameter of about 4 cm. Our results indicate that you should not assume a threshold value obtained from studies of traditional magnetic stimulation if you are using a microcoil. Figure 6 indicates that the threshold value of \(dE_{x}\)/\(dx\) is elevated by a factor of three hundred or more when using a microcoil (note the log-log scales on this plot). In our spatial frequency analysis, summarized in Equation (5) and Figure 4, the spatial frequency \(k\) affects the threshold. When the spatial frequency is small compared to the reciprocal of the length constant, the membrane potential \(V_{0}\) is proportional to the activating function \(dE_{x}\)/\(dx\). In that situation, using a threshold value of \(dE_{x}\)/\(dx\) from another study with a different sized coil would be reasonable. However, once the spatial frequency increases so it is large compared to the reciprocal of the length constant, the threshold value of \(dE_{x}\)/\(dx\) rises, so it is not reasonable to use a threshold value from an experiment with \(k\lambda<<1\) to analyze an experiment with \(k\lambda>>1\). This conclusion is reinforced by Figure 6, which shows the threshold value of \(dE_{x}\)/\(dx\) as a function of parameter \(b\), which is similar to the coil size \begin{table}
\begin{tabular}{c c c} \hline \hline \(b\) **(cm)** & **Q (mV/cm)** & \(dE_{x}\)/\(dx_{peak}\) **(mV/cm\({}^{2}\))** \\ \hline
100 & 285,197 & 1853 \\
30 & 85,763 & 1858 \\
10 & 28,911 & 1877 \\
3 & 9674 & 2094 \\
1 & 4089 & 2665 \\
0.3 & 2415 & 5229 \\
0.1 & 2445 & 15,881 \\
0.03 & 4166 & 90,195 \\
0.01 & 9393 & 610,084 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The calculations of \(dE_{x}\)/\(dx_{peak}\) for a wide range of \(b\). Figure 6: A log-log plot of the threshold electric field gradient as a function of spatial length \(b\). For large values of \(b\) (greater than about 1 cm), the threshold value of \(dE_{x}\)/\(dx\) is constant, but for smaller values of \(b\) (less than about 1 cm), the threshold increases as \(b\) decreases. or the coil-axon distance. As \(b\) decreased from 100 to 0.01 cm, the threshold value of \(dE_{x}/dx\) increased by a factor of over 300, suggesting that this variation of the threshold value is not a small effect. Based on their plots of \(dE_{x}/dx\) versus \(x\), the value of \(b\) that corresponds to the microcoil used by [PERSON] et al. [10] and [PERSON] et al. [15] is probably 0.01 cm or even smaller (perhaps as low as 0.001 cm = 10 \(\upmu\)m). We do not have good data for the length constant \(\lambda\) of their neurons, so it is difficult to predict exactly by how much the threshold value should be multiplied when using a microcoil, but clearly, the increase will be significant. There are similarities between our results and the traditional strength-duration curve often considered in electrical neural stimulation. A generic equation describing the strength-duration curve is  
\[I=I_{\text{rheobase}}\bigg{(}1+\frac{d_{\text{chroaxaie}}}{d}\bigg{)}, \tag{20}\]  
where \(I\) is the threshold stimulus current strength, \(d\) is the pulse duration, \(I_{\text{rheobase}}\) is the rheobase current (the threshold stimulus current strength for a long-duration pulse), and \(d_{\text{chronaxie}}\) is the chronaxie (the minimum stimulus duration that can excite a neuron with a stimulus of twice the rheobase). A short-duration pulse (\(d<<d_{\text{chronaxie}}\)) requires a much stronger stimulus than a long-duration pulse (\(d>>d_{\text{chronaxie}}\)). The strength-duration curve arises because the neuron has its own time constant, \(\tau\).