## 4 Discussion
If the stimulus duration \(d\) is longer than \(\tau\) (that is, if \(d>>\tau\)), then the threshold value of the stimulus (the rheobase) is a constant, independent of duration. However, for short-duration stimuli (\(d<<\tau\)), the membrane voltage does not have time to respond to the brief stimulus, and the threshold value rises. Our strength-spatial extent curve in Figure 6 for magnetic stimulation is analogous to the strength-duration curve for electrical stimulation if we replace the stimulus duration \(d\) by the spatial extent of the stimulus \(b\) and the time constant \(\tau\) by the length constant \(\lambda\). Our results in Figure 6 have a \"spatial rheobase\" \(dE_{x}/dx\) value (1853 mV/cm\({}^{2}\)) for large values of spatial extent \(b\). At small values of \(b\), the value of \(dE_{x}/dx\) rises. If we wanted to define a \"spatial chronaxie\" (the value of \(b\) for which the threshold value of \(dE_{x}/dx\) rises by a factor of two), it would be about half a centimeter (Table 2). In a broader context, our results suggest that magnetic stimulation using a microcoil may be more difficult than that suggested in previous studies. When this result is combined with our prior work showing that the electric field during microcoil stimulation may have been overestimated in early studies [22], it casts doubt on the feasibility of microcoil magnetic stimulation, at least for stimulation using relatively weak current pulses (less than an amp). We have previously suggested that capacitive coupling may explain stimulation with a microcoil better than magnetic induction [24]. However, our results suggest that capacitive stimulation can also be suppressed when applied over a short distance. Nerve axons have their own characteristic lengths and time constants. When stimuli are applied over short distances compared to the length constant or in pulse durations that are short compared to the time constant, excitation becomes more difficult. One limitation of our analysis is that we assume that the stimulus during magnetic stimulation depends on \(dE_{x}/dx\), as suggested by [PERSON] and [PERSON]'s theoretical analysis of the cable equation [6] and [PERSON] et al. 's experiments using long nerves and large coils [7]. [PERSON] et al. 's threshold value for \(dE_{x}/dx\) was for magnetic stimulation of a long, straight, uniform peripheral nerve (like that modeled by [PERSON] and [PERSON]). However, their experiments also studied excitation in bent peripheral nerves during magnetic stimulation, and their findings suggested that the excitation mechanism is different at a bend compared to a straight nerve segment. Their experiments implied that an inhomogeneity caused by, say, the bending of the nerve may have led to an excitation \"hot spot\" where the threshold was determined by the electric field rather than its gradient, as was the case for a straight nerve. [PERSON], [PERSON], and [PERSON] analyzed the influence of magnetic stimulation on neuronal structures of finite length and varying geometries, including bending, branching, and termination [25]. They found that shorter axons were more easily excited than longer ones, and other factors, such as the truncation of an axon at its distal or proximal end, could also lead to \"hot spots\". Our analysis assumes the axon is uniform, straight, and long, so there are no such \"hot spots\". If this is not the case (as might not be the case for neurons in the brain), the idea of using \(dE_{x}\)/\(dx\) to characterize the threshold may be erroneous. Our analysis in this paper does not resolve the issue of \(E_{x}\) versus \(dE_{x}\)/\(dx\) for the stimulus of a neuron in the brain. We merely claim that if you assume that \(dE_{x}\)/\(dx\) is the relevant parameter for neural stimulation, then the threshold value of \(dE_{x}\)/\(dx\) depends on the spatial extent of the electric field, especially during microcoil stimulation. Our analysis has several additional limitations. We used the Hodgkin\(-\)Huxley model to represent the behavior of a nerve axon.