## 1 Introduction  
For years, magnetic stimulation has been studied and used to stimulate brain neurons, monitor brain function [1], and treat neurological diseases such as dementia [2]. [PERSON] et al. invented \"transcranial magnetic stimulation\" (TMS), and their device was able to stimulate the human motor cortex [3]. It consisted of a large coil held a few centimeters outside the scalp and required a kiloamp pulse of current to pass through the coil, which induced an electric field in the brain by electromagnetic induction. This non-invasive method caused less pain than electrical stimulation using electrodes on the scalp. Transcranial magnetic stimulation has been utilized to improve the neurological symptoms of depression [4] and Alzheimer's disease [5]. [PERSON] and [PERSON] developed a mathematical model that explains the physics behind nerve stimulation through electromagnetic induction [6]. Their model incorporated several key components: (1) circuit analysis: the stimulating coil and current source were modeled as a series resistor-inductor-capacitor circuit, allowing for the prediction of the coil's current over time; (2) electromagnetic induction: [PERSON]'s equations of electromagnetism were used to calculate the induced electric field along the axon, \(E_{x}\), and its gradient based on the coil geometry and the time course of the coil current; (3) cable theory: they derived a source term for the cable equation representing the nerve fiber that contained the gradient of the induced electric field; and (4) the Hodgkin\(-\)Huxley model: this model of ion channel kinetics simulated the nerve fiber's response to the induced electric field, including the membrane's voltage-dependent conductance. By integrating these four components, they could predict the transmembrane potential of a nerve fiber in response to magnetic stimulation. Fundamentally, they showed that the electric field gradient \(dE_{x}/dx\) (called the activating function, where \(x\) is the distance along the fiber) is responsible for neural excitation [6]. This gradient can either hyperpolarize or depolarize the membrane. Once the gradient depolarizes the axon's transmembrane potential to a threshold value, the nerve fires an action potential that then propagates along the fiber. [PERSON] et al. performed experiments using a peripheral nerve and verified that the cause of nerve excitation is the spatial derivative of the induced electric field [7]. Recently, an implanted microcoil was proposed for the magnetic stimulation of the brain. The microcoil uses the same mechanism as transcranial magnetic stimulation (electro-magnetic induction), except that the coil is smaller, the current is weaker, and the distance between the coil and the targeted nerve is smaller. The development of microcoils began when [PERSON] et al. constructed a 500 \(\upmu\)m diameter and 1 mm high coil composed of 21 turns of copper wire positioned 300 \(\upmu\)m above the soma of an isolated ganglion cell [8]. This method has sparked significant interest in the research community. Subsequently, researchers have developed and designed many microcoils with different geometries, shapes, and sizes [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. Understanding the electric field generated by a coil through electromagnetic induction is crucial for predicting the threshold for stimulation. The induced electric field can be influenced by many factors, such as the number of coil turns, the distance between the coil and the target neuron, and the frequency of the current [22]. Some researchers have used the electric field \(E_{x}\) to determine the membrane threshold [13], while others have used the electric field gradient \(dE_{x}/dx\)[9; 10; 12; 15]. In this study, we assume that the electric field gradient is responsible for stimulation. [PERSON] and [PERSON] analyzed magnetic stimulation of a long, straight axon using a large coil, for which a threshold value of \(dE_{x}/dx\) characterized excitation [6].