# Magnetoelectric Nanoparticles for Wireless Peripheral Nerve Stimulation: A Computational Study
## 2 Materials and Methods
### Modeling of the Neuronal Dynamics  
The dynamic model of the nerve behavior was implemented in a MATLAB environment to simulate the response of the neural fibers. For its realistic characterization of the human axons, the MOTOR (MRG) model, a double-cable structure model, was chosen to describe the neural dynamic of a peripheral fiber. In its equivalent circuit (Figure 3), both nodal and internodal sections are explicitly represented. The latter consist of 2 paranodal myelin attachments (MYSA), 2 paranodal main segments (FLUT), and 6 internodal segments (STIN), and the coexistence of both the myelin sheath and the underlying axolemma in the internodal sections characterizes the double-cable structure of the model [28]. The Ranvier's nodes are represented as a parallel combination of nonlinear fast Na (Naf), persistent Na (Nap), slow K conductance (Ks), linear leakage conductance (Lk), and membrane capacitance (Cn), whereas the two layers of each internodal section include a linear conductance in parallel with the membrane capacitance. The motor equations allow us to describe the neuronal dynamics characteristic of all the nodes of Ranvier, and both linear and nonlinear membrane dynamics model the electrical behavior of the axon [20]. The fibers' diameter was fixed to 5.7 \(\upmu\)m, and the geometry and the electrical properties of the segments were assigned as in the literature [28]. When simulating the application of an external AC magnetic field directed towards the MENPs, the electrical modeling of the phenomenon is performed by superimposing an AC electrical stimulus on the dipolar configuration with maximum amplitude. Thus, the total potential delivered by the MENPs (\(\Phi\)) was a sinusoid of one period defined as follows:  
Figure 3: Motor (MRG) model of the neural fibers. On the top, the sequence of the nodal and non-nodal segments of the model. At the bottom, the electrical equivalent circuit of the MRG model. Each [PERSON]’s node is surrounded by 10 internodal sections: 2 paranodal myelin attachments (MYSA), 2 paranodal main segments (FLUT), and 6 internodal segments (STIN), split on the two sides of the nodal section. The non-nodal segments consist of a double-cable structure explicitly representing the myelin sheath and the underlying axolemma and both including a linear conductance in parallel with the membrane capacitance. Vr is the membrane potential at rest of the neural fiber. The non-linear dynamic of the [PERSON]’s node includes fast (Naf) and persistent (Nap) sodium, slow potassium (Ks), and linear leakage (Lk) channels in parallel with the nodal capacitance Cn. \(\Phi^{\mathrm{e}}\), \(\Phi^{\mathrm{P}}\) and \(\Phi^{\mathrm{i}}\) represent the extracellular potential, the peri-axonal potential, and the intracellular potential, respectively, while \(i^{int}\) corresponds to the equivalent injected current in the intracellular space. \[\Phi=\varphi\cdot MF\cdot f(t), \tag{2}\]  
where \(\varphi\) represents the potential obtained by the electromagnetic simulation, \(MF\) is the multiplication factor, defined as a dimensionless number estimating the minimum pulse amplitude able to elicit an action potential, and \(f(t)\) is the one-period sinusoidal pulse with normalized amplitude modulating the potential \(\varphi\). Due to the ME effect, the frequency of the one-period sinusoidal electric stimulus delivered results in the double (i.e., 100 Hz) of the hypothesized externally applied magnetic field frequency, not modeled here. As reported in the literature [29], this frequency is in the range for neural stimulation. The direction of the external magnetic field and the direction of the dipolar distribution are the same. The orientations considered in our work are presented in the followings. According to [30; 31], the contribution of the extracellular stimulation can be introduced to the axon model as equivalent current \(I^{int}\) injected into the intracellular space of nodal and non-nodal segments (see Figure 3).