Electromagnetic Stimulation of Neuronal Tissue

A model of a coil with metal terminals was positioned inside a homogeneous volume conductor representing the saline solution or the retinal tissue. The model was implemented using FEM simulations to determine whether such small coils could generate electric fields strong enough to evoke action potentials in neuronal tissue, provided the coils were positioned close to excitable cells. All of the electromagnetic quantities introduced in this work are summarized in Table 1. An inductor is the ideal magnetic field generator, and it stores the magnetic field energy W generated by the supplied electric current i. Simulations were performed by considering low frequencies (that is, where l is the maximum dimension of the object and f₀ is the maximum current frequency) and ignoring the contribution of the displacement currents (that is, ). The optimal μMS coil is an inductor with magnetic energy W: where A is the magnetic potential, and the curl is the magnetic flux density (that is, B = × A). The portion of energy W lost reduces the Q-factor or efficiency and the inductance of the coil. Part of the magnetic energy W in equation (1) is therefore available to elicit neuronal activity. The electric fields and magnetic flux densities were found by solving numerically the following magnetostatics equation39 : where σ is the electrical conductivity expressed in S/m.
The induced currents and electric fields in the tissue are expressed by Faraday's law: where Φ is the scalar potential. Let us now consider the following cylindrical coordinates (r, z, φ), where the coil is in the rz plane (that is, (r, z₀, φ)) and each turn of the coil has coordinates r_i and φ [0;2π]. We assumed that Φ = 0 (ref. 40 (link)), because in an unbounded medium, Φ is only due to free charges and no such sources are present. Furthermore, we consider the frequency domain by assuming time harmonic fields with angular frequency ω and we will perform simulations with the maximum frequency (70 kHz) of the class D amplifier used in the experiments, as the pulse can be represented as 1/2 of a sinusoidal/cosinusoidal function. The induced electric fields E = −jωA (or equation (3) transformed in frequency domain) were found by solving the following quasistatic equation39 : where J^e is the external current, u_φ is the unit vector in the φ-direction and each turn of the coil, approximated by a circle with radius r and potential V_r, has an electric current amplitude derived from:
FEM numerical simulations were conducted to study the microscopic magnetic flux density generated by the MEMS microinductor (Fig. 6). The FEM simulations were performed in Multiphysics 4.2a with the AC/DC module (COMSOL, Burlington MA, USA) using the emqa model or the electromagnetic quasi-static approximation.
The model of induced electric fields was solved for the magnetic vector potential A in equation (4). There were no weak constraints, and all constraints were ideal. The overall geometry consisted of a cylinder of 3.0 mm in radius and 3.0 mm in height and was chosen to study the induced electric field around the coil which contained four different types of objects: a solenoid coil, two terminals, a quartz core and physiological solution. The solenoid consisted of 21 turns (rings) and was 500 μm in height and 500 μm in internal diameter as well as a 5μm×10 μm trace section. The terminals were two cylinders of 200 μm in radius and 200 μm in height. The quartz core consisted of 500 μm diameter and 500 μm in height on top and on the bottom copper terminals. The physiological solution was a cylinder of 3.0 mm in radius and 3.0 mm in height. Table 1 describes the material properties of the coil and the surrounding physiological solution/tissue and the constant values used in the simulations. Further details are provided in Supplementary Methods.