Modeling Intracochlear Magnetic Stimulation: a Finite-Element Analysis

Abstract

This study models induced electric fields, and their gradient, produced by pulsatile current stimulation of submillimeter inductors for cochlear implantation. Using finite-element analysis, the lower chamber of the cochlea, scala tympani, is modeled as a cylindrical structure filled with perilymph bounded by tissue, bone, and cochlear neural elements. Single inductors as well as an array of inductors are modeled. The coil strength (~100 nH) and excitation parameters (Peak current of 1 – 5 A, Voltages of 16 – 20 V) are based on a formative feasibility study conducted by our group. In that study, intracochlear micro-magnetic stimulation achieved auditory activation as measured through the auditory brainstem response in a feline model. With respect to the finite element simulations, axial symmetry of the inductor geometry is exploited to improve computation time. It is verified that the inductor coil orientation greatly affects strength of the induced electric field and thereby the ability to affect the transmembrane potential of nearby neural elements. Furthermore, upon comparing an array of micro-inductors with a typical multi-site electrode array, magnetically excited arrays retain greater focus in terms of the gradient of induced electric fields. Once combined with further in-vivo analysis, this modeling study may enable further exploration of the mechanism of magnetically induced, and focused neural stimulation.

I. Introduction

SCIENTISTS have been actively studying magnetic stimulation of excitable tissue (nerves and muscles) and in 1985 Barker et al. demonstrated activation of the human motor cortex [1]. This technique relies upon electromagnetic induction and the ability of the body, a charge-carrying medium, to respond to changing magnetic fields applied externally. In such applications a generator is placed external to the region of interest and time-varying currents are pulsed through a coil thereby creating magnetic fields in the body. In turn, electrical currents are induced in the nearby region of interest and serve to modulate the activity of excitable tissue [2]. Clinically, transcranial magnetic stimulation (TMS) is a non-invasive method employed for diagnostic and therapeutic purposes to treat depression, migraine, as well as improve motor signals in those suffering from Parkinson’s disease [3]-[7]. There have been many studies exploring optimal stimulation times, pulse shapes, coil geometries and the relationship between induced electric fields and distances [8]-[14]. While noninvasive, TMS can only induce currents locally, a few mm in depth, and therefore is limited. Furthermore, there have been concerns regarding patient discomfort, loss of focus, high power consumption, as well as a host of undesirable secondary effects [15],[16].

In an effort to improve selectivity and reach deeper targets, there has been a sustained interest in implantable inductors, or micro coils, to induce electric fields locally in excitable tissue. Studies have focused on modeling membrane properties, finite element modeling to study induced electric fields, as well as in-vitro studies [17]-[22]. Most notably, Bonmassar et al. presented a finite element model of the induced electric fields along with in-vitro stimulation at the single neuron level revealing highly selective activation of neural tissue with small, implantable micro coils, micromagnetic stimulation [21].

Encouraged by the results of Bonmassar et al., our group sought to explore the application of micromagnetic stimulation to the cochlea. While cochlear implants (CIs) are one of the most successful neural prosthesis in the world, many of the patients who undergo the implantation require rigorous post-surgical rehabilitation to reap the benefits of this technology [23]. In contemporary implants, patients’ surviving neural population is stimulated using electric currents. However, the presence of highly conductive perilymph results in spread of excitation [24]. At times, significant crosstalk between channels limits the number of independent channels [25] and thereby limits the function of present-day cochlear implants. In particular, users have a limited ability to appreciate music and difficulty in having conversations in presence of background noise [26].

Efforts have been made to improve this performance by improving the spectral content through current focusing, restricting the spread of neural activation; current steering, creating additional virtual channels; or through high-density electrode arrays [27]-[33]. Parallel efforts pursued improving temporal content through signal processing schemes that retain rapid variations in the acoustic signal [27]. However, the impact of these approaches on speech perception is mixed, and all rely upon electrodes, metal contacts, to exchange charge with tissue and thereby modulate auditory nerve activity.

Considering micromagnetic stimulation as an attractive alternative, our group pursued an in-vivo study where silicone-coated surface mount inductors (1.0 mm×0.5 mm×0.5 mm) were surgically introduced into the cochlea of anesthetized felines to a depth of 8-10 mm [34]. After verifying the deafened subjects’ unresponsiveness to acoustic stimulation due to the lack of auditory brain response (ABR), electrical and magnetic stimuli were applied and ABRs were recorded. ABR threshold revealed successful activation of the auditory system for inductors varying from 1-2.5 μH.

While additional in-vivo studies are essential for further validation, our group sought to conduct a finite-element analysis to study the fields generated in the cochlea. Our motivation was to analyze the local induced fields as a function of inductor size and stimulation parameters to investigate the potential for focused activation by micromagnetic stimulation. Additionally, we introduce a method for comparing direct electrical with micromagnetic stimulation via the activating function. Fig. 1 shows an overview of existing magnetic and electrical stimulation and the method of implanted magnetic stimulation proposed in this paper.

Fig. 1.

Overview of existing electrical cochlear implants, existing external magnetic stimulation techniques and proposed implanted magnetic stimulation of the cochlea. (a) In the case of Existing Electrical stimulation (b), the complete system is depicted including the external microphone and speech processor, transmitter and implanted receiver unit along with the electrode array (Image used with permission from Med El Corporation, Durham, North Carolina). For the Proposed Magnetic Stimulation (c) an array of implanted inductors is illustrated to convey the general principle. This work is modeling the induced electric fields, hence excitation capability, of these implanted inductors, which are in close proximity to the target neurons (within ~0.5 mm). The external components as well as implanted driving circuitry are not shown.

II. Methods

In order to assess the effect of magnetic stimulation on neural elements, three major factors are important to consider: (1) spatial distribution of the induced electric fields, (2) temporal resolution of the induced electric fields, and (3) a compartmentalized model of the target neuron [35]. This paper uses finite-element analysis to study the temporal resolution of electric fields induced by submillimeter coils, and examines at spatial distribution of the fields in terms of gradient of the electric field with respect to distance.

To compare direct electrical activation of the auditory system, with micro magnetically induced activation, it is noteworthy to highlight essential differences. In electrical stimulation, the electrostatic potential [V] generated through local charge transfer facilitated by implanted electrodes is of prime importance. In contrast, for magnetic stimulation, the induced electric field [V/m] is of importance. Hence, to compare the effects of both modalities on the same scale, the first derivative of the magnetic field and second derivative of the electrical potential (also called activating functions (AFs)) are plotted and examined.

COMSOL (Version 5.1, Burlington, MA) a finite element modeling software package tool was used to complete the finite element analysis (FEA) study. The tool solves partial differential equations and represents them graphically. To begin, a user can specify space dimensions, modeling physics and a study type (e.g., time-dependent or frequency domain) for the problem. Once these steps are completed, a graphical user interface may be utilized to define the geometry. The next steps are to assign material properties to the geometry, followed by defining the boundary conditions and excitation parameters. Finally, the geometry is meshed and solved.

A. Finite Element Analysis and Simulations

To calculate induced electric fields due to a sub-millimeter inductor, 2-D axis-symmetry was selected as the space dimension. This utilizes the symmetry of the geometry (a wire-wound inductor) thereby reducing the computation time and resources considerably.² Low-frequency magnetic fields physics was used (AC/DC module). The inductor (~100 nH) dimensions were 0.5 mm×0.5 mm consisting of a quartz core surrounded by a 21-turn copper coil. Copper contacts for electrical connections were also modeled on either ends of the coil [21]. A layer of Parylene-C was modeled around the entire assembly for representing insulation. The surrounding domains were modeled to represent the cochlear environment comprising of perilymph, laminar tissue (including tissue that comprises the membranous labyrinth and the cells lining the scala tympani), bone (osseous spiral lamina), and cochlear nerve (spiral ganglion cell bodies). Electrical properties of all these layers can be found in Table I [36]-[40]. These layers were modeled to be homogeneous, isotropic, non-dispersive and linear. As the relative permeability of all biological material is the same, discontinuities at anatomical tissue junctions were not modeled.

TABLE I.

Electrical Properties of Materials

Material	Electrical Conductivity [S/m]	Relative Permittivity	Relative Permeability
Quartz	1e-12	4.2	1
Copper	5.998e7	1	1
Parylene-C	1e-14	3.1	1
Perilymph	1.8	30	1
Tissue	0.16	80000	1
Bone Tissue	0.01	8000	1
Nerve	0.028	69911	1

The core was modeled to be quartz in the simulations like [21] as opposed to ferrite (magnetic) core in the in-vivo study because the in-vivo study used multi-layered inductors that are complicated, more prone to error and difficult to model in COMSOL versus a wire-wound inductor used in [21].

Based on the typical stimulation paradigm for cochlear implants, using a time-dependent study was considered most appropriate. Single-turn coil physics (COMSOL input parameter) with the coil group feature was used to introduce the stimulation signal. Input current was applied in the form of pulses designed using rectangle and analytic features of the global definition functions in COMSOL Multiphysics.

As a constraint, the product of pulse width and current amplitude was kept constant at 85 μC per pulse (Q = I × Δt). The value of Q served as an estimate of the net charge being injected during the in-vivo pilot study. Fig. 2 illustrates the range of pulse widths (17 - 85 μs) applied in order to examine the effect of pulse width, and accompanying current amplitude, on magnetic stimulation.

Fig. 2.

Input pulses to the submillimeter coils with constant charge of 85μC. The largest allowable rise times required to induce electric fields of at least 10 V/m [51] in the cochlea were used.

In this parametric study, the pulse width, maximum current amplitudes and rise times were varied. Steeper rise times induce larger fields since the rate of change of magnetic flux density is higher in accordance with equation (4) in Appendix). The in-vivo study suggested that pulses shorter than 20 μs were not effective. The various parameters that were chosen are listed in Table II.

TABLE II.

Stimulation Parameters

Case	Pulse Width [μs]	Current Amplitude [A]	Rise Time [μs]
1	85	1	0.75
2	68	1.25	1
3	53.125	1.6	1.2
4	40	2.125	1.5
5	34	2.5	2
6	21.25	4	4
7	17	5	5

For the time dependent study, time steps were kept sufficiently small (0.1 μs) to resolve the slope of input pulses. This improved resolution of instantaneous fields³ (see Appendix for equations).

Fig. 3(a) schematically depicts of the orientation of coil placed inside the scala tympani. Three colored sections represent laminar tissue (cyan), bony labyrinth (dull yellow) and the cochlear nerve (light purple). Each of the layers is modelled to be 50 μm thick [49]. Here the axis of the coil is parallel to the tissue surface, hence parallel to the axis of the cochlea. The inset illustrates a 3-D image of fields for better visualization when inductors are placed parallel to the tissue surface. A rotation of 90-degrees is applied to the inductor in the top inset to assist in visualizing the electrical field.

Fig. 3.

(a) Schematic picture for placement of the inductor inside Scala tympani parallel to the tissue surface (colored sections); Inset: COMSOL simulation of fields induced by inductor into surrounding material when inductor is placed parallel to the tissue surface. (b) Placement of the inductor inside scala tympani perpendicular to the tissue surface (colored sections); Inset: COMSOL simulation of fields induced by inductor into surrounding material when placed perpendicular to the tissue surface. For image (a) the inductor in inset is turned 90 degrees into the plane to appear as shown. For image (b), the inductor in inset is shown as it is. A version of these FEM simulations was included in a poster presented at the Association for Research in Otolaryngology Midwinter Meeting, San Diego, CA, USA (Feb 2016). 4While this assumption is a good approximation of the electrical stimulation experiments, in case of magnetic stimulation this may not hold true universally [45].

Fig. 3(b) illustrates the coil placed with its axis perpendicular (orthogonal) to the tissue surface. The inset illustrates 3-D images of fields for this configuration. This is drawn to scale and shows relative field strengths in the adjacent neural tissue.

B. Activating Function

In this paper, in order to perform comparison of both the stimulation techniques-electrical and magnetic, second derivative of the external electrostatic potential and first derivative of the electric field with respect to distance along the axon (Activating Function introduced by F. Rattay) are plotted. The Activating Function (AF) for an external electrical stimulation is given by

$$A(x)=\frac{{\partial }^2{V}_e}{\partial z^2}.$$ (1)

Here V_e is the external stimulation potential. The derivation of this function can be found in several references [41]-[44]. This function is based on several simplifications. One of them is cylindrical symmetry of the electrode, implying neglecting the transverse fields.⁴ As per Equation (5) in the Appendix, A(x) may also be written in terms of E:

$$A(x)=-\frac{\partial \mathbf{E}}{\partial z}$$ (2)

An important point to note here is that the right-hand side of (1) and (2) are opposite in sign. Thus, while maxima in (1) denotes peak depolarization, maxima in (2) denotes peak hyperpolarization, unless the sign is accounted for while modeling [46].

Once a single inductor model was solved for different pulse widths, an array of inductors was simulated. As a first step towards assessing the potential for channel interaction (crosstalk and collective spread of fields) for both magnetic and electric arrays, the centers of any two consecutive inductors were placed 2 mm apart similar to contemporary arrays [47]. In order to show that magnetic stimulation is more localized than electric stimulation, an array of electrodes used for electric stimulation of the cochlea was also simulated using the “Electric Currents Physics” module in COMSOL, with the distance between centers of two consecutive electrodes being 2 mm (end-to-end spacing of 1 mm) [47]. Both these arrays were solved for their respective Activating Functions.

III. Results

As illustrated in Fig. 3, when the axis of the coil is orthogonal to the tissue surface, most of the fields are directed parallel to the tissue bundle and do not contribute towards excitation of the neurons present in the modiolus—the spiral ganglion neurons.

Fig. 4 and Fig. 5 illustrate values obtained from the tissue surface adjacent to the coil, that is, after passing through the perilymph layer. The distance between the edge of the coil and this tissue surface was kept at 100 μm, consistent with the distance between the comparison electrode array and the modiolus as measured via micro x-ray computed tomography imaging [48].

Fig. 4.

Electric field produced by submillimeter coils when placed parallel to the tissue surface (axial) as a function of radial distance from the coil for excitation parameters from Table II.

Fig. 5.

Electric field produced by submillimeter coils when placed perpendicular to the tissue surface as a function of vertical distance from edge of the coil for excitation parameters from Table II.

Fig. 4 plots the electric field along a line defined at the center of the geometry shown in Fig. 3(a) inset. This line starts 100 μm from edge of the coil and runs radially outward till physical boundary of the FEM model. This boundary is magnetically insulated. Here, the orientation of the major axis of the coil is parallel to the tissue surface (axial orientation). It can be seen that the fields attenuate non-linearly with distance. Three zones showing laminar tissue, bony labyrinth and cochlear nerve are also represented in Fig. 3(a), and Fig. 3(b), to emphasize the extent of effect of the fields. Parameters from Table II were used as input to obtain these results.

Fig. 5 plots the induced electric field in case the inductor is placed orthogonal (perpendicular) to the tissue surface (Fig. 3(b) and inset). The maximum fields induced (100 μm from edge of the coil) in this case were at least 10 times lower than the previous case. It is to be noted here that the copper contacts on either ends of the inductor further increase the distance from laminar tissue in this case, however, this additional distance is a practical assumption. Apart from that all the parameters, other than orientation of the coil are the same. Therefore, the color bars in insets look identical. Parameters from Table II were used as input to obtain these results as well.

Fig. 6 shows Activating Function for the proposed magnetic array and a conventional electrically stimulated electrode array. It is clearly seen from the figure that the regions likely to be depolarized and hyperpolarized are much smaller in case of the magnetic array. It is to be noted here that the Activating Function provides a fair basis of comparison between the two technologies.

Fig. 6.

(a) Activating Function for a conventional electrically simulated electrode array with two adjacent electrodes activated. (b) Activating Function for an array of inductors. Local depolarization and hyperpolarization regions can be clearly seen for the two active inductors adjacent to each other. (For equations, refer Appendix).

In the electrically stimulated array, one electrode is the source and another electrode acts as the sink. Injected current flows between these two electrodes through the surrounding tissue, thereby stimulating the neurons. It can be seen from Fig. 6 that the regions likely to be depolarized using this technology span more than the size of the electrode itself.

Fig. 7 shows the field lines of the submillimeter inductor when an array of such inductors is modeled. It is observed that most of the field is concentrated in the vicinity of the inductor, suggesting reduced probability of excitation of nearby nerves and tissues, such as the facial nerve.

Fig. 7.

Contour plot of electric fields induced in the nearby neurons. As distance from the inductor increases, the probability of being activated by these fields reduces.

Fig. 8 shows the activating function of a single inductor. It is observed that the potential region of activation is confined to the approximate length of the coil and not beyond it. This function is plotted on a line 0.3 mm from the edge of the coil, corresponding to the region where Spiral Ganglions are present in the cochlea.

Fig. 8.

Activating function for a single inductor corresponding to Fig. 3 (a). It can be seen that the peak depolarization and hyperpolarization regions are very close to the center of the coil at 1.5 mm. This function is plotted on a line 0.3 mm from the edge of the coil, corresponding to the region where Spiral Ganglions are present in the cochlea.

Fig. 9 plots the Activating Function for the electrically stimulated array on a line defined along the entire length of the array in the region of the cochlear nerve (light purple in Fig. 3). The y-axis shows the amplitude of the activating function normalized to the peak value [7]. The x-axis shows the distance along the array. It is seen that the entire space is divided in regions more likely to depolarized or hyperpolarized, i.e., there is minimal spatial resolution. As seen from Fig. 6, two electrodes adjacent to each other are activated here and red lines denote the point of peak depolarization.

Fig. 9.

Plot of Activating Function along a line defined along the entire length of the conventional electrically stimulated array at a distance of 300 micrometers from the edge of the electrodes. Vertical red lines illustrate the sites of peak depolarization.

The electric fields induced by time varying current in the inductor are generated due to time varying magnetic field. The source of this field is the current flowing through the windings of the inductor and it does not have a physical sink located outside the inductor. As seen in Fig. 6, the resulting region of depolarization is localized close to one end of the inductor. This region can be moved to the other end of the inductor by changing the pulse polarity. Thus, we can generate compact stimulation pulses each 1 mm given that we have 1 mm coils separated by 1 mm. This implies that just using inductors would dramatically change selectivity of stimulation.

Fig. 10 plots the Activating Function at a particular time-instant for the magnetic array on a line defined along the entire length of the array in the region of the cochlear neural elements. The y-axis shows the amplitude of the activating function normalized to the peak value at the particular time-instant. The x-axis shows the distance along the array. It is also observed from Fig. 10 that for two adjacent inductors, a valley of hyperpolarization separates the regions of depolarization. Therefore, if realized, a magnetic array can hypothetically provide CIs with improved frequency resolution, leading towards better sound perception for the implant patients.

Fig. 10.

Plot of Activating Function along a line defined along the entire length of the magnetic array at a distance of 300 micrometers from the edge of the coils. Vertical red lines illustrate the sites of peak depolarization and hyperpolarization.

Peak depolarization and hyperpolarization were plotted along the black lines in Fig. 6(b) to see the distance from the inductor where peaks occur. This is shown in Fig. 11 and Fig. 12.

Fig. 11.

Plot of hyperpolarization decay radially away from the coil shows that region between 0.2-0.8 mm of the center of the coil is more likely to be affected by this stimulation signal.

Fig. 12.

Plot of depolarization decay radially away from the coil shows that region between 0.2-0.8 mm of the center of the coil is more likely to be affected by this stimulation signal.

Fig. 13 plots the maximum magnetic energy produced as a result of the stimulation parameters in Table II against the pulse width. In our case, the maximum magnetic energy is proportional to the square of the current flowing through the circuit. The maximum magnetic energy is independent of coil orientation and the inductance is constant over all input parameters.

Fig. 13.

Maximum magnetic energy of the coil for parameters listed in Table II.

IV. Discussion

Stimulation of neurons using sub-millimeter coils is a relatively new field of research and stimulation standards are yet to be established. For the in-vivo studies mentioned in earlier sections of this paper, voltage levels between 16-20 V were used and surface measurement (non-invasive) of the neural activity was performed on the feline subjects. The inductor providing stimulation was placed very close to the basal turn inserted through the round window to a depth of 8-10 mm. It was ensured that the amplifier operated at the highest slew rate and the pulse width, therefore, was kept constant at 34 μs.

Another study conducted by [50] excited the exposed dorsal cochlear nucleus and measured response at the contralateral inferior colliculus in an animal model (hamster). For these experiments, voltages between 100 mV – 600 mV and pulse widths between 25 μs – 100 μs were used and the investigators reported a pulse width of 50 μs at voltage level of 300 mV to be most effective.

In our paper, the charge per pulse is kept constant and pulse width, peak current amplitude and rise time are varied. Even under these conditions, the pulse width of 40 μs is seen to generate the largest induced fields, in both parallel and perpendicular orientations. This is because the combination pertaining to 40 μs pulse produces the steepest slope. This finding is consistent with the fact that rise time in itself is not an absolute parameter but depends on the peak value being achieved. This result is consistent with the in-vivo findings of [7],[50].

The Activating Function for electrically stimulated and magnetically stimulated arrays is plotted in Fig. 6. In both the arrays, two of the electrodes/inductors are active, consistent with parallel stimulation where more than one electrode needs to be activated in order to encode a particular sound. As seen from the line graphs, spatial resolution of the region being depolarized or hyperpolarized is much better in case of the magnetic array. In Fig. 9 for example, when two adjacent electrodes are active, two depolarization peaks are observed that are 2.5 mm apart. In Fig. 10 on the other hand, when two adjacent inductors are active, the depolarization peaks are separated by 2 mm with a peak hyperpolarization sandwiched between them. This promises better spectral resolution, thereby improving the quality of sound heard by implant patients.

When considering the magnitude of Activating function, the average values for magnetic array excited with voltages in range 16-20 V were found to be an order of magnitude higher than those of conventional electric array, or the magnetic array excited with 300 mV signals. This suggests that magnetic arrays have the potential of being less power hungry; however, confirming this hypothesis will be a part of our future work. It must also be noted that the power requirements scale down with increases in inductance, demonstrating potential for use of larger inductors, or perhaps, magnetic core.

Consistent with our analysis, it can be inferred that several parameters such as pulse shape, pulse duration, stimulation voltage, and inductance and resistance values of the coils can be varied in a host of ways to elicit neural activity in neural tissues as seen in [50], [51]. Moving forward, it would be important to establish a set of standard values and to clearly establish a stimulation threshold. This will require integrating a compartmentalized neural model to our existing model, or using in-vivo experiments to confirm our hypothesis.

It is also observed that even though the maximum charge content is same for all the parameters, inductor with largest pulse width and lowest current values store the least magnetic energy. This implies lower levels of inductive heating suggesting use of longer pulses as our stimulation parameter and is consistent with the findings of [35].

Other major considerations for practical application of magnetic arrays include power consumption and heating. In order to have an accurate measure of the power consumed in magnetic arrays, a standard stimulation threshold needs to be established first. This will allow circuit-level experimentation and development of appropriate driving circuit. Once the circuit is available for testing, heating can be systematically studied and tackled. Some of these problems will be addressed in our future work.

V. Conclusion

This paper evaluated electric fields produced by submillimeter coils using finite element analysis. For pulses similar to those applied for in-vivo and in-vitro experiments, electric fields in the vicinity of 10 V/m were observed [52]. At the depth of 100 μm into the modiolus, the electric field was not found to attenuate significantly, and remained at 7 V/m.

As these fields were seen to penetrate into the modiolus without much attenuation, a potential for direct stimulation of the cochlear nerve, in case the peripheral dendrites are completely damaged, is possible.

The fields induced in the cochlea were dependent on the orientation of the coil. They were up to 10 times stronger when the axis of the coil was placed parallel (10 V/m) to the modiolus, as opposed to when the axis was placed perpendicular (1 V/m) it.

It was also shown that if an array of these inductors were used for stimulation, there could be negligible degrees of cross talk and highly focused stimulation (spatial resolution of 100 μm), due to directional nature of the fields produced.

Biographies

Sagarika Mukesh (M’15) received the B.E. degree in Instrumentation and Control engineering from Manipal Institute of Technology, Manipal, India in 2011, and Masters’ in Embedded Electronic Systems from Rouen, France, in 2014. She is currently pursuing her Ph.D. degree in Electrical and Computer Engineering from Georgia Institute of Technology. Her research is in the field of novel stimulation techniques for cochlear implants.

David T. Blake is a clinical faculty at Augusta University. He received his BSE from Duke University in 1990 and his PhD from John Hopkins University in 1995. His research interest is in brain plasticity caused by learning in adults and his clinical interests include focal dystonia, age-related cognitive decline and autism.

Brian McKinnon is Board Certified in Neurotology, received his B.S. from Boston College, his M.D. from Boston University School of Medicine, his M.B.A. from Johns Hopkins University, and is currently pursuing his M.P.H. at the University of Memphis. He is an experienced auditory Osseo-integrated implant and cochlear implant surgeon. His area of research currently is focused on novel cochlear implant electrode designs, and the financial sustainability and economics of auditory Osseo-integrated implants and cochlear implants.

Pamela T. Bhatti (M’05) received the B.S. degree in bioengineering from the University of California, Berkeley, CA, USA, the M.S. degree in electrical engineering from the University of Washington, Seattle, WA, USA, and the Ph.D. degree in electrical engineering, with an emphasis on micro-electro mechanical systems (MEMS), from the University of Michigan, Ann Arbor, MI, USA, in 1989, 1993, and 2006, respectively. Currently, she is an Associate Professor in the School of Electrical and Computer Engineering, Georgia Institute of Technology (Georgia Tech), Atlanta, GA, USA.

Appendix

To broadly define our problem, we consider the two Maxwell’s equations pertinent to micromagnetic stimulators. In their most basic form they are as follows:

$$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$ (3)

$$\nabla \times \mathbf{H}=-\frac{\partial \mathbf{D}}{\partial t}+\mathbf{J}.$$ (4)

Here D represents the displacement current or the electric flux density, H represents magnetic field, B represents magnetic flux density, E represents electric field and J represents the current density [53]-[55]. ∇ is the differential vector operator used to represent the curl (cross product) of these field vectors. In homogeneous, isotropic, non-dispersive and linear materials, the relationship between E and D is simply D = εE (where ε = ε₀ε_r). As per (3) and (4) a changing magnetic field induces electric field, whereas electric current and changing electric flux produce magnetic field.

In this study, an electric current changing in time is provided in form of a pulse, and the resulting electric field is calculated. The electric field is related to the scalar electric potential as follows:

$$\mathit{E}=-\nabla V$$ (5)

Maxwell’s equations highlight the symmetry of electromagnetism and the fact that electromagnetic (EM) waves travel at a finite speed. This implies propagation delays between the changes at the source and its effects seen down the transmission line. However, in cases where the distance between source and the target excited by induced electric field is much smaller than the wavelength (λ) of excitation (< λ/10), the effects of retarded time can be neglected. Studies where this delay may be neglected fall under the quasi-static domain [56].

In magnetoquasistatic fields, the Ampere-Maxwell law (3) takes the following form:

$$\nabla \times \mathbf{H}=\mathbf{J}$$ (6)

That is, the rate of change of electric flux density with time can be ignored. Inductors used for stimulating the cochlea along with their stimulation parameters, fall under this category of quasi-statics. Here, the current is given by

$$\mathbf{J}=\sigma \mathbf{v}\times \mathbf{B}+{\mathbf{J}}^e$$ (7)

J^e represents the external current density and v is the charge velocity [57]. This equation basically shows the dependence of magnetic flux density on moving charges or electric current.

Similar to the definition of electric field in terms of scalar potential V, the magnetic flux density vector is defined in terms of the magnetic vector potential A. This convention is adopted for convenience. The magnetic flux density is now defined as B = ∇ ×A. From (6) and (7), using the constitutive relation B = μH (where μ = μ₀μ_r) in conjunction with the expression for A gives us the final equation in magnetostatics

$$\nabla \times \left({\mathrm{\mu }}_0^{-1}{\mathrm{\mu }}_r^{-1}\nabla \times \mathbf{A}\right)-\sigma \mathbf{v}\times (\nabla \times \mathbf{A})={\mathbf{J}}^e.$$ (8)

The above expression is a direct mathematical representation of the external current density in terms of the magnetic vector potential A. In our case, to estimate the induced electric fields produced by the inductors, we set this external current density to zero.

Moreover, no electromagnetic problem is complete without a set of boundary conditions at various interfaces between materials and physical boundaries. For this study, magnetic insulation was applied at the boundary of physiological components, given by n × A = 0. The geometrical symmetry of inductors was also used to further simplify the solution. Thus a 2-D axial symmetry condition was applied.

In addition to the above set of formulations, the time varying current was modeled using the following equations:

$$I_{\mathit{coil}}={\displaystyle \int \mathbf{J}·e_{\mathit{coil}}}$$ (9)

$${\mathbf{J}}^e=\sigma ·\frac{V}{2\pi r}·e_{\mathit{coil}}.$$ (10)

Here e_coil is the unit vector in direction of current flow. These formulae essentially convey the Ohm’s law in electromagnetics, which is applicable in case of all conductive media, including the human body [54]. This may also be written simply as J = σ E. Once, the induced electric field is converted into a current density using the Ohm’s law, it may be applied to standard Hodgkin-Huxley models to check for the generation of action potentials.

In this paper we have applied the Activating Function (AF) to more accurately examine the extent and effect of magnetic excitation. The AF relates to the membrane potential as follows [37]-[40]:

$$r_i\frac{\partial v_m}{\partial t}=\frac{1}{c_m}\left(-i_{\mathit{ion}}{r}_i+\frac{{\partial }^z{v}_m}{\partial z^2}+\frac{{\partial }^z{V}_e}{\partial z^2}\right)$$ (11)

Here c_m is the membrane capacitance per unit length [μF/cm], v_m is the transmembrane voltage [mV], i_ion is the ionic current flowing across the membrane [μA/cm²], r_i is the internal membrane resistance times unit length (Ωcm) and z is the distance along the axon [cm] [41].

While solving the above equation can be complicated, under sub threshold conditions v_m is equal to zero, so the second derivative of v_m and i_ion are also zero leaving behind the AF multiplied by (1/c_m). Therefore, plotting the solution for AF provides information about which regions are more likely to be excited than others. This is due to positive values of the AF corresponding to positive values of ∂v_m/∂t, which represent the region that will depolarize. This is shown in the previous sections.