High inductance magnetic-core coils have enhanced efficiency in inducing suprathreshold motor response in rats

Objective. Transcranial magnetic stimulation (TMS) coil design involves a tradeoff among multiple parameters, including magnetic flux density (B), inductance (L), induced electric (E) field, focality, penetration depth, coil heating, etc. Magnetic materials with high permeability have been suggested to enhance coil efficiency. However, the introduction of magnetic core invariably increases coil inductance compared to its air-core counterpart, which in turn weakens the E field. Our lab previously reported a rodent-specific TMS coil with silicon steel magnetic core, achieving 2 mm focality. This study aims to better understand the tradeoffs among B, L, and E in the presence of magnetic core. Approach. The magnetic core initially operates within the linear range, transitioning to the nonlinear range when it begins to saturate at high current levels and reverts to the linear range as coil current approaches zero; both linear and nonlinear analyses were performed. Linear analysis assumes a weak current condition when magnetic core is not saturated; a monophasic TMS circuit was employed for this purpose. Nonlinear analysis assumes a strong current condition with varying degrees of core saturation. Main results. Results reveal that, the secondary E field generated by the silicon steel core substantially changed the dynamics during TMS pulse. Linear and nonlinear analyses revealed that higher inductance coils produced stronger peak E fields and longer E field waveforms. On a macroscopic scale, the effects of these two factors on neuronal activation could be conceptually explained through a one-time-constant linear membrane model. Four coils with different B, L, and E characteristics were designed and constructed. Both E field mapping and experiments on awake rats confirmed that inductance could be much higher than previously anticipated, provided that magnetic material possesses a high saturation threshold. Significance. Our results highlight the novel potentials of magnetic core in TMS coil designs, especially for small animals.


Introduction
Transcranial magnetic stimulation (TMS) is a non-invasive neuromodulation technique that the US FDA has approved for a number of neuropsychiatric disorders, including treatment-resistant major depression, obsessivecompulsive disorder, and nicotine smoking cessation (O'Reardon et al 2007, George et al 2010, Carmi et al 2019, Zangen et al 2021).During TMS administration, a brief (typically <400 μs) but strong electric current (typically >3000 A) pulse is applied to a coil placed in close proximity to the patient's head.The coil generates magnetic fields that pass through the scalp and skull, inducing an electric field in the brain.It is generally assumed that an E field of 100 V m −1 is necessary to induce action potentials in neurons (Jalinous 1991, Thielscher and Kammer 2002, Roth et al 2007, Lu and Ueno 2017, Valero-Cabré et al 2017), although a lower E field, which induces subthreshold stimulation on neuronal tissue, could have biological effects as well (Todd et al 2009, Makowiecki et al 2014).
Animal models offer unique opportunities for TMS research.First, animal models allow for invasive manipulations using modern molecular and cellular tools, which are critical for better understanding the mechanism of TMS actions.Second, animal models can serve as a testbed for novel TMS methods.Once proven safe and effective, such methods can then be translated to humans (Vahabzadeh-Hagh et al 2012).TMS pulses are expected to cause complex effects in the stimulated loci and in the interconnected network.For animal TMS to be translational and ultimately to inform TMS treatment in humans, it is necessary that similar brain networks, such as the motor system in rats and in humans, should be modulated to minimize off-target effects.Based on known neuroanatomy, a focality of a few millimeters is necessary to stimulate rat motor cortex (Paxinos andWatson 2013, Seong et al 2015).Designing rodent-specific TMS coils with such a focality, however, presents special technical challenges.First, a small-diameter coil has low efficiency.An early study by Cohen and Cuffin (1991) using a spherical model demonstrated that coil efficiency was reduced by a factor of 25 when the coil diameter was scaled from 5 to 1 cm, thus requiring exceedingly high current to induce suprathreshold stimulation.Overheating associated with high current is another practical challenge for TMS coils of small sizes (Wilson et al 2018).
There have been consistent efforts to design TMS coils tailored for animals (Tischler et  Parthoens and colleagues reported a TMS coil specific for rats that featured active cooling (Parthoens et al 2016).However, this coil induced motor responses that lacked laterality, suggesting limited focality of the induced E field.More recently, Tang et al reported a miniature TMS coil for rodent animals, but this coil produced an E field that was insufficient to induce suprathreshold motor response (Tang et al 2016).By carefully positioning a small figure-of-eight TMS coil to the rat brain, Rotenberg and colleagues were able to induce lateralized motor response (Rotenberg et al 2010).This coil is unlikely to achieve a focality of a few millimeters (Deng et al 2013, Li et al 2017, Liu et al 2023), which is necessary for rodent brains.Jiang et al introduced a miniature C-shaped TMS coil design for rodents and verified its functionality through electrophysiological recording and stimulation experiments (Jiang et al 2023).
Magnetic core has been proposed to enhance the efficiency of TMS coils for humans (Epstein andDavey 2002, Salvador et al 2009).We adopted this approach and reported a rodent-specific TMS coil featuring cylindrical winding with a silicon steel magnetic core (Meng et al 2018a).The underlying physics principle is that a magnetic material has a permeability (m) much greater than that of air, which effectively guides, enhances, and focuses magnetic flux to the end of the magnetic core.We further developed a new winding method that breaks the symmetric distributions of the magnetic field in conventional ring-type coils, creating a focused E field (Bagherzadeh et al 2022).In more recent studies, we directed the coil to the motor cortex of awake rats and mapped glucose uptake (Cermak et al 2020) and motor-evoked potentials (Meng et al 2022).These in vivo data further demonstrated that the focality of this TMS coil was about 2 mm.
Magnetic core inevitably increases coil inductance L compared with its air-core counterpart.Conventional wisdom suggests that L should not be too high because the induced E field is inversely related to L; on the other hand, L should not be too low because of the requirement for high current and coil overheating (see equations (1a) and (2a) in theoretical section 2.4.1).Perhaps for these reasons, a typical human TMS coil has an inductance in the range of 10-35 μH (Peterchev et al 2008).Nevertheless, our lab recently constructed various rodent-specific TMS coils with inductance ranging from 35.1 to 74.5 μH.During TMS administration on awake rats, somewhat to our surprise, the coils with higher inductance are more efficient in inducing suprathreshold motor response.Here, efficiency is operationally defined as the minimal energy (or minimal voltage on the energy-storing capacitor) required to induce suprathreshold stimulation.
The purpose of this article is to better understand the tradeoffs among coil inductance, magnetic field strength, electric field strength, and energy necessary to induce suprathreshold stimulation in rats.To this end, we first performed linear modeling.This approach assumes a weak current condition, and the magnetic core is not saturated.A classical monophasic TMS pulse waveform was adopted.We also performed nonlinear modeling to account for a strong current condition.This is the realistic TMS situation in which the magnetic core experiences variable degrees of saturation.The boundary element fast multipole method (BEM-FMM) developed by Makaroff et al (2023) was applied in this analysis.Results revealed that coils with higher inductance also had stronger peak E fields and longer E field waveform w(t).On a macroscopic scale, the effects of these two factors on neuronal membrane potential could be conceptually explained through the convolution of w(t) with neuronal membrane impulse response function h(t).Both the E field mapping conducted on coil prototypes and the TMS experiments involving awake rats confirmed that coil inductance can be significantly higher than previously anticipated when magnetic materials with a high saturation threshold are utilized.These findings underscore novel potentials offered by magnetic cores in the design of TMS coils, particularly for use with small animals.

Materials and methods
2.1.TMS coil designs tailored for rat brains 2.1.1.Neuroanatomical considerations Adult rat brains have dimensions of ∼20 mm along the rostro-caudal axis and 14 mm left-right (Paxinos and Watson 2013).The hindlimb motor cortex is about 2 × 1 mm 2 , and the center of rat motor cortex is about 1.5 mm below bregma (along the dorsal-ventral axis); prelimbic cortex-the medial prefrontal cortex in rats and an interesting brain area in neuromodulation (West et al 2021), spans about 4 mm rostro-caudally, 3-4 mm dorsal-ventrally.It is thus expected that the TMS coil should have a focality of about 2 mm to achieve unilateral hindlimb motor response, and it should be able to reach 4 mm when higher TMS power is delivered.

Coil geometry and design parameters
Conventional TMS coils for humans, such as the circular coil and the figure-of-eight coil, have a planar geometry: that is, the electric current is distributed along horizontal planes with a total height of 2 cm or less.Simply reducing coil size, but keeping identical planar geometry, is unlikely to achieve focal stimulation of the rat brains because of low coil efficiency.As documented in the theoretical analysis by Cohen and Cuffin (1991), human TMS coils operate at a pulsed current of 3000 A or higher; for a small-diameter coil, an exceedingly high current is necessary to induce suprathreshold motor response, which in turn causes high Ohmic heating and electromagnetic stress, especially under high-frequency TMS situations (Huang et al 2005).
In a proof-of-concept study (Meng et al 2018a), we previously proposed a new coil geometry: electric current is distributed vertically, and use magnetic core to enhance, guide, and focus magnetic flux to the end of the coil; we also proposed to tilt the coil winding patterns to break circular symmetry of the magnetic field, further improving the focality of the TMS coil.Two follow-up biological experiments further prove that this coil achieves a focality of 2 mm (Cermak et al 2020, Meng et al 2022).Since a coil prototype with this geometry has a bar shape, we call it a 'bar coil' for simplicity.We aim to further improve bar coil design by parametrically adjusting the following parameters: (1) Magnetic core dimensions: the magnetic core of the first bar coil had a cross-sectional area of 4.2 × 4.2 mm 2 (Meng et al 2018a).Considering the rat brain geometry described above and the successes in our recent in vivo experiments (Cermak et al 2020, Meng et al 2022), we came to the empirical conclusion that this cross-sectional area was reasonable for our purpose.We thus kept the cross-sectional area at either 4.2 × 4.2 or 5 × 5 mm 2 for easy construction.However, the length of the magnetic core (L1) is a variable to be explored.
(2) Coil windings: the cross-sectional area (A1) of the current-carrying element is closely related to skin effect, coil resistance, and ohmic heating.TMS operates at a frequency range of 5 kHz to 10 kHz, and the resulting skin depth is about 0.6-1 mm.The initial bar coil consisted of 100 copper wires, each with an American Wire Gauge (AWG) of 29, roughly equivalent to a single AWG 9. We decided to use the same wires and keep A1 constant.The number of coil winding layers (N1) and the number of turns of windings in each layer (N2) are variables to be explored.The tilt angle (α) has a large impact on the focality of the TMS coil (Bagherzadeh et al 2022), and is also explored.

Metrics to compare TMS coils
Under the constraint of maximum TMS machine output, being able to induce suprathreshold motor response is a rudimental requirement for a successful TMS coil.The TMS machine can be modeled as a classic inductancecapacitance (L-C) circuit (see section 2.4.1 for more details).The capacitor We define the minimum V 0 necessary to induce suprathreshold motor response as the key metric to evaluate the performance of a TMS coil.Additionally, a TMS coil with a smaller I max is preferable because I max causes maximum electric stress and Ohmic heating in the TMS coil and in the driving circuit.For example, the power switching transistors (insulated-gate bipolar transistor or IGBT) in our driver circuit had electrical ratings of 4500 V and 1200 A.

Construction of coil prototypes and bench assessment
Prototype coils were constructed as previously reported (Meng et al 2018a).We 3D-printed a template that had the proper magnetic core dimensions, coil length, tilt angle, total windings, and winding width.Silicon steel sheets were cut to proper dimensions and were inserted into the center of the template.Copper wires (100 strands, AWG 29) were bundled together and wrapped onto the template.Since it was unsafe to deliver pulsed current of high amplitude to the coil at this stage, the characteristics of the coil were initially assessed by measuring: (1) the inductance with and without the core (LCR Meter @ 10 kHz, Model #: U1732C, Keysight Technologies, California, USA); and (2) the static magnetic field strength at the end of the coil by passing a direct current to the coil (Gaussmeter Model #: GM2, AlphaLab Inc., Utah, USA).The bench assessments were repeated iteratively by changing L1, N1, N2, and α.The coil was subsequently epoxied for electrical insulation and structural strength.Finally, we interfaced the coil with the TMS driver developed in the lab (Meng et al 2022).To identify the coil hotspot, we affixed a grid paper with a 5 mm grid spacing directly beneath the coil's surface.We used an in-house developed E field probe to measure the E x and E y values at various points on the grid (Meng et al 2018b).The probe had a Rogowski coil structure (inner diameter: 3 mm; outer diameter: 12 mm, thickness: 5 mm).The probe was carefully positioned vertically under the coil's surface to measure E x and E y , maintaining a minimal separation of about 0.5 mm.The field of measurement was 30 × 30 mm 2 .Before each measurement, we carefully calibrated the coil's current to 300 A (peak-to-peak).

Animal experiments 2.3.1. Headpost and coil guide for consistent TMS targeting
We measured motor threshold on awake rats (3 males and 2 females) using the coils described above.Given the small surface area of the rat motor cortex as described above, and the high focality of the coils, it was a challenge to consistently direct the hotspot of the TMS coil to rat motor cortex.We employed the same methods previously reported to achieve consistent TMS targeting across animals and across sessions (Cermak et al 2020), and are briefly described below: Using aseptic procedures and isoflurane anesthesia, we surgically implanted a headpost on the rat skull.The coordinates of the implantation were carefully calculated based on the neuroanatomy established via intracranial microelectrode stimulation (Seong et al 2015).With the headpost as an anatomical reference, a coil guide was used to direct the hotspot of the coil to the rat hindlimb motor cortex.Both the headpost and coil guide were 3D-printed and TMS-compatible.Since the distance between the hotspots and headpost was coilspecific, we designed coil guides with specific thickness to ensure that the hotspot of a specific TMS coil was reliably directed to the hindlimb motor cortex of the rat brain.

Motor threshold measurement in awake rats
After recovery from surgery for one week, rats were habituated to experimental handling and the TMS environment, including the acoustic noise, for 5 days.We have developed a TMS stimulator that was able to deliver single pulse as well as high-frequency TMS pulses, such as the theta bursts (Meng et al 2022).This stimulator employed two energy-storing capacitors (C 1 and C 2 ), which were charged by a positive and negative power supply unit, respectively, and delivered bipolar TMS pulses.This stimulator allowed for user-defined TMS current and voltage for individual TMS coils.We were able to measure these parameters on the fly using a high voltage differential probe (Model #: THDP0100, differential voltage range of 6000 V, Tektronix Inc., Beaverton, USA) and a Rogowski current waveform transducer (Model #: CWT30, peak current rating of 6000 A, Power Electronic Measurements Ltd, Nottingham, UK).The recorded electrical parameters were displayed and analyzed using a 5 Series Mixed Signal Oscilloscope (Model #: MSO54, 4 channel inputs, Tektronix Inc., Beaverton, USA).
During TMS experiments, the TMS power levels were adjusted, and voltages on the energy-storing capacitors, as well as coil current were monitored.TMS pulses were delivered once every 5 s.Motor responses (hindlimb twitch) induced by TMS were carefully observed by 3 independent experimenters.The motor threshold was defined as the power level that induced motor switches 5 out of 10 repetitions, consistent with human TMS conventions.We previously measured motor evoked potential (MEP) by longitudinally implanting microwire electrodes to the biceps femoris and gastrocnemius of the hindlimb muscle and established an excellent correspondence between motor response and MEP signal, consistent with observations in humans (Wassermann 2002).Since this study aimed to establish motor thresholds using different coils instead of a quantitative assessment of neuroplasticity induced by repeated TMS of the motor cortex, no attempt was made to measure MEP.All procedures were approved by the Animal Care and Use Committee at the National Institute on Drug Abuse, NIH.
We performed in vivo experiments to determine which of the 4 coils was most efficient in inducing suprathreshold motor responses.Here efficiency was defined as the minimum initial energies (Q s ) in the energystoring capacitors (C 1 and C 2 ) that were required to induce motor responses in the rats.We computed voltage V , 2 which represented Q 1 and Q 2 under each experimental condition.We refer to the term of machine output (MO) as the initial voltage on C 1 and C .
2 In our machine setup, 100% MO corresponds to 930 V in C 1 and −870 V in C . 2

Theoretical simulation
We conducted both linear and nonlinear analyses to assess the performance of the TMS coils.Through these analyses, we aimed to gain a better understanding of how the coils function and behave under different conditions.

Linear modeling under the weak current condition
For simplicity and as a first step, we start with a weak current condition in which the magnetic core does not experience saturation, and we assume a constant inductance.We analyze the relationship between electric current I, coil inductance L, and induced E field strength in the context of a monophasic TMS pulse waveform.Figure 1 illustrates the circuit diagram of a monophasic TMS stimulator, which involves charging the capacitor C with a power supply unit (PSU) and transferring the energy from C to the coil by turning on the IGBT unit, the resulting circuit is analogous to the classical L-C circuit.Once IGBT is turned off, energy in L is discharged through resistor R 2 and diode D. S 2 and R 1 are for discharging C as desired.
Time-varying current in L induces the electric field in the brain.The theoretical waveforms of the coil current and induced E field can be described as follows (Peterchev et al 2008) where V 0 is the initial charging voltage; ω and σ are defined as 2 , respectively; δ is a scaling factor; R 2 and r indicates the dissipation resistor and the combined series resistance in the L-C circuit loop, respectively; and t p indicates the oscillatory response which distinguishes the initial positive phase and subsequent negative phase.
In the simulation, several parameters were set based on the monophasic TMS implemented in our lab.V 0 was set to 200 V to avoid the core saturation at high voltage and high current.Other representative parameters in equations (1a) and (2a) are set based on the system implemented in our lab: C = 190 μF, R 1 = 50 kΩ, R 2 = 1Ω, and r = 20 mΩ.The measured inductance of four coils was 35.1, 51.4,60.0, and 74.5 μH, respectively.The δ is a scaling factor which depends on coil geometry, including winding patterns and magnetic core sizes.d was determined experimentally as follows: we carefully adjusted the peak coil current (sinusoidal waveform) to be 300 A for each coil and measured the E field amplitudes at the hotspot of the TMS coils (1 mm below coil surface).Note that all four coils did not saturate at 300 A current.Consequently, the values of the scaling factor d for the four coils were: 3.20 × 10 -6 , 4.96 × 10 −6 , 6.97 × 10 −6 , and 9.56 × 10 −6 (V/m)/(A/s), respectively.
Theoretically, TMS induces a suprathreshold motor response by charging/discharging the membrane potentials of the neuronal elements (axons, dendrites, etc) through the TMS-induced E field in the brain (Basser and Roth 1991, Nowak and Bullier 1998, Geddes 2004).On a macroscopical scale, change in membrane potential ( ) DV t can be modeled as a convolution of the E field waveform w(t) and the neural membrane dynamics represented by its impulse response function h(t) (Alavi et al 2022): Here h(t) is approximated by a first-order low-pass filter with time constant τ: ( ) Here τ was set to be 150 μs based on literature reports (Barker et al 1991, Alavi et al 2022).

Nonlinear modeling under the strong current condition
Under strong current condition, the magnetic core experiences variable degrees of saturation.The electric field generated by the coil-core combination includes the primary field and the secondary field.The combined electric field E can be described as = + E E E.
p s Where E p and E s denote the primary and secondary E fields generated by the coil windings and magnetic core, respectively.The relative permeability of the magnetic core μ r is a function of H field generated by the coil windings.We used M3 silicon steel as the core material in this study.Figure 2 where m 0 was the magnetic permeability of vacuum ( ) ´-1.25663706 10 6 H m −1 .a 1 and a 2 were set to 2.5 Tesla (T) and ´-1 3 10 2 m A −1 , respectively.For the coil modeling, we constructed the models comprising coil windings and cores based on the physical dimensions of the TMS coils.Specifically, the TMS coil was constructed with two main components: Litz wire and the silicon steel core.In Makaroff's software package (Makaroff et al 2023), copper Litz wire was modeled as a computation wire grid containing straight and thin electric current filaments.Differently, the silicon steel core is regenerated by the tetrahedral mesh generator from the imported stereolithography file.Moreover, the tilted inner and outer helical spirals of the Litz wire were consistently replicated in the simulation to match the tilt angle observed in the actual physical coil's winding pattern.Figure 3 illustrates the simulated combination of the core and coil, which accurately replicates the geometrical shape and winding pattern of the physical coils depicted in figure 4. Specific steps along with examples on how to build the model, how to compute inductance, charges, B field, and E field were detailed in Supplemental Materials of our recent publication (Makaroff et al 2023).Computations to compare the electric field strengths across the four coils were performed with the Makaroff software package (Makaroff et al 2023).A specific point (−0.01,0, −0.005) was selected to record the E field strengths across the four coils for comparison.

Prototype TMS coils tailored for rat brain
Figure 4 shows four prototype TMS coils, each featuring unique winding patterns and magnetic core lengths, as detailed in table 1.We mapped E field distribution for each coil.As an example, figure 5 shows a current waveform along with the induced E field (E x ).In this case, the current was 1000 A (peak-to-peak).There were multiple peaks in the E field waveform resulting from the nonlinear behavior of the magnetic core, which were otherwise not expected from an air-core coil ( ( )/ ¶ ¶ I t t).For example, at the initial rising phase of the current waveform (figure 5(C)), the E field reached a peak followed by a second peak separated by 15 μs (figure 5

(D)).
There were two sharp peaks in the E field when current transitioned from positive to negative phase (green arrow).Similarly, the last E field peak is not expected in conventional air-core coil either.
To compare the performance of the 4 coils, we measured the peak E field at the hotspot.The coil current was carefully adjusted to 300 A across coils.Readings at the second peak of the E field waveforms (red arrow) were used to compare the E field strength across coils.Supplemental figure 1 shows the current waveform along with the induced E field waveform at 300 A.
Table 1 yields several noteworthy observations: Although coil 3 possessed 1.71 times the inductance of coil 1, its static B field strength was only 1.46 times greater, and its E field strength was only 1.27 times higher.Similarly, coil 4 exhibited 2.11 times the inductance of coil 1, but its static B field strength was only 1.7 times higher, and its E field strength was 1.8 times greater.These observations highlight intriguing relationships between inductance, static B field strength, and E field strength among these coils.

Animal experiments
We recorded minimal MO values for inducing motor responses with the four TMS coils.Figure 6 shows MO values for each rat.Statistical analyses reveal that, compared with coils 3 and 4, coil 1 required significantly higher MO to induce suprathreshold responses (two-tailed, paired t-test, p < 0.011).
It is notable that, in several of the rats, the MO values for the adjacent coils were equivalent.However, the currents required to reach the MO values were quite different for these coils, as shown in supplemental table 1.Compared with coil 1, coils 3 and 4 operated at substantially lower coil currents.

Linear modeling under the weak current condition
Depolarizing neuronal membrane potential to a threshold level (∼50 mV) is necessary to induce motor response, we next simulated the changes in neuronal membrane potential ( ) DV t during a monophasic TMS pulse based on equation (3).The coil currents and corresponding E fields for the four TMS coils are depicted in  As anticipated, for a given initial voltage on the energy-storing capacitor (V 0 ), a coil with lower inductance L exhibited a higher current.However, the relationship between induced E field and inductance was not a simple inverse one due to the influence of different weighting factors (δ) among the four coils, as defined in equation (2a).It is worth noting that coil 4, characterized by the highest inductance (L = 74.5 μH), also exhibited the highest peak E field and peak ( ) DV t among the four coils.

Nonlinear modeling under the strong current condition
As shown in supplementary table 1, our TMS coils operated at high electrical current (I peak > 3000 A).The magnetic core materials experienced different levels of saturation.The nonlinear process was modeled using the software package developed by Makaroff et al (2023).Figure 8(a) illustrates the coil inductance as a function of current.The initial inductances of the 4 coils at a small current were 35.4, 48.1, 64.4, and 72.8 μH, respectively; they were in good agreement with experimental measurements shown in table 1. Coil inductances decreased with increasing current up to 800 A and approached a plateau with a higher current.While the purpose of the above nonlinear modeling was to investigate coil performance under high coil current conditions, we also applied these analyses to weak current conditions.Results are depicted in the zoomin plot of figure 8(d).These results demonstrate that even at low current levels, coil 4, which had the highest inductance, also induced the highest E field.

Discussion
The coil efficiency, i.e. the energy required to induce suprathreshold stimulation, is a primary concern in designing TMS coils tailored for small animals.We constructed 4 coils with magnetic cores using the design strategy we reported previously (Meng et al 2018a).These coils had inductances ranging from 35.1 to 74.5 μH, and the maximum B field strengths ranged from 41 to 70 Gauss at 1 A DC current.We found that the coils with higher inductance were more efficient in inducing suprathreshold stimulation during in vivo experiments, which was somewhat contrary to the conventional wisdom suggesting that L should not be too high or too low (human TMS coils (air core) have an inductance typically in the range of 10-30 μH).
Two key observations are apparent from the linear analysis shown in figure 7. First, coil 4 had the highest inductance; it also had the highest peak E field; second, coil 4 had the longest pulse duration (E field waveform w (t)).By convolving w(t) with the neural membrane impulse response function h(t) (49), coil 4 induced the greatest changes in membrane potential.
4.1.E field strength, pulse duration, and current requirement for TMS Both the peak E field and pulse duration were changed in our experiments.The peak E field is perhaps a defining variable in inducing suprathreshold motor response.Using the same TMS coil, but changing the capacitors in the L-C circuit of the TMS system, (Barker et al 1991) demonstrated shorter TMS pulses required less energies in the energy-storing capacitors to induce suprathreshold motor response.This was presumably due to the fact that shorter TMS pulses induced a stronger peak E field (higher / ¶ ¶ B t). On the other hand, pulse durations are known to have profound effects on energy requirement for inducing suprathreshold TMS responses as well.By using large capacitors in the L-C circuit to produce long time constant in the circuit, and cutting off the current pulse at arbitrary times, Peterchev et al (2008) were able to produce E field waveforms close to rectangular shapes as opposed to cosine shape in conventional TMS.Monophasic TMS pulses of variable widths were applied to the human motor cortex (Peterchev et al 2013), the motor threshold (measured in percentage of maximum machine output) decreased with increasing pulse duration up to 120 μs (the longest pulse duration in this study); and the trailing trend did not plateau, suggesting a further increase in pulse width may be able to reduce motor threshold even more.Alavi et al (2022) explained this phenomenon through linear analysis using equation (3).From a biophysical point of view, neuronal elements, including cell bodies, dendrites, axons, and synapses, have different time constants (Vöröslakos et al 2018).The dynamic processes of voltage-dependent ion channels, such as sodium and potassium channels, can be well modeled using the classical Hodgkin-Huxley equations (Hodgkin and Huxley 1952).Thus, the neuronal response depends not only on the peak amplitude but also on the temporal waveforms of the E field.Indeed, a recent study by Goetz et al optimized pulse dynamics for magnetic stimulation (Goetz et al 2013).
Note that the above studies applied air-core coils.With the introduction of magnetic core, the induced E field is the summation of the primary field E p generated by coil windings and the secondary field E s by the magnetic core: = + E E E.
p s The high relative permeability (m r ) of silicon steel substantially changed the E field dynamics during a TMS pulse.As shown in figure 2, m r is nonlinearly related to H, which in turn is linearly proportional to I; the effective coil inductance is also a nonlinear function of I. Coil 4 had the highest E field but also the highest inductance (thus the longest pulse duration), as shown in figure 7. The effects of these two factors on neuronal membrane potential could be conceptually explained through linear analysis using equation (3).
In addition to E field strength, the current required to induce suprathreshold neuronal response is also an important metric in coil evaluation, because coil overheating is a concern in TMS with high pulse rate, such as the theta burst stimulation (Huang et al 2005, Meng et al 2022).As shown in supplemental table 1, compared with coil 1, coils 3 and 4 required significantly lower current to induce motor response.

Optimal coil inductance
What is the optimal inductance for a TMS coil?This is related to membrane properties of the neuronal elements and stimulator hardware configuration, in particular, the energy-storing capacitance (C).Our stimulator generated biphasic pulses with the pulse width in the range of 160-185 μs.We recently constructed another TMS coil using a different magnetic core material (low carbon steel AISI 1008), and we were able to further increase pulse width to 220 μs.The energy required to induce motor response was reduced by 20% compared to coil 3 (silicon steel core).However, the magnetic properties of carbon steel and silicon steel are quite different; factors other than pulse width may have contributed to this observation as well.

4.
3. E field temporal dynamics induced by the magnetic-core coil Magnetic materials, such as silicon steel, have a nonlinear permeability as a function of the H field (see figure 2).Such nonlinearity has a profound impact on the temporal dynamics of the induced E field.As shown in figure 5, even a marginal increase in current from 0 to 50 A at the initial phase of TMS pulse produced the first peak in E field; the second peak emerged 15 μs later.With a sinusoidal current input, conventional air-core coils would induce a classical cosine E field waveform (5).The E field plot depicted in figure 5(B) deviates significantly from a conventional cosine waveform.As discussed above, both the peak E field and the E field temporal dynamics influence membrane potential changes.TMS coil design could benefit from magnetic materials with high saturation value and high permeability.

Conclusion
In summary, we have performed theoretical analyses employing linear and nonlinear models in an attempt to explain an experimental observation: for rodent-specific TMS coils employing magnetic cores, the ones with a high inductance are more efficient in inducing suprathreshold motor responses.Further study is warranted to find optimal coil inductance for a given stimulator configuration and pulse paradigm.

Figure 1 .
Figure 1.Illustration of the monophasic TMS circuit.
illustrates the anhysteretic curve (| |) m H r of this material based on manufacture datasheet.Since the magnetic field intensity H and relative permeability m r from the datasheet of M3 silicon steel are discrete numbers, the anhysteretic curve (| |) m H r was modeled based on the arctangent function as follows (Vladimirov and Schroeck 1971, Shane and Sudhoff 2010):

Figure 4 .
Figure 4. (a)-(d): four prototype TMS coils tailored for rat brain.(e) shows a coil being epoxied for electrical insulation and structural strength.

Figure 5 .
Figure 5.An illustration of recorded coil current I and the induced E field using coil 4 (L = 74.5 μH) at a current of 1000 A.

Table 1 .a
figures 7(a)-(b).Additionally, figure 7(c) illustrates the ( ) DV t values.As anticipated, for a given initial voltage Figure 8(b) shows relative permeability (m r ) as a function of coil current.The E fields at the same point below the coil surface (see figure 8(c)) are shown in figure 8(d).Coil 4, despite its high inductance, also had the highest peak E field.

Figure 6 .
Figure6.Comparison of minimal machine output (MO) to induce motor response in five rats.MO value for coil 1 was significantly higher than that for coils 3 and 4 (p < 0.011).

Figure 7 .
Figure 7.Comparison of the 4 TMS coils through linear modeling.(a) Coil currents under V 0 = 200 V; (b) induced electric fields; and (c) changes in neural membrane potential caused by the E field (abbreviation: a.u.arbitrary unit).

Figure 8 .
Figure 8. Nonlinear modeling of coil inductance (a), average core permeability (b) and induced E field strength (c)-(d) as a function of coil current.