The fast and efficient transmission of electrical pulses in the nervous system is mediated by myelinated nerve fibers. Removal of the myelin sheath or demyelination leads to a delay, inconsistency, or failure of AP propagation in demyelinated axons. This accounts for the cellular mechanisms underlying CNS demyelinating diseases, such as MS and NMOSD, and PNS demyelinating diseases, such as GBS and CMT. Local neural injury can also lead to focal demyelination, as in the case of SCI when the local spine was crushed (Eftekharpour et al 2007, Liu et al 2013). Peripheral nerve injuries are commonly observed during sports and occupational activity. Shoulder dislocation can cause spinal accessory nerve injuries. Repetitive forearm pronation causes injury to the superficial branch of the radial nerve. Under mild injury (neuropraxia), these conditions are characterized by local damage to myelin fibers around the axon (Neal and Fields 2010).

The biophysics of axonal failure in demyelinated axons have been a central topic of demyelination studies for decades. Axonal failure in the demyelinated region is largely due to the failure of activation of the nodes of Ranvier by the invading AP. In a healthy, myelinated axon, the invading AP in the upstream node generates an axial current that depolarizes and activates the downstream node, which warrants the saltatory propagation of nerve pulses. When demyelination occurs, there is insufficient longitudinal current spreading in front of the impulse to the next node (Dale Purves et al 2011). Supporting this notion, about three-quarters of people with MS find their symptoms worsen in response to heat (the Uhthoff phenomenon (Frohman et al 2013)), as higher temperatures could shorten the invading AP and decrease the axial current. A shortened internode length, which allows less leakage current alongside increased axial current spreading and node activation, was observed in patients with MS when spontaneous remyelination occurred (Chari 2007).

Currently, there is no effective pharmacological treatment to completely restore the functionality of myelin and nerve conduction in demyelinated axons. Therefore, demyelination research that targets conductance failure focuses on ensuring a large axial current flow through the demyelinated region by increasing node activity or via remyelination. Increasing node activity can directly rescue axonal conductance during demyelination. Node excitability can be increased by pharmacological methods, such as 4-AP, a potassium channel blocker. In vitro studies have shown that 4-AP can improve the conduction of APs in demyelinated nerve fibers, thereby increasing the release of neurotransmitters in synapses and at neuromuscular junctions (Sherratt et al 1980, Bostock et al 1981). 4-AP is currently approved for the treatment of patients with MS and has been shown to improve walking ability (Goodman and Stone 2013, Baird et al 2018). Although pharmacological approaches that can enhance axonal conductance have significant benefits for the treatment of demyelinating diseases, these approaches are typically associated with adverse side effects. For example, 4-AP can cause paresthesia, dizziness, nausea/vomiting, etc (Jensen et al 2014).

Remyelination is the process of restoring demyelinated nerve fibers with new myelin (Franklin and Ffrench-Constant 2008). Remyelination can prevent the leakage of axial currents and restore axonal conductance. Several small molecules have been identified with the potential to promote remyelination by targeting diverse cellular and molecular pathways (Najm et al 2015, Rankin et al 2019, Chen et al 2021). These molecules typically enhance the differentiation of oligodendrocyte precursor cells into myelinating oligodendrocytes or protect remyelinating oligodendrocytes from an inflammatory environment. Stem cell transplantation could lead to remyelination, restoration of node structure, ion channel clustering, and functional recovery of axonal conductance in animal models of demyelination (Eftekharpour et al 2007, Ruff et al 2013). Electrical stimulation can also induce myelin formation by enhancing oligodendrocyte maturation (Lee et al 2017). Recently, electrical stimulation was used to restore the impaired myelin membrane in the mouse dorsal root ganglion via upregulation of lipid biosynthesis (Intisar et al 2022). Clinically, TMS can promote remyelination of neurons by activating axonal fibers and increasing the number of oligodendrocytes (Cullen et al 2019). In these examples, the electromagnetic fields were harnessed to promote remyelination.

To the best of our knowledge, there are no previous studies that have investigated whether electromagnetic stimulation of the demyelinated region can enhance node activation and rescue conductance failure after demyelination. However, several recent findings have suggested that such stimulation can be used to control and activate axonal elements, such as the nodes of Ranvier (Ye and Steiger 2015, Ye et al 2022b). For example, electromagnetic pulses cause immediate and transient activation of voltage-gated sodium channels in cultured neurons and acute rat brain slices (Banerjee et al 2017). Electromagnetic fields also affect potassium channels by changing their activation and inactivation properties (Tan et al 2013). Consequently, it is critical to investigate whether electromagnetic stimulation can be used to directly promote axonal conductance by activating nodes in the demyelinated region. In this proof-of-concept study, we tested the hypothesis that magnetic stimulation of the demyelinated area, by providing excitation to the nodes of Ranvier, could rescue locally demyelinated axons from conductance failure.

However, two technical challenges must be addressed. First, although it is possible to directly monitor node activity using advanced electrophysiology technology, such as patch clamp on the node (Kanda et al 2019), experimentally monitoring node activity is extremely challenging. This is because the externally applied electromagnetic field can interfere with the delicate node signal. Therefore, we performed computational simulations. We implement a multi-compartment model of a locally demyelinated axon under magnetic stimulation, using the NEURON simulation environment (Hines and Carnevale 1997).

Secondly, the externally applied electric field must be sufficiently focused on the demyelinated area. This could be achieved with either electrode or miniature coil. For electric stimulation, the electrodes were positioned close to the targeted neural tissue to deliver electric current. In magnetic stimulation, electric current is generated via electromagnetic induction, which does not require the stimulating coil to be in direct contact with the target tissue (Maccabee et al 1991, 1993, Ye et al 2010, 2011, Ye and Steiger 2015). This mitigates numerous problems that can arise at the brain–electrode interface, such as charge transfer, electrode surface modification, and corrosion (Polikov et al 2005, Cogan 2008, Koivuniemi et al 2011). Recent development of µMS technology significantly improved the specificity of coil stimulation (Bonmassar et al 2012, Park et al 2013). These submillimeter coils are smaller than the dimension between two adjacent nodes on a single myelinated axon and can, therefore, be positioned to target a specific node. Furthermore, the coil can be implanted under the cover of soft biocompatible materials. This largely prevents the neural response to implantation (Saxena et al 2013, Canales et al 2015), including inflammatory and immune responses (Kim et al 2004, Lee et al 2016, Liu et al 2017). With such an implantation, focal stimulation of the targeted deep structure is feasible (Bonmassar et al 2012, Saha et al 2022b).

In this study, we applied electric stimulation using microcoil technology to ensure focal stimulation of the demyelinated area in a demyelinated axon model. We demonstrated, for the first time, that by activating the nodes in the demyelinated area, microcoil stimulation could rescue conductance failure in the locally demyelinated axons. These results suggest a potential novel intervention strategy for the treatment of local demyelinating conditions.

2.1. Multi-compartment, myelinated axon model

The modeled myelinated axon was 100 000 µm in length. It contained 101 nodes and 100 internodal segments. The internode length was 1000 µm. The diameter of the entire fiber was 10 µm and the axon diameter was 7 µm (figure 1(A)). The nodes were described (appendix A, figure 1(B)) using the FH model. Equations and conductances that define the Na, K, and P channels at the node were given in (Frankenhaeuser and Huxley 1964), (Hines and Shrager 1991), and (Reilly 2016). The model parameters displayed in figure 1(B) are listed in table 1. The simulation code was modified from a myelinated model previously developed (Reilly 2016).

Figure 1. Multi-compartment model of a myelinated axon under microcoil stimulation. (A). The modeled axon contains 101 nodes and 100 internodes. (B). Each segment is characterized by its membrane conductance (node: gL; internode: gi) and membrane capacitance (node: Cm; internode: Ci). Node of Ranvier segments have additional voltage-dependent elements for sodium (Na), potassium (K), and non-specific (P) channels. Electrodynamics within each segment are expressed by relating axial, membrane leak, membrane capacitive, and ionic currents using Kirchhoff's law (equation (A2)). The myelin sheath is represented by a linear conductance (gin) in parallel with capacitance Cin. Longitudinally, the model contains cytoplasmic (Ra) and periaxonal (Ra) resistivities (see table 1).

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Table 1. Parameters of the modeled myelinated axon.

ParametersValueReferencesFiber diameter (including myelin)10 µm(Reilly 2016)Number of nodes of Ranvier101(Reilly 2016)Number of internodes100(Reilly 2016)Width of node of Ranvier2.5 e−4 cm(Reilly 2016)Axon diameter7 µm(Reilly 2016)Length of internode segments1000 µm(Reilly 2016)Total axon length100 000 µm(Reilly 2016)Cytoplasmic resistivity (Ra)100 ohm cm(Reilly 2016)Extracellular resistivity300 ohm cm(Reilly 2016), not usedNode membrane capacitance (Cm)2 µF cm−2(Frankenhaeuser and Huxley 1964, Hines and Carnevale 1997)PNa8 e−3 cm s−1(Frankenhaeuser and Huxley 1964, Hines and Carnevale 1997)PK1.2 e−3 cm s−1(Frankenhaeuser and Huxley 1964, Hines and Carnevale 1997)PP0.54 e−3 cm s−1(Frankenhaeuser and Huxley 1964, Hines and Carnevale 1997)gL0.0303 S cm−2(Frankenhaeuser and Huxley 1964, Hines and Carnevale 1997)Reversal potential of leakage channel (EL)−69.74 mV(Reilly 2016)[Na]i13.74 mM(Frankenhaeuser and Huxley 1964, Hines and Carnevale 1997)[Na]o114.5 mM(Frankenhaeuser and Huxley 1964, Hines and Carnevale 1997)[K]i120 mM(Frankenhaeuser and Huxley 1964, Hines and Carnevale 1997)[K]o2.5 mM(Frankenhaeuser and Huxley 1964, Hines and Carnevale 1997)Internodal capacitance (Ci)2 µF cm−2 (single lamellae)(Mcintyre et al 2002)Internodal leak conductance (gi)0.001 S cm−2 (single lamellae)(Mcintyre et al 2002)aMyelin capacitance (Ci)0.1 µF cm−2(Mcintyre et al 2002)Myelin conductivity (gi)0.001 S cm−2(Mcintyre et al 2002)Resting potential (Er)−70 mV(Reilly 2016)Temperature20 Celsius(Reilly 2016)

a-STIN conductance in the MRG model (Mcintyre et al 2002).

To adapt the model to simulate pathological alterations in myelinated axons, we implemented a modified version of the core conductor model of myelinated axons (Koles and Rasminsky 1972, Basser 2004, Resnick et al 2018). The core model treated the internodal segment of the myelinated axons as a single layer of resistive–capacitive medium separating resistive compartments. We also assumed that demyelination did not affect the nodal membrane, physiological properties of the node, or the axon diameter, as in (Koles and Rasminsky 1972).

The myelin sheath is formed by the repeated wrapping of a glial cell membrane around the axon, creating a multi-layered structure that insulates the axon and allows for rapid signal transmission. Demyelination causes the reduction of myelin thickness while increasing the conductance (gin) and capacitance (cin) of the myelin sheath (Koles and Rasminsky 1972, Mcintyre and Grill 1999, Mcintyre et al 2002, Resnick et al 2018). Numerical increase of the myelin conductance and capacitance can be used to simulate demyelination process (Sleutjes et al 2019). Others have linked myelin thickness with changes in myelin conductance and capacitance. For example, in the MRG model, the lamella membrane conductance (gi = 1 × 10−3 S cm−2) and lamella membrane capacitance (ci = 0.1 μF cm−2) (Mcintyre et al 2002) were divided by the number of lamella to produce overall myelin conductance and capacitance (Mcintyre et al 2002). In our simulation, since individual layers were not modeled, we scaled the overall myelin conductance and capacitance with changes of the myelin thickness, as reported previously (Koles and Rasminsky 1972, Resnick et al 2018), using the following equations,

where gin(d%) represents the overall myelin conductance and cin(d%) the overall myelin capacitance when the myelin thickness was reduced to d%, respectively.

A shortcoming of this approach is that the myelin conductance and capacitance approach infinity in the case of complete myelin loss. As in (Resnick et al 2018), we address this problem by incorporating the internodal plasma membrane conductance and capacitance in series with the myelin sheath's, creating a core-dual conductor model (figure 1(B)). The total internodal conductance and capacitance can be calculated by adding plasma and myelin contributions in series. This procedure was implemented in all subsets of internodes, allowing us to simulate any forms of partial and complete demyelination of any internodes.

2.2. The microcoil model that generates focal stimulation

The microcoil model was developed in our previous work (Ye 2022) and was used to stimulate the demyelinated axon (appendix B). Briefly, we modeled the submillimeter microcoil as a multiple-loop circular structure (figure 2(A)). The coil parameters are listed in table 2. These parameters closely match the miniature coils that were used in our experiments to stimulate neurons from invertebrates (Skach et al 2020, Ye et al 2022a, 2024) and rodents (Ye et al 2020).

Figure 2. Distribution of induced electric field generated by a microcoil and its gradient along the axon. (A). Electric field distribution around the axon. The coil is located at the center of the axon, 50 µm away from the axon. (B): magnetic field; E: induced electric field. B. Gradient of the induced electric field in the axon direction (x-direction) and its dependence on the coil-axon distance (y). (C). Electric field gradient along the axon.

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Table 2. Coil parameters.

ParameterValueReferencesInductance (L)100 nH(Skach et al 2020)Maximal current (Imax)200 mA(Skach et al 2020)Resistance (R)2 Ω(Skach et al 2020)Coil length (l)0.5 mm(Skach et al 2020)Radius (Rc)0.25 mm(Skach et al 2020)Number of loops (N)20(Skach et al 2020)

Orientation of the coil: Several recent experiments have suggested that microcoil orientation played a pivotal role in axonal stimulation (Golestanirad et al 2018, Saha et al 2022a). In this work, the coil was positioned to induce an electric field parallel to the axons to generate effective stimulation (Jefferys 1981, Gluckman et al 1996). During the simulation, the distance between the coil and the axon was equal to or greater than 50 µm (thickness of the biocompatible coating of the coil during experiment).

Spatial profile of the stimulation: Previously, we have calculated the intensity of the electric field induced by the coil ((Ye 2022), also see appendix B), and found that the small size of the coil ensured that a relatively large electric field was generated locally around the coil. Furthermore, we calculated the gradients of the induced electric field along the axon (figures 2(B) and (C)) and used this data to provide the most effective activation of the node of interest.

Temporal profile of the magnetically induced electric field: When a current pulse was delivered into the coil for neural stimulation, it induced the electric field at the onset and offset of the pulse, as confirmed by our calculation (appendix B). The induced electric field was biphasic in shape (figure 3(A)), as previously confirmed by our group (Skach et al 2020, Ye and Barrett 2021) and others (Minusa et al 2018) under various stimulation conditions, such as dorsal cochlear nucleus (Golestanirad et al 2018) and cortical neuron (Lee and Fried 2017) stimulation. In the model, we displayed the induced electric field as biphasic pulses (figure 3(B)).

Figure 3. Magnetic stimulation with the microcoil causes changes in local node membrane potential. (A). Voltage pulses (500 Hz, 1 ms pulse width) in the coil induce an electric field around the coil (experimentally measured). Electromagnetic induction happens at the rising and falling phases of the voltage pulses, producing a biphasic electric field. (B). In NEURON simulation, the biphasic waveform is programmed and applied to the myelinated axon model for stimulation. Red arrows: direction of the induced electric field around the coil. Note the polarity of the field corresponds with the direction of the induced electric field (E). (C). Membrane potential in several nodes during a subthreshold stimulation by the microcoil (NEURON simulation). Node [56] is targeted and its membrane potential change is most obvious. The membrane potential is generated at the onset (top panel) and offset (bottom panel) of the coil current, which induced electric field along the myelinated axon. The membrane potential in this area oscillates at the frequency of the coil current during stimulation.

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The multi-compartment, myelinated axon model and the microcoil model were implemented together using the NEURON (v7.8) simulation environment package (Hines and Carnevale 1997). Simulations used NEURON's built-in adaptive integrator with DASPK, a differential algebraic solver with preconditioned Krylov method (Brown et al 1994). The model was operated at room temperature (20 °C).

To study the effect of demyelination on axonal conductance, a physiological AP was simulated by activating the node 0 with a pulse (2 nA in amplitude and 0.1 ms in pulse width). Timing of the AP to invade the demyelinated area was adjusted by altering the delay of the injected pulse. Local demyelination was simulated by reducing the myelin thickness to different levels, and the deficit in axonal conductance was characterized by measuring the delay or capability for the AP to travel through the demyelinated area.

To study the capability of the microcoil stimulation in recovering/reinstalling axonal conductance in the demyelinated axon, biphasic short pulses in alternating directions were utilized to simulate the waveform of the coil-induced electric field in NEURON (figure 3(B)). The extracellular stimulus by the coil was implemented directly, i.e. by controlling the potential at the outer surface of each model compartment. This potential is calculated by integrating the scalar component of the electric field along the axon path (appendix B, equations (B8) and (B9)). Microcoil stimulation caused oscillation of local membrane potential (figure 3(C)), whose profile matches perfectly with the gradient of the electric field (figure 2(C)).

During the simulation, the program first searched for a threshold for AP initiation by the coil stimulation alone. Threshold of axonal activation was identified as the lowest stimulus intensity that could initiate an AP in the myelinated axon. We then adjusted the stimulation intensity for subthreshold stimulation. The intensity for subthreshold stimulation was normalized to this identified threshold.

Internode segment length, diameter, and axonal resistance remained constant for all described experiments, while myelin thickness, capacitance, and resistance were experimentally manipulated. The impact of the coil on various locations of the demyelinated axons was tested, guided by the analysis of the induced electric field and gradient (appendix B). Two frequencies (100 Hz and −5000 Hz) were tested and compared to investigate the frequency-dependent effects. To demonstrate the underlying ion channel responses to the microcoil stimulation during conductance restoration, membrane currents and state variables or the targeted nodes in the demyelinated area were analyzed.

To investigate the possibility of rescuing axonal conductance failure after demyelination with electromagnetic stimulation, we built a multi-compartment NEURON model of a demyelinated axon under focal electromagnetic stimulation with a miniature coil. We computed the electric field generated by the coil and applied it to the axonal model. We simulated AP propagation in the axon when local demyelination occurred. We positioned the coil next to the demyelinated area to activate nodes in the demyelinated region. We analyzed the impact of coil location, stimulus frequency and degree of demyelination on the success of functional rescuing of the axon conductance. We also investigated ion channel dynamics during magnetic stimulation and recovery of axonal function.

3.1. Microcoil stimulation depolarizes node membrane potential, and the location of depolarization is predicable by the gradient of the magnetically induced electric field

Under time varying magnetic stimulation, neural tissue is stimulated by the magnetically induced electric field via magneto-electric induction (Ye et al 2011, 2022b). Due to its miniature size, the microcoil can generate a very focal electric field distribution around the coil, to provide local activation of the subcellular structure (Bonmassar et al 2012, Lee et al 2016, Ye and Barrett 2021). Previously, we have computed the induced electric field distribution generated by a circular miniature coil (Ye 2022), and have validated the effects of the focal stimulation using various preparations, such as mouse hippocampal neurons (Ye et al 2020) and invertebrate ganglion cells (Ye and Barrett 2021, Ye et al 2022a, 2024). This coil has a diameter of 0.5 mm, and contains 20 loops of circular wires (table 2), representing a commercially available coil used for neural activation in our experiments (Ye et al 2020, 2024, Ye and Barrett 2021). Therefore, the parameters that describe the microcoil closely match those used in the experiments.

In our model, we positioned the microcoil close to the myelinated axon (assuming 50 µm thickness of the coated material, figure 1(A)) and computed the magnetically induced electric field. The node contained F–H ion channel mechanisms (figure 1(B), appendix A). Figure 2(A) demonstrates the spatial distribution of the induced electric field around the myelinated axon. The axon could experience a local electric field intensity as large as 50–80 V m−1 (figure 2(A)). Previously, a 10 V m−1 threshold was reported for neuronal activation (Chan and Nicholson 1986). Fields at 10–20 V m−1 are sufficient to modulate neuron firing in Purkinje and stellate cells in vitro (Chan and Nicholson 1986), or in the guinea pig hippocampus (Jefferys 1981).

As the miniature size coil allows subcellular stimulation, finding the location of neural activation is critical to target the fine structure, such as a node on the axon. Previous studies have established that the gradients of the electric field along the axon define the location and speed of depolarization or hyperpolarization by the extracellular stimulation (Rattay 1986, Lee and Fried 2017). Figure 2(B) plots the field gradient (dEx/dx) along the axon, and its dependency on the coil-axon distance (y). When the coil is located 50 µm away from the axon, it can produce a field gradient as large as 250 000 V m−2 (figure 2(C)), well above the value that can cause neural activation (50 000 V m−2 in (Lee et al 2016)). As the coil moves away from the axon, dEx/dx decreases, suggesting a decreased field gradient and decreased efficacy of stimulation.

Previous work has theoretically (Ye 2022) and experimentally (Bonmassar et al 2012) confirmed that the rising and decaying phases of a coil current can lead to a large change in the magnetic field and the induced electric field (figure 3(A)). Therefore, in this study, we modeled the induced electric field using biphasic voltage (figure 3(B)) and applied it to locally demyelinated axons. To confirm that the field gradient indeed predicts the location of membrane polarization, we applied a subthreshold stimulus to the NEURON model (to avoid triggering of the AP). The coil stimulation depolarized several nodes next to the coil (figure 3(C)), and the amount of depolarization in each node is defined by the profile of the field gradient (figure 2(C)). Because the induced electric field is biphasic, we observed a depolarization/hyperpolarization oscillation in the membrane potential (figure 3(C)). Therefore, it is possible to position the coil along the axon to selectively target a specific node for stimulation with various frequencies.

3.2. Demyelination causes delay and blockage of AP, depending on the severity of demyelination

When simulating AP propagation in the modeled axon, we injected an intracellular current to node 0 to initiate an AP. In the healthy, myelinated axon, the AP propagated to the other side of the axon at a constant speed 22 m s−1 (supplementary figure 1), confirming the results from previous work (Reilly 2016). To monitor the behavior of the nodes during AP propagation, we plotted the membrane potentials in nodes 53–57 (figure 4(A)). In the subsequent experiments, the microcoil was positioned close to these nodes for magnetic stimulation. Among these nodes, node 56 was designated as the 'targeted node' by coil stimulation, whose ion channel dynamics will be investigated in greater detail.

Figure 4. Demyelination causes failure of axonal conductance. Action potential is initiated at node 0. Illustrated here is the propagation of an action potential in the simulated axon at nodes 53–57. Node [53].v(0.5): membrane potential in node 53. (A). normal fiber with 100% myelin thickness. (B). Type I demyelination. Myelin of internode 55 is reduced to 20%, 10%, 5%, 2%, and 1.3% original thickness. Decreased myelin thickness causes prolonged delay for AP propagation. (C). Type II demyelination. Myelin of internode 55 is significantly reduced (below 1.1% original myelin thickness). Type II axon demonstrates complete failure of axonal conductance.

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To simulate local demyelination, the capacitance and resistance of the myelin are linearly scaled by the loss of myelin thickness (Mcintyre et al 2002