Mathematically modeling action potentials in myelinated neurons to examine the role of myelin, ion channel density, and myelinated lengths on conduction

Since the seminal work of Hodgkin and Huxley, which quantitatively described the propagation of electrical signals through neurons, there has been much investigation into the electrical and geometrical properties of neurons and how they affect conduction velocity along a neuron’s length. To study human neuron behaviors, mathematical models have expanded upon Hodgkin and Huxley’s models to incorporate the effects of neurons that are myelinated by modeling myelinated portions of neurons as passive cables. Here, we present a developed mathematical model that discretizes a myelinated axon length and finely allows for control over a number of important electrical and geometrical properties. Using this model, we present and compare how myelin, inter-node length, and ion channel density affect conduction velocity in two different lengths of axons. We confirm that myelination, internode-length, and ion channel density correlate positively with conduction velocity, and propose potential mechanisms of this effect at lower node length and inter-node length values.

can depolarize. Thus, a strong enough input will allow for action potential generation in an "all-ornone" manner. During depolarization, Na+ channels rapidly open up to allow sodium ions to enter the membrane, increasing the voltage. Once the Na+ equilibrium potential is reached, potassium channels open up to allow potassium to flow out to restore the voltage back to negative during repolarization [4]. The refractory period that follows allows the neuron to restore its chemical gradient while preventing another action potential from firing. To ensure reliable action potential propagation speed, amounts of myelinated sections of axons are of key importance. In fact, the thickness of myelin has been shown to be a dynamic trait of neuronal axons, whereby neurons can control their action potential propagation speed through activity dependent mechanisms [5,6].
However, when myelin sheaths are depleted or partially damaged in people with diseases such as Multiple Sclerosis and Guillain-Barré syndrome, the transmission of electrical impulses is less efficient, which results in ineffectiveness or inability to control physical actions [7,8]. In recent years, many studies have investigated the role of myelin on action potential conduction velocity [9,10]. But in order to do so, the HH model is not sufficient. While the HH model identified the rate and amplitude of depolarization is proportionally related to the concentration of sodium ions and that sodium and potassium gates as well as their ionic conductances are dependent on time and membrane voltage, it did only examine the action potential in an unmyelinated giant squid axon, but can also be utilized for studying unmyelinated sections of axons. For the purpose of examining the effects of myelin on action potential propagation, a discretized model of the axon using the HH model at unmyelinated nodes and modeling myelinated nodes as passive cables [9,10], would prove fruitful. Using this combination of HH model and wave equation model, scientists investigated the effects of different physical parameters (node of ranvier length, internode length, myelin thickness, axon radius) on conduction velocity. Studies found that conduction velocity has a positive correlation with myelin thickness and axon radius has a negative correlation with node length and internode length. Using a simpler, yet powerful discretized model of the axon, where the axon characteristics and myelination pattern can be easily controlled, this paper examines the effects of a number of physical axon parameters on conduction velocity and compares those effects in two distinct axon lengths.

2．Material and Methods
Expanding upon the classical Hodgkin Huxley model, which examines the unmyelinated giant squid axon, a discretized axonal model was generated, whereby individual nodes of the model could be assigned with myelinated or unmyelinated status, and the corresponding electrical properties for the corresponding section are assigned. Moreover, the two types of sections are modeled as passive cable equations (unmyelinated) and the HH model equations (myelinated). Model parameter values and assumptions for myelin thickness, axon radius, membrane and myelin capacitance, membrane and myelin resistivity, and ionic conductances for Na+, K+, and the leak channel were obtained from literature review [10,11,12]. Tabulated values of model parameters are shown in the appendix. Simulation equations were solved using ode23s solver in MATLAB with resting membrane potential and baseline channel gating parameters as initial conditions . Different myelination patterns were chosen to examine three relationships: 1. The effect of myelinated node percentage on action potential conduction velocity, 2. The effect on inter-node length (length between unmyelinated nodes) on action potential conduction velocity, and 3. The effect of node length (length of unmyelinated nodes) on action potential conduction velocity. To ensure proper conduction, the first and last sectioned node of the model was set to unmyelinated parameters, and the input current stimulus amplitude at the first node was set to an above-threshold value of 0.4 µA to ensure action potential generation in all cases examined. The input current was introduced in the model after 40 ms to ensure stabilization of the model and was kept active for a duration of 40 ms. To determine the conduction velocity, response wave-forms of the voltage at the last section of the axonal model were analyzed and the time of first action potential peak was determined post stimulus. The conduction velocity was calculated using the distance from first to last node of the axon divided by the amount time post-stimulus for the first peak to generate at the end node.

Effect of myelination on conduction velocity in 1 cm axon and 0.25 cm axon
To examine the effect of percent myelin on conduction velocity, an axon was discretized into 13 sections and the node length was set to be a constant length of 1/13 of the axon length. Different internode lengths (myelinated regions of an axon) were constructed by assigning sections of the axon as a myelinated or unmyelinated section. For example, in a 1 cm axon discretized into 13 sections, an inter-node length of 11/13 cm could be constructed by assigning the middle 11 sections as myelinated.
For additional myelination patters, axon myelination patterns were constructed in a uniform pattern, where internode length was conserved for the myelination percentage tested. As shown in Figure 2, conduction velocity increases with increased myelination. For the 1 cm axon, if the percent myelination increase from 46.15% to 84.62%, there is a 76.39% increase in conduction velocity of the neuron. Similarly, when the same percentages of myelin were examined in the 0.25 cm axon, there was a 67.57% increase in conduction velocity of the neuron. Interestingly, this seems to suggest that a decrease in myelination has a more drastic percent change in conduction velocity for larger axons.

Effect of inter-node length on conduction velocity in 1 cm axon and 0.25 cm axon
To examine the effect of percent myelin on conduction velocity, an axon was discretized into 13 sections and the node length was set to be a constant length of 1/13 of the axon length. Different internode lengths (myelinated regions of an axon) were constructed by assigning sections of the axon as a myelinated or unmyelinated section. For example, in a 1 cm axon discretized into 13 sections, an inter-node length of 11/13 cm could be constructed by assigning the middle 11 sections as myelinated. For additional myelination patters, axon myelination patterns were constructed in a uniform pattern, where internode length was conserved for the myelination percentage tested. As shown in Figure 2, conduction velocity increases with increased myelination. For the 1 cm axon, if the percent myelination increase from 46.15% to 84.62%, there is a 76.39% increase in conduction velocity of the neuron. Similarly, when the same percentages of myelin were examined in the 0.25 cm axon, there was a 67.57% increase in conduction velocity of the neuron. Interestingly, this seems to suggest that a decrease in myelination has a more drastic percent change in conduction velocity for larger axons. Fig. 5 The effects of myelination on action potential (AP) propagation. Simulations for different myelination patterns, which conserved node length for each simulation and inter-node length within an axon, were conducted for a 1 cm axon (red) and a 0.25 cm axon (blue). The calculated conduction velocities after a 40ms current pulse of 0.4 µA are shown by each marker.

Effect of node length on conduction velocity in 1 cm axon and 0.25 cm axon
To examine the effect of inter-node length (INL) on conduction velocity, the node length was again set to a constant of 1/13 of the axon length, while other parameters were kept constant. The axon was separated into 13 sections of equivalent pieces in order to achieve a number of variations of internode length while fixing the node length. It should be indicated that the INL for each stimulation was held constant by generating different myelination patterns on the axon to keep each INL of an axon simulated constant. As shown in Figure   thirteenth of the axon length. Thus, each section of the axon alternated between unmyelinated and myelinated categories and the AP makes twelve jumps from a section of one category to another, with myelinated sections of 1/13 of the axon length. The next INL examined was 2/13 of the axon length. Such axons were constructed by alternating one unmyelinated section followed by two myelinated sections. While the percent myelin increased from approximately 46 to 61% from the previous INL case, the conduction velocity increase by 10.7% for the 1cm axon and 19.2% for the 0.25cm axon. This observation would hint that the effect of myelinated lengths along an axon, the INL length, on conduction velocity is not a relative effect of the length of the axon as the INL values examined are equal fractions of the total length. Fig. 6 Effects of inter-node length on conduction velocity. Simulations for different internode-lengths, which conserved node length and axon length for each simulation, were conducted for a 1 cm axon (red) and a 0.25 cm axon (blue). The calculated conduction velocities after a 40ms current pulse of 0.4 µA are shown by each marker.

Effect of node length on conduction velocity in 1 cm axon and 0.25 cm axon
To examine the effect of ion channel density on conduction velocity, simulations were conducted to vary node length and calculate the predicted conduction velocity. Given that our model fixes ion channel density indirectly through parameter values of the electrical properties of the myelinated category sections, modulation of node length would allow for perturbations of density. The inter-node length for all simulations was set constant as 8/10 of the axon length. The node length was altered by cutting the axon into different numbers of pieces. As shown in Figure 4, increasing node length (thereby decreasing ion channel density), decreases action potential propagation for both length axons examined. In the 1 cm axon, as the node length increase from 0.01 cm to 0.1 cm, there is a 44.51% decrease in conduction velocity. When the node length increased from 0.0025 to 0.025 cm in the 0.25 cm axon, there was a 41.56% decrease in conduction velocity. Thus, decreasing ion channel density at unmyelinated nodes decreases conduction velocity.  In an unmyelinated axon, the Na+ channels are distributed uniformly across the length of the axon. But when myelin are introduced, the channels get moved around and gather in the nodes of Ranvier, which increases the conduction velocity because the higher density of ion channels allow signals to travel faster. Our model assumes that electrical properties of each myelinated section and each unmyelinated section are similar based on their category, and that the number of ion channels, represented by the electrical properties of each node are constant. Therefore, as we decrease node lengths, increase INL, or increase percent myelination, the constant number of ion channels are condensed into a smaller space, which allows a more rapid conduction velocity. Thus, by increasing the ion channel density in a unmyelinated node, the conduction velocity can be increased. Oppositely, when we increase node lengths, decrease INL, or decrease percent myelination, the same number of ion channels will be more spread out across the nodes, slowing down the conduction velocity. By looking at a 1cm and a 0.25 cm axon, the effects of different sized axons are investigated. Given the variety of axon lengths that exist in our CNS and PNS, understanding how these correlations can be different in different regions of the nervous system would prove beneficial. From the results, the percent change in conduction velocity of the 0.25 cm axon is smaller compared to the 1 cm axon. This shows that deterioration of myelin in longer neurons in the body may possibly cause more severe symptoms than in shorter neurons. While there are a significant number of axon parameters that can affect conduction velocity, the developed model presents a valuable tool for examining further questions on the effects of myelin patterning, and in particular non-uniform patterning, electrical properties of myelin and nonmyelinated sections, and how different dimensions (length and radii) of axons can perturb conduction velocity.

ACKNOWLEDGMENTS
I would love to thank Dr. Archer Hamidzadeh for guiding and supporting me through every stage of this research paper.