Intracellular Sodium Concentration and Membrane Potential Oscillation in Axonal Conduction Block Induced by High-Frequency Biphasic Stimulation

Abstract

Objective.

A new axonal conduction model was used to analyze the interaction between intracellular sodium concentration and membrane potential oscillation in axonal conduction block induced by high-frequency (kHz) biphasic stimulation (HFBS).

Approach.

The model includes intracellular and extracellular sodium and potassium concentrations and ion pumps. First, the HFBS (1 kHz, 5.4 mA) was applied for a duration (59.4 seconds) long enough to produce an axonal conduction block after terminating the stimulation, i.e., a post-stimulation block. Then, the intensity of HFBS was reduced to a lower level for 4 seconds to determine if the axonal conduction block could be maintained.

Main results.

The block duration was shortened from 1363 ms to 5 ms as the reduced HFBS intensity was increased from 0 mA to 4.1 mA. The block was maintained for the entire tested period (4000 ms) if the reduced intensity was above 4.2 mA. At the low intensity (<4.2 mA) the membrane potential oscillation disrupted the post-stimulation block caused by the increased intracellular sodium concentration, while at the high intensity (>4.2 mA) the membrane potential oscillation was strong enough to maintain the block and further increased the intracellular sodium concentration.

Significance.

This study indicates a possibility to develop a new nerve block method to reduce the HFBS intensity, which can extend the battery life for an implantable nerve stimulator in clinical applications to block pain of peripheral origin.

Introduction

Axonal conduction block by high-frequency (kHz) biphasic stimulation (HFBS) has been known for 80 years [1]. Recently it has been successfully applied in clinical applications to treat obesity and amputation limb pain [2,3] or in preclinical study to restore bladder function after chronic spinal cord injury [4]. However, currently the mechanism of action is still unclear. Thus, further study of the mechanism of HFBS nerve block is needed to improve the therapy and/or to develop new nerve block methods.

It is well known that HFBS can block axonal conduction during the stimulation at a frequency ≥4 kHz [5,6]. However, recent animal studies discovered that axonal conduction block can also occur after termination of the HFBS, i.e., a post-stimulation block that can be induced by HFBS at ≥4 kHz [7,8] as well as at 100–1000 Hz [9]. A recent modeling study [10] further revealed that the post-stimulation block is due to the accumulation of intra-axonal sodium during a long-duration stimulation. This blocking mechanism is very different from the acute block mechanism that can occur quickly after the HFBS is applied [5,6]. Previous modeling studies [11,12] also showed that the acute block is hypothetically due to high-frequency membrane potential oscillation causing the potassium channel to open constantly at a level sufficient to prevent the generation of an action potential. Since low frequency (100–1000 Hz) stimulation only produces post-stimulation block but not the acute block, it is not clear how the low frequency membrane potential oscillation interacts with the post-stimulation blocking mechanism mediated by high levels of intra-axonal sodium ions. For example, if the HFBS is not turned off after it has been applied for a long period to increase the intra-axonal sodium concentration and cause a post-stimulation block but instead is continued at a lower intensity, does it alter the mechanism underlying post-stimulation block? Since the post-stimulation block always has a limited duration, it would be beneficial in clinical applications to apply HFBS at a reduced stimulus intensity to maintain the increased level of intra-axonal sodium concentration and produce a long-lasting block of axonal conduction. Reducing HFBS intensity would save the battery life of the fully implanted stimulator in clinical applications [2,3].

In this study the interaction between intracellular sodium concentration and membrane potential oscillation during axonal conduction block induced by HFBS was investigated. A newly developed axonal conduction model [10,13] that includes intracellular and extracellular sodium and potassium concentrations and ion pumps was used. The model analysis reveals the possibility to reduce HFBS intensity to a minimal level for maintaining the axonal conduction block once it has been applied at a high intensity for a period long enough to cause a post-stimulation block. This study revealed a new method for blocking nerve conduction with HFBS that might be useful for many potential clinical applications [2,3].

Materials and Methods

Axonal Conduction Model

The axonal conduction model is shown in Fig. 1. A 9-mm long unmyelinated axon is segmented into many small cylinders of length Δx = 0.25 mm, each of which is modeled by a membrane capacitance, a variable membrane resistance and sodium and potassium pumps. Two monopolar electrodes (with the indifferent electrode at infinity) are placed at 1 mm distance distance from the unmyelinated axon. One is the test electrode at the 1 mm location, which delivers a uniphasic single pulse to evoke an action potential that can propagate along the axon in both directions. The other is the block electrode at the 5 mm location along the axon. The test electrode is always cathodal (negative pulse). The block electrode always delivers HFBS with a cathodal pulse followed by a symmetric anodal pulse (see the waveform shown in Fig. 1) to block the propagation of the action potential (AP) generated by the test electrode.

Fig.1.

Unmyelinated axon model including sodium and potassium pumps to simulate conduction block induced by high-frequency biphasic pulses. The unmyelinated axon is segmented into many small cylinders of length Δx, each of which is modeled by a resistance-capacitance circuit based on the modified Hodgkin–Huxley model. R_a: Axoplasm resistance. R_m: Membrane resistance. C_m: Membrane capacitance. V_a: Intracellular potential. V_e: Extracellular potential. Na_P: 𝑆odium pump. K_P: Potassium pump.

It is assumed that the axon is in an infinite homogeneous medium (resistivity ρ_e = 300 Ωcm). After neglecting the small influence of the axon in the homogeneous medium, the extracellular potential V_e,j at the j^th segment along the axon can be calculated by:

$${\text{V}}_{e,j}(\text{t})=\frac{{\text{ρ}}_{\text{e}}}{4\pi }\left[\frac{{\text{I}}_{\text{block}}(\text{t})}{\sqrt{{\left(\text{j}\mathrm{\Delta }\text{x}-{\text{x}}_0\right)}^2+{\text{z}}_0^2}}+\frac{{\text{I}}_{\text{test}}(\text{t})}{\sqrt{{\left(\text{j}\mathrm{\Delta }\text{x}-{\text{x}}_1\right)}^2+{\text{z}}_1^2}}\right]$$

Where I_block(t) represents the biphasic pulses delivered to the block electrode (at location x₀ = 5 mm, z₀ = 1 mm); I_test(t) is the single test pulse delivered to the test electrode (at location x₁ = 1 mm, z₁ = 1 mm).

The membrane potential E_j at the j^th segment of the unmyelinated axon is calculated by:

$$\frac{d{E}_j}{dt}=\left[\frac{d}{4{\rho }_{\text{i}}}\left(\frac{E_{j-1}-2E_j+E_{j+1}}{\mathrm{\Delta }{\text{x}}^2}+\frac{V_{e,j-1}-2V_{e,j}+V_{e,j+1}}{\mathrm{\Delta }{\text{x}}^2}\right)-I_{i,j}\right]/c_m$$

Where E_j = V_a,j − V_e,j; V_a,j is the intracellular potential at the j^th segment; V_e,j is the extracellular potential at the j^th segment; d is the unmyelinated axon diameter; ρ_i is the resistivity of axoplasm (34.5 Ωcm); c_m is the capacity of the membrane (1 uF⁄cm²); I_i,j is the total ionic current at the j^th segment described by the modified Hodgkin-Huxley (HH) equations [11,14–16] in the Appendix.

The model equations were solved by Runge-Kutta method [17] using MATLAB programing language with a time step of 4 μsec for an axon of 2 μm diameter at the temperature T = 291.5 K (18.5 °C). The simulation was always started at the initial condition E_j = E_rest = −60.08 mV. At the two end segments of the modeled axon the sealed boundary conditions were implemented by setting the longitudinal axonal current to zero. This new axonal conduction model has been successfully validated in previous studies in terms of conduction velocity, refractory period, maximal firing rate, DC block, and post-stimulation block by HFBS [10,13].

Simulation Protocol

HFBS of 1 kHz frequency was chosen in this study to analyze the axonal conduction block because previous animal and simulation studies [7–10] showed that 1 kHz HFBS was more effective than >4 kHz HFBS to change intracellular and extracellular sodium and potassium concentrations and required a much shorter duration of stimulation to induce a post-stimulation block. The 1 kHz HFBS was first applied for 59.4 seconds at a constant intensity of 5.4 mA. These initial HFBS parameters were chosen based on a previous study [10] showing that these parameters increased the intracellular sodium concentration enough to cause a post-stimulation block. Then, the HFBS was changed to an intensity ranging between 0 mA and 5.4 mA for an additional 4 seconds during which the axonal conduction block was tested on a propagating action potential initiated at the test electrode (see Fig. 1) by a single stimulus pulse (0.1 ms, 10 mA). Initially, the single stimulus pulse was tested every 5 ms during the 4-second period to detect a change in axonal conduction (block or loss of block). Once a change was observed, the interval between test pulses was reduced to 1 ms to determine the latency for the onset of the change.

Results

Axonal Conduction or Block after HFBS Intensity Reduction

HFBS (1 kHz and 5.4 mA) can produce axonal conduction block when it is applied for 59.4 seconds as shown in a previous study [10]. After turning off the stimulation (indicated by “OFF” in Fig. 2), the axonal block continued for 1363 ms (termed post-stimulation block) (Fig. 2A). Axonal conduction recovered 1 ms later (at 1364 ms) as shown in Fig. 2B where the AP successfully passed the block electrode with a delay of about 5 ms. However, if the HFBS was continued beyond 59.4 seconds at a reduced intensity of 4 mA, axonal conduction recovered almost immediately (within 5 ms after reducing HFBS intensity, see Fig. 2C). On the other hand, if HFBS was maintained at the 5.4 mA intensity longer than 59.4 seconds the axonal block was also maintained during the entire period of stimulation (4000 ms in Fig. 2D).

Fig.2.

After 59.4 seconds of 1 kHz high frequency biphasic stimulation (HFBS) at a constant intensity of 5.4 mA, reducing the stimulation intensity to different levels can maintain the axonal block for different durations. The action potential induced by a test pulse (0.1 ms, 10 mA) at 1359 ms after reducing the HFBS intensity to 0 mA (OFF) is blocked at 1363 ms when it arrives at the block electrode (A), but it can propagate through the block electrode when the test pulse is delivered 1 ms later with the action potential arriving at the block electrode at 1364 ms (B). Reducing the intensity from 5.4 mA to 4 mA causes the axonal conduction to occur immediately (within 5 ms) after the change (C), while the axonal block can be maintained for at least 4000 ms if the intensity is not reduced (D).

The effect of reduced HFBS intensity was analyzed by comparing the effects of two intensity ranges: A and B (Fig. 3). In range A (0–4.1 mA) the duration of post-stimulation block was shortened as the HFBS intensity increased, and eventually axonal conduction occurred immediately after the HFBS intensity reached the level of 3–4.1 mA. In range B (4.2–5.4 mA) the post-stimulation block was maintained except that an initial short period of axonal conduction occurred during the intensity range of 4.2-.4.5 mA. Therefore, the HFBS could maintain a continuous axonal block when the stimulation intensity was reduced from 5.4 mA to a level ≥4.6 mA (Fig. 3). At 4.6 mA, the block was maintained during the maximal tested period of 80 seconds.

Fig.3.

Axonal conduction or block is dependent on the reduced intensity and the time after the reduction. The 1 kHz high frequency biphasic stimulation (HFBS) is applied for 59.4 seconds at a constant intensity of 5.4 mA and then reduced in intensity to different levels (0.01–5.4 mV) or turned off.

Interaction between Intracellular Sodium Concentration and Membrane Potential Oscillation

A previous study [10] showed that post-stimulation block is due to a significant increase in intracellular sodium concentration caused by the long duration HFBS and that the duration of post-stimulation block is determined by the activity of ion pumps that slowly restore the intracellular sodium concentration. Therefore, after HFBS was turned off, the ion pumps required 1363 ms to reduce the increased intracellular sodium concentration to a level allowing axonal conduction (Fig. 4A). However, when the HFBS intensity was reduced to 0.2–2 mA (see Fig. 4A) the intracellular sodium concentration recovered at almost the same rate as at 0 mA intensity, but the axonal conduction occurred earlier at a higher level of intracellular sodium concentration (Fig. 4A and Fig. 3). This was mainly because the membrane potential oscillated with a larger amplitude at the higher HFBS intensity (Fig. 4B), which disrupted the post-stimulation block earlier allowing the propagation of the action potential through the block electrode. However, when the reduced HFBS intensity increased to ≥4.2 mA as shown in intensity range B (Fig. 3), the intracellular sodium concentration increased fast enough to prevent the disruption by the membrane potential oscillation (see range B in Fig. 5) or the membrane potential oscillation alone was strong enough to maintain the block, or both. Also shown in Fig. 5, the minimal intensity to maintain the level of intracellular sodium concentration is about 3.6 mA. However, at this intensity the block is lost quickly (Fig. 3) indicating that it is not the intracellular sodium concentration alone that determines the block during the stimulation. Rather, it is the combination of stimulation intensity that controls the magnitude of the membrane potential oscillation and the intracellular sodium concentration that determines the block as shown in Fig. 4.

Fig.4.

After reducing the intensity of the 1 kHz high frequency biphasic stimulation (HFBS), the axonal block is determined by a combination of intracellular sodium concentration and the magnitude of the membrane potential oscillation. A. After HFBS intensity is reduced from 5.4 mA to three levels between 0 mA and 2 mA, the intracellular Na⁺ concentration gradually declines almost at the same rate. However, when the reduced HFBS intensity is higher, the time to the disappearance of axonal block is shorter as indicated by the downward arrows (also see Fig. 3). B. At a higher HFBS intensity, the membrane potential oscillation is stronger, which can disrupt the axonal block allowing the action potential to pass through the block electrode. The membrane potential oscillations are plotted 5 ms before and after the time when the axonal block is lost (indicated as 0 ms time) for different HFBS intensities as shown in A. The vertical axis represents the change of membrane potential relative to the resting membrane potential. The resting membrane potentials at the different times when the axonal block is lost are −54 mA for 2 mA, −57 mA for 0.2 mA, and −58 mA for 0 mA.

Fig.5.

Change of intracellular sodium concentration after the intensity of 1 kHz high frequency biphasic stimulation (HFBS) is reduced to different levels. A – the reduced HFBS intensity range where the membrane potential oscillation dominates over the change of intracellular sodium concentration causing the loss of axonal block earlier than turning off the HFBS (see Fig. 3). B – the reduced HFBS intensity range where the increase of intracellular sodium concentration dominates over the membrane potential oscillation thereby maintaining the axonal block (see Fig. 3).

Eliminating the Initial Axonal Conduction Period

The initial short period of axonal conduction observed when the HFBS intensity was reduced to 4.2–4.6 mA (Fig. 3) was due to a transient change of the gating properties of sodium and potassium channels (Fig. 6). The AP arriving during the transient period propagated through the region of the block electrode (Fig. 6 A and B), while it was blocked at this site when it arrived at the end of this transient period (Fig. 6 C and D). This transient change of ion channel gating properties was eliminated by linearly and slowly reducing the HFBS intensity over a longer period (1000 ms, Fig. 7A) so that the AP could be blocked during the linear transition period (Fig. 7B).

Fig.6.

Initial short duration of axonal conduction after the intensity of 1 kHz high frequency biphasic stimulation (HFBS) is reduced from 5.4 mA to a level between 4.2–4.5 mA (see Fig. 3) is due to a transient change of the ion channel gating properties. At the block electrode the ion channel gating properties (m/h/n) experience a transient partial recovery over an 80-ms period after the HFBS intensity is reduced to 4.4 mA (A), which allows the action potential (AP) arriving at 55 ms during this transient period (A) to propagate through the block electrode (B). However, the AP arriving at 90 ms (C) fails to propagate though the block electrode (D) because the transient change of the ion gating properties has ended.

Fig.7.

The initial conduction period occurs when the intensity of 1 kHz high frequency biphasic stimulation (HFBS) is reduced from 5.4 mA to a level between 4.2–4.5 mA (see Fig. 3) can be eliminated by linearly reducing the HFBS intensity. A. At the block electrode the ion channel gating properties (m/h/n) change gradually without a transient recovery when the HFBS intensity is linearly reduced from 5.4 mA to 4.3 mA during a 1000-ms period. B. An action potential (AP) arriving during the 1000-ms period as indicated by the arrow in (A) is blocked.

Discussion

This study using a new axonal conduction model (Fig. 1) shows that a post-stimulation block can be induced after a long-duration HFBS is turned off (Fig. 2 A and B). However, if the HFBS intensity is only reduced instead of completely turned off, the block duration is either shortened or prolonged depending on the reduced HFBS intensity (Fig. 2 and Fig. 3). There is a critical level for the reduced HFBS intensity, below which the block duration is shortened, but above which it is prolonged (see ranges A and B in Fig. 3). These intensity-dependent effects are determined by an interaction between membrane potential oscillation and the increase in intracellular sodium concentration that is responsible for the post-stimulation block (Fig. 4 and Fig. 5). Linearly reducing the HFBS intensity slowly over a longer period increases the range of HFBS intensities that elicit block (Fig. 7) and also eliminates the initial transient recovery of axonal conduction that precedes the block elicited at certain HFBS intensities (Figs. 3 and 6).

Previous modeling studies [11,12] revealed a different hypothetical blocking mechanism that can produce axonal conduction block quickly after HFBS is turned on. This acute block is due to HFBS-induced membrane potential oscillation that opens the potassium channel constantly and eliminates the delay between sodium and potassium current generation during a depolarization, thereby preventing the generation of an action potential [11,12]. However, when HFBS is applied for a long period, a post-stimulation block can occur as revealed by this (Fig. 2 A and B) and previous modeling studies [10,18] as well as animal studies [7–9]. The post-stimulation block is due to the increase in intracellular sodium concentration by a long-duration HFBS as revealed by previous modeling studies [10,18]. Therefore, the question becomes which mechanism is playing the role in axonal conduction block immediately before terminating a long-duration HFBS. Is it due to the increased intracellular sodium concentration or the constantly opened potassium channel? The current study indicates that if HFBS stimulation is strong enough to induce an acute block at the beginning of the stimulation, then it is probably playing a critical at the time immediately before terminating the long-duration HFBS. If the increased intracellular sodium concentration were playing the major role at the end of the HFBS, then the block should not be sensitive to the change of HFBS intensity. At least, the post-stimulation block duration should not be shortened by reducing the HFBS intensity (see the A range in Fig. 3) because HFBS at the reduced intensity does not change the rate of recovery of the intracellular sodium concentration (Fig. 4A). In addition, at the end of HFBS the minimal HFBS intensity required to maintain the block, i.e., 4.2 mA (Fig. 3), is not the intensity that maintains the level of intracellular sodium concentration but rather is a higher intensity that progressively increases the intracellular sodium concentration (Fig. 5). This result indicates that the increased intracellular sodium concentration alone is not sufficient to maintain nerve block during the HFBS, although it is enough to produce a post-stimulation block after HFBS is turned off.

Similarly, the increased intracellular sodium concentration alone is also not sufficient to maintain the axonal block when the intensity of HFBS is reduced below a critical level (see the A range in Fig. 3). Rather, it is the combination of the membrane potential oscillation and the increased intracellular sodium concentration that determine axonal conduction or block. A larger membrane potential oscillation requires a higher intracellular sodium concentration resulting in a shorter block duration as the membrane pump slowly returns the intracellular sodium concentration to normal (Fig. 4).

The transient recovery of axonal conduction when the intensity of HFBS is reduced below 4.5 mA (see range B in Fig. 3 and Fig. 6) further illustrates the interaction between the increased intracellular sodium concentration and the membrane potential oscillation in maintaining axonal block. During this short period (<100 ms) at the beginning of the changed HFBS intensity (Fig. 6), the increased intracellular sodium concentration is minimally changed due to the slow recovery by the ion pump (Fig. 4A). However, the axonal block is lost briefly due to a transient change in membrane gating parameters (m, h, and n, see Fig. 6) due to the evolution functions (m, h, and n) as described in appendix [14]. When this change in gating parameters ends, the axonal block is restored. This indicates that the magnitude of the membrane potential oscillation at the reduced HFBS intensity in combination with the increased concentration of intracellular sodium is sufficient to produce axonal block as long as the membrane gating parameters become stable. However, even a transient deviation from the stable levels can eliminate the axonal block.

This study raises the possibility of improving the kHz HFBS method for blocking nerve conduction. For example, in clinical applications after applying HFBS for a long period the HFBS intensity could be reduced while still maintaining the nerve block. If the intensity can be reduced from 5.4 mA to 4.6 mA as shown in this study, this could save about 13% battery power of a chronically implanted stimulator thereby extending the life of the implant or the time interval for electrical charging. Furthermore, recent model analysis [18] has shown that by adaptively stepwise increasing the HFBS intensity the initial nerve firing produced by HFBS can be eliminated, which is beneficial in clinical applications to block chronic pain of peripheral origin [3]. Once the HFBS intensity is increased to a level that produces nerve block, this could be followed by a linear reduction as suggested by this study to reach a final HFBS intensity to maintain the nerve block for a longer time. A similar nerve block method has been used in a previous animal study [19] to reduce the initial nerve firing by first applying HFBS at a very high intensity and then linearly reducing the intensity to a lower level to maintain the nerve block. In this situation it is possible that the initial application of a very high intensity HFBS might have accelerated the increase in intracellular sodium concentration. Therefore, the blocking mechanism revealed by our modeling study (Fig. 4 and Fig. 5) might have played a role in the animal experiments [19] to allow a linear reduction of the HFBS blocking intensity. It is worth noting that the animal study [19] used 10 kHz with an initial stimulation intensity much higher than the acute block threshold, while this study used 1 kHz that is lower than the minimal frequency required to produce an acute block, i.e., acute block does not occur in current study. The significance of this modeling study lies in the discovery of a possible strategy to reduce HFBS intensity to a low maintenance level for nerve block, although additional animal and human studies are still needed to confirm this stimulation strategy.

In addition, this study also reveals a method to quickly reverse the block and avoid the post-stimulation block after a prolonged stimulation by partially reducing the stimulation intensity rather than completely turning off the stimulation. This method could find its clinical application in blocking a nerve that consists of both sensory and motor axons, so that the motor function could be restored quickly at the end of the sensory axon block such as in an acute sensory block application.

HFBS of 1 kHz frequency was analyzed at 18.5 °C in this study because previous animal and simulation studies [7–10] showed that 1 kHz HFBS was more effective than 10 kHz HFBS to induce a post-stimulation block. A previous animal study [20] also showed that 1 kHz HFBS could induce acute nerve block during the stimulation at room temperature. More experiments are needed to determine how the results presented in our study change at a higher frequency (10 kHz) or a higher temperature (37 °C).

In summary, this study reveals that long-duration HFBS can produce a post-stimulation block by increasing the intracellular sodium concentration. The HFBS-induced membrane potential oscillation can interact with the increased intracellular sodium concentration, resulting in a minimal HFBS intensity below which the block cannot be maintained. These results indicate a possibility to develop a new nerve block method that can reduce the HFBS blocking intensity and extend the battery life for implantable nerve stimulators used to block pain of peripheral origin.

Appendix

The total ionic current I_i,j at the j^th segment is described as follows:

$$I_{i,j}=I_{Na}+I_k+I_L+I_{Na-p}+I_{K-p}$$

I_Na is the sodium (Na⁺ current; I_K is the potassium (K⁺) current; I_L is the leak current; I_Na−p is the Na⁺ pump current. I_K−p is the K⁺ pump current.

The Na⁺ current at j^th segment is described as:

$$I_{Na}=G_{Na}{m}^{3}h\left(V_j-V_{Na}\right)$$

$$V_j=E_j-E_{rest}$$

$$V_{Na}=E_{Na}-E_{rest}$$

$$E_{Na}=\frac{R*T}{F}ln\frac{N{a}_{ps}}{N{a}_{in}}$$

where G_Na is the maximal Na⁺ conductance (120 kΩ⁻¹ cm⁻²); m is the activation of Na⁺ channel; h is the inactivation of Na⁺ channel; V_j is the change of the membrane potential relative to the resting membrane potential E_rest; V_Na is reduced equilibrium membrane potential of Na⁺ ions after subtracting the resting membrane potential E_rest; E_Na is the Na⁺ equilibrium potential; R is the gas constant (8.3 J/(mol * K)); T is the temperature in Kelvin; F is the Faraday constant (96485 C/mol). Na_in is the Na⁺ concentration inside the axon with an initial value of 50 mmol/L; Na_ps is the Na⁺ concentration outside the axon in the periaxonal space with an initial value of 440 mmol/L that equals to the constant Na⁺ concentration in the large extracellular space (Na_o). The periaxonal space consists of a very thin layer of fluid (thickness of 1.45 × 10⁻² μm) that separates the axonal membrane from the large extracellular space [21–23]. Ions can diffuse between the periaxonal space and the large extracellular space where the ion concentrations are maintained constant.

The K⁺ current at j^th segment is described as:

$$I_K=G_K{n}^4\left(V_j-V_K\right)$$

$$V_K=E_K-E_{rest}$$

$$E_K=\frac{R*T}{F}ln\frac{K_{ps}}{K_{in}}$$

where G_K is the maximal K⁺ conductance (36 kΩ⁻¹ cm⁻²); n is the activation of K⁺ channel; V_K is reduced equilibrium membrane potential of K⁺ ions after subtracting the resting membrane potential E_rest; E_K is the K⁺ equilibrium potential; K_in is the K⁺ concentration inside the axon with an initial value of 400 mmol/L; K_ps is the K⁺ concentration outside the axon in the periaxonal space with an initial value of 20 mmol/L that equals the constant K⁺ concentration in the large extracellular space (K_o).

At resting membrane potential (V_j = 0 mV), the total ionic current across the membrane (I_i,j) should be zero. Since at resting membrane potential I_Kp = −I_K, I_Nap = −I_Na, the leakage current at j^th segment is described as follows:

$$I_L=G_L*V_j$$

where G_L is the maximal conductance (0.3 kΩ⁻¹ cm⁻²) of the leakage current.

The resting membrane potential E_rest is described by modified Goldman equation as follows [22,24]:

$$E_{rest}=\frac{R*T}{F}ln\frac{r{K}_{ps}+eN{a}_{ps}}{r{K}_{in}+eN{a}_{in}}$$

where r is the Na-K pump ratio (Mullins and Brinley, 1969) and e is a constant (0.0566) representing the ratio of Na⁺ and K⁺ permeability.

$$r=c_{r}N{a}_{in}+d_r$$

where c_r and d_r are constants; In the squid axon c_r has a value of 0.05 (Mullins and Brinley, 1969) and d_r has a value of −1 since at initial resting state r = 1.5 [25] and Na_in = 50.

The currents of Na⁺ and K⁺ pumps are described as [22,26]:

$$I_{K-p}=-\frac{a}{{\left(1+b_1/K_{ps}\right)}^2\left(1+b_2/N{a}_{in}\right)}$$

$$I_{Na-p}=-r{I}_{K-p}$$

where a is a constant (0.0954). b₁ is the dissociation constant for K_ps, and b₂ is the dissociation constant for Na_in. The constants b₁ and b₂ are estimated to be 1 mmol/L and 30 mmol/L, respectively [26,27].

The evolution equations for m, h, n are described as following [14,22]:

$$\frac{dm}{dt}={\alpha }_m(1-m)-{\beta }_{m}m$$

$$\frac{dh}{dt}={\alpha }_h(1-h)-{\beta }_{h}h$$

$$\frac{dn}{dt}={\alpha }_n(1-n)-{\beta }_{n}n$$

$${\alpha }_m=\mathrm{\Phi }\cdot \frac{2.5-0.1\left({\text{V}}_{\text{j}}-ms\right)}{\exp \left(2.5-0.1{\text{V}}_{\text{j}}\right)-1}$$

$${\beta }_m=\mathrm{\Phi }\cdot 4\exp \left(-\frac{{\text{V}}_{\text{j}}-ms}{18}\right)$$

$${\alpha }_h=\mathrm{\Phi }\cdot 0.07\exp \left(-\frac{{\text{V}}_{\text{j}}-hs}{20}\right)$$

$${\beta }_h=\mathrm{\Phi }\cdot \frac{1}{\exp \left(3.0-0.1\left({\text{V}}_{\text{j}}-hs\right)\right)+1}$$

$${\alpha }_n=\mathrm{\Phi }\cdot \frac{0.1\left(1-0.1\left({\text{V}}_{\text{j}}-ns\right)\right)}{\exp \left(1-0.1\left({\text{V}}_{\text{j}}-ns\right)\right)-1}$$

$${\beta }_n=\mathrm{\Phi }\cdot 0.125\exp \left(-\frac{V_j-ns}{80}\right)$$

$$\mathrm{\Phi }=3^{(T-279.3)/10}$$

where ms, hs, ns are constants that equal 7.8, 3.1, 18.5, respectively. These small voltage shifts (ms, hs, ns) are used to produce a properly shaped action potential and allow the system to be stable at the initial resting potential (−60.08 mV). The initial values for m, h, n are 0.0204, 0.6988, and 0.1, respectively. Φ is the coefficient of temperature.

The evolution equations for Na_in, K_in, Na_ps, K_ps are described as follows [21,22]:

$$\frac{dN{a}_{in}}{dt}=\frac{-4\times \left(I_{Na}+I_{Na-p}\right)}{F\times d}$$

$$\frac{d{K}_{in}}{dt}=\frac{-4\times \left(I_K+I_{K-p}\right)}{F\times d}$$

$$\frac{{dNa}_{ps}}{dt}=\frac{\left({\text{I}}_{\text{Na}-\text{p}}+{\text{I}}_{\text{Na}}\right)/\text{F}-{\text{D}}_{\text{Na}}\left({\text{Na}}_{\text{ps}}-{\text{Na}}_{\text{o}}\right)}{\text{θ}}$$

$$\frac{d{K}_{ps}}{dt}=\frac{\left({\text{I}}_{\text{K}-\text{p}}+{\text{I}}_{\text{K}}\right)/\text{F}-{\text{D}}_{\text{K}}\left({\text{K}}_{\text{ps}}-{\text{K}}_{\text{o}}\right)}{\text{θ}}$$

where D_Na and D_K are diffusion constants for Na⁺ (0.1 μm/s) and K + (0.1 μm/s) to diffuse between the periaxonal space and the large extracellular space, respectively. θ (1.45 × 10⁻² μm) is the thickness of the periaxonal space [21,22].