The Role of Slow Potassium Current in Nerve Conduction Block Induced by High-Frequency Biphasic Electrical Current

Abstract

The role of slow potassium current in nerve conduction block induced by high-frequency biphasic electrical current was analyzed using a lumped circuit model of a myelinated axon based on Schwarz-Reid-Bostock (SRB) model. The results indicate that nerve conduction block at stimulation frequencies above 4 kHz is due to constant activation of both fast and slow potassium channels, but the block at stimulation frequencies below 4 kHz could be due to either anodal or cathodal DC block depending on the time of the action potiential arriving at the block electrode. When stimulation frequency was above 4 kHz, the slow potassium current was about 3.5 to 6.5 times greater than the fast potassium current at blocking threshold, indicating that the slow potassium current played a more dominant role than the fast potassium current. The blocking location moved from the node under the blocking electrode to a nearby node as the stimulation intensity increased. This simulation study reveals that in mammalian myelinated axons the slow potassium current probably plays a critical role in the nerve conduction block induced by high-frequency biphasic electrical current.

I. INTRODUCTION

Reversible nerve conduction block induced by high-frequency biphasic electrical current has many potential clinical applications, for example, alleviating chronic pain [1], stopping unwanted muscle movements (muscle spasms and spasticity) [2], and improving voiding efficiency [3]. Since high-frequency biphasic electrical stimulation causes less tissue damage than uniphasic stimulation due to electro-chemical reactions [4], it is a very promising approach for long term clinical application. Although many experiments [2],[3],[5]–[12] have been performed recently to investigate this type of nerve conduction block, the underlying mechanisms are still uncertain.

It has been known for more than 60 years that high-frequency biphasic electrical current could block nerve conduction [8],[9]. However, due to the electrical noise induced by the high-frequency blocking stimulation, it is very difficult to investigate the blocking mechanisms in animal experiments using traditional electrophysiological methods. Therefore, recent studies [2],[12]–[18] have focused on computer simulation using axonal models. Our previous studies [14]–[18] using both amphibian and mammalian axon models indicated that the fast potassium current might play a role in nerve conduction block induced by high-frequency biphasic electrical current. However, there is a significant difference in nodal potassium current between mammalian and amphibian axons. The node of amphibian (frog) myelinated axons has a large fast potassium current [19], whereas in mammalian myelinated axons the slow potassium current is significantly larger than the fast potassium current [20],[21]. The role of slow potassium channels in the conduction block of mammalian myelinated axons has not been investigated.

This study used the Schwarz-Reid-Bostock (SRB) axonal membrane model, which was derived from human axonal data and incorporates both fast and slow potassium channels. The goal of this study was to further examine the mechanisms underlying nerve conduction block induced in mammalian myelinated axons by high-frequency biphasic electrical current with a particular focus on the possible role of slow potassium current.

II. METHODS

The nerve model used in this study is shown in Fig. 1. A 60 mm long myelinated axon is modeled with an inter-node length Δx = 100d (where d is the axon diameter) and d = 0.7D (where D denotes the axon external myelin diameter). The nodal length is denoted by L. Each node of the axon is modeled by a membrane capacitance (C_m) and a variable membrane resistance (R_m). R_a is the inter-node axoplasm resistance. The ionic currents passing through the variable membrane resistance are described by the SRB model (see Appendix and [22]). Two monopolar point electrodes (with the indifferent electrode at infinity) are placed at 1 mm distance to the axon (Fig. 1). One is the block electrode at the 30 mm location along the axon, where the high-frequency biphasic rectangular pulses is delivered (as shown in Fig. 1). The other is the test electrode at the 10 mm location, which delivers a uniphasic single pulse (pulse width 0.1 ms and intensity 3 mA) to evoke an action potential and test whether this action potential can propagate through the site of the block electrode. The test electrode is always a cathode (negative pulse), and the block electrode always delivers biphasic square pulses with the cathodal phase first (see Fig. 1).

Fig. 1.

Myelinated axonal model used to simulate conduction block induced by high-frequency biphasic electrical current. The inter-node length Δx = 100d; d is the axon diameter. L is the nodal length. Each node is modeled by a resistance-capacitance circuit based on the SRB model. R_a: inter-nodal axoplasmic resistance; R_m: nodal membrane resistance; C_m: nodal membrane capacitance; V_{i, j}: intracellular potential at the j^th node; V_{e, j}: extracellular potential at the j^th node.

We assume that the axon is in an infinite homogeneous medium (extracellular resistivity ρ_e= 0.3 k Ω·cm). After neglecting the small influence induced by the presence of the axon in the homogeneous medium, the extracellular potential V_{e, j} at the j^th node along the axon can be calculated by:

$$V_{e,j}(t)=\frac{{\rho }_e}{4\pi }\left[\frac{I_{\mathit{block}}(t)}{\sqrt{{(j\mathrm{\Delta }x-x_0)}^2+z_0^2}}+\frac{I_{\mathit{test}}(t)}{\sqrt{{(j\mathrm{\Delta }x-x_1)}^2+z_1^2}}\right]$$

where I_block (t) is the high-frequency biphasic pulse current delivered to the block electrode (at location x₀ = 30 mm, z₀ = 1 mm); I_test (t) is the single test pulse delivered to the test electrode (at location x₁ = 10 mm, z₁ = 1 mm).

The change of the membrane potential V_j at the j^th node is described by:

$$\frac{d{V}_j}{dt}=\frac{1}{c_m}\left[\frac{d}{4{\rho }_{i}L\mathrm{\Delta }x}(V_{j-1}-2V_j+V_{j+1}+V_{e,j-1}-2V_{e,j}+V_{e,j+1})-i_{i,j}\right]$$

where V_j = V_{i, j} − V_{e, j} − V_rest; V_{i, j} is the intracellular potential at the j^th node; V_{e, j} is the extracellular potential at the j^th node; V_rest is the resting membrane potential; c_m is the membrane capacitance per unit area; ρ_i is the intracellular axoplasm resistivity; i_{i, j} is the ionic current density at the j^th node described by SRB equations (see Appendix and [22]).

The SRB model was derived from the total nodal current instead of current density. In order to normalize the model, the axon diameter used to develop SRB model needs to be estimated, which was not given in the original SRB model [22]. It is known that the relationships between axonal diameter and the duration and conduction velocity of action potential is affected by temperature [23]. The duration of action potential recorded at 25 °C is 1.4 ms in SRB model [22]. According to Paintal’s experimental data [23], the conduction velocity v could be estimated to be 64 m/s in SRB model. Based on v = D ×5.7×10⁶ given by Boyd and Kalu [24], the estimated axon external myelin diameter D is 11.23 μm in SRB model. Therefore, the axon diameter d = 0.7D = 7.86 μm was used in this study to convert the parameters in original SRB model to the values per unit area (see Table 1).

Table 1.

The parameters used in the SRB model

Symbol	Description	Value
Vrest (mV)	resting membrane potential	−84
EK (mV)	potassium equilibrium potential	−84
EL (mV)	unspecific ion equilibrium potential	−84
gKf (mS/cm2)	fast potassium conductance per unit area	60.75
gKs (mS/cm2)	slow potassium conductance per unit area	121.51
gL (mS/cm2)	leakage conductance per unit area	121.51
cm (μF/cm2)	membrane capacitance per unit area	5.67
PNa (cm/s)	sodium permeability	0.01426
[Na]o (mmole/l)	extracellular sodium concentration	154
[Na]i (mmole/l)	intracellular sodium concentration	35
L (cm)	nodal length	1.0*10−4
F (C/mole)	Faraday constant	96485
R (mJ/K/mole)	gas constant	8314.4
ρe(kΩ·cm)	extracellular resistivity	0.3
ρi (kΩ·cm)	intracellular resistivity	0.11
T (Kelvin)	temperature	310.15

The axonal model was solved by the Runge-Kutta method [25] with a time step of 0.001 ms. The simulation was always performed with the initial condition V_j = 0 and the temperature T = 37 °C. The membrane potentials at the two end nodes of the modeled axon were always equal to the membrane potentials of their closest neighbors, which implemented sealed boundary conditions (no longitudinal currents) at the two ends of the modeled axon.

III. RESULTS

A. Nerve conduction block induced by high-frequency biphasic electrical current

The SRB model successfully simulated the conduction block induced by high-frequency biphasic electrical current. As an example, Fig. 2 shows a typical nerve firing pattern and conduction block at different stimulation intensities. The locations of the test and block electrodes are marked by short arrows along the axon (see Fig. 2A). The single test pulse and the high-frequency biphasic blocking pulses are schematically plotted on the side in Fig. 2A to show the timing of the two different stimulations. With the stimulation intensity below the block threshold as shown in Fig. 2A, the high-frequency blocking stimulation generated an initial action potential (at location 3 cm) that propagated in two directions. Then, the high-frequency stimulation alternatively depolarized and hyperpolarized the axon membrane without generating any action potential. At the time of 30 ms, an action potential was initiated by the test electrode (at location 1 cm) and it propagated through the site of the block electrode (at location 3 cm). When the intensity of high-frequency stimulation was increased above the block threshold as shown in Fig. 2B, the propagation of action potential evoked by the test electrode was blocked. However, further increasing the intensity of high-frequency stimulation caused repetitive firing as shown in Fig. 2C, although the high-frequency stimulation induced only an initial action potential at other intensities (Fig. 2A, B, and D). This repetitive firing disappeared and nerve conduction could be blocked again if the stimulation intensity increased further (Fig. 2D).

Fig. 2.

Propagation of action potentials along an axon induced by high-frequency biphasic stimulation at different intensities. The short arrows mark the locations of test and block electrodes along the axon in each figure. Stimulation: 7 kHz. Axon diameter: 5 μm.

Fig. 3 shows the pattern of nerve conduction block and repetitive firing at different stimulation frequencies (1–10 kHz) and intensities (0–10 mA) for axons of different diameters (2 μm, 5 μm, 10 μm, and 20 μm). At low frequencies (< 4 kHz), the axon could fire repetitively over a large range of stimulation intensities, and nerve block only occurred in a narrow range of stimulation intensities. However, when the frequency was increased above 4 kHz, the nerve conduction could be blocked over a large range of stimulation intensities. The blocking threshold became lower as the axonal diameter increased.

Fig. 3.

Pattern of nerve block and repetitive firing at different stimulation frequencies and intensities for axons of different diameters. The dark areas represent the stimulation intensity ranges causing nerve block as shown in Fig. 2B or D. The hatched areas represent repetitive firing as shown in Fig. 2C. The white areas represent nerve conduction block failure as shown in Fig. 2A.

B. Mechanism of nerve conduction block

Three different blocking mechanisms were identified. At stimulation frequencies below 4 kHz, anodal or cathodal block occurred. However, at stimulation frequencies above 4 kHz the constant activation of potassium channels played a critical role in the nerve conduction block.

Fig. 4 shows that an anodal or a cathodal block occurred at a stimulation frequency of 1 kHz. The legend in Fig. 4B indicates the distance of each node from the block electrode. The node at 0 mm was under the block electrode. As shown in Fig. 4A the propagating action potential initiated by the test electrode arrived at the block electrode when the biphasic blocking current was in the anodal phase. The propagating action potential from the test electrode was marked by a “*”. The strong hyperpolarization induced by the anodal pulse under the blocking electrode caused the conduction failure of the action potential. In Fig. 4B, the propagating action potential arrived at the block electrode when the biphasic blocking current was in the cathodal phase. The strong depolarization at the block electrode induced by the cathodal pulse caused the conduction block of the action potential.

Fig. 4.

Propagation of membrane potentials near the block electrode when anodal (A) or cathodal (B) block occurs. The legend in B indicates the distance from the block electrode for both A and B. The thinnest dashed line (0 mm) corresponds to the node under the block electrode (i.e. at 30 mm location). The thickest solid line (4 mm) corresponds to the node 4 mm away from the block electrode (i.e. at 26 mm location). The anodal block (A) occurred at 1 mA stimulation intensity for a 10 μm axon. The cathodal block (B) occurred at 1.6 mA stimulation intensity for a 5 μm axon. Stimulation frequency: 1 kHz. The asterisks mark the propagating action potential from the test electrode.

At stimulation frequencies above 4 kHz, a completely different blocking mechanism was identified. Fig. 5A shows the nerve conduction block at a stimulation frequency of 7 kHz. The black frame in Fig. 5A circled the area where nerve block occurred, which included the data from nine consecutive nodes at distances 0–4 mm away from the block electrode. The area is enlarged and redrawn in Fig. 5B to show the details. The activity of the nearest five nodes to the block electrode are shown in Fig. 5C–J. The propagation of the action potential, as well as the sodium and potassium currents associated with the action potential were completely abolished at the node (0 mm) under the blocking electrode where axonal membrane was alternatively depolarized and hyperpolarized (Fig. 5C–F). As the action potential propagated toward the blocking electrode, the activation (m) of sodium channels became oscillatory at the node under the blocking electrode (Fig. 5G) while the inactivation (h) remained at a relatively constant level (Fig. 5H), resulting in a pulsed inward sodium current (Fig. 5D). Therefore, the sodium channels were never completely blocked when conduction block occurred. The activation of both fast (n) and slow (s) potassium channels became constant at the node under the block electrode (Fig. 5I and J) resulting in large pulsed outward fast and slow potassium currents (Fig. 5E and F). The large outward potassium currents opposed the large inward sodium current, which caused the node under the block electrode to become unexcitable leading to the block of nerve conduction. Therefore, the mechanism of conduction block at frequencies above 4 kHz was due to the activation of both fast and slow potassium channels under the blocking electrode. However, note that the maximal slow potassium current was much larger (about 3.5 times) than the fast potassium current at the node under the block electrode (Fig. 5E and F), which indicated that slow potassium current might be more critical than fast potassium current in blocking nerve conduction. For the purpose of discussion in this paper, this blocking mechanism is termed “potassium block”.

Fig. 5.

Propagation of membrane potential, ionic current, and activation/inactivation of ion channels near the block electrode when potassium block occurs. The legend in C indicates the distance from the block electrode (0 mm is under the block electrode). The propagation of membrane potential near the block electrode is shown in detail in B. Stimulation: intensity 2.2 mA, frequency 7 kHz. Axon diameter: 5 μm. The asterisks in C–F mark the propagating action potential from the test electrode, and its corresponding ionic currents.

The “potassium block” could occur not only at the node under the block electrode but also at nodes adjacent to the block electrode as the stimulation intensity increased. As shown in Fig. 6, at a low stimulation intensity (2.2 mA) the conduction block occurred at the node under the block electrode at the 30 mm location (Fig. 6A), but at a high stimulation intensity (8 mA) the conduction block occurred at the node 2 mm away from the block electrode (i.e., at the 28 mm location, see Fig. 6B). This is due to the fact that at 8 mA current intensity the range of membrane potential oscillations at the 28 mm location was approximately equivalent to that at the 30 mm location when stimulation intensity was 2.2 mA. The activating function (Δ²V_{e, j}/Δx²) [26],[27] during the anodal or cathodal phase of the high-frequency blocking stimulation was also plotted in Fig. 6 for stimulation intensities of 2.2 mA and 8 mA respectively. A positive value of the activating function depolarizes the axonal membrane, whereas a negtive value hyperpolarizes the membrane [26],[27]. It can be seen that the range between the anodal and cathodal activating functions at the 28 mm location (i.e. the “side lobe”) when stimulation intensity is 8 mA (Fig. 6B) is about the same as the range at the 30 mm location when stimulation intensity is 2.2 mA (Fig. 6A). This causes the node at the 28 mm location to be alternatively depolarized and hyperpolarized by the 8 mA stimulation to the same extent as the node at the 30 mm location during 2.2 mA stimulation. The “side lobe” of the activating function explains why at a high stimulation intensity the block location moves to the node adjacent to the block electrode.

Fig. 6.

Block locations and activating functions at 2.2 mA (A) and 8 mA (B) stimulation intensities. The upper trace showing the membrane potentials at different nodes shares the same horizontal axis as the lower trace showing the activating functions. The block electrode is located at 30 mm. The activating function for both anodal and cathodal pulses are shown. A: Nerve conduction was blocked at the node under the block electrode. B: The block occurred at the node 2 mm away (at 28 mm) from the block electrode. Stimulation frequency: 7 kHz. Axon diameter: 5 μm.

C. The role of slow potassium current

Figs. 7A–D show that at different frequencies and intensities the maximal activation of fast potassium channels is almost the same (Fig. 7A and B), whereas the activation of slow potassium channels is increased as the frequency decreases (Fig. 7C) or as the intensity increases (Fig. 7D). The conduction block occurs only when the activation of slow potassium channels is above 0.56 (Fig. 7C and D). Fig. 8 is similar to Fig. 7 except that the axonal diameter is 20 μm instead of 5 μm. Again, the activation of fast potassium channels has almost the same maximal values for different frequencies (Fig. 8A) or intensities (Fig. 8B), but the activation of slow potassium channels is increased as the intensity increases (Fig. 8D). However, it is different from the 5 μm axon (see Fig. 7C) that the activation of slow potassium channels in the 20 μm axon is increased as the frequency increases (Fig. 8C). This difference is consistent with the decrease of the blocking threshold as the frequency increases as shown in Fig. 3D, whereas the blocking threshold increases as the frequency increases as shown in Fig. 3A. The fact that the slow potassium channels have to be activated above a certain level to induce a block (see Fig. 7C–D and Fig. 8C–D) indicates that the slow potassium channels may play a critical role in nerve conduction block. This is further supported by the observation that at different blocking thresholds the activation of slow potassium channels remains in a relatively narrow range although the frequency and intensity are very different (Fig. 9A), while the inactivation of sodium channels is oscillating around 0.1 and can be further inactivated when action potential arrives (Fig. 9B).

Fig. 7.

Activation of fast (n, in A and B) and slow (s, in C and D) potassium channels at different stimulation frequencies (A and C) and intensities (B and D) for the node under the block electrode. In A and C, the stimulation intensity is 2 mA. In B and D, the stimulation frequency is 7 kHz. The parameters in the legend enclosed by a square box are above the blocking threshold. Axon diameter: 5 μm.

Fig. 8.

Activation of fast (n, in A and B) and slow (s, in C and D) potassium channels at different stimulation frequencies (A and C) and intensities (B and D) for the node under the block electrode. In A and C, the stimulation intensity is 1 mA. In B and D, the stimulation frequency is 7 kHz. The parameters in the legend enclosed by a square box are above the blocking threshold. Axon diameter: 20 μm.

Fig. 9.

Activation of slow potassium channels (A) and inactivation of sodium channels (B) at the node under the block electrode for different blocking thresholds. The legend in A shows the intensity thresholds at different frequencies. Axon diameter: 5 μm.

Based on the SRB model (see Appendix and [22]), the ratio between slow and fast potassium currents is i_Ks/i_Kf = 2s/n⁴. Further quantitative analysis of fast and slow potassium currents shows that at blocking thresholds the activation of fast potassium channels (n) has a range of about 0.65 to 0.75, and the activation of slow potassium channels (s) has a range of about 0.55 to 0.57 for a 5 μm axon. Therefore, it can be estimated that the ratio (2s/n⁴) between slow and fast potassium currents is about 3.5 to 6.5 at blocking thresholds (see Fig. 10). This indicates a dominant role of the slow potassium current over the fast potassium current in nerve conduction block.

Fig. 10.

Contribution of fast (n) and slow (s) potassium currents to nerve conduction block at the blocking threshold level. The single solid line represents the n⁴. The double solid lines indicate the range of fast potassium activation at blocking thresholds and the corresponding range for n⁴. The single dashed line represents the 2s. The double dashed lines indicate the range of slow potassium activation at blocking thresholds and the corresponding range for 2s. At blocking thresholds, 2s/n⁴ is about 3.5 to 6.5 indicating that slow potassium current is more dominant than fast potassium current. Axon diameter: 5 μm.

IV. DISCUSSION

This simulation study employing a mammalian myelinated axonal model based on SRB equations investigated the nerve conduction block induced by high-frequency biphasic electrical current. Several possible mechanisms underlying nerve block were identified including anodal block (Fig. 4A), cathodal block (Fig. 4B), and potassium block (Fig. 5). At stimulation frequencies below 4 kHz anodal or cathodal block occurred due to membrane hyperpolarization or depolarization, but at stimulation frequencies above 4 kHz the constant activation of both slow and fast potassium channels played a critical role in blocking nerve conduction. When stimulation frequency is above 4 kHz, more slow potassium current (about 3.5–6.5 times the fast potassium current) was induced at blocking thresholds, indicating that in mammalian myelinated axons the slow potassium current played a more prominent role in nerve conduction block than the fast potassium current.

The block location moved from the node under the block electrode to a nearby node as stimulation intensity increased due to the “side lobe” effect of the activating function (see Fig. 6B). The blocking stimulation also induced repetitive firing of action potentials at the “side lobe” region at stimulation intensities above the blocking threshold for stimulation frequencies from 4 kHz to 10 kHz in a 5 μm axon as shown in Fig. 3B. This “side lobe” induced firing divided the blocking intensity range into two parts (see Fig. 3B). The lower intensity range produced nerve block under the blocking electrode, but the higher intensity range blocked nerve conduction at the “side lobe” region close to the blocking electrode. The “side lobe” induced firing was not seen to divide the blocking intensity range in other axons of different diameters (see Fig. 3A, C, and D). This indicates that in a compound nerve consisting of many axons of different diameters the blocking stimulation might block the majority of axon fibers, but cause a small percent of axons to fire repeatively. Only a monopolar blocking electrode is investigated in this study. Different geometry of the blocking electrode (monopolor, bipolor, or tripolor) will produce a significantly different shape of the activating function [26],[27]. This may result in a “side lobe” effect that is very different from what is shown in Fig. 6. Therefore, the geometry of the blocking electrode needs to be considered when applying high-frequency biphasic stimulation to block nerve conduction. It is also worth noting that the blocking electrode is positioned on top of the anxonal node at a fixed distance (1 mm) and the electrical properties of the internodal region are neglected in this study. Changing the electrode position or incoporating the internodal region into the model could influence the blocking thresholds resulting in a shift of the blocking intensity range in Fig. 3. However, the minimal blocking frequency as shown in Fig. 3 and the blocking mechanisms identified in Fig. 4–5 should remain the same. Since a small time step of 0.001 ms was used in this simulation study, the numerical error of the calculated blocking threshold was estimated to be less than 0.1 mA.

The node of amphibian (frog) myelinated axons has mainly a fast potassium current [19], whereas mammalian myelinated axons have both fast and slow potassium currents, and the slow potassium current is dominant over the fast potassium current [20],[21]. Our previous simulation studies [16],[17] using the Frankenhaeuser-Huxley model [19] based on amphibian axons showed that the fast potassium current played a critical role in nerve conduction block. However, this study using a mammalian myelinated axonal model showed that the slow potassium current had a more important role than the fast potassium current. Although previous studies [2],[3],[5]–[12] in cats, rats, and frogs all reported nerve conduction block at stimulation frequencies above 4 kHz, the underlying mechanisms may be very different for amphibian and mammalian.

The results obtained in this study need to be further confirmed by both animal studies and simulation analysis. Simulation analysis with other axonal models may further verify the role of slow potassium currents in nerve conduction block. The MRG model [28] also incorporates slow and fast potassium currents. However, a previous study [13] employing the MRG model did not investigate the role of slow potassium current in nerve conduction block. Therefore, a simulation study based on MRG model should be performed to further verify the role of slow potassium channels. It is worth noting that this study used a McNeal type axonal model that did not incorporate a detailed representation of the internodal region. However, this study and previous studies using McNeal type models [16]–[18] have sucessfully simulated the nerve conduction block phenomena observed in animal studies [2],[3],[5]–[11]. They also produced very similar results as the study [13] using MRG model that incorporated a detailed representation of the internodal region, indicating that the internodal region might not be involved in the possible mechanisms underlying the nerve conduction block induced by high-frequency biphasic electrical stimulation. Simulation studies using more complex axon models [29]–[31] that incorporate more realistic extracellular space, glial buffering, and ion pumps may further reveal the role of potassium channels in nerve conduction block.

Previous studies [32]–[34] in rat hippocampal slices showed that the neuronal epileptiform activity and axonal conduction could be blocked by high-frequency (<500 Hz) sinusoidal electrical field stimulation. The block was always coincident with a stimulus-induced rise in extracelluar potassium concentration, suggesting the opening of the potassium channels and potassium outflow from the neurons/axons during the stimulation. The stimulation frequency to block the hippocampal neuron/axon is relatively low (<500 Hz) compared to the minimal stimulation frequency required to block the conduction of peripheral nerve (1–10 kHz) [2],[3],[5]–[12]. However, this frequency discrepancy might be caused by the slow membrane dynamics of the hippocampal neuron/axon. The duration of the action potential generated in neurons can be several milliseconds long [35], but it is less than 1 ms in mammalian myelinated axons of peripheral nervous system. Our previous simulation study [18] has shown that a lower stimulation frequency is required to block the nerve when the membrane dynamics become slower. Whether the slow potassium channels also play an important role in blocking the hippocampal neuron/axon needs to be further investigated.

A reversible nerve conduction block method will find many applications in both clinical medicine and basic neuroscience [1]–[11]. Understanding the biophysics and mechanisms underlying the nerve conduction block induced by high-frequency biphasic electrical current could promote its clinical application and possibly the design of new stimulation waveforms using less current to block nerve conduction. Simulation analysis using computer models provides a tool to reveal mechanisms underlying the nerve conduction block and may help to design new animal experiments to further test new nerve blocking methods.

APPENDIX: The SRB Model

The ionic current density i_{i, j} at the j^th node is described as:

$$\begin{array}{l}{i}_{i,j}=i_{Na}+i_{Kf}+i_{Ks}+i_L\\ i_{Na}=m^3{hP}_{Na}\frac{{EF}^2}{RT}\frac{{[Na]}_o-{[Na]}_{i}exp(EF/RT)}{1-exp(EF/RT)}\\ i_{Kf}=n^4{g}_{Kf}(E-E_K)\\ i_{Ks}={sg}_{Ks}(E-E_K)\\ i_L=g_L(E-E_L)\end{array}$$

The evolution equations for variables m,h,n and s are:

$$\begin{array}{l}dm/dt=[{\alpha }_m(1-m)-{\beta }_{m}m]k_m\\ dh/dt=[{\alpha }_h(1-h)-{\beta }_{h}h]k_h\\ dn/dt=[{\alpha }_n(1-n)-{\beta }_{n}n]k_n\\ ds/dt=[{\alpha }_s(1-s)-{\beta }_{s}s]k_s\end{array}$$

where

$$\begin{array}{ll}{\alpha }_m=\frac{1.86(E+18.4)}{1-exp[(-18.4-\text{E})/10.3]}\hfill & {\alpha }_n=\frac{0.00798(E+93.2)}{1-exp[(-93.2-\text{E})/1.1]}\hfill \\ {\beta }_m=\frac{0.086(-22.7-E)}{1-exp[(E+22.7)/9.16]}\hfill & {\beta }_n=\frac{0.0142(-76.0-E)}{1-exp[(E+76.0)/10.5]}\hfill \\ {\alpha }_h=\frac{0.0336(-111.0-E)}{1-exp[(E+111.0)/11.0]}\hfill & {\alpha }_s=\frac{0.00122(E+12.5)}{1-exp[(-12.5-\text{E})/23.6]}\hfill \\ {\beta }_h=\frac{2.30}{1+exp[(-28.8-E)/13.4]}\hfill & {\beta }_s=\frac{0.000739(-80.1-E)}{1-exp[(E+80.1)/21.8]}\hfill \end{array}$$

and

$$\begin{array}{l}{k}_m={2.2}^{(T-293.15)/10}\\ k_h={2.9}^{(T-293.15)/10}\\ k_n={3.0}^{(T-293.15)/10}\\ k_s={3.0}^{(T-293.15)/10}\end{array}$$

where T is temperature (310.15 °K i.e. 37°C) and E = V_j +V_rest. The initial values for m,h,n and s are 0.0382, 0.6986, 0.2563, and 0.2011 respectively, which correspond to the initial condition V_j =0, or E = V_rest = −84 mV.