Model Analysis of Post-Stimulation Effect on Axonal Conduction and Block

Abstract

Objective

To reveal the possible contribution of changes in membrane ion concentration gradients and ion pump activity to axonal conduction/block induced by long-duration electrical stimulation.

Methods

A new model for conduction and block of unmyelinated axons based on the classical Hodgkin-Huxley (HH) equations is developed to include changes in Na+ and K+ concentrations and ion pumps. The effect of long-duration stimulation on axonal conduction/block is analyzed by computer simulation using this new model.

Results

The new model successfully simulates initiation, propagation, and block of action potentials induced by short-duration (multiple milliseconds) stimulations that do not significantly change the ion concentrations in the classical HH model. In addition, the activity-dependent effects such as action potential attenuation and broadening observed in animal studies are also successfully simulated by the new model. Finally, the model successfully simulates axonal block occurring after terminating a long-duration (multiple seconds) direct current (DC) stimulation as observed in recent animal studies and reveals 3 different mechanisms for the post-DC block of axonal conduction.

Conclusion

Ion concentrations and pumps play an important role in post-stimulation effects and activity-dependent effects on axonal conduction/block. The duration of stimulation is a determinant factor because it influences the total charges applied to the axon, which in turn determines the ion concentrations inside and outside the axon.

Significance:

Despite recent clinical success of many neurostimulation therapies, the effects of long-duration stimulation on axonal conduction/block are poorly understood. This new model could significantly impact our understanding of the mechanisms underlying different neurostimulation therapies.

I. Introduction

With recent advances in electronics and computer technology, fully implantable stimulators are now applying electrical stimulation chronically to the brain, spinal cord, and peripheral nerves to treat different neurological disorders including Parkinson disease [1], chronic pain [2], or urinary dysfunctions [3]. However, the effects of electrical stimulation on axonal conduction are poorly understood, which impedes research aimed at elucidating the neural mechanisms underlying different types of neurostimulation therapies.

It is well known that the initiation, propagation, and block of an action potential are dependent on changes in the ion channels of the axonal membrane, the ion concentrations inside and outside the axon, and the ion pumps in the axonal membrane. Classical axonal conduction models are very successful in analysis of nerve responses to short-duration stimulation that only lasts for several milliseconds [4–7], but they are not suitable for analysis of nerve responses to long-duration stimulation that lasts for seconds, minutes, or hours. These classical models are based on Hodgkin-Huxley (HH) equations [8] that assume the ion concentrations are not changed significantly within several milliseconds. Therefore, the ion concentrations and ion pumps are not included in these models [4–7]. However, it is expected that long-duration stimulation can significantly change the ion concentrations inside and outside the axon and elicit prolonged block of axonal conduction. This requires ion pumps to slowly restore the ion concentrations and reestablish normal axonal conduction. Therefore, to better understand the effects of electrical stimulation in various clinical applications where the stimulation is applied for a long duration, an axonal conduction model that includes the ion concentrations and ion pumps is needed.

In this study, we developed an axonal conduction model for unmyelinated axons based on the classical HH model but included estimates of changes in membrane ionic gradients and ion pump activity. This new model successfully simulated the initiation, propagation, and block of action potentials generated by both short-duration (multiple milliseconds) and long-duration (multiple seconds) of electrical stimulation, revealing a critical role of ion concentrations and ion pumps in axonal responses to long-duration electrical stimulation or block. This axonal conduction model will be very useful for a better understanding of nerve responses to different types of electrical stimulation that are applied chronically to treat many neurological disorders [1–3].

II. Methods

The axonal conduction model developed in this study is shown in Fig.1. A 20-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 electrodes at infinity) are placed at 1 mm distance from the unmyelinated axon. One is the test electrode at the 4 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 12 mm location along the axon. The test electrode is always cathodal (negative pulse). The block electrode always delivers cathodic direct current (DC) to block the propagation of the action potential generated by the test electrode.

Fig.1.

Unmyelinated axon model including sodium and potassium pumps to simulate conduction block induced by direct current (DC). The unmyelinated axon is segmented into many small cylinders of length Δx, each of which is modeled by a resistance-capacity 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.

We assume 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,n, at the n^th segment along the axon can be calculated by:

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

where I_block(t) is the DC stimulation delivered to the block electrode (at location x₀ = 12 mm, z₀ = 1 mm); I_test(t) is the single test pulse delivered to the test electrode (at location x₁ = 4 mm, z₁ = 1 mm).

The membrane potential E_n at the n^th segment of the unmyelinated axon is calculated by:

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

where E_n = V_a,n − V_e,n; V_a,n is the intracellular potential at the n^th segment; V_e,n is the extracellular potential at the n^th segment; d is the unmyelinated axon diameter; ρ_i is the resistivity of axoplasm (34.5 Ωcm); c_m is the capacitance of the membrane (1 uF/cm²); I_i,n is the total ionic current at the n^th segment described by the modified HH equations [4–9] as follows:

$$I_{i,n}=I_{\mathit{\text{Na}}}+I_k+I_L+I_{\mathit{\text{Na}}-p}+I_{K-p}$$

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 n^th segment is described as:

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

$$V_n=E_n-E_{\mathit{\text{rest}}}$$

$$V_{\mathit{\text{Na}}}=E_{\mathit{\text{Na}}}-E_{\mathit{\text{rest}}}$$

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

where G_Na is the maximal Na⁺ conductance (120 kΩ⁻¹cm⁻²); m is the activation of Na⁺ channel; h is the inactivation of Na⁺ channel; V_n 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_sp is the Na⁺ concentration outside the axon in the periaxonal space with an initial value of 440 mmol/L that equals 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 [9]. Ions can diffuse between the periaxonal space and the large extracellular space where the ion concentrations are maintained as constant.

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

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

$$V_K=E_K-E_{\mathit{\text{rest}}}$$

$$E_K=\frac{R*T}{F}\mathit{\text{ln}}\frac{K_{\mathit{\text{sp}}}}{K_{\mathit{\text{in}}}}$$

where G_K is the maximal K⁺ conductance (36 kΩ⁻¹cm⁻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_sp 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).

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

$$I_L=G_L*V_n$$

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

The resting membrane potential E_rest is described by modified Goldman equation as follows [9–10]:

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

where r is the Na-K pump ratio [11] and e is a constant (0.0566) representing the ratio of Na⁺ and K⁺ permeability.

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

where c_r and d_r are constants; In the squid axon c_r has a value of 0.05 [11] and d_r has a value of −1 since at initial resting state where r = 1.5 [12] and Na_in = 50.

The currents of Na⁺ and K⁺ pumps are described as [9, 11]:

$$I_{K-p}=-\frac{a}{\underset{I_{\mathit{\text{Na}}-p}=-r{I}_{K-p}}{{\left(1+b_1/K_{\mathit{\text{sp}}}\right)}^2\left(1+b_2/N{a}_{\mathit{\text{in}}}\right)}}$$

where a is a constant (0.0954). b₁ is the dissociation constant for K_sp, 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 [11, 13].

The evolution equations for m, h, n are described as following [8–9]:

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

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

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

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

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

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

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

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

$${\beta }_n=\Phi \cdot 0.125\text{exp}\left(-\frac{\left({\text{V}}_n-\mathit{\text{ns}}\right)}{80}\right)$$

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

where ms, hs, ns are constants that equal 7.8, 3.1, 18.5, respectively. 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_sp, K_sp are described as follows [9, 14]:

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

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

$$\frac{dN{a}_{\mathit{\text{sp}}}}{dt}=\frac{\left({\text{I}}_{\text{Na}-\text{p}}+{\text{I}}_{\text{Na}}\right)/\text{F}-{\text{D}}_{\text{Na}}\left({\text{Na}}_{\text{sp}}-{\text{Na}}_{\text{o}}\right)}{\theta }$$

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

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 × 10⁻²μm) is the thickness of the periaxonal space [9, 14].

The model equations were solved by Runge-Kutta method [15] with a time step of 10 μsec for an axon of 2 μm diameter at the temperature T = 291.5 K (18.5 °C). The simulation was always started at initial condition V_n = 0 mV or E_n = E_rest = −60.08 mV. The membrane potentials at the two end segments 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. Initiation and propagation of action potential

The axonal conduction model successfully simulates the initiation and propagation of the action potential (Fig.2). As shown in Fig.2A, an action potential is initiated by a test pulse (6.5 mA, 0.1 ms) that is applied at the 4 mm location and then propagates in both directions along the 20-mm long axon. The details of this action potential propagating through the 12 mm location are presented in Figs.2 B–H. The action potential (Fig.2B), the Na⁺ and K⁺ ionic currents (Fig.2C) and activation/inactivation of Na⁺ and K⁺ channels (Fig.2D) are similar to the simulation results produced by classical HH models [4–8]. The action potential shifts the resting potential from −60.08 mV to −58.69 mV following the action potential (Fig.2E) mainly by increasing the K⁺ concentration and decreasing the Na⁺ concentration in the periaxonal space (Fig.2G) because the changes in Na⁺ and K⁺ concentrations inside the axon by a single action potential are minimal (Fig.2F). The Na⁺ and K⁺ pumps are not activated and remain at their resting state (Fig.2H) due to the small changes in Na⁺ and K⁺ concentrations. The relatively large changes in Na⁺ and K⁺ concentrations in the periaxonal space by a single action potential is due to the fact that the thickness (1.45 × 10⁻² μm) of the periaxonal space is much smaller compared to the diameter of the axon (2 um).

Fig.2.

A. Propagation of an action potential induced by a test pulse (0.1 ms pulse width, 6.5 mA intensity) that is applied at 4 mm location and at the time of 1 ms. B-H. Detail presentations of the change of membrane potential (B), ionic currents (C), activation and inactivation of Na⁺ and K⁺ channels (D), resting potential before and after the action potential (E), ion concentrations of Na⁺ and K⁺ inside (F) and outside the axon in the periaxonal space (G), and ion pump currents (H) at the 12 mm location of the axon as indicated by the arrow in A. m - Na⁺ channel activation; h - Na⁺ channel inactivation; n - K⁺ channel activation.

B. Refractory period and repetitive firing of action potentials

The refractory period is determined by applying double test pulses of 0.1 ms pulse width (Fig.3). After applying the first test pulse at threshold (T = 6.5 mA) intensity that initiates an action potential, the second test pulse at 1.3T intensity and applied at 6 ms after the first test pulse fails to initiate another action potential (Fig.3A). However, the second test pulse successfully induces the second action potential when its intensity increases to 1.4T (Fig.3B), indicating a refractory period after the first test pulse that increases the threshold for the second test pulse to induce an action potential. Fig.3C shows the absolute and relative refractory periods. The threshold intensity during the refractory period is measured with a resolution of 0.1T. The absolute refractory period is defined as the second test pulse fails to initiate an action potential at intensity of 4T.

Fig.3.

Refractory period determined by double pulse testing. A. The first pulse is applied at the intensity threshold (T) to induce an action potential. The second pulse at intensity 1.3T fails to induce an action potential when it is applied at 6 ms after the first pulse. Pulse width = 0.1 ms; T = 6.5 mA. B. However, the second pulse successfully induces an action potential when the intensity increases to 1.4T. C. The absolute and relative refractory period determined by the double pulse testing. The absolute refractory period is defined as the second pulse fails to induce an action potential at intensity 4T.

A 300-ms train of high frequency (247 Hz) stimulus pulses (0.1 ms) at a super-threshold intensity (2T = 13 mA) can generate action potentials in response to every pulse at the test electrode (4 mm location, Fig.4A) and every action potential can propagate along the axon to the 12 mm location (Fig.4B). However, when the frequency is increased to 248 Hz, one action potential at the end of the 300-ms train is lost (Fig.4C). Therefore, the highest firing frequency during a 300-ms testing period is 247 Hz which is very close to the estimation of 250 Hz based on the absolute refractory period of 4 ms (Fig.3C). It is noted that the amplitude of the action potentials gradually attenuates with time while the width of the action potential is broadened (Fig.4B and Fig.4G). These changes are due to the cumulative changes in Na⁺ and K⁺ concentrations both inside and outside the axon (Fig.4E and Fig.4F), which results in the resting potential changing from −60.08 mV to about −45 mV (Fig.4D). The change in resting potential is mainly due to the changes in Na⁺ and K⁺ concentrations in the periaxonal space because the changes in Na⁺ and K⁺ concentrations inside the axon are relatively small when compared to those in the outside periaxonal space (Fig.4E vs Fig.4F). The changes in Na⁺ and K⁺ concentrations in the periaxonal space and the resting membrane potential plateau (Fig.4F) when the diffusion of Na⁺ and K⁺ ions between the periaxonal space and the large extracellular space reaches an equilibrium with the Na⁺ and K⁺ currents across the axonal membrane (Fig.4D).

Fig.4.

Repeated firing induced by high frequency (247 Hz or 248 Hz) stimulation. Pulse width = 0.1 ms; Intensity = 13 mA that is 2 times threshold (T). High frequency (247 Hz) stimulation induces repeated firing of action potentials that propagate from the test electrode at the 4 mm location (A) to the recording electrode at 12 mm location (B) without failure of any action potentials. However, one action potential is dropped after about 300 ms stimulation at 248 Hz (C). During the high frequency firing, the resting membrane potential (D) is gradually depolarized from −60 mV to about −45 mV due to the accumulative changes in ion concentrations both inside (E) and outside the axon (F), which results in the action potential attenuation and broadening (G).

C. Conduction block by direct current (DC)

Application of DC (3.5 mA) at the block electrode (12 mm location) produces a constant depolarization after initiating an action potential that propagates in both directions (Fig.5A). The constant depolarization (about 30 mV, Fig.5B) blocks conduction of the action potential induced by a test pulse (10 mA, 0.1 ms) that is applied at the test electrode (4 mm location) after 20 ms of DC application (Fig.5A). The details of the conduction block are shown in Fig.5 B–G. As the action potential is propagating toward to the block electrode, its amplitude is gradually reduced and finally blocked by the constant depolarization (Fig.5B). The Na⁺ current is very small at the block electrode (Fig.5C) because the constant depolarization causes the inactivation of the Na⁺ channel (h ≈ 0, Fig.5F) that blocks the axonal conduction. The K⁺ channel is constantly activated under the block electrode (Fig.5G), which results in a constant K⁺ current (Fig.5D).

Fig. 5.

Conduction block during direct current (DC). A. The DC (3.5 mA) generates an initial action potential propagating in both directions. The test pulse (10 mA, 0.1 ms) applied at the 4 mm location after 20 ms of DC generates another action potential that fails to propagate through the site of the DC block electrode at 12 mm location. B-G. Graphs showing the temporal changes in axonal ionic mechanisms at different axonal locations during DC conduction block (B), Na⁺ and K⁺ currents (C and D), and activation/inactivation of the ion channels (E, F, and G) near the block electrode as indicated in A. The legends in C indicate the locations where the 12 mm is under the block electrode.

The constant depolarization during a long-lasting (20 seconds) DC (Fig.6A) produces a constant K⁺ current (about 100 μA/cm²) and a very small constant Na⁺ current (Fig.6B) due to the activation/inactivation of the ion channels (Fig.6C). These constant ion currents over a long period (20 seconds) produce a large shift in resting potential from −60.08 mV to almost 0 mV (Fig.6D), which is due to the dramatic changes over time in Na⁺ and K⁺ concentrations inside (Fig.6E) and outside of the axon in the periaxonal space (Fig.6F). Due to the significant changes in Na⁺ and K⁺ concentrations, axonal conduction will be blocked if the DC is terminated at a time point during the 20-second application. Meanwhile, the Na⁺ and K⁺ pumps are also activated due to the large change in Na⁺ and K⁺ concentrations (Fig.6G) and will gradually restore the normal Na⁺ and K⁺ concentration gradients and reestablish axonal conduction.

Fig.6.

Axonal responses to long-lasting (20 secs) direct current (DC, 3.5 mA) at the site of DC block electrode. A. Change of membrane potential. B. Na⁺ and K⁺ currents. C. Activation/inactivation of the ion channels. D. Resting potential. E. Na⁺ and K⁺ concentrations inside the axon. F. Na⁺ and K⁺ concentrations outside the axon in the periaxonal space. G. Na⁺ and K⁺ ion pump currents.

Our simulation analysis identifies 3 different types of conduction block (Figs.7–9) that can occur after termination of the DC depending on the duration of DC application. If the DC is applied for a duration less than 6 seconds, the conduction block can only persist for 5 ms after termination of the DC when activation/inactivation of the ion channels are still recovering from the constant DC depolarization. A post-DC block is shown in Fig.7 for a 1-second long DC application. The action potential induced by a test pulse (10 mA, 0.1 ms, and applied at 994 ms) arrives at the block electrode (12 mm location) about 3 ms after termination of the 1-second (1000 ms) DC and it fails to propagate through the site of the block electrode (Fig.7A). However, the same test pulse that is applied 1 ms later (at 995 ms) induces an action potential that can successfully propagate through the site of the block electrode (Fig.7B). The details of the post-DC block are shown in Fig.7 C–H where the incomplete recovery of ion channel activation/inactivation under the block electrode (12 mm location, Fig.7 F–H) eliminates almost all Na⁺ and K⁺ currents (Figs.7 D–E), thereby producing block of the action potential (Fig.7C).

Fig.7.

Conduction block occurring 3 ms after termination of 1000-ms direct current (DC, 3.5 mA). A. The action potential induced by a test pulse (10 mA, 0.1 ms) at 994 ms arrives at the DC block electrode 3 ms after termination of the DC and it is blocked. B. The action potential propagates through the DC block electrode when the test pulse is delivered 1 ms later at 995 ms. C-H. Graphs showing the temporal changes in axonal ionic mechanisms at different axonal locations during DC conduction block (C), Na⁺ current (D), K⁺ current (E), and activation/inactivation of the ion channels (F, G, and H) near the block electrode as indicated in A. The legends in C indicate the locations where the 12 mm is under the block electrode.

Fig.9.

Conduction block occurring 3.64 seconds after termination of 18.35-second direct current (DC, 3.5mA). A-B: The 25-ms recording starts from 21.98 seconds, i.e. 3.63 seconds after terminating DC. The action potential induced by a test pulse (10 mA, 0.1 ms) at 2 ms is blocked (A), but it propagates through the DC block electrode when the test pulse is delivered 1 ms later (B). C-F. During the 5-second period after terminating the 18.35-second DC, the changes produced at the DC block site are rapidly reversed including the ion concentrations outside the axon in the periaxonal space (C), resting potential (D), and ion pump currents (F), but the ion concentrations inside the axon (E) are recovering slowly.

If the DC is applied for a duration longer than 6 seconds but less than 17 seconds, the post-DC block of conduction can last longer than 5 ms after termination of the DC (Fig.8). The action potential induced by a test pulse (10 mA, 0.1 ms) that is applied at 19 ms after termination of a 10-second long DC arrives at the block electrode (12 mm location) about 27 ms after termination of the DC and fails to propagate through the site of the block electrode (Fig.8A). However, the same test pulse that is applied 1 ms later (at 20 ms after the DC) induces an action potential that can successfully propagate through the site of the block electrode (Fig.8B). The details of the post-DC block are shown in Figs.8 C–H. Although ion channel activation/inactivation under the block electrode are fully recovered to their resting levels before the arrival of the action potential (Fig.8E), the membrane can only be depolarized to about 30 mV during the action potential (Fig.8C) due to the partial activation of the Na+ channels (Fig.8E) and the small sodium current (Fig.8D) that are caused by the significantly changed resting membrane potential (Fig.8F) and the changed ion concentrations (Fig.8 G–H). The recovery of axonal conduction after the 10-second long DC is dependent on the gradual recovery of the resting potential (Fig.8F) that is mainly determined by the recovery of ion concentrations outside the axon in the periaxonal space (Fig.8H) since the ion concentrations inside the axon recover at a much slower rate (Fig.8G). Therefore, the recovery of axonal conduction is dependent on the slow diffusion of ions between the periaxonal space and the large extracellular space.

Fig.8.

Conduction block occurring 27 ms after termination of 10-second direct current (DC, 3.5mA). A. The action potential induced by a test pulse (10 mA, 0.1 ms) at 19 ms after terminating DC is blocked. B. The action potential propagates through the DC block electrode when the test pulse is delivered 1 ms later at 20 ms. C-H. Detail presentations for the change of membrane potentials (C), Na⁺ and K⁺ currents (D and E), resting potential (F), ion concentrations inside (G) and outside the axon in the periaxonal space (H) at the block electrode as indicated in A.

If the DC is applied for a duration longer than 17 seconds, the post-DC block of conduction can last for much longer time after termination of the DC. Fig.9 shows a post-DC block occurring in a 25-ms recording period (Fig.9A). The recording starts from 3.63 seconds after termination of the 18.35-second long DC. The action potential induced by a test pulse (10 mA, 0.1 ms) that is applied at 2 ms during the recording fails to propagate through the site of the block electrode (Fig.9A). However, the same test pulse that is applied 1 ms later (at 3 ms) induces an action potential that can successfully propagate through the site of the block electrode (Fig.9B). The details of the post-DC block are shown in Figs.9 C–F. After termination of the DC, the ion concentrations outside the axon in the periaxonal space come to equilibrium with the large extracellular space by diffusion within about 0.5 seconds (Fig.9C), which results in a quick partial recovery of the resting potential (Fig.9D). Then, the resting potential starts a very slow recovery (not visible in the 5-ms period in Fig.9D) that follows the slow recovery of the ion concentrations inside the axon (Fig.9E) because ion pump currents are relatively small (Fig.9F). Therefore, the long-lasting post-DC block is mainly due to the changes in the ion concentrations inside the axon and the slow restoration of normal concentrations mediated by the ion pumps.

The 3 different types of post-DC block shown in Figs.7–9 result in 3 different segments in the curve showing the relationship of post-DC block duration to the DC duration (Fig.10). It is noted that there is a sharp increase in post-DC block duration once the block mechanism starts to involve the ion concentrations inside the axon and the ion pumps (Fig.10).

Fig.10.

The post-DC block duration increases with the DC duration. The 3 different mechanisms of post-DC block as shown in Figs.7–9 result in 3 segments in the relationship between post-DC block duration and DC duration.

IV. Discussion

This study establishes an axonal conduction model for unmyelinated axons that includes both ion concentrations and ion pumps (Fig.1). This new model successfully reproduces the initiation, propagation, and block of the action potential conduction (Figs.2–5) as described in previous reports using classical models that assume no change in ion concentrations [4–8]. In addition, this new model can also simulate post-stimulation effects on axonal conduction/block (Figs.7–10) induced by a long-duration stimulation that causes significant changes in ion concentrations (Fig.6). The key advantage of the new model over the classical models is the ability to analyze the effects induced by long-duration stimulation. This ability is important because most clinical applications use long-term electrical stimulation, which certainly changes the ion concentration both inside and outside the axons.

Animal studies in rats and guinea pigs [16–17] have determined that the unmyelinated axons in brain hippocampal area have an absolute refractory period of 2–6 ms and relative refractory period of 15–40 ms. Our model calculates an absolute refractory period of 4 ms and relative refractory period of 17 ms (Fig.3), which is in the range of animal data. The relative refractory period is mainly caused by the hyperpolarization after the action potential due to the slow recovery of potassium channel activation and the sodium channel inactivation (Fig.2D). The changes in ion concentrations by a single action potential are small, which only result in a small change in the resting potential (Fig.2 E–G). Therefore, the effect of ion concentration change by a single action potential on relative refractory period is minimal. Furthermore, the model shows that at high frequency (247 Hz) the amplitude of action potential is broadened, and the duration is widened (Fig.4G). The activity-dependent attenuation and broadening of the action potential are well documented by animal studies of unmyelinated axons in the rat hippocampal brain region [18–19]. The model further reveals that the activity-dependent effect on the action potential is mainly due to the change of ion concentrations outside the axon in the periaxonal space (Fig.4F) because the change in ion concentrations inside the axon is relatively small (Fig.4E) when compared to those occurring on the outside. The ion concentration changes in the periaxonal space determines the time course of the resting potential depolarization or repolarization (Fig.4D) that in turn determines the shape of the action potential (Fig.4G).

It is well known that axonal conduction can be blocked during the application of DC [20–22]. This new model successfully simulates the cathodic DC block (Fig.5) and reveals the same blocking mechanism as the classical model [23], i.e., the Na⁺ channels are completely inactivated during the DC (Fig.5 C and Fig.5F). In addition to the conduction block occurring during DC, the new model also simulates the conduction block after termination of the DC (Figs. 7–9) showing that the duration of post-DC block increases with the duration of DC application (Fig.10). Post-DC block has also been observed in recent animal studies using rat sciatic nerve [24–25] showing that the block duration increases with the duration and amplitude of the DC, i.e., the total charges delivered by the DC. This observation from animal studies agrees well with our simulation result (Fig.6) showing that the total charges delivered by the DC can significantly change the ion concentrations inside and outside the axon, which can cause conduction block after termination of the DC. Furthermore, our model analysis reveals three mechanisms underlying the post-DC block: 1. Block within 5 ms after termination of the DC is mainly due to the incomplete recovery of activation or inactivation of Na⁺ and K⁺ channels (Fig.7), i.e., the axon is still in the absolute refractory period (Fig.3); 2. During the period of 5–400 ms (Fig.10) after termination of the DC, the conduction block is mainly due to the recovery of ion concentrations in the periaxonal space (Fig.8F) because the activation/inactivation of Na⁺ and K⁺ channels is fully recovered during the post-DC period (Fig.8E); 3. The post-DC block lasting longer than 400 ms (Fig.10) is mainly due to the slow restoration of the ion concentrations inside the axon by the ion pumps (Fig.9) because the ion concentrations in the periaxonal space have fully recovered due to diffusion communication with the constant ion concentrations in the large extracellular space within 400 ms (Fig.9C). It is worth noting that our previous study [5] using the classical axonal model can only simulate the first block mechanism but not the second and third mechanisms because the classical model does not include the changes in ion concentration and resting potential. These different mechanisms for post-DC block could be further verified in animal studies by manipulating the ion channels, the ion concentrations, or the ion pumps pharmacologically to increase/decrease the post-DC block duration. Interestingly, a recent study in rats [25] also showed that the post-DC block could be short-lasting or long-lasting depending on the total electrical charge applied by the DC stimulation. The current study has clearly shown the ability of the new axonal conduction model in simulating the phenomena observed in animal studies and in predicting the possible mechanisms underlying these phenomena.

High frequency (kHz) electrical stimulation has been applied clinically for spinal cord stimulation to treat chronic back pain [26]. In addition, kHz stimulation has also been used to block peripheral nerves to treat extreme obesity [27] or amputation limb pain [28]. Despite these clinical successes, the mechanism underlying the effects of kHz stimulation on axonal conduction/block is still not fully understood. Previous animal studies [29–31] have shown that long-duration kHz stimulation can produce prolonged post-stimulation block of axonal conduction. Our recent animal study further shows that the post-stimulation nerve block (not muscle fatigue) can also be produced by stimulation at frequencies less than 1 kHz [32] indicating a common blocking mechanism for repetitive stimulation across a wide range of frequencies. The mechanisms underlying post-stimulation block are certainly involved in the conduction block during the stimulation, i.e., immediately before the stimulation is turned off. Thus, the new axonal conduction model developed in this study provides an opportunity to further analyze the post-stimulation effects of repetitive stimulation at various frequencies to reveal the possible mechanisms underlying post-stimulation block. We expect that the ion concentrations could be significantly changed by repetitive stimuli of different frequencies in a manner similar to DC as shown in this study, which results in axonal conduction block both during and after termination of the stimulation.

Our model is mainly based on the classical HH model of the squid axon. When a parameter in our model is not available from studies using the squid axon, we used the parameter obtained from other species and included a reference to the literature. Due to this limitation, our model may not be able to simulate some phenomena observed in different animal species or even in squid axon. The usability of our model will have to be tested in future studies with the expectation that adjustments to the model parameters may be necessary for a successful simulation. Our current study has shown that this new model is successful in simulating many phenomena observed in a small diameter (2 μm) mammalian unmyelinated axon. Since the classical HH model is derived from the squid axon but has been successful in simulation of many phenomena observed in different species, we expect that our model mainly based on the HH model could also be useful in analysis of axonal responses in different species. Certainly, our model will require more studies to establish its broader applicability. The other limitation of this model includes the assumption that the ion pump is not influenced by the ATP consumption. However, during long-lasting stimulation, the consumption and in turn the availability of ATP can also affect the ion pump, ion concentration and action potential conduction. Our model could certainly be improved further by including more biological processes such as ATP consumption, blood supply, and oxygen exchange, etc.

V. Conclusion

This study reveals that changes in ion concentrations and ion pump activity play an important role in post-stimulation effects and activity-dependent effects on axonal conduction/block. The new axonal conduction model developed in this study not only successfully simulates the initiation, propagation, and block of the action potential as revealed using the classical HH model, but it can also simulate the effects of long-duration stimulation on axonal conduction and block. This new model which includes estimates of axonal ion concentrations and ion pump activity will be useful in analyzing the effects and possible mechanisms of long-duration stimulations that are currently employed in many clinical applications to treat different types of neurological diseases [1–3] [26–28].