The periodic axon membrane skeleton leads to Na nanodomains but does not impact action potentials

Abstract

Recent work has established that axons have a periodic skeleton structure comprising of azimuthal actin rings connected via longitudinal spectrin tetramer filaments. This structure endows the axon with structural integrity and mechanical stability. Additionally, voltage-gated sodium channels follow the periodicity of the active-spectrin arrangement, spaced ∼190 nm segments apart. The impact of this periodic arrangement of sodium channels on the generation and propagation of action potentials is unknown. To address this question, we simulated an action potential using the Hodgkin-Huxley formalism in a cylindrical compartment, but instead of using a homogeneous distribution of voltage-gated sodium channels in the membrane, we applied the experimentally determined periodic arrangement. We found that the periodic distribution of voltage-gated sodium channels does not significantly affect the generation or propagation of action potentials but instead leads to large, localized sodium action currents caused by high-density sodium nanodomains. Additionally, our simulations show that the distance between periodic sodium channel strips could control axonal excitability, suggesting a previously underappreciated mechanism to regulate neuronal firing properties. Together, this work provides a critical new insight into the role of the periodic arrangement of sodium channels in axons, providing a foundation for future experimental studies.

Significance

We simulated an action potential using the Hodgkin-Huxley formalism in a cylindrical compartment, but instead of using a homogeneous distribution of voltage-gated sodium channels in the membrane, we applied the experimentally determined periodic arrangement. We found that the periodic distribution of voltage-gated sodium channels does not significantly affect the generation or propagation of action potentials but instead leads to large, localized sodium action currents caused by high-density sodium nanodomains. Additionally, we found that the distance between periodic sodium channel strips and the axon diameter also regulate action potentials in simulations.

Introduction

The action potential, a rapid all-or-none change in the membrane potential $V_m$, is a defining feature of excitable cells like neurons. Hodgkin and Huxley first described the fundamental principles of action potentials 70 years ago, informing generations of scientists on the inner workings of axon currents and their relationship to the generation and propagation of action potentials. In the canonical model, an action potential is triggered when a stimulus in the form of an excitatory event passively reaches the axon initial segment, depolarizing it sufficiently to reach the activation threshold of voltage-gated sodium $\left(N{a}_v\right)$ channels (1). This kindling then leads to an eruption of electrical activity that generates an action potential. The membrane potential returns to its resting state due to a combination of $N{a}_v$ channel inactivation and delayed activation of voltage-gated potassium channels. In subsequent years, researchers discovered and cloned the multiple sodium and potassium channels that contribute to the action potential as well as the scaffolding and anchoring proteins that tether these channels to the axonal plasma membrane (2).

Although much progress had been made over the last few decades to understand not only the electrical properties of the axon but also its cell biology, it was not until 2013 that the nanoscopic organization of the actin-spectrin skeleton of the axon was revealed (3). Deviating from previous models, the actin-spectrin skeleton is composed of ring-like F-actin proteins connected longitudinally by αII/βII- or αII/βIV-spectrin tetramers. The periodicity of the membrane-associated periodical skeleton (MPS) is ∼190 nm, set by the spectrin filaments that are under entropic tension (3,4). The spatial arrangement of the actin-spectrin proteins differs from the one found in the soma, where the actin-spectrin skeleton is arranged as a two-dimensional polygonal lattice (5) similar to the actin-spectrin network observed in erythrocytes (6,7). Computational and experimental work has now shown that the MPS bestows the axon with distinct mechanical properties. For instance, atomic force microscopy experiments revealed that the axon is substantially stiffer than the soma and dendrites of hippocampal neurons (5). Moreover, a coarse-grain molecular dynamics model for the axon membrane skeleton showed that the axonal MPS has two distinct moduli of elasticity that are correlated with its geometric structure, characterized by stiff actin rings connected by extended spectrin filaments oriented along the axon (5). The role of the MPS in defining the mechanical properties of the axon is also supported by work showing that deletion of spectrin proteins leads to axon deformation and a higher propensity to form kinks and breaks (8,9). More recent work has shown that the MPS might restrict axonal protein and lipid diffusion, although the functional implications of these effects on the action potential are not fully understood (10, 11, 12).

Despite our understanding of the cell biology of the axonal MPS and its role in setting the mechanical properties of the axon, its role in the action potential has remained unexplored. As mentioned above, the current model of action potential generation and propagation is based on Hodgkin and Huxley’s mathematical description of the action potential (13). Consequently, multiple investigators have applied the Hodgkin-Huxley (H-H) formalism to understand how sodium and potassium channels control neuronal excitability of different neurons and to determine the $N{a}_v$ channel surface density necessary to stimulate an action potential in neurons. However, an inherent assumption of the H-H model is that $N{a}_v$ channels are uniformly and homogeneously distributed on the axonal plasma membrane. This assumption is inconsistent with recent super-resolution microscopy studies that showed that $N{a}_v$ channels follow the MPS periodicity and are spaced ∼190 nm segments apart (3). Therefore, to examine the effects of the $N{a}_v$ channel periodicity on the action potential, we simulated the action potential in a cylindrical compartment using the H-H model, with the $N{a}_v$ channels following the experimentally established periodic arrangement rather than being evenly distributed in the membrane. We note that earlier studies have reported that the density of membrane channels, especially sodium and potassium channels, can affect the action potential (14,15). However, the periodic distribution of sodium channels has not yet been considered, providing a gap in our knowledge.

Materials and methods

Here, we discuss the methodology we followed to solve the H-H equation numerically:

$$\frac{d}{4R_{ax}}\frac{{\partial }^{2}V}{\partial x^2}=I_m,$$ (1.1)

where $d$ is the axon diameter; $Rax$ is the axon core resistivity; $V=E_m-E_{res}$, where $E_m$ is the actual membrane potential and $V$ is the membrane potential relatively to the resting potential $E_{res}$; and $I_m$ is the membrane current density (13,16). In the active cable model, the membrane current density is the sum of the potassium current $I_K(x,t)$, the sodium current $I_{Na}(x,t)$, the leak current $I_L(x,t)$, and the capacitive current $I_C(x,t)$.

$$I_m\left(x,t\right)=I_C\left(x,t\right)+I_{Na}\left(x,t\right)+I_K\left(x,t\right)+I_L\left(x,t\right)$$ (1.2)

The currents are given by the expressions:

$$I_{Na}\left(x,t\right)={\overline{g}}_{Na}{m}^{3}h\left(V-V_{Na}\right),$$

$$I_K\left(x,t\right)={\overline{g}}_K{n}^4\left(V-V_K\right),$$ (1.3)

$$I_L\left(x,t\right)=g_L\left(V-V_L\right),$$

and

$$I_C\left(x,t\right)=C_m\frac{\partial V}{\partial t},$$

where ${\overline{g}}_{Na}$ and ${\overline{g}}_K$ are the maximum sodium and maximum potassium conductances, respectively, and $g_L$ is the nonspecific leak conductance (Table 1). $V_{Na}=E_{Na}-E_{res}$, $V_K=E_K-E_{res}$, and $V_L=E_L-E_{res}$ are the sodium, potassium, and leak reversal potentials measured relatively to the resting potential, respectively (Table 1). For the calculation of $V_{Na},V_K$ we used E_Na = +55 mV, E_K = −85 mV, and E_res = −75 mV (17). $m(x,t)$, $h(x,t)$, and $n(x,t)$ are dimensionless variables that are used to describe the kinetics of voltage-dependent ionic currents and obey the following equations:

$$\frac{dm}{dt}={\alpha }_m\left(1-m\right)-{\beta }_{m}m,\frac{dh}{dt}={\alpha }_h\left(1-h\right)-{\beta }_{h}h,$$ (1.4)

and

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

Table 1.

Parameters used in the Hodgkin-Huxley model

Parameters	Symbol	Value	Units
Membrane capacitance per unit area	$C_m$	0.9	μF/cm2
Na maximum conductance	${\overline{g}}_{Na}$	300	mS/cm2
K maximum conductance	${\overline{g}}_K$	30	mS/cm2
Leak conductance	$g_L$	0.3	mS/cm2
Na reversal potential	$V_{Na}$	130	mV
K reversal potential	$V_K$	−10	mV
Leak reversal potential	$V_L$	10.07	mV
Axon diameter	d	3	μm
Axial resistivity	$R_{ax}$	100	$\Omega \text{cm}$

Note: $V_{Na}=E_{Na}-E_{res}$ and $V_K=E_K-E_{res}$ were calculated based on the following values: E_Na = +55 mV, E_K = −85 mV, and E_res = −75 mV (17).

The rate functions that appear in the first-order kinetic Eq. 1.4 are functions of the membrane potential and are given by the expressions

$${\alpha }_m=0.1\left(25-V\right)/\left[\exp \left(\frac{25-V}{10}\right)-1\right],$$ (1.5)

$${\beta }_m=4\exp \left(\frac{-V}{18}\right),$$

$${\alpha }_h=0.07\exp \left(\frac{-V}{20}\right),{\beta }_h=1/\left[\exp \left(\frac{30-V}{10}\right)+1\right],$$

and

$${\alpha }_n=0.01\left(10-V\right)/\left[\exp \left(\frac{10-V}{10}\right)-1\right],$$

$${\beta }_n=0.125\exp \left(\frac{-V}{80}\right),$$

Hodgkin and Huxley obtained the expressions of the rate coefficients by fitting data from several experiments in Eq. 1.4 (13,18).

Combining Eqs. 1.2 and 1.3, we wrote the membrane current as

$$I_m=C_m\frac{\partial V}{\partial t}+{\overline{g}}_{Na}{m}^{3}h(V-V_{Na})+{\overline{g}}_K{n}^4\left(V-V_K\right)+g_L\left(V-V_L\right).$$ (1.6)

The axonal membrane contains a voltage-independent leak conductance $g_L$, which is not dependent on the applied voltage and remains constant over time. The leak conductance $g_L$ is associated with the reversal potential $V_L$, which was adjusted so that the total membrane current at the resting potential $V_{res}$ was I_res = 0. The $V_L$ was defined via the equation $g_{Na}\left(0\right)V_{Na}+g_K\left(0\right)V_K+g_L{V}_L=0$, in which $g_{Na}\left(0\right)$= 0.0106 mS/cm² and g_K(0) = 0.3666 mS/cm², and found to be 10.07 mV in our simulation (19). When the sodium channels were periodically distributed along the longitudinal direction of the axon, $g_L$ was adjusted to be periodically distributed accordingly so that the membrane current was zero at resting potential along the axon.

After combining Eqs. 1.1 and 1.6, we obtained the H-H active cable equation:

$$\frac{d}{4R_{ax}}\frac{{\partial }^{2}V}{\partial x^2}=C_m\frac{\partial V}{\partial t}+{\overline{g}}_{Na}{m}^{3}h(V-V_{Na})+{\overline{g}}_K{n}^4\left(V-V_K\right)+g_L\left(V-V_L\right).$$ (1.7)

The independent variables are the longitudinal distance along the axon $x$ and time $t$. The dependent variables are the membrane potential V(x,t) and the dimensionless variables m(x,t), h(x,t), and n(x,t).

We used the Neumann boundary conditions for the initial-boundary value problem of the H-H equation, which corresponds to injecting a variable current at one end of an axon obeying H-H kinetics:

$$\frac{\partial V(0,t)}{\partial x}=-\frac{RI\left(t\right)}{\pi a^2},\frac{\partial V(L,t)}{\partial x}=0,$$ (1.8)

where $I\left(t\right)$ is the injected current at $x=0$.

The initial conditions are the following:

$$\underset{t\to 0^{+}}{\lim }V(x,t)=V_0\left(x\right),$$ (1.9)

$$\underset{t\to 0^{+}}{\lim }m(x,t)=m_0\left(x\right),$$

$$\underset{t\to 0^{+}}{\lim }h(x,t)=h_0\left(x\right),$$

and

$$\underset{t\to 0^{+}}{\lim }n(x,t)=n_0\left(x\right).$$

Hence, given the above initial conditions for the rate variables and the potential, we can calculate the values of the rate variables $m(x,t)$, $h(x,t)$, and $n(x,t)$ as a function of time.

To solve the H-H system of the equations discussed above using the initial and boundary conditions above, we employed the backward Euler scheme in MATLAB (20). We defined a computational grid with time step Δt and number of spatial grid points $N$. The spatial discretization along the axon has a uniform unit length Δx = L/N. The grid contained $N+1$ points numbered from zero (corresponding to $x=0$) to $N$ (corresponding to $x=L$).

As described in the introduction, a $N{a}_V$ channel can bind to ankyrin, which is tethered to the 15^th repeat of β-spectrin near its carboxyl terminus located in the middle area of a spectrin tetramer (21). This means that a $N{a}_V$ channel connected to ankyrin is also located in the middle area of two consecutive actin rings (Fig. 1). Spectrin tetramers that are not under tension have an end-to-end equilibrium distance of 75 nm. In the axon, however, spectrin tetramers are under tension with a reduced range of thermal motion. After considering the size of actin particles, researchers have determined the end-to-end distance of spectrin tetramers in axons to be approximately 150 nm (5). The mean distance between two consecutive ankyrin proteins along the longitudinal direction when the spectrin is under tension is L_t = 185.78 nm (5). Because the spectrin tetramers are under entropic tension, the trajectory of ankyrin particles describes a small area around the equilibrium position; consequently, the connected $N{a}_v$ channels maintain an ordered configuration, as they are distributed in azimuthal bands between the actin rings (3,5,10). Work has shown that the distance of an ankyrin particle from its mean position during thermal motion along the longitudinal direction follows a Gaussian distribution with a standard deviation $\sigma =0.027L_t$, where L_t = 185.78 nm in the actual axon, which gives $\sigma \cong 5\text{nm}$ (5). We considered that the width of the band within which the $N{a}_v$ channels are distributed is $0.108L_t\cong 20\text{nm}$, which is four times the standard deviation of the ankyrin particles and almost 1/9 of the distance between two consecutive actin rings.

Figure 1.

Periodic Gaussian distribution of Na_v channels on the axon. (A) Illustration of an axon membrane skeleton model based on super-resolution microscopy results. (B) Two-dimensional illustration of the periodic Gaussian distribution of $N{a}_v$ channels on the axonal plasma membrane considered in this paper. The $N{a}_v$ channel bandwidth is 20 s, approximately 1/9 of the equilibrium distance between consecutive actin rings. To see this figure in color, go online.

To investigate the effect of the periodic distribution of $N{a}_v$ channels on the action potential compared with the conventional assumption of homogeneously distributed $N{a}_v$ channels, we implemented two types of $N{a}_V$ channel conductances $g_{Na}$ in our simulation. In the first case, we assumed that $N{a}_v$ channels are distributed homogeneously along the axon. Thus, we adopted a homogeneous $N{a}_V$ channel conductance ${\overline{g}}_{Na\_H}=G_{Na}$, where G_Na = 300 mS/cm². In the second case, we considered that the location of $N{a}_V$ channels follows a Gaussian distribution in bands located in the middle areas between actin rings with a width of 1/9 of the distance between two consecutive actin rings, as mentioned above. It is extremely computationally expensive to use the actual scale L_t = 185.78 nm for the numerical solution of the H-H equations. Instead, we adopted a much larger distance between consecutive actin rings and maintained the proportions of the width of the $N{a}_v$ channels’ azimuthal band with respect to the distance between consecutive actin rings. In particular, we chose 20 s as the width of the $N{a}_V$ channel band and 185 s as the distance between two consecutive actin rings. Then, we chose $s=n\epsilon$, where $\epsilon =1\text{nm}$ is the length scale and $n$ is an integer, and showed that the numerical solutions converge as $n$ increases. We obtained plots for $n=20$. Consequently, the maximum conductance ${\overline{g}}_{Na\_G}$ for periodically distributed $N{a}_v$ channels is given by the expression

$${\overline{g}}_{Na\_G}\left(x\right)=\{\begin{array}{l}0ifx\in (kl,\left(k+\frac{4}{9}\right)l]\cup x\in (\left(k+\frac{5}{9}\right)l,\left(k+1\right)l]\\ G_{Na}{e}^{\frac{-{\left(x-k·185s\right)}^2}{2{\sigma }^2}}ifx\in (\left(k+\frac{4}{9}\right)l,\left(k+\frac{5}{9}\right)l]\end{array},$$ (1.10)

where $k=1,2,3,\dots$, $\sigma =5\text{s}$, and $l=185\text{s}$ is the distance between two consecutive actin rings.

Results

H-H model in the case of a periodic actin membrane skeleton

To address whether the periodic arrangement of $N{a}_v$ channels (Fig. 1) impacts action potential properties, we applied the H-H model to the action potential and compared the results generated by a constant versus periodic maximum conductance using the rates and conditions described previously (17,22,23). To numerically solve the H-H model, we developed a MATLAB-based code. First, we simulated the case of a constant maximum conductance, which corresponds to homogeneously distributed $N{a}_v$ channels. Based on our simulations, we made several observations: 1) the propagating action potential preserved its waveform as it traveled along the axon (illustrated in Figs. 2 A and S1 A). 2) The action potential peak amplitude was independent of the distance traveled; thus, the peak did not attenuate as it traveled across the axon. The difference between the resting membrane potential and the peak voltage of the action potential was 108 mV, similar to values obtained in pyramidal neuron recordings in the cortex (17). 3) The action potential propagated at a constant speed of 2.1 m/s (Fig. 2 A). The period of the action potential was 1.2 ms, and the wavelength (peak-to-peak distance) was 2.52 mm. Our numerical results are very similar to the solution obtained by Kole and colleagues (22,23) using the NEURON simulation environment. This confirms that our numerical method is able to accurately solve the H-H model and simulate an action potential and its propagation (17).

Figure 2.

Simulation of action potential propagation for homogeneous and periodic Na_v channel distributions with a variable maximum sodium conductance. All potentials are computed relatively to the resting potential. (A) Illustration of action potential propagation for homogeneously distributed $N{a}_v$ channels. The homogeneous $N{a}_v$ channel conductance is ${\overline{g}}_{Na\_H}=G_{Na}$, where G_Na = 300 mS/cm². (B–E) Illustration of action potential propagation for cases in which the $N{a}_v$ channels are periodically distributed with (B) ${\overline{g}}_{Na\_G}=4G_{Na}$, (C) ${\overline{g}}_{Na\_G}=7G_{Na}$, (D) ${\overline{g}}_{Na\_G}=12G_{Na}$, and (E) ${\overline{g}}_{Na\_G}=15G_{Na}$. The maximum conductance ${\overline{g}}_{Na\_G}$ follows the Gaussian distribution of Eq. 1.10. To see this figure in color, go online.

Next, we investigated the impact of distributing $N{a}_v$ channels in a periodic configuration, matching what has been observed using super-resolution microscopy. Specifically, we positioned $N{a}_V$ channels in azimuthal bands repeated periodically along the axon. Each of these bands was positioned in the middle area between two consecutive actin rings and was approximately 20 s wide (see materials and methods and Fig. 1). The position of $N{a}_v$ channels within the azimuthal strips followed a Gaussian distribution around the center line of each strip. The corresponding maximum conductance ${\overline{g}}_{Na\_G}\left(x\right)$ is given by Eq. 1.10. We also set the $N{a}_v$ channel maximum conductance $G_{Na}$ of the Gaussian distribution to be the same as the maximum conductance $G_{Na}$ in a homogeneous distribution (300 mS/cm²).

Using this scenario, we ran the same simulations as we did for the homogenous distribution of $N{a}_v$ channels. We found that the resulting action potential was different than the action potential generated by homogeneously distributed $N{a}_v$ channels. For instance, the peak amplitude of the action potential was 19 mV, five- to sixfold smaller than in the homogenous conditions, the propagation speed was 0.8 m/s (compared with 2.1 m/s), and the wavelength was 1.2 mm, two times smaller than the corresponding value (2.52 mm) in homogeneously distributed $N{a}_v$ channels.

We reasoned that the difference between the two scenarios was due to the lower number of $N{a}_v$ channels per area (surface density) between two consecutive actin rings for periodically distributed $N{a}_v$ channels compared with homogeneously distributed $N{a}_v$ channels. As mentioned above, periodically spaced $N{a}_v$ channels are positioned within an azimuthal band following a Gaussian distribution (see Eq. 1.10). Integrating the surface density of the $N{a}_v$ channels, we found that the total number of $N{a}_v$ channels between two consecutive actin rings was approximately 60% of the homogeneously distributed $N{a}_v$ channels when applying the same maximum conductance for simulations in the homogenous (${\overline{g}}_{Na\_H}=G_{Na}$) and periodic (Eq. 1.10) configurations.

Thus, we repeated our simulations with increasing maximum conductances in the Gaussian distribution from 4 to 15 $G_{Na}$ (see Fig. 2 B–E) to identify the maximum conductance value that produces an action potential similar to the action potentials derived under the homogenous conditions. We found that increasing the maximum conductance G_Na of $N{a}_v$ by 15 times reproduced the action potential properties of the homogenous configuration. The maximum conductance in this case is given by the expression:

$${\overline{g}}_{Na\_G}(x)=\{\begin{array}{l}0ifx\in (kl,\left(k+\frac{4}{9}\right)l]\cup x\in (\left(k+\frac{5}{9}\right)l,\left(k+1\right)l]\hfill \\ 15G_{Na}{e}^{\frac{-{(x-k·185s)}^2}{2{\sigma }^2}}ifx\in \left(\left(k+\frac{4}{9}\right)l,\left(k+\frac{5}{9}\right)l\right]\hfill \end{array}.$$ (2.1)

Thus, by solving the H-H equation using the maximum conductance shown in Eq. 2.1, we obtained the action potential illustrated in Figs. 2 E and S1 B. Under these conditions, the generated action potential had very similar properties (i.e., peak amplitude, period, wavelength, propagation speed) as the action potential generated by homogeneously distributed $N{a}_v$ channels. The reason for this similarity is that by increasing the $N{a}_v$ maximum conductance to 15 G_Na, we achieved the same number of $N{a}_v$ channels between two consecutive actin rings as with the homogeneously distributed $N{a}_v$ channels, for which ${\overline{g}}_{N{a}_H}=G_{Na}$. This can be also shown by recognizing that the integrals ${\iint }_{0,0}^{185,\pi d}{G}_{Na}dxdy\cong$ $185\pi d{G}_{Na}$ (homogenous) and ${\iint }_{-3\sigma ,0}^{3\sigma ,\pi d}Q{G}_{Na}{e}^{\frac{-{(x-185)}^2}{2{\sigma }^2}}dxdy\cong$ $Q\pi d{G}_{Na}\sqrt{2\pi }\sigma$ with $\sigma =5\text{nm},$ (periodic Gaussian), which are proportional to the number of $N{a}_v$ channels between two actin rings, are approximately equal when $Q\cong 14.8$, a value similar to the factor 15 that we used in Eq. 2.1. Thus, as long as the number of $N{a}_v$ channels is the same between the two consecutive actin rings, their distribution has no impact on the generation or propagation of the action potential. We note that since the wavelength of the action potential is on the order of 2.5 mm, which covers an area of approximately 1.4 × 10⁴ actin rings, the effect of $N{a}_v$ stripes is averaged over the entire wavelength.

An important question is whether the required number of $N{a}_v$ channels necessary to simulate the action potential could fit in the area between two actin rings when they are homogenously distributed and when their distribution follows a Gaussian. To examine this question, we considered that the radius of a $N{a}_v$ channel is approximately r_c = 5 nm (24) and that the surface membrane density of sodium channels is between ${\rho }_1=110$ and ${\rho }_2=300$ channels per μm² (22). In the case of homogeneous distribution, the maximum number of channels is $N_m^h={\rho }_2\pi d\ast 0.185\mu \text{m}\cong 174$ channels. The percentage of the maximum area covered by those channels compared with the available membrane area is only $\frac{N_m^h\pi r_c^2}{\pi dh}\times 100\%\cong 2.36\%$. When the $N{a}_v$ channel distribution follows a Gaussian, the maximum number of channels in a strip with a width $(-2\sigma ,2\sigma )$ is $N_{4\sigma }^G=\frac{{\iint }_{-2\sigma ,0}^{2\sigma ,\pi d}15G_{Na}{e}^{\frac{-{(x-185)}^2}{2{\sigma }^2}}dxdy}{185nm\pi d{G}_{Na}}\ast 174\cong 169$ channels, occupying $\frac{N_{4\sigma }^G\pi r_c^2}{\pi d4\sigma }\times 100\%\cong 21.1\%$ of the central strip (width $4\sigma$). Similarly, the maximum number of channels in the strip $(-\sigma ,\sigma )$ is $N_{2\sigma }^G=\frac{{\iint }_{-\sigma ,0}^{\sigma ,\pi d}15G_{Na}{e}^{\frac{-{(x-185)}^2}{2{\sigma }^2}}dxdy}{185nm\pi d{G}_{Na}}\ast 174\cong 121$ channels, which would occupy $\frac{N_{2\sigma }^G\pi r_c^2}{\pi d2\sigma }\times 100\%\cong 30.2\%$ of the central strip (width $2\sigma$). Based on these calculations, we conclude that the number of $N{a}_v$ channels required to simulate action potential propagation fit within the available surface membrane space independent of whether the channels are homogeneously or periodically distributed.

Although the $N{a}_v$ channel periodic distribution did not directly impact action potential generation or propagation, our model suggests substantial effects on the sodium action currents. This is best shown in Fig. 3, which plots the sodium currents across the axon as they drive the traveling action potential. The sodium current amplitude per unit area was much larger in the periodic configuration. This is because the surface density of $N{a}_v$ channels in the nanostrips is $15$ times greater in the periodic configuration than in the homogeneous configuration. We also plotted the sodium current as a function of time, averaged across areas corresponding to 5, 10, and 20 actin rings and centered on a 1.6 mm distance from the left end of the simulated axon (Fig. 4). This step is necessary, as our knowledge on sodium currents in neurons is through whole-cell recordings that record the activity across a large area. We found that the overall currents for different numbers of rings had very similar properties (i.e., peak amplitude, time course) independent of whether the $N{a}_v$ channels were distributed homogeneously or periodically. We found the same result whether we measured the current at 1.2, 2.0, or 2.4 mm from the left end of the axon. Thus, macroscopic measurement of sodium currents cannot distinguish whether the channels are arranged in a periodic or homogenous manner.

Figure 3.

Simulation of the sodium current propagation for homogeneous and periodic Na_v channel distributions. (A and B) Illustrations of the axonal plasma membrane with (A) homogeneously and (B) periodically distributed $N{a}_v$ channels. (C and D) Plots of sodium current density propagation for (C) homogeneously and (D) periodically distributed $N{a}_v$ channels. The inset figure is a magnification of the peak of the sodium current where the sawtooth shape of the current is clearly visible. (E and F) Three-dimensional plots of the sodium current density propagation for (E) homogeneously and (F) periodically distributed $N{a}_v$ channels. Blue, red, magenta, green, and yellow lines in (C) and (D) are the sodium current densities as a function of position $x$ at 1.2, 1.5, 1.8, and 2.0 ms, respectively, from the initiation of the action potential propagation. To see this figure in color, go online.

Figure 4.

Simulation of the average sodium current as a function of time. (A and B) Sodium current as a function of time averaged over areas corresponding to 5, 10, and 20 rings centered on a location 1.6 mm from the left end of the axon for (A) homogeneous and (B) periodic distributions of $N{a}_v$ channels. To see this figure in color, go online.

Effect of varying the distance between actin rings on action potentials

As discussed above, $N{a}_v$ channels are distributed in periodic azimuthal bands that co-localize with ankyrin-G and ankyrin-B proteins alternating with actin rings (Fig. 1 B). This periodic distribution of $N{a}_v$ channels does not affect the propagated action potential. However, this does not exclude the possibility that the distance between actin rings might play a role in the generation of the action potential, as it might alter the local sodium channel surface membrane density. Thus, we next investigated the impact of a larger distance between the $N{a}_v$ stripes and determined whether it would modify the action potential properties.

We performed simulations in which we increased the distance between actin rings, as spectrin repeats might sequentially unfold during axonal extension (i.e., 2, 3, 4, and 4.5 times larger than the physiological distance l (Fig. 5) (25). A larger spacing of $N{a}_v$ channels in the axon decreased the amplitude and its propagation speed as the distance between $N{a}_v$ stripes increased to 4 l (Fig. 5 C). When the distance between the sodium channels stripes was larger than 4.5 l, we did not observe action potential propagation (Fig. 5 D). The reason for this result is that the generated membrane voltage does not reach the excitation threshold (∼17 mV), i.e., the critical level at which a membrane potential is depolarized to initiate an action potential. We also decreased the distance between two consecutive actin rings to 3/4 and 1/2 times the normal distance. In these cases, we found that the action potential peak increased (Fig. 6). This increase is likely the result of an increased homogeneous density of $N{a}_v$, due to a decrease in the distance between two consecutive rings with a constant density of $N{a}_v$ channels in each strip.

Figure 5.

Illustration of action potential propagation with larger spacing of $N{a}_v$ stripes. All potentials are computed relatively to the resting potential. (A–D) Illustration of action potential propagation for cases in which the distance between two consecutive stripes of $N{a}_v$ channels is (A) 2, (B) 3, (C) 4, and (D) 4.5 l, where l is the original distance between consecutive stripes of $N{a}_v$ channels. To see this figure in color, go online.

Figure 6.

Illustration of action potential propagation with narrower spacing of $N{a}_v$ stripes. All potentials are computed relatively to the resting potential. (A and B) Illustration of action potential propagation for cases in which the distance between two consecutive stripes of $N{a}_v$ channels is (A) 3/4 and (B) 1/2 l, where l is the original distance between consecutive stripes of $N{a}_v$ channels. To see this figure in color, go online.

To further clarify the effect of modifying the distance between two consecutive rings on the action potential, we performed simulations in which we doubled the distance between two consecutive rings while doubling the density of $N{a}_v$ in the periodic stripes from ${\overline{g}}_{Na\_G}$ to $2{\overline{g}}_{Na\_G}$ in order to hold the number of sodium channels between two consecutive rings constant and therefore keep the homogenous $N{a}_v$ density constant. With an unchanged overall density, we expected that the action potential would remain the same as in the case of a homogeneous distribution of ion channels. Indeed, Fig. 7 A shows that the resulting action potentials were approximately identical. In addition, we decreased the distance between two consecutive rings to half of the original distance and similarly decreased the density of $N{a}_v$ channels to $0.5{\overline{g}}_{Na\_G}$ to keep the number of $N{a}_v$ channels between two consecutive rings constant. Again, we found that the action potential was approximately the same as with the homogeneous distribution of sodium and potassium channels (see Fig. 7 B).

Figure 7.

Illustration of action potential propagation for different spacing but with the same effective homogeneous surface density of $N{a}_v$ channels. All potentials are computed relatively to the resting potential. (A) Illustration of action potential propagation when the distance between consecutive stripes of $N{a}_v$ channels is 2 l and ${\overline{g}}_{Na}^{\ast }=2{\overline{g}}_{Na\_G}$. (B) Illustration of action potential propagation when the distance between consecutive stripes of $N{a}_v$ channels is 0.5 l and ${\overline{g}}_{Na}^{\ast }=0.5{\overline{g}}_{Na\_G}$. l is the original distance between consecutive stripes of $N{a}_v$ channels. To see this figure in color, go online.

Effects of axon diameter on action potential simulations

Axons come in different diameters, some as small as 0.5 μm (26). A model of action potential in mammalian C-fiber axons proposed that nonuniform sodium channel surface membrane density could cause saltatory conduction (27). Thus, we tested the effect of smaller axon diameters when sodium channels were homogenously or periodically distributed. We selected two axonal diameters: 1.0 and 0.5 μm. We found that the action potential propagation speed was decreased, and the period of the action potential was increased, independent of the $N{a}_v$ model (Fig. 8). The action potential propagation speed was 1.25 m/s when the axonal diameter was 1.0 μm (Fig. 8 C and D) and decreased to 0.89 m/s when the axonal diameter was 0.5 μm (Fig. 8 E and F). Finally, we note that our model did not produce saltatory conduction.

Figure 8.

Illustration of action potential propagation for homogeneous and periodic $N{a}_v$ channel distribution with 1.0 and 0.5 μm axonal diameters. All potentials are computed relatively to the resting potential. (A and B) Illustrations of the axonal plasma membrane with (A) homogeneously and (B) periodically distributed $N{a}_V$ channels. (C and D) Plots of action potential propagation with a 1.0 μm axonal diameter for (C) homogeneously and (D) periodically distributed $N{a}_v$ channels. (E and F) Plots of action potential propagation with a 0.5 μm axonal diameter for (E) homogeneously and (F) periodically distributed $N{a}_v$ channels. To see this figure in color, go online.

Discussion

One of the major breakthroughs in our understanding of axon biology was the recognition that the actin-spectrin scaffold network in the axon follows a unique anatomical arrangement. In particular, super-resolution microscopy revealed that the plasma membrane skeleton has a periodic structure comprising of actin rings connected by spectrin tetramers (3). These experiments also showed that $N{a}_V$ channels exhibit a periodic ring-like distribution pattern that alternates with actin rings and co-localizes with ankyrin-G and ankyrin-B proteins (3,4). A coarse-grain molecular dynamics model of the axon membrane skeleton showed that the spectrin tetramers are under entropic tension and that the distance of a thermally moving ankyrin particle from its mean position follows a Gaussian distribution (5). As a result, the ankyrin proteins and connected $N{a}_V$ channels are distributed in azimuthal bands following a Gaussian distribution.

Although the role of the periodic actin ring structure in establishing the mechanical properties of the axon has been well established, it is unclear whether this structure also impacts the generation and propagation of the action potential along the axon. Our current understanding of the influence of $N{a}_v$ channels on action potentials stems from the seminal work of Hodgkin and Huxley (13,16,18). However, an assumption of the H-H model is that $N{a}_V$ channels are distributed in a homogenous manner, which is now understood to be incorrect for mammalian axons. Thus, in this study, we tested whether the periodic distribution of the $N{a}_v$ channels in the axon influences the generation and propagation of action potentials. Based on our simulations, we propose that 1) $N{a}_V$ channel periodicity does not affect action potential properties. A key parameter for an action potential is the number of $N{a}_v$ channels per surface area between two consecutive actin rings rather than their periodic spatial arrangement. We note that the characteristic scale of the action potential is its wavelength, which is on the order of 2.5 mm. Based on this, one wavelength corresponds to approximately 13,500 actin rings, which means that the electrophysiological measurements cannot provide an insight for the spatial arrangement of $N{a}_v$ in the axon. 2) Periodic $N{a}_v$ channels lead to the formation of high surface density $N{a}_v$ channel nanodomains. Such nanodomains result in large, localized action sodium currents, leading to high sodium concentrations and current densities similar to the nanodomains seen with calcium channels in presynaptic terminals (28,29).

Additionally, the periodic $N{a}_v$ arrangement greatly affects the $N{a}_v$ channel surface density. In particular, the $N{a}_v$ channel surface density must be 15 times higher than previous estimates that assumed $N{a}_v$ homogeneity. The large surface density of $N{a}_v$ channels in the nanostrips is required to equalize the average number of $N{a}_v$ channels per surface area in both the periodic and homogeneous models of $N{a}_v$ channel surface distribution. This density is necessary to generate the same action potential, which ultimately depends on the average sodium current density and the number of $N{a}_v$ channels. Of course, considering that action potential amplitudes are not uniform across the different neuronal cell types as well as the role the resting membrane potential and synaptic activity might play in setting sodium channel availability in axons, the precise density of sodium channels contributing to an action potential in axons at any given moment might vary substantially (30,31).

If the unique arrangement of the $N{a}_v$ channel in the axon does not contribute to the action potential, what might be its role? We considered a few possibilities. First, a sodium nanodomain might increase the likelihood of capturing sodium by Na-K ATPases. For instance, a large sodium flux is more likely to drive the occupancy of Na-K ATPase to a sodium-bound state followed by slow translocation to the outside. Considering the energetic cost of clearing sodium from the cytosol, increasing the likelihood of Na-K ATPase pumps to capture sodium will lead to more efficient transport. Similarly, a large sodium nanodomain would also increase the probability of activating low affinity sodium-activated potassium channels (KCNT1, KCNT2) following an action potential or a train of action potentials (32). Thus, the periodic sodium channel arrangement might be optimized for sodium signaling and for reducing the energetic costs associated with action potentials.

By performing simulations in which the distance between actin rings was altered, we investigated whether the hypothetical larger distance between the $N{a}_v$ stripes would modify the action potential and its parameters such as the amplitude, propagation speed, and morphology of the action potential change. We observed that the larger spacing of the $N{a}_v$ channels in the axon maintained the morphology of the action potential but decreased its amplitude and propagation speed as the distance between $N{a}_v$ stripes increased to 4 l. We also decreased the distance between two consecutive actin rings to 3/4 and 1/2 times the normal distance. In these cases, we found that the action potential peak increased. The reason for this result is that by decreasing the distance between two consecutive actin rings and keeping $g_{Na}$ the same, we actually increased the effective density of $N{a}_v$ channels along the axon, thus changing the action potential amplitude. We also note that when the distance between the sodium channel stripes was larger than 4.5 l, we did not observe action potential propagation. This is because the generated membrane voltage with a 4.5 l spacing would be less than ∼17 mV, which is below the excitation threshold to initiate an action potential. Thus, we suggest that any manipulation that alters the interring actin distance would also change neuronal excitability since it would change the surface membrane density of $N{a}_v$ channels. Such changes might occur during mechanical stress and subsequent unfolding of the spectrin tetramers.

To further clarify the effect of modifying the distance between two consecutive rings on the action potential, we performed simulations in which we doubled the distance between two consecutive rings and, at the same time, doubled the density of $N{a}_v$ channels in the periodic stripes from ${\overline{g}}_{Na\_G}$ to $2{\overline{g}}_{Na\_G}$. In addition, we decreased the distance between two consecutive rings to half of the original distance and similarly decreased the density of the $N{a}_v$ channels from ${\overline{g}}_{Na\_G}$ to $0.5{\overline{g}}_{Na\_G}$. As the overall density did not change, we found that the action potential remained the same as in the case of the homogeneous distribution of ion channels. Thus, the spacing of the actin rings would determine the density of sodium channels.

Overall, our simulations are in agreement with the repeated nanodomains of the axon membrane skeleton structure, which is based on previously published super-resolution microscopy results (3) and mesoscale particle model results (5,10). These repeated nanodomains of the axonal $N{a}_v$ channel distribution do not affect the generation or propagation of the action potentials but do alter the time course of the sodium currents locally and might constrain the distribution of potassium channels with respect to sodium channels. However, we predict that these changes can be measured not at a micrometer length scale but at the nanodomain level.

Author contributions

Z.C. wrote the code and ran the simulations. G.L. and A.V.T. conceptualized the study and interpreted the results. All authors wrote and edited the paper. All authors have approved the final version of the manuscript.

Editor: Baron Chanda.