Sodium channel slow inactivation normalizes firing in axons with uneven conductance distributions

Summary

The Na⁺ channels that are important for action potentials show rapid inactivation, a state in which they do not conduct, although the membrane potential remains depolarized^1,2. Rapid inactivation is a determinant of millisecond scale phenomena, such as spike shape and refractory period. Na⁺ channels also inactivate orders of magnitude more slowly, and this slow inactivation has impacts on excitability over much longer time scales than those of a single spike or a single inter-spike interval^3–10. Here, we focus on the contribution of slow inactivation to the resilience of axonal excitability^11,12 when ion channels are unevenly distributed along the axon. We study models in which the voltage-gated Na⁺ and K⁺ channels are unevenly distributed along axons with different variances, capturing the heterogeneity that biological axons display^13,14. In the absence of slow inactivation, many conductance distributions result in spontaneous tonic activity. Faithful axonal propagation is achieved with the introduction of Na⁺ channel slow inactivation. This “normalization” effect depends on relations between the kinetics of slow inactivation and the firing frequency. Consequently, neurons with characteristically different firing frequencies will need to implement different sets of channel properties to achieve resilience. The results of this study demonstrate the importance of the intrinsic biophysical properties of ion channels in normalizing axonal function.

eTOC Blurb

Zang et al. model spike propagation in axons with uneven distributions of Na⁺ and K⁺ channels. Including slow inactivation of the Na⁺ channel enhances the reliability of spike propagation, and can compensate for uneven channel distributions.

Graphical Abstract

Results and Discussion

Ion channels have a complex and rich set of slow processes that can capture them in inactivated states, removing them from the easily activated pool for as long as minutes and hours^15–24. We explore the hypothesis that slow inactivation of axonal Na⁺ channels may be a normalizing factor, contributing to resilience of spike propagation in axons^11,12,25 with heterogeneous channel densities.

Slow inactivation.

Slow inactivation reflects an activity-dependent transition of an ion channel between two pools of states: a pool that is available for activation by membrane voltage within the millisecond time scale of spike generation (depicted $A$ in Figure 1A), and a pool that is not available for activation, depicted ($1-A$). The states of a channel in the unavailable pool are coupled by transition rates operating in timescales extending from tens of milliseconds to many minutes. Recent studies demonstrate that at the protein population level, the unavailable (i.e. slowly inactivated) pool is a dynamic reservoir that regulates membrane excitability via control of the effective number of ion channels that participate in action potential generation^9,26.

Figure 1. The effect of the Na⁺ channel slow inactivation gate on axonal excitability in a single compartment model.

(A) A schematic of Na⁺ channel slow inactivation gate (left) and its time constant (right, black). The blue trace shows the time constant of the h gate in the H-H model. (B) Arrows on the x-axis indicate stimulation times (10 Hz). The top panel has no slow inactivation (no SI). Slow inactivation (+SI) was included in the shaded panels. At $1.8\times g_{Na}$ (left) with no SI, the axon fires more rapidly than the stimulation frequency, but with SI, the axon faithfully follows the stimuli (normalization). At $3.5\times g_{Na}$ (middle) the axon again fires more rapidly than the stimuli, but the addition of SI produces burst-like interruptions of activity. At $0.6\times g_{Na}$ (right) the axon faithfully follows the stimulation frequency in the absence of SI, but with SI it fails to follow repeated stimuli.

Parameter variability across cells and within a cell.

There is a growing appreciation that neurons of the same cell type can be degenerate, that is, produce similar activity with variable sets of conductance densities^25,27–29. These degenerate solutions can arise naturally from the self-organizing homeostatic processes that govern the development and maintenance of the intrinsic electrical properties of neurons^30–32. Less attention has been paid to understanding the variance of ion channel densities along axons. Ion channel densities are not evenly distributed down the length of axons, and changes in axonal distribution of ion channels undoubtedly occur in disease ¹⁴. The purpose of this study is to understand the tolerance of action potential propagation to uneven distributions of ion channels, and the role of Na⁺ channel slow inactivation in compensating for variability in channel distributions.

Na⁺ channel slow inactivation in a single compartment model.

The impact of slow inactivation on a single compartment canonical Hodgkin-Huxley (H-H) model¹ is demonstrated in Figure 1. The slow $A\leftrightarrow (1-A)$ kinetics is modelled with forward $\left(\gamma \right)$ and backwards $\left(\delta \right)$ rate constants that are scaled versions of the rates governing the standard rapid inactivation in the H-H model. Thus, $\delta (V)=0.1*{\alpha }_h(V)$ and $\gamma (V)=0.1*{\beta }_h(V)$ (see Methods). The resulting time constant of the $A\leftrightarrow (1-A)$ transition as a function of membrane potential is shown in Figure 1A (right). Figure 1B shows examples of the effects of slow inactivation on spiking in this toy model. As expected, at high maximal sodium conductance the model produces tonic spiking, which may be normalized by the introduction of slow inactivation (Figure 1B, left). The effect of the slow inactivation gate depends on the extent of deviation from the standard H-H maximal sodium conductance (see, for example, middle and right panels of Figure 1B). Note that while Hodgkin and Huxley’s standard parameter for the sodium conductance is 120 mS/cm², their measured range was 65–260 mS/cm².

Spike propagation in multicompartmental axonal models with even and uneven distributions of conductances.

We built axonal multicompartment models that differed in their Na⁺ and K⁺ channel densities, as illustrated in Figure 2A (top). In any given compartment, the Na⁺ (red) and K⁺ (dark brown) conductance densities could be either above or below the values in the H-H model. In these studies, we stimulated the axon at one end at a fixed frequency, and determined the extent to which the entire axon followed the stimulus. Many sets of parameters generated “faithful propagation” (Figure 2A. bottom left), in that each stimulus generated a propagating action potential. Other parameters resulted in “propagation failure” (Figure 2A, bottom center). Still other sets of parameters produced “tonic activity” when the axon was active either spontaneously or between stimuli. This occurs when the maximal sodium conductance in one or more compartments is high enough to cause tonic spiking. In the simulation shown in Figure 2A (bottom right), the axon was stimulated at 1Hz at one end, but it was tonically active at ~ 50 Hz in this case from an ectopic site (high Na⁺ conductance around this location) of action potential generation that triggered spikes that then propagated in both directions.

Figure 2. Abnormal spike generation and propagation in axons with varied ratios of channel conductance densities.

(A) Examples of faithful propagation (left), propagation failure (middle) and tonic spiking (right) in axon models with uneven channel conductance densities. In the left two plots, models were stimulated at 1 Hz, but there is no stimulus during the time range shown in the right plot. The above panels show the relative distribution of Na⁺ (red) and K⁺ (dark brown) conductance density scaling factors at corresponding sites in the bottom panel (0.5, 1.5, 2.5, 3.5, and 4.5 mm distant from axonal starting end). The dashed line represents a uniform H-H model (${\overline{g}}_{Na}=1$; ${\overline{g}}_K=1$). (B) The proportion of model behaviors with increased variance of channel conductance densities. For each variance value, the left (right) bar corresponds to axon models with uneven (even) distributed conductance densities. Na⁺ and K⁺ conductance density scaling factors were varied by randomly sampling scaling factors from Gaussian distribution functions with different variances (right).

To compare directly the effect of uneven and even channel distributions, we randomly sampled scaling factors for the Na⁺ and K⁺ channel conductance densities (baseline is the H-H model) from Gaussian probability density functions with different variances (0.04, 0.09, 0.16, 0.25). For each variance we simulated a large population of axons (Figure 2B). The mean of the Gaussian function was 1 (i.e., the H-H value) for all groups (Figure 2B, right). The histograms in Figure 2B show that at low variances, most axon models showed faithful propagation, but at the lowest variance, the uneven case was more reliable than the even case, presumably because there were axons in which neighboring compartments were able to compensate for compartments with Na⁺ conductances that were insufficient to maintain faithful propagation. As the conductance density variations increased, the number of cases that showed faithful propagation decreased, and the number of cases that showed tonic spiking dramatically increased, especially in axons with uneven distribution of conductances. At higher variances, cases of propagation failure appeared, most notably in the axons with even distributions of conductances.

Slow Na⁺ channel inactivation in multicompartmental models of axons with even and uneven conductance distributions.

Next, we explored whether incorporating a slow inactivation gate can normalize spike generation and propagation by reducing axonal over-excitability. Because slow inactivation is intrinsically complex, we adopted the simplified formulations of Migliore ³³ (see Methods) to manipulate systematically the inactivation properties and study their effects on the normalization of axonal propagation. Using this model, the steady state inactivation level, and the time constants of inactivation and recovery from inactivation, can be varied relatively independently (Figure 3A).

Figure 3. The normalizing effect of the Na⁺ channel slow inactivation gate.

(A) The steady state inactivation curve (left) and time constants (right) of the $i$ gate. Arrows show parameters whose variations are explored in Figure 4. (B) Typical examples of spike propagation pattern changes before and after switching on the $i$ gate. In each plot, black traces represent Vms at different distances from the stimulation site (as in Figure 2). Gray trace shows the $i$ gate at the distally recorded axonal site. Scale bar of $i$ gate is 0.6 and scale bar of Vm is 100 mV. (C) The proportion of propagation patterns in axon models with varying degrees of conductance density variances, after including the slow inactivation gate. For each variance value, the left (right) bar corresponds to axon models with uneven (even) distributed conductance densities. Color codes the propagation patterns.

We used the same set of models as in Figure 2 to test the impact of slow inactivation on axonal spiking. Slow inactivation obviously can’t increase spiking reliability for those neurons showing faithful propagation or propagation failure (examples not shown). Figure 3B shows three types of responses following the introduction of slow inactivation to axon models that produce tonic spiking. Initially the gate was clamped to 1 in all three examples (slow inactivation was turned off), and they all produced tonic spiking. Once the slow inactivation gate is allowed to change dynamically with spikes, the axonal excitability consistently decreases. If the reduction is in the functional range, tonic spiking will not occur, and axons faithfully spike and propagate only when stimulated (Figure 3B, top). However, if the reduction is insufficient or excessive, the axons may produce either irregular firing (Figure 3B, middle, qualitatively similar phenomena have been observed experimentally³) or propagation failure (Figure 3B, bottom). Statistical results include neurons showing faithful propagation and propagation failure before incorporating the slow inactivation gate.

As shown in Figure 3C, incorporating the slow inactivation gate makes most axon models faithfully spike in response to stimuli. By comparison to Figure 2C, incorporating the slow inactivation gate did not substantially improve the propagation reliability of axon models with even conductance density distributions.

The normalization of spiking by Na⁺ channel slow inactivation is frequency-dependent.

We explored how the three parameters (Figure 3A) of the slow inactivation gate affect its capacity to normalize axonal spike propagation. These parameters represent the steady-state availability of the slow inactivation gate $\left(i_{\min }\right)$, the inactivation time constant (${\tau }_{\text{inact}}$, time constants at depolarized membrane potentials), and the recovery time constant (${\tau }_{\text{recov}}$, time constants at hyperpolarized membrane potentials).

Each model was stimulated at 1 Hz and 10 Hz. The normalizing effect of the slow inactivation gate depends on the stimulation frequency, as seen in the example traces in Figure 4A. This model fired tonically in the absence of slow inactivation. When the slow inactivation was turned on and the axon was stimulated at 1 Hz, the model faithfully followed. However, when the same model was stimulated at 10Hz, propagation continued for about the first second, and then it failed. This occurred as a consequence of the slow inactivation that accumulated because of the 10 Hz firing. Interestingly, after about 2 seconds of silence a single spike was propagated, as the inactivation started to wane.

Figure 4. Frequency-dependent normalizing effect of the slow inactivation gate.

(A) The effect of the slow inactivation gate on the propagation of spikes at different frequencies. Top to bottom, 1 Hz and 10 Hz, respectively. Traces represent Vms at different sites with distance from the stimulation site (as in Figure 2). Before the dashed vertical line, the $i$ gate was clamped to be 1. (B) The complex relationships among the $i$ gate parameters, firing frequency and the normalizing effect. The top and bottom row correspond to stimulation signals at 1 Hz and 10 Hz, respectively. (i) ${\mathbf{\tau }}_{\text{inact}}$ was varied, while $i_{\text{min}}=0.2$ (norm), ${\mathbf{\tau }}_{\text{recov}}=0.5$ (scaled); (ii) $i_{\min }$ (norm) was varied, while ${\mathbf{\tau }}_{\text{inact}}=20\text{ms}$, ${\mathbf{\tau }}_{\text{recov}}=0.5$ (scaled); (iii) ${\mathbf{\tau }}_{\text{recov}}$ (scaled) was varied, while ${\mathbf{\tau }}_{\text{inact}}=20\text{ms}$, $i_{\min }=0.2$ (norm). The axon models have a conductance density variance of 0.25. Color codes propagation patterns.

We randomly generated models that produced tonic spiking without a slow inactivation gate and then systematically explored how inactivation gate parameters affect their normalizing effect on these over-excitable axons. Figure 4B presents a comparison of the effects of frequency on a population of axons with high variance. The first panel shows that as ${\tau }_{\text{inact}}$ increases, the rate of faithful spike propagation decreases for stimuli at 1 Hz, but increases for 10 Hz stimulation. The 10 Hz stimulation results in variable amounts of propagation failure, while the 1 Hz stimuli results in a relatively high proportion of irregular firing. Similar results are seen when $i_{\min }$ is increased. In contrast, when the ${\tau }_{\text{recover}}$ scaling factor increases, faithful propagation increases at 1 Hz but decreases at 10 Hz.

The frequency-dependent normalizing effects essentially reflect a non-trivial trade-off between the kinetics of slow inactivation and the excitability status of the axon. This trade-off is activity-dependent over a wide range of time scales, as the time constant of recovery from the unavailable (slow-inactivated) pool is related to the duration of previous activation by a power law^7–9,21,23.

Concluding remarks.

Different neuronal types show characteristic firing rates, with some that usually fire at very slow rates and others much more quickly. They also have different spike shapes and constellations of ion channels ^34–36. Essentially, slow inactivation reduces membrane over-excitability. In this work, we tested the impacts of slow inactivation on axons with different degrees of uneven channel distribution¹⁴ in the Hodgkin-Huxley model¹. Figure 4 suggests that the role of slow inactivation in normalization of firing rates must match the specifics of the action potential mechanisms and ion channels in each type of neuron ^37–39. Thus, cell type definition should include the specification of the channel properties that can optimally implement normalization by slow inactivation. If a similar study of the role of slow Na⁺ channel inactivation were to be implemented in axonal models with fast transient, A type K⁺ currents, or other types of voltage-gated channels that are sometimes found in axons ¹³, we would predict that the effects of Na⁺ channel inactivation on propagation would be qualitatively similar, but quantitatively different. Despite data indicating that ion channels are unevenly distributed ¹⁴, for most neuronal types detailed descriptions of the spatial distributions of ion channel densities are lacking.

Axonal over-excitability will be influenced by channel distributions as well as axonal diameter⁴⁰. In thin unmyelinated axon models, spontaneous spiking elicited by uneven channel distributions occurs frequently and slow inactivation can efficiently normalize the firing (data not shown). It should be noted that this normalizing effect is not only limited to the uneven distribution caused local over-excitability explored here, but will also apply to neuronal over-excitability because of varied ratios of sodium/potassium channels at the cellular level^25,27–29. In this paper we focused on the effects of slow Na⁺ channel inactivation on the reliability of spike propagation, but it may also function as a leaky integrator to constrain neurons at lower firing rates ⁴¹.

Neuronal resilience can be enhanced in numerous ways, including the specifics of neuronal morphology^35,42. Here, we show another example of enhanced neuronal resilience that depends on the features of ion channel proteins but does not require idiosyncratic tuning. Because we looked at populations of neurons with similar behaviors, the principle illustrated here holds across degenerate, or multiple, solutions. The effects of variable channel densities across extended morphologies can be complex¹⁴, and slow channel inactivation is likely to be an important mechanism that is employed by neurons to enhance normalization and thus resilience. These results demonstrate that robust neuronal firing activities may be a direct result of the properties of proteins, and not demand regulation mechanisms that involve signal transduction, transcription, and translation.

STAR★METHODS

RESOURCE AVAILABILITY

Lead contact

Further information and requests for resources and model codes should be directed to and will be fulfilled by the lead contact, Yunliang Zang (ylzang@brandeis.edu).

Materials availability

No materials were generated for this study.

Data and code availability

• The model parameter data will be automatically generated when running the simulation code.

• All original code has been deposited at ModelDB and is publicly available as of the date of publication. Simulation code is available at https://senselab.med.yale.edu/modeldb/enterCode?model=267610.

• Any additional information required to reanalyze the simulation data reported in this paper is available from the lead contact upon request.

METHOD DETAILS

In this work, all the simulations were implemented in NEURON⁴³ and visualized in MATLAB. The model formulations were the same as in the original Hodgkin-Huxley model and with a new added $i$ gate.

$$I_{Na}={\overline{g}}_{Na}*g_{Na}*m^3*h*i*\left(V-E_{Na}\right)$$ (1)

$$I_K={\overline{g}}_K*g_K*n^4*\left(V-E_K\right)$$ (2)

$$I_{leak}=g_{leak}*\left(V-E_{leak}\right)$$ (3)

$${\alpha }_m=0.1*((V+40)/(1-\exp (-(V+40)/10)))$$ (4)

$${\beta }_m=4*\exp (-(V+65)/18)$$ (5)

$$m_{ss}={\alpha }_m/\left({\alpha }_m+{\beta }_m\right)$$ (6)

$${\tau }_m=1/\left({\alpha }_m+{\beta }_m\right)$$ (7)

$$dm/dt=\left(m_{ss}-m\right)/{\tau }_m$$ (8)

$${\alpha }_h=0.07*\exp (-(V+65)/20)$$ (9)

$${\beta }_h=1/(1+\exp (-(V+35)/10))$$ (10)

$$h_{ss}={\alpha }_h/\left({\alpha }_h+{\beta }_h\right)$$ (11)

$${\tau }_h=1/\left({\alpha }_h+{\beta }_h\right)$$ (12)

$$dh/dt=\left(h_{ss}-h\right)/{\tau }_h$$ (13)

$${\alpha }_i=0.007*\exp (-(V+65)/20)$$ (14)

$${\beta }_i=0.1/(1+\exp (-(V+35)/10)$$ (15)

$$i_{ss}={\alpha }_i/\left({\alpha }_i+{\beta }_i\right)$$ (16)

$${\tau }_i=1/\left({\alpha }_i+{\beta }_i\right)$$ (17)

$$di/dt=\left(i_{ss}-i\right)/{\tau }_i$$ (18)

$${\alpha }_n=0.01*((V+65)/(1-\exp (-(V+55)/10)))$$ (19)

$${\beta }_n=0.125*\exp (-(V+65)/80)$$ (20)

$$n_{ss}={\alpha }_n/\left({\alpha }_n+{\beta }_n\right)$$ (21)

$${\tau }_n=1/\left({\alpha }_n+{\beta }_n\right)$$ (22)

$$dn/dt=\left(n_{ss}-n\right)/{\tau }_n$$ (23)

$g_{Na}=120\text{mS}/{\text{cm}}^2$; $g_K=36\text{mS}/{\text{cm}}^2$; $g_{leak}=0.3\text{mS}/{\text{cm}}^2$; $E_{Na}=50\text{mV}$; $E_K=-77\text{mV}$; $E_{leak}=-54.4\text{mV}$. ${\overline{g}}_{Na}$ and ${\overline{g}}_K$ are scaling factors sampled from Gaussian probability density function. V is in mV and reaction rates are in ms⁻¹. In all models, axial resistivity Ra is 100 Ωcm and Cm is 1 μF/cm². We used two different formulations for the slow inactivation gate. In Figure 1, the slow inactivation gate shown above was simulated by scaling the h-gate kinetics by 0.1. In Figures 2–4, all the formulations are the same except new $i$ gate formulations are according to Migliore ³³. The impact of slow inactivation is independent of the specific model formulations.

$$i_{ss}=1/(1+\exp ((V+58)/2))+i_{\min }*(1-1/(1+\exp ((V+58)/2)))$$ (24)

$${\tau }_i=\exp (0.09*(V+60))/(0.0003*(1+\exp (0.45*(V+60))))$$ (25)

$$di/dt=\left(i_{ss}-i\right)/{\tau }_i;\text{If}{\tau }_i\le {\tau }_{inact},{\tau }_i={\tau }_{inact}$$ (26)

${\tau }_{inact}$ is 20 ms except in Figure 4. In Figure 4, $i_{ss}$ was varied by changing $i_{min}$; the inactivation time constant at depolarized membrane potentials was varied by changing ${\tau }_{inact}$; the recovery time constant at hyperpolarized membrane potentials were varied by a scaling factor, because of its voltage dependence implemented in the model. The axon model has a length of 5 mm. We simulated axonal diameter in the range of 0.1 – 1 μm (only 1 μm data were shown), with the segment length of 6 – 120 μm. To avoid the boundary effect, in the beginning part of the model, we added an axon initial segment with the standard Hodgkin-Huxley model channel densities evenly distributed and the stimuli were exerted at the starting point of the axon initial segment. We only analyzed spike propagation in the first 4.5 mm of the axon to avoid potential boundary effect in the distal end.

QUANTIFICATION AND STATISTICAL ANALYSIS

All statistical analyses were conducted using MATLAB and are detailed in the Results.

KEY RESOURCES TABLE.

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Deposited code
Code for performing simulations	This paper	Available via ModelDB at https://senselab.med.yale.edu/modeldb/enterCode?model=267610
Software and algorithms
NEURON 8.1.0	https://www.neuron.yale.edu/neuron/	version 8.1.0
MATLAB R2022a	Math Works	RRID: SCR_001622

Highlights.

• Variation in ion channel density in axons may compromise axonal spike propagation.

• Na⁺ channel slow inactivation increases the reliability of spike propagation.

• Normalization by slow inactivation can compensate for uneven channel distributions.

• The normalization effect depends on inactivation kinetics and firing frequency.