A novel HH-based mathematical model for resurgent Na⁺ current

As noted earlier, the total I_Na in our model has the novel resurgent component, I_NaR; the transient and persistent components are similar to our previous report [8]. Fig 3A (left panel) illustrates a well-accepted mechanism of Na⁺ resurgence [30], wherein a putative blocking particle occludes an open channel following brief depolarization such as during an action potential; subsequently upon repolarization, a voltage dependent unblock results in a resurgent Na⁺ current. Our I_NaR formulation recapitulates this unusual behavior of Na⁺ channels using nonlinear ordinary differential equations for a blocking variable (b_r) and a competing inactivation (h_r) (see Methods). Different from a traditional activation variable of an ionic current in HH models, we formulated the I_NaR gating using a term (1−b_r) to enable an unblocking process (see I_NaR equation in Methods and in Fig 3A). Here, the model variable b_r reflects the fraction of channels in the blocked state at any instant, and ‘1’ denotes the maximum proportion of open channels, such that (1−b_r) represented the fraction of unblocking channels. Such a formulation enabled mimicking the open-channel unblocking process as follows: Normally the b_r is maintained ‘high’ such that (1−b_r) term is small and therefore no I_NaR flows, except when a spike occurs, which causes b_r to decay in a voltage-dependent manner. This turns on the I_NaR which peaks as the membrane voltage repolarizes to ~ -40 mV, following which the increasing inactivation variable h_r gradually turns off the I_NaR. The steady-state voltage dependency of unblock, (1−br_∞(V)), and the competing inactivation (hr_∞(V)) in the model are shown in Fig 3A (middle panel), along with the equation for I_NaR; the magenta shaded region highlights the voltage-dependency of I_NaR activation as posited to occur during open-channel unblocking. In Fig 3A (right panel), we show simulated I_NaR (in magenta), peaking during the recovery phase of spikes (in black). In Fig 3B (I.), we reproduced experimentally observed I_Na during voltage-clamp recording and highlight the resurgent component in both model (left) and experiment (right), (inset shows experimental protocol; also see legend and Methods). A comparative current-voltage relationship for the model and experiments is shown in Fig 3B (II.); also see S3 Fig for detailed kinetics of model I_NaR. Taken together, the above tests and comparisons ensured the suitability of our model for further investigation of I_NaR mediated burst control.

Resurgent and persistent Na⁺ currents offer push-pull modulation of spike/burst intervals

Given that I_NaR is activated during the recovery phase of an action potential, physiologically, any resulting rebound depolarization may control the spike refractory period, and increase spike frequency and burst duration [22, 37]. We tested this by selectively increasing the maximal resurgent conductance g_NaR in our model neuron simulation and validated the predictions using dynamic-clamp experiments as shown in Fig 4A and 4B. In parallel, we also exclusively modified the maximal persistent conductance g_NaP using model simulations and verified the effects using dynamic-clamp experiments as shown in Fig 4C and 4D. We only focused on rhythmically bursting Mes V neurons and quantified the burst timing features including the inter-burst intervals (IBIs), burst duration (BD) and inter-spike intervals (ISIs) as illustrated in the boxed inset in Fig 4. Figure panels 4e –j show a comparison of the exclusive effects of g_NaR versus g_NaP on each of these burst features. These experimental manipulations using model currents and quantification of resulting burst features revealed significant differences and some similarity between the action of g_NaR and g_NaP in burst control. First, increases in g_NaR resulted in longer IBIs, which was in contrast with the effects of the persistent Na⁺ conductance, g_NaP, which decreased IBIs (effect highlighted with red double arrows in Fig 4A–4D and quantified using box plots in Fig 4E and 4H). Alternatively, increasing g_NaR reduced ISIs, while g_NaP had the opposite effect on these events, resulting in an overall increase in ISIs with g_NaP increases (see Fig 4F and 4I); whereas g_NaR and g_NaP had similar effect in increasing BDs (see Fig 4G and 4J). Box plots in Fig 4E–4J show 1^st, 2^nd (median) and 3^rd quartiles; error bars show 1.5x deviations from the inter-quartile intervals. For each case (or cell), n values represented events (ISI, IBI and BD) and reported here are within-cell effects for different g_NaR and g_NaP applications during 20 sec step-current stimulation. For the g_NaR applications (or series), IBI mean ± std were 210.49 ± 69.33 (control, n = 7), 450.29 ± 116.56 (1x, n = 26) and 1074.06 ± 199.17 (2x, n = 11), and for g_NaP applications, these values were 1448.71 ± 450.92 (control, n = 4), 910.10 ± 527.27 (1x, n = 8) and 644.35 ± 234.19 (1.5x, n = 6). For g_NaR series, ISI mean ± std were 20.26 ± 1.65 (control, n = 59), 12.56 ± 2.25 (1x, n = 646) and 10.56 ± 1.81 (2x, n = 783), and for g_NaP series these values were 12.76 ± 1.13 (control, n = 159), 12.57 ± 1.05 (1x, n = 611) and 13.56 ± 1.42 (1.5x, n = 503). For g_NaR series, BD mean ± std were 176.36 ± 69.57 (control, n = 8), 300.56 ± 77.90 (1x, n = 27) and 671.16 ± 285.96 (2x, n = 12), and for g_NaP series these values were 346.59 ± 88.39 (control, n = 4), 571.30 ± 197.56 (1x, n = 8) and 1334.20 ± 592.59 (1.5x, n = 7). Treatment effects and group statistics for all the replicates showing the above effects across six Mes V bursting neurons, each from a different animal are summarized in S2 Table. A one-way ANOVA was used to test the treatment effects of g_NaR and g_NaP applications, and when significant, a post hoc two-sample Student t-test was used for group comparisons between control and 1x, 1x and 2x, and control and 2x for g_NaR cases, and between control and 1x, 1x and 1.5x, and control and 1.5x for g_NaP cases. Asterisks above box plots between groups in Fig 4E–4J indicate p<0.05 using two-sample Student t-tests for post-hoc comparisons. Furthermore, as shown in Fig 4, the model simulations predicted consistent effects (note white circles denote predicted values in panels e—j) with dynamic-clamp experiments, making the model suitable for further analysis of I_NaR/I_NaP mediated mechanism of burst control (see white circles in panels 4e - j). In two additional bursting neurons, we also conducted g_NaR and g_NaP subtraction experiments which showed consistent reverse effects of additions (see S4 Fig and legend). In the g_NaP subtraction experiment, note that a -2x g_NaP resulted in the abolition of bursting and subthreshold oscillations as shown in the figure. Such an effect was reproduced in the neuron model by setting g_NaP = 0, as shown in S4C Fig.

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Fig 4. Physiological consequences of I_NaR and I_NaP on burst discharge.

a, c) Simulated membrane voltage demonstrating the effect of g_NaR addition (a), and g_NaP increase (c) on burst discharge. b, d) Dynamic-clamp experimental traces showing effects of real-time addition of g_NaR (b) and g_NaP (c) in intrinsically bursting sensory neurons; the red double arrows highlight the opposite effects of increases in g_NaR and g_NaP on inter-burst intervals in panels (a–d). The boxed inset (left) indicates color coding of traces for panels e–j. The boxed inset (right) highlights burst features quantified in panels (e–j): inter-burst intervals (IBI), burst duration (BD), and inter-spike intervals (ISI. e–j) Box plots showing IBIs (e, h), BDs (f, i), and ISIs (g, j) for application of a series of g_NaR or g_NaP conductance values (see Results for statistics). Model values of 1x g_NaR and g_NaP are adjusted to match experimental data for 1x application of the corresponding conductance tested; C is control, values of g_NaR used for dynamic-clamp are 2 and 4 nS/pF for 1x and 2x respectively, and, values of g_NaP used are 0.25 and 0.375 nS/pF for 1x and 1.5x respectively. Asterisks indicate a p < 0.05 using a Student t-Test for group comparisons.

https://doi.org/10.1371/journal.pcbi.1007154.g004

To test whether I_NaR and I_NaP differentially modulate the regularity and precision of spike timing we observed the inter-event intervals (or IEIs) during real-time addition of g_NaR and g_NaP. As shown in Fig 5A and 5B, addition of g_NaR improved the regularity of the two types of events: the longer IBIs and the shorter ISIs, whereas, addition of g_NaP did not show such an effect. Note that application of real-time g_NaR improved both the timing precision and regularity of IBIs (compare left and right panels of Fig 5A), while application of dynamic-clamp g_NaP did not seem to affect either of these features (compare left and right panels of Fig 5B). Furthermore, in Fig 5C we highlight that g_NaR application also improved spike-to-spike regularity of ISIs within bursts (shown are two representative bursts for g_NaR (Fig 5C, left panel) versus g_NaP (Fig 5C, right panel) application from Fig 5A and 5B (see figure legend). Taken together, these opposite effects of persistent and resurgent Na⁺ currents arising from a single Na⁺ channel may act in concert to offer a push-pull modulation of burst timing regulation.

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Fig 5. Spike timing regularity monitored by I_NaR.

a, b) Time series graphs showing inter-event intervals (IEIs) under control condition (left panels in both a, b), and dynamic-clamp addition of g_NaR (right panel in a), and g_NaP (right panel in b). The green and cyan square brackets in each panel highlight the IBI and ISI range respectively. The y-axis shows IEIs on a log scale. Magenta boxes demarcate a representative burst in each case, which is expanded in the graphs in c), corresponding with the bold letters, A, B, C and D. The solid lines in (c) are power regression fit in each case; in all the panels, black open diamonds are control, maroon open diamonds correspond with g_NaR application and blue open diamonds correspond with the g_NaP application.

https://doi.org/10.1371/journal.pcbi.1007154.g005

Resurgent Na⁺ aids stabilization of burst discharge

The effect of persistent g_NaP in reducing IBIs can be explained by its subthreshold activation [11], wherein increases in g_NaP promotes STO and burst initiation which in turn increases burst frequency and reduces IBIs. Additionally, a high g_NaP together with its slow inactivation helps maintain depolarization which can increase BDs. During spiking, I_NaP inactivation slowly accumulates and can contribute to any spike frequency adaptation as a burst terminates. Further increases in g_NaP can accentuate such an effect and lead to increases in spike intervals within a burst. In contrast, the rebound depolarization produced due to I_NaR can decrease ISIs and promote spiking which prolongs the BDs. An effect which is not immediately obvious is an increase in IBI due to increases in g_NaR since IBIs are order of magnitude slower events than rise/decay kinetics of I_NaR. Moreover, this current is not active during IBIs. It is likely that I_NaR which promotes spike generation produces its effect on lengthening the IBIs by further accumulation of I_NaP inactivation during spiking and in turn reducing channel availability immediately following a burst. We examined this possibility using model analysis of I_NaR’s effect on modulating the slow I_NaP inactivation/recovery variable, h_p.

The simulated membrane potential (grey traces) and the slow I_NaP inactivation/recovery variable, h_p (overlaid magenta traces) of the model neuron under three conditions shown in Fig 6A–6C: 1) with default values of g_NaR and g_NaP (Fig 6A), 2) an 1.5x increase in g_NaP (Fig 6B), and, 3) a 2x increase in g_NaR (Fig 6C). The peak and trough of the slow inactivation/recovery variable, h_p correspond to burst onset and offset respectively. Comparing these traces in the three panels, we note that an increase in g_NaR effectively lowers the h_p value at which burst terminates (see curvy arrow in Fig 6C and legend). This observation was further supported by estimations of theoretical thresholds for burst onset and offset for increasing values of g_NaR (Fig 6D) and, similar thresholds for increasing values of g_NaP are provided for comparison in Fig 6E (see legend and S1 Text for details). Note that changes in g_NaR did not alter the burst onset thresholds, consistent with a lack of resurgent current before spike onset (see brown arrow indicating burst onset threshold in Fig 6D). In contrast, increasing g_NaR, consistently lowered the threshold values of slow inactivation/recovery for burst offset (see highlighted dashed box with arrows pointing to the burst offset thresholds decreasing with increasing g_NaR values in Fig 6D). The net effect is longer recovery time between bursts and therefore prolonged IBIs. Additionally, in Fig 6D, an increase in g_NaR extended the range of slow inactivation/recovery for which stable spiking regime exists (marked by the green circles). This gain in stability is indicative of a negative feedback loop in the Na⁺ current gating variables which model the slow inactivation and the open-channel unblock process. As shown in Fig 6F (see boxed inset), during a burst, presence of a channel unblocking process and the resulting resurgent Na⁺ can lead to further accumulation of slow inactivation (a positive effect) which eventually shuts off the unblocking events with further inactivation (negative feedback), and terminates a burst. The schematic on the left in Fig 6F summarizes a negative feedback loop between the unblocking and slow inactivation processes of Na⁺ currents which could mediate the stabilization of burst discharge as described above.

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Fig 6. Analysis of a mechanism of action of I_NaR and I_NaP.

a-c) The slow I_NaP inactivation/recovery variable, h_P is overlaid (magenta) on membrane voltage traces (grey) under default (a), 1.5x g_NaP (b), and 2x g_NaR (c) conditions; the dark blue dashed lines in (a-c) indicate the maximum and minimum values of the persistent inactivation variable under default condition; the light green arrow in (b), highlights a reduced peak recovery required for burst onset; the light blue curvy arrows in (b, c) indicate further accumulation of slow inactivation before burst termination. d, e) Bifurcation diagrams showing the steady states and spiking regimes of the membrane voltage (V) and slow I_NaP inactivation/recovery (h_p). The red solid lines represent resting/quiescence states consistent with low values of h_p. The meeting points of stable equilibria (red) and unstable equilibria (black solid lines) represents the theoretical threshold for burst onset, the Hopf bifurcation point (see S1 Text). The blue open circles are the unstable periodics that form region of attraction on either side of the stable equilibria for sub-threshold membrane voltage oscillations; the meeting point of the curve of unstable periodics with the stable periodics (green filled circles) represents the theoretical threshold for burst offset/termination, saddle node of periodics (see S1 Text). The dashed boxes in (d) and (e) magnify the burst offset thresholds due to increases in g_NaR (d) and g_NaP (e); brown dashed arrows in (e) highlight shifts in burst onset thresholds due to g_NaP increases; 1X g_NaR = 3.3 nS/pF and 1X g_NaP = 0.5 nS/pF. f) Schematic showing a negative feedback loop between the unblocking process and the slow inactivation of the Na⁺ currents. Inset shows membrane potential (top), blocking variable, b_r (middle) and slow inactivation, h_r (bottom), illustrating the negative feedback mechanism.

https://doi.org/10.1371/journal.pcbi.1007154.g006

Resurgent Na⁺ current offers noise tolerance

Next, we examined whether the stability of burst discharge offered by the presence of I_NaR can also contribute to noise tolerance. During quiescence/recovery periods between bursts, the membrane voltage is vulnerable to perturbations by stochastic influences, which can induce abrupt spikes and therefore disrupt the burst timing precision. We introduced a Gaussian noise to disrupt the rhythmic burst discharge in the model neuron when no g_NaR was present as shown in Fig 7A and 7B. Subsequent addition of g_NaR restored burst regularity (Fig 7C). Model analyses in the (h_p,V) plane provides insight into the mechanism of I_NaR mediated noise tolerance. Briefly, we project a portion of a (h_p,V) trajectory corresponding to the termination of one burst until the beginning of the next (see expanded insets in Fig 7A, 7B and 7C) to the (h_p,V) diagrams shown in Fig 7D, 7E and 7F respectively (see S1 Text for details). In Fig 7D, beginning at the magenta circle, the (h_p,V) trajectory (magenta trace) moves to the right as h_p recovers during an IBI, until a burst onset threshold is crossed; point where the blue circles meet the red and black curves (see S1 Text for details), and eventually bursting begins; see upward arrow marking a jump-up in V at the onset of burst. During a burst, while V jumps up and-down during spikes, h_p moves to the left as slow inactivation accumulates during bursting (left arrow). Finally, when h_p reduces sufficiently, (h_p,V) gets closer to the burst offset threshold (points at which the green and blue circles meet), and the burst terminates (down arrow). What is key in this figure is that the IBI is well-defined as the time period in which the (h_p,V) trajectory moves along the red curve of steady states during the recovery process and moves past the burst onset threshold until a burst begins. However, when stochastic influences are present, the recovery period near-threshold is subject to random perturbations in V and can cause abrupt jump-up/spikes during the recovery period (see expanded inset in Fig 7B). Projecting (h_p,V) during this period on to Fig 7E, we note that the near-threshold noise amplitudes can occasionally push the (h_p,V) trajectory (magenta) above a green region of attraction and this results in such abrupt spikes. Now, when g_NaR is added, the apparent restoration of burst regularity (see Fig 7C) can be attributed to an expansion in this green shaded region as shown in Fig 7F (see arrow pointing to a noise-tolerant region). In this situation, near-threshold random perturbations have less of an effect during the recovery process to induce abrupt spikes. This way, the net effect of I_NaR on slow Na⁺ inactivation prevents abrupt transitions into spiking regime following burst offset and in turn contributes to burst refractoriness and noise tolerance. We suggest that such a mechanism can make random fluctuations in membrane potential less effective in altering the precision of bursts and therefore aid information processing.

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Fig 7. Resurgent Na⁺ current offers burst stability and noise modulation.

a—c) Simulated membrane voltage shows neural activity patterns without any g_NaR (a), with additive Gaussian noise input (b), and with subsequent addition of g_NaR (c). Expanded regions in (a-c) show membrane voltage sub-threshold oscillations (STO) during an inter-burst interval. Also overlaid is the evolution of the slow sodium inactivation/recovery variable; arrow indicates recovery during IBI, and magenta circle marks an arbitrary time point used to track these trajectories in (d—f). d—f) Bifurcation diagrams with projected trajectories of (h_∞,V), shown in magenta to highlight the effect of addition of noise near sub-threshold voltages (e) and then the enlarged region of noise tolerance (highlighted green shaded region) due to addition of g_NaR in (f). The magenta circle marks the beginning time points of each trajectory.

https://doi.org/10.1371/journal.pcbi.1007154.g007

Resurgent Na⁺ moderates burst entropy

The spike/burst intervals, their timing precision and order are important for information coding [38–42]. Given our prediction that I_NaR can offer noise tolerance and stabilize burst discharge, we examined whether it can reduce uncertainty in spike/burst intervals and restore order in burst discharge. We tested this using model simulations and dynamic-clamp experiments as shown in Fig 8A–8D. In both cases, as shown by spike raster plots in Fig 8E and 8F, we disrupted the inter-event intervals (IEIs) by additive White/Wiener noise input while driving rhythmic burst discharge using step depolarization (also see Methods). Subsequent addition of g_NaR conductance restored the regularity of rhythmic bursting. We used Shannon’s entropy as a measure of uncertainty in IEIs and show that increases in entropy due to noise addition was reduced to control levels by subsequent increases in g_NaR as shown in Fig 8G (see Methods). We also quantified the Coefficient of Variation (CV) and noted that adding noise which shortened IBIs, indeed decreased the CV, due to a reduction in the standard deviation (s.d.) of the IEI distribution. Subsequent addition of I_NaR, which significantly lengthened the IBIs, resulted in increases in CV values due to an increase in IEI s.d. Taken together, I_NaR moderated burst entropy and improved the regularity of spike/burst intervals.

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Fig 8. Resurgent Na⁺ current moderates the entropy of burst discharge.

a) Schematic showing the experimental setup for real-time application of I_NaR and a stochastic input, I_Noise (see Methods); the dynamic-clamp current, I_dyn is the instantaneous sum of I_NaR, a step current (I_step) and I_Noise. b) Simulated time series of two stochastic noise profiles used to disrupt the rhythmic burst discharge in Mes V neurons; I_{white−noise} was generated from a uniformly distributed random number and I_{wiener−noise} was generated from a normally distributed random number (see Methods). c–d) Raster plots showing patterns of inter-event intervals (IEIs) for the different conditions shown in the model (e), and during dynamic-clamp (f). e–f) Time series of IEI shown on a log scale for the different conditions shown in the model (e), and during dynamic-clamp (f). g–h) Shannon entropy (H) and coefficient of variability (CV) measured for IEIs under the different conditions presented in (c) and (d). Plotted circles for the model represent an average across 10 trials, while individual trials are presented for the data points from two cells. In both (g) and (h), C: control, N: after addition of random noise, 1x and 2x are supplements in g_NaR values.

https://doi.org/10.1371/journal.pcbi.1007154.g008

Stabilization of burst discharge by I_NaR

In contrast with I_NaP, which drives near threshold behavior and burst generation, I_NaR facilitated slow channel inactivation as bursts terminate (also note [11]). Increased channel inactivation due to I_NaR in turn prolonged the recovery from inactivation required to initiate subsequent burst of activity. Such an interaction between open-channel unblock process underlying I_NaR, and, the slow inactivation underlying I_NaP, offer a closed-loop push-pull modulation of ISIs and IBIs. Specifically, presence of I_NaR facilitates slow Na⁺ inactivation as shown by our theoretical analyses of model behavior; such enhanced slow channel inactivation can eventually shut off channel opening and unblocking. This resulted in burst stabilization. Theoretically, this represented an enlarged separatrix (or boundary) for transitioning from a sub-threshold non-spiking behavior to spiking behavior (see enlarged green shaded region in Fig 7F), and the neuron becomes refractory to burst generation and hence offers noise tolerance.

Is this apparent effect of I_NaR physiologically plausible? Biophysical studies indicate that recovery from fast inactivation is facilitated in sodium channels that can pass resurgent current [30]; as shown here, this appears to be true for recovery from slow inactivation as well. Consistently, in the SCN8a knockout Med mouse, which lack the Nav 1.6 sodium channel subunit, recordings from mutant cells showed an absence of maintained firing during current injections, limited recovery of sodium channels from inactivation, and failure to accumulate in inactivated states. This is attributed to a significant deficit in I_NaR [11, 20, 37]. Furthermore, maintained or repeated depolarization can allow a fraction of sodium channels in many neurons to enter inactivation states from which recovery is much slower than for normal fast inactivation (reviewed in [43]). Here, our simulations and model analyses predict that the presence, and increase in I_NaR conductance, provides for a such a physiological mechanism to maintain sustained depolarization and promote fast and slow Na⁺ inactivation.

Sodium currents and signal processing in neurons

Neuronal voltage-gated Na⁺ currents are essential for action potential generation and propagation [5]. However, to enable fight-or-flight responses, an overt spike generation mechanism must be combined with noise modulation to extract behaviorally relevant inputs from an uncertain input space. Here we show that, the voltage-gated Na⁺ currents can serve an important role in neural signal processing (see summary in Fig 9). As shown in the figure, a sub-threshold activated persistent Na⁺ current contributes to membrane resonance, a mechanism of bandpass filtering of preferred input frequencies [9]. We call this type of input gating, which is widely known to be important for brain rhythms [9, 41], a tune-in mechanism (see figure legend). In some cases ambient noise or synaptic activity can amplify weak inputs and promote burst generation [44, 45]. This way, a tune-in mechanism such as the persistent Na⁺ current can contribute to weak input detection and promote burst coding [46, 47]. Then again, during rhythmic bursting, presence of resurgent Na⁺ maintains the order and precision of the timing events of bursts while preventing abrupt transitions into spiking phase due to stochastic influences as shown here. During ongoing sensory processing, such burst timing regulation can provide for noise cancellation or what we call a tune-out mechanism, which can mitigate random irregularities encoded in bursts (see Fig 9 and legend). Whether this leads to improved sensory processing in the presence of natural stimuli and/or sensorimotor integration during normal behaviors needs to be validated. Our biological prediction here that a sensory neuron can utilize these voltage-gated Na⁺ currents as a tune-in-tune-out mechanism to gate preferred inputs, attenuate random membrane fluctuations and prevent abrupt transitions into spiking activity supports such a putative role.

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Fig 9. A consolidated role for persistent and resurgent Na⁺ currents in information processing.

The schematic shows the broadband input space of a sensory neuron which can produce rhythmic bursting activity. The Nav1.6-type Na⁺ channels mediating the persistent Na⁺ current (schematized as a listening ear) enhance neuronal resonance and tune in relevant input frequencies by bandpass filtering (f_resonant refers to the resonant frequency of the neuron). Such selected inputs carrying behaviorally relevant stimuli are encoded as bursts but may be prone to uncertainty due to intrinsic or extrinsic stochastic influences. The resurgent Na⁺ current mediated by the Nav1.6 Na⁺ channels improve the regularity of spike/burst timing by mitigating the effects of noise by tuning out random spikes between bursts (schematized as a noise cancelling headphone), thus aiding in neural signal processing.

https://doi.org/10.1371/journal.pcbi.1007154.g009

Neuron model for bursting activity

The conductance-based Mes V neuron model that we used to investigate the physiological role for I_NaR and I_NaP components of I_Na in burst discharge, incorporates a minimal set of ionic conductances essential for producing rhythmic bursting and for maintaining cellular excitability in these neurons [8]. These include: 1) a potassium leak current, I_leak, 2) sodium current, I_Na as described above, and, 3) a 4-AP sensitive delayed-rectifier type potassium current (I_K) [8, 48]. The model equations follow a conductance-based Hodgkin-Huxley formalism [5] and are as follows. In what follows, we provide the formulation for each of the ionic currents and describe in detail, the novel I_NaR model.

Voltage-gated sodium currents

In vitro action potential clamp studies in normal mouse Mes V neurons, and voltage-clamp studies in Nav1.6 subunit SCN8a knockout mice have demonstrated existence of three functional forms of the total sodium current, I_Na, including the transient (I_NaT), persistent (I_NaP) and resurgent (I_NaR) components [11, 14]. Each of these currents is critical for Mes V electrogenesis including burst discharge, however, their exclusive role is yet unclear. Lack of suitable experimental model or manipulation to isolate each of these TTX-sensitive components, led us to pursue an alternative approach involving computational model development of the physiological I_Na. To further allow model-based experimental manipulation of individual components of the I_Na, we designed a conductance-based model as follows. Although a single Nav1.6 channel can produce all three I_Na components observed experimentally, we used a set of three HH-type conductances, one for each of the transient, the persistent, and the resurgent components. This allowed us to easily manipulate these components independently to test their specific role in neural burst control. The equation for the total sodium current can be written as: where,

The maximal persistent conductance, g_NaP was set 5–10% of the transient, g_NaT [49] and the resurgent was set to 15–30% of g_NaT, based on the relative percentage of maximum I_NaR and I_NaT as revealed by voltage-clamp experiments shown in Fig 3; E_Na is the Na⁺ reversal potential.

Based on experimental data, the gating function/variable, m_t∞(V), and h_t, for I_NaT, and, m_p∞(V), and, h_P, for I_NaP are modeled as described in [8]. The rate equations for the inactivation gating variables h_t, and, h_P, model the fast and slow inactivation of the transient and persistent components respectively. The activation gates are steady-state voltage-dependent functions, consistent with fast voltage-dependent activation of I_Na.

Steady-state voltage-dependent activation and inactivation functions of transient sodium current respectively include: Steady-state activation, inactivation and steady-state voltage-dependent time constant of inactivation for persistent sodium current respectively include:

The novel I_NaR formulation encapsulates the block/unblock mechanism using a block/unblock variable (b_r), and, a second hypothetical variable for a competing inactivation, which we call, h_r. We call this a hybrid model, to highlight the fact that the model implicitly incorporates the history or state-dependent sodium resurgence, following a transient channel opening, and combines this into a traditional Hodgkin-Huxley type conductance-based formulation. In the and rate equations for b_r, and, h_r, the block/unblock variable, b_r increases or grows according to the term, α_b(1−b_r)br_∞(V), and decays as per the term, k_bβ_br(V)b_r, described as follows:

α_b(1−b_r)br_∞(V): In this growth term, we incorporate state-dependent increase in b_r, as follows; we assume that the rate of increase in b_r is proportional to the probability of channels currently being in the open state, with a rate constant, α_b which we call ‘rate of unblocking’; such probability is a function of the membrane voltage given by, br_∞(V), defined as below:

The term (1−br_∞(V)), models the steady-state voltage-dependency guiding the unblocking process. The channels being in open state is represented by the term, (1−b_r). Note that if (1−b_r) is close to 1, this means that larger proportion of channels are in an open state, and therefore b_r grows faster, promoting blocking. We modeled br_∞(V) as a decreasing sigmoid function, such that, at negative membrane potentials, channels have a high probability to enter future depolarized states and therefore, (1−b_r)~0, in turn, b_r does not grow fast.

k_bβ_br(V)b_r: In this decay term, we assume that the rate of decay of b_r, is proportional to the probability of channels being in the blocked state, with a constant of proportionality k_b, and, this probability is given by a voltage-dependent function, β_br(V), defined as below: Note that, β_br(V) gives a high probability at depolarized potentials, indicating a blocked state and enables decrease in b_r in subsequent time steps.

Taken together, b_r, represents a phenomenological implementation of a previously described block/unblock mechanism of a cytoplasmic blocking particle [19] (see schematic of channel gating in Fig 3A). Additionally, a hypothetical competing inactivation variable, h_r, sculpts the voltage-dependent rise and decay times and peak amplitude of sodium resurgence at -40 mV following a brief depolarization (i.e., transient activation), as observed in voltage-clamp experiments (see Fig 3B). The functions, α_hr(V),β_hr(V) and hr_∞(V) are defined as voltage-dependent rate equations that guide the voltage-dependent kinetics and activation/inactivation of the I_NaR component as given below.

The steady-state voltage-dependency of the competing inactivation necessary to generate a resurgent Na⁺ current is defined as follows: The voltage-dependent rate functions of such inactivation is defined by two functions as follows: The steepness of the voltage-dependent sigmoid functions for activation and inactivation were tuned to obtain the experimentally observed I_NaR activation (see Fig 3; also see [11, 14, 30]). To obtain the kinetics (rise and decay times) of I_NaR comparable to those observed during voltage-clamp experiments (see S3 Fig), the model required three units for the blocking variable ((1−b_r)³) and five units for the inactivation variable () (see I_NaR equation). Together, the modeled I_Na reproduced the key contingencies of the Nav1.6 sodium currents (see S2 Fig) [18, 30, 50].

Sensitivity analyses was conducted for the key parameters of I_NaR gating including α_b, and, k_b. Note that these two parameters control the rate of blocking. As expected, increasing α_b, that controls rate of increase in b_r, decreased the peak amplitude of I_NaR, similar to an experimental increase in block efficacy by a β-peptide (e.g., [19]). On the other hand, k_b also moderates b_r, and increasing k_b, enhances b_r decay rate, that significantly enhanced I_NaR, and, therefore burst duration (not shown). Large increases in k_b significantly enhanced I_NaR, and indeed transformed bursting to high frequency tonic spiking. However, the effects of I_NaR on bursting described in the results section were robust for a wide range of values of these parameters (>100% increase from default values), and, for our simulations, the range of values, α_b = 0.08 to 0.1, k_b = 0.8 to 1.2, were used to reproduce Mes V neuron discharge properties. To reproduce experimentally observed spike width, we additionally tuned the inactivation time constant, τ_t = 1.5±0.5, for I_NaT.

Potassium and leak currents

The 4-AP sensitive delayed-rectifier type potassium current, I_K, and the leak current, I_leak were modeled similar to [8] as below; also see [48]. where, the steady-state voltage-dependent activation function for the gating variable, n is given as: E_K and E_leak are K⁺ and leak reversal potentials respectively. Model parameter values used are provided in S1 Table.

Brain slice preparation

All animal experiments were performed in accordance to the institutional guidelines and regulations using protocols approved by Animal Research Committee at UCLA. Experiments were performed in P8-P14 wild-type mice of either sex. Mice were anesthetized by inhalation of isofluorane and then decapitated. The brainstem was extracted and immersed in ice-cold cutting solution. The brain-cutting solution used during slice preparation was composed of the following (in mM): 194 Sucrose, 30 NaCl, 4.5 KCl, 1.2 NaH₂PO₄, 26 NaHCO₃, 10 glucose, 1 MgCl₂. The extracted brain block was mounted on a vibrating slicer (DSK Microslicer, Ted Pella) supported by an agar block. Coronal brainstem sections consisting of rostro-caudal extent of Mes V nucleus, spanning midbrain and pons were obtained for subsequent electrophysiological recording.

Voltage-clamp electrophysiology

To obtain direct experimental data to drive I_NaR model development, we performed voltage-clamp experiments on Mes V neurons and recorded Na⁺ currents by blocking voltage-gated K⁺ and Ca²⁺ currents similar to [11]. The pipette internal solution contained the following composition (in mM): 130 CsF, 9 NaCl, 10 HEPES, 10 EGTA, 1 MgCl₂, 3 K₂-ATP, and 1 Na-GTP. The external recording solution contained the following composition (in mM): 131 NaCl, 10 HEPES, 3 KCl, 10 glucose, 2 CaCl₂, 2 MgCl₂, 10 tetraethylammonium (TEA)-Cl, 10 CsCl, 1 4-aminopyridine (4-AP), and 0.3 CdCl₂. The voltage-clamp protocol consisted of a holding potential of -90 mV followed by a brief voltage pulse (3 ms) of +30 mV, to remove voltage-dependent block, followed by voltage steps between -70 mV to -10 mV, in steps of 10 mV for ~ 100 ms to activate I_NaR, and then returned to -90 mV. A 1 μM TTX abolished the Na⁺ current and the residual leak current was subtracted to isolate evident sodium currents. Recordings with series resistance R_series>0.1R_m were discarded, where R_m is the input resistance of the neuron; we did not apply any series resistance compensation.

Dynamic-clamp electrophysiology

Dynamic-clamp electrophysiology and in vitro current-clamp recording were used for testing the physiological effects of Na⁺ currents on burst discharge as well as noise-mediated entropy changes corrected by I_NaR [51]. We selected neurons responding with a bursting pattern in response to supra-threshold step current injection in the Mes V nucleus in brainstem slice preparation for our study; >50% of neurons showing other patterns (e.g., tonic or single spiking cells) were discarded. Dynamic-clamp was successfully performed in bursting cells (n = 10). For dynamic-clamp recording, slices were placed in normal ACSF at room temperature (22–25°C). The ACSF recording solution during patch-clamp recording consisted of the following (in mM): 124 NaCl, 4.5 KCl, 1.2 NaH₂PO₄, 26 NaHCO₃, 10 glucose, 2 CaCl₂, 1 MgCl₂. Cutting and recording solutions were bubbled with carbogen (95% O₂, 5% CO₂) and maintained at pH between 7.25–7.3. The pipette internal solution used in current clamp experiments was composed of the following (in mM): 135 K-gluconate, 5 KCl, 0.5 CaCl_2, 5 HEPES (base), 5 EGTA, 2 Mg-ATP, and 0.3 Na-ATP with a pH between 7.28–7.3, and osmolarity between 290 ± 5 mOsm. Patch pipettes (3–5 MΩ) were pulled using a Brown/Flaming P-97 micro pipette puller (Sutter Instruments). Slices were perfused with oxygenated recording solution (~2ml/min) at room temperature while secured in a glass bottom recording chamber mounted on an inverted microscope with differential interface contrast optics (Zeiss Axiovert 10). Current clamp (and dynamic-clamp) data were acquired and analyzed using custom-made software (G-Patch, Analysis) with sampling frequency: 10 kHz; cut-off filter frequency: 2 kHz.

The Linux-based Real-Time eXperimental Interface (RTXI v1.3) was used to implement dynamic-clamp, running on a modified Linux kernel extended with the Real-Time Applications Interface, which allows high-frequency, periodic, real-time calculations [52]. The RTXI computer interfaced with the electrophysiological amplifier (Axon Instruments Axopatch 200A, in current-clamp mode) and the data acquisition PC, via a National Instruments PCIe-6251 board. Euler’s method with step size 0.05 ms was used for model integration resulting in a computation frequency of 20 kHz.

The model I_NaR current used for dynamic clamping into Mes V neuron in vitro was developed as discussed above. The ionic conductance g_NaR was set to suitable values to introduce model I_NaR current into a Mes V neuron during whole-cell current-clamp recording. For dynamic-clamp experiments involving g_NaR mediated noise modulation, two approaches were used to model random noise I_noise, generated in RTXI and injected as pA current:

1. Uniform-distributed random values (white noise): where U(t) was a uniformly-distributed random number between 0 and 1 generated by the C⁺⁺ rand() function, and A was a scaling factor, in this case determining peak-to-peak noise amplitude. It was varied between 3 and 5, to adjust noise amplitudes to produce discernable burst irregularities. The mean injected noise was 0 pA, and its effective standard deviation varied between 0.87 pA (A = 3) and 1.44 pA (A = 5).
2. Normally distributed random values (Wiener noise). where t indicates the value generated at the current computation cycle, and t−1 to the value of the previous computation cycle. T was the computation period in seconds (0.00005 s for 20 kHz computation) and was discretized to mimic a Wiener process (random-walk); A was a scaling factor, in this case affecting the noise increment each computation cycle, varying between 3 and 5. N(t) was a normally-distributed random number with mean, μ = 0, and standard deviation, σ = 1, N(t) was generated from C⁺⁺-generated uniformly-distributed random numbers using the central limiting theorem: where each U_j is a uniformly-distributed random number between 0 and 1 generated using the C⁺⁺ rand() function. In three 1-second simulations, this noise had mean −0.31±0.19 pA (A = 3), −0.21±0.62 pA (A = 4) or −0.59±0.33 pA (A = 5); and standard deviation 0.77±0.13 pA (A = 3), 1.07±0.15 pA (A = 4) or 1.68±0.44 pA (A = 5).

We used stochastic current injection as an external input (additive noise) in order to produce irregularities/uncertainties in burst discharge. Our choice of the noise model was to experimentally disrupt spike timing regularity [45] and is not directly based on any known noise characteristics in Mes V neurons. The jaw muscle spindle afferent Mes V neurons are not known to have spontaneous synaptic events and we did not characterize Na⁺ channel fluctuations in these neurons [53]. Nonetheless, the stochastic noise we used as shown in the representative example in Fig 8 most closely matched a diffusive synaptic noise model with Gaussian distribution [54]; Fig 8C illustrates the temporal features of the noise inputs described above. A Gaussian white noise generated in MATLAB with zero mean and unit standard deviation or a Wiener noise generated in XPPAUT were used to disrupt firing patterns in the model neuron.

Model simulation and Data analyses

Model simulation and all the analyses were performed using MATLAB (Mathworks) Model code available up on request. Model bifurcation analyses were performed using XPPAUT/AUTO [55]. A variable step Runge-Kutta method ‘ode45’ was used for current-clamp simulations and ‘ode23s’ was used for voltage-clamp simulations.

Inter-event intervals (IEI) between spikes in dynamic-clamp recordings were detected using Clampfit 9.0 software and were classified post hoc as ISIs and IBIs based on a bi-modal distribution of IEIs. Typically, IEI values < 40 ms were considered as ISIs within bursts and IEI values ≥ 40 ms were considered as IBIs. Any occasional isolated spikes were eliminated from analyses for burst duration calculations.

To calculate Shannons’ entropy [56] in the inter-event intervals (IEIs), we generated histograms IEI and calculated the probabilities for each bin of the underlying IEI distributions for each 10 sec spike trains. The probability of k^th IEI bin from a distribution of n equal size bins was calculated from the bin counts, N(k) as: The entropy, H was calculated using the following formula: where, n is the total number of IEI bins, each with probability, p_k.

The coefficient of variation (CV) in IEIs was calculated as follows: where, s is the IEI sample standard deviation, and, is the sample mean.