Resurgent Na+ Current Offers Noise Modulation in Bursting Neurons

Neurons utilize bursts of action potentials as an efficient and reliable way to encode information. It is likely that the intrinsic membrane properties of neurons involved in burst generation may also participate in preserving its temporal features. Here we examined the contribution of the persistent and resurgent components of voltage-gated Na+ currents in modulating the burst discharge in sensory neurons. Using mathematical modeling, theory and dynamic-clamp electrophysiology, we show that, distinct from the persistent Na+ component which is important for membrane resonance and burst generation, the resurgent Na+ can help stabilize burst timing features including the duration and intervals. Moreover, such a physiological role for the resurgent Na+ offered noise tolerance and preserved the regularity of burst patterns. Model analysis further predicted a negative feedback loop between the persistent and resurgent gating variables which mediate such gain in burst stability. These results highlight a novel role for the voltage-gated resurgent Na+ component in moderating the entropy of burst-encoded neural information.

The nervous system extracts meaningful information from natural environments to guide precise behaviors. Sensory neurons encode and relay such complex peripheral information as electrical events, known as action potentials or spikes. The timing intervals between the spikes carry stimulus-relevant information. Therefore, disruption of spike timing by random perturbations can compromise the nervous system function. In this study we investigated whether the widely-distributed voltage-gated sodium (Na + ) ion channels important for spike generation can also serve as noise modulators in sensory neurons. We developed and utilized mathematical models for the different experimentally inseparable components of a complex Na + channel current. This enabled phenomenological simplification and examination of the individual roles of Na + components in spike timing control. We further utilized real-time closed-loop experiments to validate model a1111111111 a1111111111 a1111111111 a1111111111 a1111111111

Introduction
Real-time signal detection in uncertain settings is a fundamental problem for information and communication systems. Our nervous system performs the daunting task of extracting meaningful information from natural environments and guides precise behaviors. Sensory neurons for instance use efficient coding schemes such as bursting that aid in information processing [1]. Mathematical models of bursting have helped explain the basic structure of an underlying dynamical system as one in which a slow process dynamically modulates a faster action potential/spike-generating process, leading to stereotypical alternating phases of spiking and quiescence [2,3]. The so-called recovery period of the slow process governs the intervals between bursts which is often susceptible to random perturbations. Uncertainty in spike/burst intervals can alter the timing precision and information in a neural code [4]. Consequently, ionic mechanisms that modulate the recovery of membrane potential during spike/burst intervals, can play a role in maintaining the stability of these timing events and aid neural information processing. Here we examined a candidate mechanism involving neuronal voltage-gated Na + currents for a role in the stabilization of burst discharge (durations and intervals) and noise modulation.
Voltage-gated Na + currents are essential for spike generation in neurons [5]. The molecular and structural diversity of Na + channels and the resultant functional heterogeneity and complexity, suggest their role beyond mere spike generation [6]. For instance, in addition to the fast/transient Na + current (I NaT ) mediating action potentials, a subthreshold activated persistent Na + current (I NaP ) participates in the generation of subthreshold membrane oscillations (STO) (e.g., see [7]). These oscillations can lead to membrane resonance by which a neuron produces the largest response to oscillatory inputs of some preferred frequency [8,9]. Neurons utilize this mechanism to amplify weak synaptic inputs at resonant frequencies [10]. The slow inactivation and recovery of I NaP further provides for the slow process required for rhythmic burst generation [11][12][13] and therefore can contribute to efficient information processing in multiple ways. However, during ongoing activity, random membrane fluctuations can alter the precision and order of burst timings which can distort/diminish the information in neural code. Here, we provide evidence that a frequently observed resurgent Na + current (I NaR ), often coexistent with I NaP might be a mechanism by which neurons stabilize burst discharge while maintaining its order and entropy. The I NaR , in neurons and other excitable cells is an unconventional Na + current which physiologically activates from a brief membrane depolarization followed by repolarization, such as during an action potential [11,[14][15][16][17]. In the well-studied neuronal Nav1.6-type Na + channels, such a macroscopic I NaR is biophysically suggested to occur from an open-channel block/unblock mechanism [18,19]. Consequently, I NaR is known to mediate depolarizing after-potentials and promote high-frequency spike discharge in neurons [14,[20][21][22][23][24]. Sodium channels containing the Nav1.6 subunits carry all three types of sodium currents and are widely distributed in the central and peripheral neurons and participate in burst generation [14,25]. Sodium channelopathy involving alteration in I NaR and I NaP , and its association with irregular firing patterns and ectopic bursting in disease (e.g., [26][27][28][29]), prompted us to investigate distinct roles for these Na + currents in regulating bursting in sensory neurons.
Lack of suitable functional markers and experimental tools to dissociate the molecular mechanism of I NaR from I NaP , led us to use computational modeling and dynamic-clamp electrophysiology to examine a role for I NaR in burst control; however see [19,23,24]. Although existing Markovian models model a single channel Nav1.6 type I Na using a kinetic scheme (e.g., [30,31]), they have limited application for studying exclusive roles of I NaR and I NaP in the control of neural bursting; however see [13,23,32,33]. Here, we developed a novel mathematical model for I NaR using the well-known Hodgkin-Huxley (HH) formalism which closely mimics the unusual voltage-dependent open-channel unblocking mechanism. We integrated the model I NaR into a bursting neuron model with I NaP that we previously reported to study their exclusive roles in burst control in the jaw proprioceptive sensory neurons in the brainstem Mesencephalic V (Mes V) nucleus [8]. To validate model predictions, we used in vitro dynamic-clamp electrophysiology, and theoretical stability analysis. Using these approaches, we identify a novel role for I NaR in stabilizing burst discharge and noise modulation in these sensory neurons (see

Results
We first developed a novel HH-based model for the resurgent component of a Nav1.6-type Na + current and combined it with our previous HH-type models for the transient and persistent Na + currents [8]. Although the total Na + current arises from a single channel [11,30], we formulated the model I Na as a sum of the three components as shown in Fig 2A (also see Methods): the transient, I NaT , the resurgent, I NaR , and, the persistent, I NaP current. This enabled easy manipulation of individual components to study their exclusive role in burst control. We ensured that each of these components mimic the distinct voltage-dependencies and Noise Modulation in Neurons kinetics observed during voltage-clamp recording in Mes V neurons [11,34]. The classic I NaT with fast inactivation kinetics (order of 1 ms) mediates spike generation, the persistent I NaP with slow inactivation/recovery (order of 1000 ms) mediates STO and provides for the slow process responsible for rhythmic bursting in Mes V neurons [8,35,36], and, the resurgent I NaR which mimics the phenomenology of an open channel unblock mechanism with a peak time of~6 ms and decay kinetics on the order of 10 ms [11,19,30] (also see Fig 3).
We compared our model I Na and its components to the macroscopic I Na observed in voltage-clamp recordings in Mes V neurons [11,14] to ensure qualitative and quantitative similarities (see Fig 3B). We further ensured that our I Na model closely resembled the I Na generated in Markovian models [30], which follow a kinetic scheme and do not formulate the three components separately (see S1 Fig). Our model I Na also satisfied previously described contingency of Nav1.6-type Na + channels carrying resurgent current: activation of I NaR only up on brief depolarization followed by hyperpolarization to~-40 mV, and the correspondence between maximum channel availability and peak time of resurgent current 30 (see S2 Fig).
We incorporated the model I Na into a conductance-based single-compartment neuron model (see schematic and membrane voltage trace in black in Fig 2B). Together with a minimal set of Na + , K + and leak conductances, the model neuron faithfully reproduced an expected rhythmic burst discharge observed in Mes V sensory neurons; the total I Na generated during action potentials is shown in expanded time in the figure (red trace). Fig 2C shows the dynamic-clamp experiment in an intrinsically bursting Mes V neuron. The control burst was generated by simply driving the neuron with a step depolarization, following which we blocked action potential generation by bath application of tetrodotoxin (1 μM TTX) (see black horizontal bar in Fig 2C). Subsequently, we introduced the real-time model I Na using dynamicclamp during TTX application and by adjusting the conductances of the three I Na components suitably, we were able to regenerate action potential bursts (see Methods on the choice of conductance values). The dynamic-clamp I Na generated during action potentials is shown in expanded time in the figure for comparison with the model simulation in Fig 2B (red trace).

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 showing total I Na ; the tree map shows each of the three components: transient I NaT , resurgent I NaR , and, persistent I NaP ; These are also highlighted by the color-matched dashed boxes in the top panel. The τ decay shows the order of magnitude of the decay kinetics of the three components. b) Left: A schematic showing a conductance-based minimal model for a bursting neuron with the I Na incorporated; other ionic currents in the model include a delayed-rectifier potassium (I K ) and a leak current (I leak ). Right: Model simulation demonstrates rhythmic burst discharge and inset highlights the I Na current in the model during action potentials, in red. c) Left: Schematic shows the dynamic-clamp experimental approach; V m is the membrane potential. Right: Membrane potential recorded from a rhythmically bursting sensory neuron; action potentials were blocked using 1μM TTX, and dynamic-clamp model I Na was applied to regenerate spikes; double slanted lines indicate break in time; inset highlights the dynamic-clamp I Na in red, during action potentials. (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 1 (V)), and the competing inactivation (hr 1 (V)) in the model are shown in Fig  3A (middle panel), along with the equation for I NaR ; the magenta shaded region highlights the shown with novel blocking (b r ), and inactivation (h r ) gating variables. Right: The simulated I NaR (in magenta) with peaks occurring during the repolarization phase of action potentials (in black) is shown. b) I. A comparison between simulated I Na and experimentally generated I Na from voltage-clamp recording is shown; boxed inset shows the experimental protocol typically used to test for voltage-dependent activation of I NaR . II. Graphs show the nonlinear current-voltage relationship of peak resurgent current in the model (magenta) and average peak resurgent currents measured from voltage-clamp experiments (black); error bars show standard deviation (n = 5 neurons from 5 animals). 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 voltageclamp 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  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.
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,  . 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.

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/ 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  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 https://doi.org/10.1371/journal.pcbi.1007154.g005 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 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 Noise Modulation in Neurons 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.

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.

Resurgent Na + moderates burst entropy
The spike/burst intervals, their timing precision and order are important for information coding [38][39][40][41][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.

Discussion
Using a combination of mathematical modeling and simulations, theory and dynamic-clamp electrophysiology, we demonstrate a novel role for voltage-gated Na + currents in burst control, noise modulation and information processing in sensory neurons. While the subclasses of Na + currents presented here are experimentally inseparable, our phenomenological simplification using the HH-formalism with in silico knock-in of each Na + current component using dynamic-clamp experiments, allowed examination of their individual contributions to burst control. Additionally, theoretical analysis suggested a putative negative feedback loop between the persistent inactivation and resurgent unblocking processes of the Nav1.6 Na + channels. These results show the apparent consequences on burst control and signal processing capacity of neurons when these currents are present.

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 subthreshold 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. 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

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.

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 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 specific role in neural burst control. The equation for the total sodium current can be written as: 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 t1 (V), and h t , for I NaT , and, m p1 (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 b 0 r and h 0 r 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 1 (V), and decays as per the term, k b β br (V)b r , described as follows: 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 1 (V), defined as below: The term (1−br 1 (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 1 (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 b β 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 voltageclamp experiments (see Fig 3B). The functions, α hr (V),β hr (V) and hr 1 (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 voltageclamp experiments (see S3 Fig), the model required three units for the blocking variable ((1 −b r ) 3 ) and five units for the inactivation variable (h 5 r ) (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: ð Þ 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 NaH 2 PO 4 , 26 NaHCO 3 , 10 glucose, 1 MgCl 2 . 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 voltageclamp experiments on Mes V neurons and recorded Na + currents by blocking voltage-gated K + and Ca 2+ currents similar to [11]. The pipette internal solution contained the following composition (in mM): 130 CsF, 9 NaCl, 10 HEPES, 10 EGTA, 1 MgCl 2 , 3 K 2 -ATP, and 1 Na-GTP. The external recording solution contained the following composition (in mM): 131 NaCl, 10 HEPES, 3 KCl, 10 glucose, 2 CaCl 2 , 2 MgCl 2 , 10 tetraethylammonium (TEA)-Cl, 10 CsCl, 1 4-aminopyridine (4-AP), and 0.3 CdCl 2 . 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 . 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 custommade 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).
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 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. 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, � x is the sample mean.
Supporting information S1 Table. Common model parameters used for various simulations presented.
(TIF) S2 Table. Statistical summary of burst analysis for dynamic-clamp application of resurgent (g NaR ) and persistent (g NaP ) Na+ conductances/currents.  Fig. a, c) Representative burst discharge showing the effects of dynamic-clamp subtraction of g NaR (a) and g NaP (c); decreasing values of 0.25 and 0.5 nS/pF, and 0.05 and 0.1 nS/pF were used for 1X and 2X g NaR and g NaP respectively. Red double arrows highlight inter-burst intervals (IBIs). The magenta rectangles and the corresponding numbers are shown in expanded time and voltage in the boxed inset to highlight abolition of STO due to g NaP subtraction (compare upper and middle traces in the box), which was further comparable to setting g NaP = 0 in neuron model simulations (lower trace). a, c) Box plots showing effects of g NaR (b) and g NaP (d) subtraction on inter-spike intervals (ISIs) within bursts; C is control. A one-way ANOVA for treatment effect on ISIs had a p<0.001 for g NaR subtraction, with no significant effect of g NaP between C and -1x; however, all spikes were abolished upon -2x g NaP application. Asterisks in (b) indicate statistical significance with p<0.001 using a Student t-test for group comparisons (Control v/s -1x and -1x v/s -2x).