In isolated neonatal (P12–P16) mouse cerebellar Purkinje neurons, the fast transient (I_NaT), persistent (I_NaP), and resurgent (I_NaR) Nav current components can be distinguished using voltage-clamp protocols that take advantage of the unique time- and voltage-dependent properties of the three current components (Figure 1A). On membrane depolarization, for example, I_NaT activates fast and subsequently decays rapidly to a steady-state (persistent) level of inward current, I_NaP (Figure 1A). I_NaR , in contrast, is revealed on membrane hyperpolarizations from the depolarized membrane potentials that evoke I_NaT (Figure 1A). In addition, the time courses of I_NaR activation and decay are much slower than I_NaT activation and decay (Figure 1A). Additional experiments revealed that, in response to membrane hyperpolarizations following brief (5 ms) depolarizing steps (to 0 mV), the amplitude of I_NaR varies as a function of the membrane potential of the hyperpolarizing voltage step (Figure 1B). The maximal amplitude of I_NaR is observed at approximately –45 mV (Figure 1C).

Figure 1

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To determine how the duration and the voltage of the depolarizing voltage step (that evokes I_NaT) affect the amplitudes and waveforms of I_NaR, voltage-clamp paradigms were designed in which either the duration or the voltage of the depolarizing step was varied (Figure 2). In initial experiments, the external and internal Na⁺ concentrations were 151 and 8 mM, respectively, resulting in a Na⁺ reversal potential of +75 mV. These experiments revealed that prolonging the duration of the +20 mV depolarizing voltage step resulted in the marked attenuation of the peak amplitudes of I_NaR measured during hyperpolarizing voltage steps to –45 mV (Figure 2A). The time course of the attenuation of I_NaR was well described by a single exponential characterized by a mean ± SEM (n = 12) time constant of 15.5 ± 0.5 ms (Figure 2B).

Figure 2

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Subsequent experiments explored the effect of the driving force on Na⁺ on the time-dependent attenuation of I_NaR. In these experiments, the extracellular Na⁺ was reduced to 50 mM and the Na⁺ concentration in the internal solution was increased to 15 mM, resulting in a Na⁺ reversal potential of approximately +30 mV. Under these recording conditions, depolarizations to +46 mV resulted in outward I_NaT (Figure 2C). Similar to the results obtained with inward I_NaT (Figure 2A and B), prolonging the depolarizing (+46 mV) voltage step when I_NaT is outward results in the rapid attenuation of the amplitudes of I_NaR evoked during the subsequent hyperpolarizations to –45 mV (Figure 2C). Under these conditions (outward I_NaT), the time course of the attenuation of I_NaR was also well described by a single exponential with a mean ± SEM (n = 6) time constant of 13.8 ± 1.1 ms (Figure 2C), a value that is very similar to that observed when I_NaT is inward (Figure 2B). Taken together, these combined results demonstrate that, following brief depolarizations, there is a time-dependent accumulation of Nav channels (which underlie I_NaR) in a non-conducting state, and that this accumulation occurs independent of the direction of the movement of permeating Na⁺ ions.

To determine how the voltage of the depolarizing step affects the amplitudes and kinetics of I_NaR, the currents recorded at –45 mV after 5 ms depolarizing steps to various membrane potentials (Figure 2D) were measured. These experiments revealed that hyperpolarizations to –45 mV following brief (5 ms) depolarizations to various membrane potentials (ranging from –45 to +10 mV) resulted in identical I_NaR amplitudes (Figure 2E). Additionally, varying the voltage of the depolarizing step did not affect the kinetics of the decay of I_NaR (Figure 2D). This is clearly illustrated in Figure 2F, in which Nav currents recorded (in the same cell) at –40 mV during a sustained voltage step (red) and following a 5 ms depolarizing voltage step to +10 mV are superimposed. The opening of Nav channels that conduct I_NaR, therefore, is not affected by the voltage of the prior membrane depolarization. Taken together, these observations suggest that there are two parallel, and kinetically distinct, Nav channel inactivation pathways: a fast inactivating pathway that is responsible for I_NaT; and, a second, slower inactivation pathway that underlies I_NaR.

The results of the voltage-clamp experiments described above suggest that there are (at least) two distinct inactivation pathways that contribute to the gating of the Nav channels expressed in mouse cerebellar Purkinje neurons, that is, one that is populated quickly on channel opening and inactivates rapidly (fast inactivation), and a second that is populated and decays much more slowly (slow inactivation). To explore this hypothesis, we developed a Markov kinetic state model that, after numerical optimization (see Materials and methods), recapitulates the range of time- and voltage-dependent properties observed experimentally for the Nav currents in mouse cerebellar Purkinje neurons. The optimized Markov model (Figure 3A) includes parallel fast (IF1, IF2) and slow inactivation (IS) pathways to reconcile the experimental findings (Figure 2) that the duration of the depolarizing voltage step that underlie Nav channel activation, and not the potential of the depolarizing voltage step or the direction (i.e., inward or outward) of the Na⁺ flux through open channels, determines the amplitudes of I_NaR recorded during subsequent membrane hyperpolarizations. The model was constrained to fit the experimental data derived from multiple voltage-clamp protocols designed to detail the properties of I_NaT, including those to determine the voltage dependence of I_NaT activation and steady-state inactivation, the time course of I_NaT recovery from inactivation (Figure 3B–D), and the time constant (tau) of decay of the peak I_NaT. The model was further constrained by the experimental data obtained using protocols designed to detail the properties of I_NaR, including the voltage dependence of the ratio of the amplitudes of I_NaR and I_NaT (I_NaR:I_NaT), the time-dependent attenuation of I_NaR amplitudes, observed as a function of the duration of the depolarizing voltage step (Figure 2A and B), and the time constants (tau) of I_NaR decay determined for the currents recorded during hyperpolarizing voltage steps to various membrane potentials (Figure 3E–G). Consistent with the experimental results presented in Figure 2, simulations using this gating model reveal that the amplitude of I_NaR is dependent on the duration, but not on the potential, of the depolarizing voltage step that evokes I_NaT (Figure 3H,I).

Figure 3

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One potential benefit of computational modeling is the ability to dissect out possible mechanisms of channel gating by examining the occupancy of the individual channel states as a function of voltage and time. Taking advantage of this benefit, we examined the proportion of Nav channels populating each gating state during a simulated voltage-clamp protocol that evoked I_NaR at –45 mV after a 5 ms depolarizing voltage step to 0 mV (Figure 4). As illustrated, fast inactivation of I_NaT (during the 0 mV step) reflects (simulated) channels exiting the open state and accumulating into the IF1/IF2 states. The activation of I_NaR (at –45 mV) reflects channel transitioning from IF1/IF2 back into the open state, and the decay of I_NaR during the –45 mV step reflects the time-dependent accumulation of (simulated) Nav channels in the secondary, slow-inactivated state, IS (see Figures 3A and 4A). The distinct pathways of inactivation are separated in time, but are not distinguished by differing voltage dependences. Additionally, Figure 4—figure supplement 1 shows that simulated channels follow a similar inactivation pathway during a sustained depolarization to 0 mV, with an initial accumulation in the IF1/IF2 states and subsequent accumulation in the IS state, a property of the model that reveals why I_NaR amplitudes are reduced as the duration of the depolarizing voltage step (that evokes I_NaT) is increased (Figure 3F and H).

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The simulations (Figure 4) indicate the fast decay of I_NaT and the much slower decay of I_NaR reflect separate, that is, fast and slow, pathways of Nav channel inactivation and, in addition, that the decay of I_NaR reflects Nav channels accumulating in an absorbing, slow-inactivated state (i.e., the IS state in Figure 3), suggesting there may be discrete Nav channel populations that accumulate in the ‘IS’ state during prolonged depolarizations. To test this hypothesis directly, we measured peak I_NaT and I_NaR amplitudes, recorded at 0 and –45 mV, respectively, during sequential voltage-clamp protocols separated by a brief (20 ms) interval at –90 mV; the paradigm is illustrated in Figure 5A below the current records. The 20 ms interval at –90 mV between the sequential protocols was determined to be sufficient for the near complete recovery of I_NaT from fast inactivation (Ransdell et al., 2017; Aman and Raman, 2007). However, if the channels that underlie I_NaR have accumulated in the second, slow-inactivated (IS) state during the first –45 mV voltage step and recovery from this state is also slow, one would expect to see differential effects on peak I_NaT and peak I_NaR amplitudes when the time interval at –90 mV is sufficient to allow complete recovery of I_NaT, but too short to allow the complete recovery of I_NaR. As illustrated in the representative records shown in Figure 5A, this voltage-clamp paradigm revealed that, when the time interval at –90 mV was reduced to 20 ms, the amplitude of I_NaR was indeed reduced to a greater extent than the amplitude of I_NaT. Recordings from five additional Purkinje neurons using this voltage-clamp paradigm yielded similar results. Plotting the relative peak amplitudes of I_NaT and I_NaR measured during the second protocol, compared with the first, reveals that the 20 ms hyperpolarizing voltage step to –90 mV was sufficient to provide nearly complete (0.95 ± 0.01; n = 6) recovery of I_NaT (Figure 5B), whereas there was a marked reduction in the amplitude of I_NaR (0.63 ± 0.05; n = 6), measured during the second, compared with the first, protocol (Figure 5B).

Figure 5

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The results presented in Figure 5 indicate that Nav channels open during membrane hyperpolarizations from depolarized potentials and that these (open) channels recover from inactivation at a rate that is distinct from the complement of Nav channels responsible for I_NaT. To determine the relationship between I_NaR and the channels that give rise to the persistent component of the sodium current, I_NaP, we used two voltage-clamp protocols, designed to allow direct measurements of I_NaP alone or I_NaP plus I_NaR. In the first protocol, a slow (dV/dt = 0.12 mV/ms) depolarizing voltage ramp (from –100 to 0 mV) was presented and inward currents, reflecting only I_NaP, were recorded (Figure 6A, blue). In the same cell, we also recorded Nav currents evoked during a slow (dV/dt = 0.12 mV/ms) hyperpolarizing (from 0 to –100 mV) voltage ramp (Figure 6A, red). In the latter case, the measured inward currents reflect both I_NaP and I_NaR, that is, channels capable of recovering from a fast-inactivated state into an open (conducting) state on membrane repolarization. It should be noted that I_NaR decays (see Figure 2B) during the hyperpolarizing voltage-ramp and that the relative amplitudes of I_NaR and I_NaP to the measured currents vary as function of time during the ramp. In addition, we recorded the currents evoked during depolarizing voltage steps to various test potentials between –75 and +10 mV from a holding potential of –100 mV, and we measured the amplitudes of I_NaP directly, at 25 ms after the onset of each depolarizing voltage step (Figure 6A, green). The current-voltage plots, derived from the data obtained in these experiments, are presented in Figure 6B; the colors correspond to those used to illustrate the current records presented in Figure 6A. As is evident, the voltage dependences and the magnitudes of the Nav currents recorded using the three voltage-clamp protocols (in the same cell) are indeed indistinguishable. Similar results were obtained in recordings from four additional Purkinje neurons (see Discussion).

Figure 6

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The novel Markov model developed here (Figure 3A) is quite different from the previously proposed model of Nav channel gating in mouse cerebellar Purkinje neurons (Raman and Bean, 2001). In this earlier model (illustrated in Figure 7A), there are two distinct competing pathways that depopulate the open state, one of which involves fast Nav channel inactivation and results in the population of the I6 state (Figure 7A) and the other, competing, pathway involves the blockade of open Nav channels and the generation of the OB state (Figure 7A). The isolation of the OB state from all of the other kinetic states except the open state is a distinctive feature in the model of Raman and Bean (Figure 7A). This configuration means that channels that have entered the OB state can only exit this state (i.e., become unblocked) by transitioning into the open/conducting state to generate resurgent Na⁺ influx, that is, I_NaR (Raman and Bean, 2001). Although it was suggested that the blocking particle responsible for producing the OB state was a protein, specifically the Nav channel accessory subunit Navβ4 (Grieco et al., 2002; Grieco et al., 2005; Bant and Raman, 2010), it was subsequently demonstrated that I_NaR is reduced, but is not eliminated, in cerebellar Purkinje neurons in (Scn4b^-/-) mice lacking Navβ4 (Ransdell et al., 2017; White et al., 2019) (see Discussion).

Figure 7

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Subsequent efforts here were focused on determining directly whether the Nav channel gating model in which I_NaR is generated by the OB mechanism (Figure 7A) could/would also reliably reproduce the detailed time- and voltage-dependent properties of the Nav currents determined experimentally in mouse cerebellar Purkinje neurons (and presented in Figures 1, 2, 5,, 6). As illustrated in Figure 7B, simulations with the OB model for I_NaR generation (Figure 7A) provided transient and resurgent Nav current components (Figure 7B, upper panel) that resemble those measured experimentally (Figure 1A). A time-locked plot of gating state occupancies with this model reveals that I_NaR activation occurs as simulated channels exit the OB state and into the open (conducting) state (Figure 7B, lower panel). Additional simulations revealed that this model also recapitulates the voltage and time dependences of I_NaT activation, inactivation and recovery from inactivation (Figure 7C–E) in mouse cerebellar Purkinje neurons, although the modeled I_NaT activates at more hyperpolarized voltages than native I_NaT. Although recapitulating the voltage dependence of I_NaR activation (Figure 7F), the model also predicts that the magnitude of I_NaR evoked at all hyperpolarized membrane potentials, relative to the peak amplitude of I_NaT (evoked at 0 mV), is much smaller than observed experimentally (Figure 7F). It should be noted, however, that the relative I_NaR amplitudes measured here were obtained in experiments conducted with a much higher (151 mM) extracellular Na⁺ concentration than the concentration (of 50 mM) used in earlier studies (Raman and Bean, 2001), which likely accounts for some (if not all) of the difference in the relative magnitudes of I_NaR measured here and previously (Raman and Bean, 2001).

In the open-channel block model of I_NaR gating (Figure 7A), more positive depolarizing voltage steps promote entry into the OB state, whereas channels are favored to undergo fast (conventional) inactivation at more hyperpolarized membrane potentials (Raman and Bean, 2001; Lewis and Raman, 2014). When the voltage-clamp protocols used to generate the data presented in Figure 2 were used in simulations with the open-channel block model (Figure 7A), however, substantial differences between the predictions of this model and our experimental data were revealed. In contrast to what is observed experimentally (Figure 2), for example, increasing the duration of the brief (5 ms) depolarizing voltage step (to 0 mV) in the open channel block model resulted in very little time-dependent attenuation of the amplitude of I_NaR recorded on membrane hyperpolarization to –45 mV (Figure 7G and H). In addition, the time course of the decay of the resurgent currents predicted by the open channel block model (Figure 7A) are slower than we observed experimentally for I_NaR in mouse cerebellar Purkinje neurons (Figure 7I).

The experimental data and the simulations using the novel Nav channel gating model developed here (Figure 3A) suggest that I_NaR is mediated by Nav channels transiting from a fast-inactivated state (IF1) into the open state, and subsequently accumulating in an absorbing, slow-inactivated state (IS). It is also possible, however, that the absorbing slow-inactivated state that underlies I_NaR decay reflects Nav channels accumulating into a long-term inactivated state, that is, a state in which channels are non-conducting for hundreds of ms, for example, by a blocking particle(s) that competes on a time scale similar to conventional fast inactivation (Dover et al., 2010; Venkatesan et al., 2014). To test this possibility directly, a voltage-clamp protocol was developed to allow direct comparison in the same cell of I_NaR recorded during a single (80 ms) hyperpolarizing voltage step to –45 mV, presented following a brief (5 ms) depolarization to 0 mV, with I_NaR recorded at –45 mV during successive brief (2 ms) hyperpolarizing voltage steps interspersed with brief (5 ms) depolarizations to 0 mV (Figure 8A). If a competing extrinsic blocking particle has fast-onset, competes with conventional inactivation and is absorbing, one would expect to see reductions in the amplitudes of I_NaR recorded during each successive hyperpolarizing voltage step to –45 mV compared with I_NaR recorded during a sustained hyperpolarizing voltage step to –45 mV. As is evident in the experimental records shown in Figure 8A, however, I_NaR waveforms evoked (in the same cell) using these two voltage-clamp protocols were quite similar.

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The representative traces presented in Figure 8A were recapitulated in simulations (Figure 8B) using the novel gating state model (Figure 3A) developed here. The current waveforms generated by the model are indistinguishable from the experimental results (compare Figure 8A and B). In addition, and without tuning any of the model parameters, the kinetic state occupancy plots generated using the two voltage-clamp protocols were also very similar (Figure 8C). This voltage-clamp protocol was also applied in simulations using the previously described (Figure 7A) open channel block model (Raman and Bean, 2001). In this case, in marked contrast with the results presented in Figure 8, the model does not reproduce the experimental data (Figure 8—figure supplement 1A). The simulations revealed that, in this model, channels did not maximally enter into the OB state on depolarizations to 0 mV and the 2 ms hyperpolarizations were not sufficient to fully activate I_NaR (Figure 8—figure supplement 1B). Additionally, in this model, during each of the successive depolarizations (to 0 mV), transient Nav currents were revealed (Figure 8—figure supplement 1A), reflecting channels exiting the OB state on membrane hyperpolarization and re-entering the OB state on membrane depolarization.

We previously reported that the targeted deletion of Scn4b in mice results in a marked (~50%) reduction in I_NaR amplitudes in cerebellar Purkinje neurons (Figure 9A) without measurable effects on I_NaR kinetics or voltage dependence (Ransdell et al., 2017). In subsequent studies conducted using a different Scn4b^-/- mouse line, it was reported that the targeted deletion of Scn4b had no effects on I_NaR kinetics, voltage dependences, or amplitudes (relative to I_NaT) (White et al., 2019; Ransdell et al., 2017). To explore the ability of the novel model of Nav gating, developed and presented here (Figure 3A), to scale the amplitude of I_NaR while leaving the time- and voltage-dependent properties of the currents unaffected, we optimized the parameters of the model (see Materials and methods) to reproduce the experimentally determined reduction in I_NaR amplitudes with the loss of Navβ4 (Ransdell et al., 2017; White et al., 2019). As illustrated in Figure 9B, although reduced in amplitude, the voltage dependence of I_NaR generated by the Scn4b^-/- Nav channel gating model is very similar to wild type I_NaR. In Figure 9C, representative I_NaR waveforms generated by the gating models of wild type and Scn4b^-/- Nav currents are superimposed; the time courses of wild type and Scn4b^-/- I_NaR are indistinguishable. Time-locked with the wild type and Scn4b^-/- Nav current traces are plots of the Nav channel gating state occupancies as a function of time (Figure 9D) in the wild type (solid lines) and the Scn4b^-/- (dashed lines) I_NaR models. The gating state occupancy plots (Figure 9D) reveal that the attenuation of the amplitude of I_NaR in the Scn4b^-/- model is the result of the reduced accumulation of channels in the IF2 state during the initial depolarization and increased occupancy in the IS state. Together, these data suggest that Navβ4 delays entry into the slow-inactivated state (IS), allowing for greater recovery from conventional inactivation and thus, larger I_NaR amplitudes (see Discussion).

Figure 9

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The experiments and the simulations presented here suggest that I_NaR reflects the transitioning of Nav channels that have undergone conventional fast inactivation, into an open/conducting state on membrane hyperpolarizations. Consistent with the slow decay of I_NaR, the model includes two parallel, but distinct, inactivation pathways: the fast (i.e., IF1 and IF2) inactivation pathway and the slow (i.e., IS) inactivation pathway, which satisfies the key experimental finding that the amplitude of I_NaR is insensitive to the voltage of the initial depolarization (Figure 2). Although it was previously reported that the slower component of the biexponential decay of I_NaT (measured at –30 mV) in Purkinje neurons is identical to the decay rate of I_NaR (also measured at –30 mV), this observation was interpreted as suggesting that all Nav channels initially undergo open channel block (Raman and Bean, 2001). The results of the experiments here involving prolonged depolarizations (Figure 2A–C), however, are not consistent with this model. In the experiments here, we found that there is still a time-dependent attenuation of I_NaR amplitudes on membrane hyperpolarizations following depolarizations to positive membrane potentials, with little or no inward driving force on Na⁺. These findings are inconsistent with the open-channel block hypothesis, in which positive membrane potentials are thought to promote and stabilize open-channel block (Raman and Bean, 2001; Aman and Raman, 2010; Lewis and Raman, 2014).

One assumption usually made when considering the gating of Nav channels is that deactivation must occur prior to recovery from inactivation. Or, to state another way, Nav channels cannot exit the conventional inactivated state directly into an open or conducting state. This idea is based on voltage-clamp experiments conducted on CA1 hippocampal neurons (Kuo and Bean, 1994) and the squid giant axon (Armstrong, 2006), which revealed that recovery from inactivation occurs with a delay and is voltage-dependent. Indeed, if recovery from inactivation requires all four of the VSDs of domains I–IV to move into the deactivated position, the model presented here is not possible because channels that have undergone conventional, fast inactivation would not be able to move directly into an open/conducting state without first transiting through an intermediate closed state. However, if deactivation of VSD IV is sufficient to release the cytosolic DIII–DIV linker peptide from the channel pore, while VSD I, II, and III remain in the activated conformation, Nav channels could transit from an inactivated state directly into an open/conducting state. There are several reports suggesting that this is possible. Schiavon et al., 2012, for example, demonstrated that scorpion beta toxins induce I_NaR in heterologously (in HEK-293 cells) expressed Nav channels. These toxins were previously shown to cause a negative shift in the voltage dependence of DI-VSD, DII-VSD, and DIII-VSD activation by trapping the DII-VSD in the activated position after depolarizing prepulses (Cestèle et al., 1998; Schiavon et al., 2006). Thus, on repolarization, the channel recovers from inactivation, allowing resurgent Na⁺ influx.

Resurgent Nav currents have also been induced in heterologously expressed (in HEK-293 cells) Nav1.5-encoded channels following application of the classical type II pyrethroid, deltamethrin. Similar to our model for the generation of I_NaR, the deltamethrin-modified Nav1.5 channels were found to recover from inactivation prior to deactivation (Thull et al., 2020). In addition, it has also been reported (Cummins et al., 2004) that heterologously expressed (in HEK-293 cells) mutant Nav1.7-encoded channels have prolonged deactivation time constants at the hyperpolarized membrane potentials associated with the activation of I_NaR. These mutant Nav1.7 channels were also reported to display an increase in the relative amplitude of the non-inactivating Nav current (i.e., I_NaP). These two observations are clearly consistent with the I_NaR gating model proposed here in which Nav channels can recover from conventional, fast inactivation directly into an open/conducting state. It is, however, worth noting that in this report (Cummins et al., 2004), I_NaR was never measured directly, instead the membrane was repolarized prior to the completion of I_NaT inactivation. As a result, it is not possible to conclude, with certainty, that the mutant Nav1.7 channels were recovering from inactivation into an open/conducting state, or simply displaying prolonged deactivation at hyperpolarized membrane potentials.

In developing the model (Figure 3A), we found that two fast-inactivated states, that is, IF1 and IF2, were required to recapitulate the experimental data. It is certainly possible, however, that IF1 and IF2 are substrates of the same channel state. We also note here that, initially, the model was developed with two slow-inactivated states (IS, IS2) for symmetry. The simulations revealed, however, that the two slow-inactivated states were redundant and that we were able to fit the experimental data reliably with only one of these slow-inactivated states.

It is important to note that we appreciate that our model topology and rate constants, while reliably recapitulating the time- and voltage-dependent properties of the currents determined experimentally, are not necessarily ‘unique’ or potentially the ‘most simple’. Indeed, in using Markov kinetic state modeling, it is not possible to tell the uniqueness or the ambiguity of the model (both with regard to the parameters and the model topology). Following the approach of Menon et al., 2009, using a state mutating genetic algorithm to vary topologies in a Markov model, Mangold et al., 2021, recently reported the development of an algorithm to enumerate all possible model structures distinctly using rooted graph theory (e.g., all possible combinations of models, rooted around a single open state). These efforts revealed that there are many model structures and parameter sets that can adequately fit some experimental datasets.

Soon after the discovery of I_NaR, it was reported that the targeted deletion of Scn8a (which encodes the Nav1.6 α subunit) in mouse cerebellar Purkinje neurons resulted in a 90% reduction in the amplitude of I_NaR, suggesting that, of the α subunits (Nav1.1 and Nav1.6) expressed in Purkinje neurons (Vega-Saenz de Miera et al., 1997; Xiao et al., 2013), Nav channels formed by Nav1.6 are the major contributors to I_NaR (Raman et al., 1997). However, it was later reported that the targeted deletion of Scn1a (which encodes the Nav1.1 α subunit) also results in ~65% reduction in I_NaR, suggesting that multiple Nav α subunits contribute to I_NaR in cerebellar Purkinje neurons and, in addition, that the contributions are non-linear. To date, all but one (Nav1.3) of the nine Nav channel α subunits have been shown to mediate, or can be induced to mediate, I_NaR in native or heterologous cells (Lewis and Raman, 2014; Jarecki et al., 2010, Do and Bean, 2004; Tan et al., 2014). It is unclear whether Nav1.3-encoded Nav channels also generate I_NaR, as it appears that voltage-clamp studies focused on exploring this possibility have not been conducted to date.

In the model proposed here, the amplitude of I_NaR is dependent on, and ultimately regulated by, two factors: the proportion of Nav channels that are non-inactivating at hyperpolarized voltages, that is, the number of channels that recover from inactivation into an open state on repolarization; and, the proportion of these non-inactivating channels that accumulate into a slow-inactivated state. In the simulations presented in Figure 9, we show that by promoting the occupancy of the IS (slow-inactivated) state, we can recapitulate the experimental observation that I_NaR amplitudes are reduced in Scn4b^-/-, compared with wild type, cerebellar Purkinje neurons, suggesting the possibility that Navβ4 expression regulates I_NaR by influencing slow inactivation. The hypothesis that I_NaR is directly affected by Nav channel slow inactivation is also consistent with previous work (Hampl et al., 2016), showing that mutations in the Nav1.6 and Nav1.7 α subunits that enhance or inhibit slow inactivation also result in reduced or increased, respectively, I_NaR amplitudes. Interestingly, it has also been reported that mutation of the IFM motif (to QQQ) in Nav1.4, in addition to completely eliminating fast inactivation, increases the proportion of channels that enter the slow inactivated state (Featherstone et al., 1996), suggesting that the model developed here (Figure 3A) in which the fast and slow components of inactivation are exclusive, may be applicable to diverse Nav channels in different cell types and encoded by different α subunits.

There are a number of additional factors, intrinsic and extrinsic, to Nav α subunits, as well as additional pre- and post-translational mechanisms, that regulate persistent Nav currents (Aman et al., 2009; Hammarstrom and Gage, 1998; Paul et al., 2016; Lin and Baines, 2015) and slow inactivation (Chen et al., 2008; Silva, 2014; Chen et al., 2006; Carr et al., 2003). In the model developed and presented here, the combined effects of these factors/mechanisms will regulate/modulate the amplitudes and the time- and voltage-dependent properties of I_NaR. In this context, it is interesting to note that the experiments presented in Figure 5 indicate that the Nav channels underlying I_NaR, compared to the sum of the Nav channels underlying I_NaT, are differentially sensitive to slow inactivation. This observation clearly suggests that there is functional (and perhaps molecular) heterogeneity in the population of Nav channels underlying the Nav currents in mouse cerebellar Purkinje neurons, with channels underlying I_NaR having properties distinct from at least a portion of the Nav channels that participate in the generation of I_NaT, but that do not produce I_NaR. This possibility is consistent with results presented by White et al., 2019, in which I_NaR was found to have a greater sensitivity to tetrodotoxin block than I_NaT in mouse cerebellar Purkinje neurons. It should also be noted that the Markov model developed here was created to reflect a single channel population, that is, Nav channels capable of generating both I_NaT and I_NaR. As such, the developed Markov model does not reproduce the differential recovery from inactivation presented in Figure 5.

Based on the experimental data and the computational modeling presented here, we propose a novel, blocking particle-independent, gating mechanism for the generation of I_NaR that involves two, kinetically distinct inactivation pathways. The modeling results suggest that two parameters are critical in determining the magnitude and the time- and voltage-dependent properties of I_NaR: (1) the relative amplitude of the persistent Nav current, I_NaP, component; and, (2) the proportion of the persistent Nav channels (channels that fail to undergo fast inactivation) that undergo slow inactivation. Interestingly, I_NaR has now been identified in over 20 types of neurons, many of which do not display the high rates of repetitive firing that are characteristics of cerebellar Purkinje neurons, suggesting that the role(s) of I_NaR in the regulation of membrane excitability are diverse, and likely distinct, in different neuronal cell types (Lewis and Raman, 2014). Recent studies conducted on serotonergic raphe neurons, for example, suggest that the accumulation of Nav channels in a slow-inactivated state functions as a homeostatic brake on repetitive firing (Navarro et al., 2020). In addition, I_NaR has been implicated in several inherited and acquired neurological diseases (Lewis and Raman, 2014), including paroxysmal extreme pain disorder, paramyotonia congenita (Jarecki et al., 2010), and chemotherapy-induced neuropathy (Lewis and Raman, 2014; Sittl et al., 2012), as well as epilepsy (Hargus et al., 2013). Given the implications of these findings, it will be of considerable interest to detail the properties of I_NaR in different types of neurons and to explore directly the hypothesis that the rate of decay of I_NaR on membrane repolarization plays a role in determining how much persistent sodium current is available to contribute to repetitive firing, as well as to define the molecular mechanisms that control I_NaR amplitudes, kinetics, and functioning in diverse neuronal cell types.

All animal experiments were performed in accordance with the guidelines published in the National Institutes of Health Guide for the Care and Use of Laboratory Animals. Protocols were approved by the Washington University Institutional Animal Care and Use Committee (IACUC). Postnatal day 12–16 (P12–P16) male and female C57BL6/J (Jackson laboratories) mice were used in the experiments reported here.

For the preparation of isolated cerebellar Purkinje neurons, postnatal day 12–16 (P12–P16) animals were anesthetized with 1.25% Avertin and brains were rapidly removed and placed in ice-cold isolation medium containing (in mM): 82 Na₂SO₄, 30 K₂SO₄, 5 MgCl₂, 10 HEPES, 10 glucose, and 0.001% phenol red (at pH 7.4). Using a scalpel, the cerebellum was removed, minced into small chunks and incubated in isolation medium containing 3 mg/ml protease XXIV at 33°C for 10–15 min. Following this incubation period, the tissue pieces were washed with enzyme-free isolation medium containing 1 mg/ml bovine serum albumin and 1 mg/ml trypsin inhibitor. The tissue pieces were transferred to artificial cerebral spinal fluid (ACSF) that was continuously bubbled with 95% oxygen/5% carbon dioxide and contained (in mM): 125 NaCl, 2.5 KCl, 1.25 NaH₂PO₄, 25 NaHCO₃, 2 CaCl₂, 1 MgCl₂, and 25 dextrose (310 mosmol/l) at 22–23°C. Tissue pieces were triturated with a fire-polished glass pipette. An aliquot of the cell suspension was placed on a coverslip in the recording chamber and superfused with fresh ACSF (at a rate of 0.5 ml/min), saturated with 95% O₂/5% CO₂, for 25 min before beginning electrophysiological experiments.

Whole-cell voltage-clamp recordings were obtained at room temperature from visually identified cerebellar Purkinje neurons using differential interference contrast optics. Data were collected using a Multiclamp 700B patch clamp amplifier interfaced to a Dell PC with a Digidata 1332 and pCLAMP 10 software (Axon Instruments, Union City, CA). In all recordings, tip potentials were zeroed prior to forming a giga-ohm membrane–pipette seal. Pipette capacitances were compensated using the pCLAMP software. Signals were acquired at 50–100 kHz and filtered at 10 kHz prior to digitization and storage.

In the experiments here, I_NaR was routinely recorded in ACSF (containing ~151 mM Na⁺; see above) with 5 mM tetraethylammonium chloride (TEA-Cl) and 250 μM cadmium chloride (CdCl₂) added. To decrease Nav currents, improve the spatial control of the membrane voltage (space-clamp) and enable reliable recordings of I_NaT or I_NaP, I_NaR was also recorded in some experiments (see Figure 6) in bath solution containing (in mM): 25 NaCl, 100 TEACl, 2.5 KCl, 1.25 NaH₂PO₄, 25 NaHCO₃, 2 CaCl₂, 1 MgCl₂, 25 dextrose, and 0.25 CdCl₂. Recording pipettes routinely contained (in mM): 110 CsCl, 15 TEACl, 5 4AP, 1 CaCl₂, 2 MgCl₂, 10 EGTA, 4 Na₂-ATP, and 10 HEPES, pH 7.25 300 mosmol/l. Alternative bath and internal solutions were used in experiments presented in Figure 2; these solutions are described in the Results section and in the legend to Figure 2. In all experiments, recording pipettes had resistances of 2–3 MΩ.

Membrane capacitances were determined by analyzing the decays of capacitive currents elicited by short (25 ms) voltage steps (±10 mV) from the holding potential (−70 mV). Whole-cell membrane capacitances (C_m) were calculated for each cell by dividing the integrated capacitive transients by the voltage. Consistent with the short time in culture and lack of extensive processes, the capacitive transients of recorded cells had single-exponential decay phases. Input resistances were calculated from the steady-state currents elicited by the same ±10 mV steps (from the holding potential). For each cell, the series resistance was calculated by dividing the time constant of the decay of the capacitive transient (fit by a single exponential) by the C_m. Series resistances were routinely compensated ≥80%. Voltage errors resulting from the uncompensated series resistance were always ≤4 mV and were not corrected. Only data obtained from cells with input resistances >50 MΩ and capacitive transients well described by single exponentials were included in the cumulative data analyses presented.

In the standard voltage-clamp studies, involving step depolarizations, peak transient Nav conductances (in each cell at each test potential) were calculated (using E_reversal = +75 mV) from measurements of peak I_NaT following digital subtraction of the non-inactivating, persistent Nav current component (I_NaP), measured during the same voltage step. These peak transient Nav conductances were then normalized to the maximal peak transient Nav conductance determined in the same cell. Mean ± SEM normalized peak transient Nav conductances were then determined, plotted as a function of the test potential, and fitted with a single (Equation 1) Boltzmann function:

where V_h is the membrane potential of half-maximal activation and k is the slope factor.

To determine the voltage dependence of steady-state inactivation of I_NaT, a two-step voltage-clamp protocol was used. From a holding potential of –70 mV, brief (20 ms) voltage steps to various conditioning voltages, between –120 and –35 mV, were presented prior to depolarizations to 0 mV to evoke I_NaT. Peak I_NaT amplitudes at 0 mV, evoked from each conditioning potential in each cell, were measured following digital subtraction of I_NaP, measured during the same voltage step, and subsequently normalized to the peak I_NaT amplitude evoked (at 0 mV) from the –120 mV conditioning step in the same cell. Mean ± SEM normalized peak I_NaT amplitudes were plotted as a function of the conditioning voltage and fitted with a single (Equation 2) Boltzmann function:

where V_h is the membrane potential of half-maximal inactivation and k is the slope factor.

The peak amplitudes of I_NaR, evoked on hyperpolarizations to various membrane potentials, following brief (5 ms) depolarizing voltage steps to 0 mV, were determined following digital subtraction of I_NaP, measured during the same voltage step (in the same cell). The time constants (tau) of the decay of I_NaR were determined from first-order exponential fits to the decay phases of these subtracted records. The amplitudes of I_NaP were routinely determined by direct measurement of the steady-state inward currents remaining at the end of depolarizing (or hyperpolarizing) voltage steps. I_NaP (or I_NaP plus I_NaR) amplitudes were also determined in additional experiments from analyses of the currents recorded during depolarizing (or hyperpolarizing) voltage ramps (see Figure 6).

The peak amplitudes of I_NaT and I_NaR, generated using our Nav channel gating model or the Raman-Bean model, were routinely determined directly from the simulated current records, that is, without subtraction of I_NaP. Determinations of I_NaT and I_NaR amplitudes following subtraction of I_NaP yielded indistinguishable results. The time constants of decay of I_NaR simulated using our model or the Raman-Bean model were determined from single exponential fits to the inactivating component of the current (i.e., I_NaR uncontaminated by I_NaP).

Sample sizes were determined based on mean data (and associated standard deviations) obtained from analyses of voltage-clamp recordings of Nav currents obtained from mouse cerebellar Purkinje neurons (Ransdell et al., 2017). As in this previous study, all of the mean data presented here reflect results obtained from biological replicates, that is, analyses of Nav current recordings, evoked using identical voltage-clamp protocols, obtained from individual isolated Purkinje neurons. Technical replicates, that is, the presentation of identical voltage-clamp protocols to the same Purkinje neuron, were occasionally acquired to ensure that the properties of the Nav currents did not change during prolonged whole-cell recordings. Technical replicates were not included in the data analysis or reported results.

Data were analyzed using ClampFit (Molecular Devices), MATLAB (Mathworks), Microsoft Excel, and Prism (GraphPad Software Inc).

The Raman-Bean model of Nav channel gating (Raman and Bean, 2001) was coded in MATLAB using the equations and schematic of the Markov model structure presented in Figure 7 of Raman and Bean, 2001; no changes were made to the published model or parameters (Raman and Bean, 2001). The model simulations were run as described below using the matrix exponential technique (Teed and Silva, 2016).

A Markov kinetic state model of Nav channel gating in mouse cerebellar Purkinje neurons was formulated based on our acquired experimental data that led to a hypothesized structure of channel gating states (see: Figure 3A). The rate constants in the model were all single exponentials derived from the acquired experimental data and were optimized numerically using described methods (Moreno et al., 2016; Teed and Silva, 2016). All simulations, numerical optimization, and data visualization were done in MATLAB 2017B. Detailed methods are below. Model definition files and MATLAB scripts used for the simulations are available at https://github.com/morenomdphd/Resurgent_INa; Moreno, 2022. The equations were verified by one of the authors (DB) who was not involved in the creation of the model.

The Nav channel gating model developed (see Figure 3) consists of nine states: three closed states (C1, C2, C3), a single open state (O), two closed-inactivated states (IC1, IC2), two inactivated states, resulting from fast channel inactivation (IF1, IF2), and an additional, slowly populated inactivated state (IS). The model accurately simulates the experimentally observed kinetics and voltage dependences of Nav channel gating, including activation, inactivation (closed and open state), deactivation, recovery from fast inactivation, as well as the voltage dependence of activation and the proportion (relative to I_NaT) of I_NaR.

All computations were done in MATLAB 2017B. We coded all voltage protocols and used the matrix exponential technique, which is described in Teed and Silva, 2016, for simulations. A modified Nelder-Mead simplex method that allows for constrained optimization (only positive rate constants) was used for simultaneous optimization of the protocols listed below. A cost function for each protocol was defined as the sum of squared differences between the experiments and the simulations. The total cost function (sum of the individual protocol errors) was then minimized and converged when a tolerance of 0.01 for the change of the cost function and 0.01 for the change in parameters was achieved. For further details about this numerical optimization method, see Moreno et al., 2016.

The protocols were as follows:

• Steady-state availability: For each voltage between –120 and –10 mV, the steady-state probabilities of the channel were found. The channel was then depolarized to 0 mV, and the open-state probability was determined. The value of the open-state probability was then normalized to the open-state probability at –120 mV.

• Steady-state activation: Channel steady state was found at –80 mV. The channel was then depolarized to voltages between –77 and 0 mV (in 3 mV increments). For each voltage, the maximum open probability of the channel was calculated, and the conductance, G_Na, at each voltage was determined. The calculated values were then normalized to G_Na at 0 mV.

• Tau of inactivation: Channel steady state was found at a holding potential of –90 mV. From steady state, the channel was depolarized to voltages between –50 and 0 mV (in 5 mV increment), and the time constant (tau) corresponding to 64% decay (1/e) of the peak current was calculated for each voltage.

• Recovery from fast inactivation: From a steady state of –90 mV, the channel was depolarized to 0 mV, and the peak current determined. The channel was then allowed to recover at –90 mV for variable time intervals, before being depolarized again to 0 mV. The peak currents during the depolarizations to 0 mV following the various recovery times were determined and normalized to the initial peak current.

• Persistent component of the Nav current: From a steady state of –90 mV, the channel was depolarized to 0 mV for 5 ms, and subsequently hyperpolarized to –45 mV for 100 ms. The persistent current recorded after the 100 ms hyperpolarizing step was measured and normalized to the initial peak current.

• Voltage dependence of the ratio of the peak resurgent to peak transient Nav current amplitude: From a steady state of –90 mV, the channel was depolarized to 0 mV for 5 ms, and subsequently hyperpolarized to potentials between –5 and –80 mV (in 5 mV increments). The peak resurgent current at each hyperpolarized voltage step was determined and normalized to the peak transient inward current evoked at 0 mV.

• Dependence of the peak I_NaR amplitude on the duration of the depolarizing voltage step duration: From a steady state of –90 mV, the channel was depolarized to +20 mV for varying times (2 to 36 ms) prior to hyperpolarization to –45 mV. The amplitude of I_NaR at –45 mV following each depolarizing voltage step to +20 mV (of varying durations) was measured and normalized to the peak I_NaR evoked following the 2 ms depolarizing voltage step.

• Tau of decay of the resurgent Nav current: From a steady state of –90 mV, the channel was depolarized to 0 mV for 5 ms, and subsequently hyperpolarized to potentials ranging from –5 to –45 mV (in 5 mV increments). The time constant (tau) of the exponential decay (1/e) of the resurgent current at each hyperpolarized test potential was determined.

• Voltage independence of peak resurgent current: Consistent with the experimental data presented in Figure 2D and E, the peak resurgent current was independent of the potential of the depolarizing voltage step. From a steady state of –90 mV, the channel was depolarized to membrane potentials ranging from –5 to –35 mV (in 5 mV increments) prior to a hyperpolarizing voltage step to –45 mV. The difference between the peak I_NaR measured at –45 mV from each depolarized potential was determined, and the mean resurgent peak current determined across all depolarized potentials was minimized. This ensured the peak resurgent current was constant without specifying the magnitude of the resurgent peak a priori.

To develop the model for Nav channels lacking Scn4b, the optimization routine was restarted from the optimized wild type rate constants (initial conditions). The ‘Voltage dependence of the ratio of the peak resurgent to peak transient Nav current amplitude’ protocol described above was fitted to the experimental data acquired from isolated cerebellar Purkinje neurons from (Scn4b^-/-) animals harboring a targeted disruption in the Scn4b locus (Ransdell et al., 2017); the data are presented in Figure 9A. All of the other protocols used in the optimization procedure for Scn4b^-/- were the same as those described above for the wild type Nav channel to ensure no other changes to the model.

Of note, in the MATLAB script, the Nav channel model developed here contains 24 parameters; these are inputted as a matrix ‘Input’. For example, Input(1) corresponds to a11_variable1, and Input(2) corresponds to a11_variable2, Input(3) corresponds to a12, and Input(4) corresponds to a13, etc. The transition rate constants are of the form denoted in the right-hand column of the table above.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

The authors thank Mr Richard Wilson for expert technical assistance. The financial support provided by the NIH (R01 NS065761 to JMN, R01 HL136553 to JRS, and F32 NS090765 to JLR) is also gratefully acknowledged; JDM was supported by an NIH institutional training grant (T32 HL007081) and a grant from the Foundation for Barnes Jewish Hospital. The authors declare no competing financial interests.

This study was performed in strict accordance with the recommendations in the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health. All of the animals were handled according to an approved institutional animal care and use committee (IACUC) protocol (20180045) of Washington University in St. Louis (Animal Welfare Assurance # A-3381-01).

© 2022, Ransdell et al.

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