Estimation of persistent sodium‐current density in rat hippocampal mossy fibre boutons: Correction of space‐clamp errors

We used whole‐cell patch clamp to estimate the stationary voltage dependence of persistent sodium‐current density (iNaP) in rat hippocampal mossy fibre boutons. Cox's method for correcting space‐clamp errors was extended to the case of an isopotential compartment with attached neurites. The method was applied to voltage‐ramp experiments, in which iNaP is assumed to gate instantaneously. The raw estimates of iNaP led to predicted clamp currents that were at variance with observation, hence an algorithm was devised to improve these estimates. Optionally, the method also allows an estimate of the membrane specific capacitance, although values of the axial resistivity and seal resistance must be provided. Assuming that membrane specific capacitance and axial resistivity were constant, we conclude that seal resistance continued to fall after adding TTX to the bath. This might have been attributable to a further deterioration of the seal after baseline rather than an unlikely effect of TTX. There was an increase in the membrane specific resistance in TTX. The reason for this is unknown, but it meant that iNaP could not be determined by simple subtraction. Attempts to account for iNaP with a Hodgkin–Huxley model of the transient sodium conductance met with mixed results. One thing to emerge was the importance of voltage shifts. Also, a large variability in previously reported values of transient sodium conductance in mossy fibre boutons made comparisons with our results difficult. Various other possible sources of error are discussed. Simulations suggest a role for iNaP in modulating the axonal attenuation of EPSPs.


Introduction
Successful modelling of how neurons integrate their synaptic inputs requires, among other things, knowledge of ion channel densities and gating kinetics.For example, Martinello et al. (2019) developed a Hodgkin-Huxley-type model of the M (Kv7/KCNQ) conductance in the axons (mossy fibres) of hippocampal dentate gyrus granule cells, specifically in the mossy fibre boutons (MFBs, varicosities).In addition, a persistent sodium conductance (g NaP ) is also present in these MFBs (Alle, Ostroumov et al., 2009).Persistent sodium conductance is an important determinant of subthreshold behaviour and can interact with the M conductance to amplify electrical resonance (Hu et al., 2002;Hu et al., 2009;Vervaeke et al., 2006).Here, we present an estimate of the voltage dependence of the stationary persistent sodium (NaP) current density and maximum specific conductance in rat hippocampal mossy fibres.
In attempting to develop a kinetic model of a voltage-gated conductance, such as g NaP , the natural experimental methodology of choice is the voltage clamp.Unfortunately, when applied to neurons with long axons and/or dendrites, whole-cell voltage clamp suffers from the problem of poor space clamp (Gurkiewicz & Korngreen, 2006;Williams & Mitchell, 2008), i.e. the intracellular voltage [V(t), where t is time] is controlled accurately only at the recording/current-injection site.Away from this site, the effects of non-zero axial resistance and membrane capacitance, in addition to finite membrane resistance, mean that V deviates from its desired value (the command voltage, V c ).This can cause large voltage errors, e.g.≤50% in the achieved membrane voltage in distal dendrites during somatic voltage-clamp experiments in large pyramidal cells.The use of membrane patches (inside out, outside out or even, for sufficiently low currents, cell attached) obviates this problem but presents its own difficulties.First, channels in excised patches (inside out or outside out) often suffer rapid rundown or other artefacts caused by loss of their native intracellular milieu (Fenwick et al., 1982;Gurkiewicz & Korngreen, 2006;Ruppersberg et al., 1991).Second, the membrane that is sucked into the recording pipette while forming seals for cell-attached or excised patches often lacks the native channel density (Kole et al., 2008).Third, the difficulty of estimating membrane-patch area in the cell-attached mode results in significant uncertainties in channel densities (Gurkiewicz & Korngreen, 2006).Fourth, the random sampling of membrane patches, each containing only a few channels, necessitates a large number of repeated trials (Gurkiewicz & Korngreen, 2006).Fifth, furthermore, low current density (owing to low channel density or small channel conductance) might cause too low a signal-to-noise ratio, preventing extraction of accurate channel gating parameters.Therefore, whole-cell recording, with an appropriate correction for space-clamp error, is still an invaluable tool for obtaining the data needed for the accurate description and modelling of intrinsic signalling processes in neurons.
The need to correct for space-clamp errors in whole-cell voltage-clamp recordings from neurons is demonstrated clearly by modelling studies.Especially relevant for the present work are the simulations by White et al. (1995), who used ball-and-stick models of neurons.They found that space-clamp errors resulted in distortion and, depending on the holding potential, even bistability of the stationary I(V) curve of persistent inward currents, such as I NaP .Bar-Yehuda and Korngreen (2008) simulated voltage-clamp experiments using a variety of reconstructed neuronal morphologies and demonstrated serious distortions of V-gated K + and Ca 2+ currents recorded from somata and dendrites.Some of the predictions of the simulations were confirmed by experimental whole-cell recordings.A method for calculating correction factors for a cylindrical neurite was described by Castelfranco and Hartline (2004), who used a 'full-trace' approach, in which a Hodgkin-Huxley (HH)-type model is fitted to the complete (distorted) current trace recorded during a voltage clamp.The resulting parameter estimates are then used to derive the correction factors.Schaefer et al. (2003) used an iterative approach to estimate voltage-dependent HH channel gating parameters from simulated V-clamp data generated with reconstructed neuronal morphologies and detailed knowledge of passive electrical properties.They applied this method successfully to the analysis of dendritic K + currents (Schaefer et al., 2003(Schaefer et al., , 2007)).Cox (2008) proposed a 'direct' method for correcting space-clamp errors in unbranched neurites based on Cole's theorem (Cole & Curtis, 1941) that does not require iteration.His analysis yields rapid, semi-quantitative information on channel kinetics and densities at the stimulus site.Parameter estimates obtained with this method are, however, adversely affected by tapering and branching, and the method is not directly applicable to recordings from essentially isopotential compartments, such as somata, axonal blebs or synaptic boutons.
In the present paper, we present a new method for correcting space-clamp errors, which shares features with the 'full-trace' approach of Castelfranco and Hartline (2004), the iterative approach of Schaefer et al. (2003) and the direct approach of Cox (2008).However, Cox's approach is extended to include isopotential compartments with several attached neurites, and the iterative fitting is performed on the full membrane current-density trace (similar to Castelfranco and Hartline, 2004) rather than by fitting each time point sequentially (as in the study by Schaefer et al., 2003).We have not attempted to estimate the gating kinetics of g NaP , which will be rapid if g NaP is a manifestation of the transient sodium conductance (Carter et al., 2012;Hsu et al., 2018;Magistretti et al., 2006;Raman & Bean, 2001;Taddese & Bean, 2002).Instead, we have used our method, which we call the Cox method with correction, to analyse the results of V-ramp experiments, where the ramp is so slow that g NaP can be considered to gate instantaneously.What comes out of this analysis is an estimate of the voltage dependence of the stationary persistent sodium-current density (i NaP ) and maximum specific conductance (ḡ NaP ) in rat hippocampal mossy fibre axons.We hope these results will inform future models of dentate gyrus granule-cell electrophysiology.In addition, some of the issues considered when interpreting the results might be relevant to electrophysiological modelling and analysis in general.
It has been suggested that g NaP might be explained in terms of the transient sodium conductance (g NaT ).Indeed, a window current generated by a Markov model of g NaT will account for i NaP in rat tuberomammillary neurons, CA1 pyramidal neurons and cerebellar Purkinje neurons R. Murphy and others J Physiol 602.8 (Carter et al., 2012;Hsu et al., 2018;Taddese & Bean, 2002).Likewise, a single Markov model will account for g NaT , g NaP and also the resurgent sodium conductance in cerebellar Purkinje neurons (Raman & Bean, 2001) and cerebellar granule cells (Magistretti et al., 2006).In the present study, we have attempted to relate our estimates of g NaP to g NaT by modifying the Hodgkin-Huxley model of Engel and Jonas (2005) to include incomplete inactivation (Taddese & Bean, 2002).We also use simulations to explore a possible functional role for g NaP in modulating the conduction of EPSPs by the mossy fibre axon from the soma to presynaptic terminals (Alle & Geiger, 2006).

Ethical approval
Ethical approval numbers were uncommon at the time of these experiments ( 2009), but the treatment of the animal complied with the local ethical requirement and guidelines at the time and conforms to the ' Animal ethics checklist' given by Grundy (2015).The male Wistar rat used stemmed from the institutional animal house and breeding facility and had free access to food and water at any time; he was housed together with littermates in an extra-large cage providing an enriched environment.Before experiments, the animal was deeply anaesthetized within a transfer cage by putting a small paper towel, onto which isoflurane was dropped, into the cage (0.8 ml isoflurane for a cage volume of ∼9 L, yielding a vapourized anaesthetic concentration of ∼1.6% by volume).Following eye closure and complete loss of the righting reflex, the animal was quickly decapitated.

Application of Cox's method to an isopotential compartment with attached neurites of varying diameters
In this and the following subsection, we describe the method used to estimate current densities in the presence of space-clamp errors.We use i to represent current density and I to represent current (both macroscopic).Membrane conductances (g) are specific, i.e. expressed per unit membrane area.Consider an infinite neurite of constant diameter, d, and uniform axial resistivity, R a .Suppose further that this neurite is equipped with a (pharmacologically isolated) voltage-dependent membrane conductance (in the present case, the persistent sodium conductance, g NaP ) and a passive linear leak (g pas ).
Then, adopting the usual formalism, the membrane current densities i NaP and i pas are given by: where V is the membrane potential, E Na and E pas are reversal potentials and t is time.The total current density is then: Now suppose that at some point, x c , along the neurite V is clamped to a test value V = V c by injecting a current I(t) at that point.Then, as shown by Cox (2008), I(V c ,t) and i(x c ;V c ,t) are related to a good approximation by Cole's theorem (Cole & Curtis, 1941): Hence, knowing I(V c , t), V c , d and R a one can determine i(x c ;V c ,t), i.e. the total membrane current density at x c .And if one also knows i pas (x c ;V c ) and E Na , then i NaP (x c ;V c ,t) and g NaP (x c ;V c ,t) can be found from eqns (3) and (1), respectively.Now consider one of the semi-infinite cables (one to the right of x c , the other to the left).The axial current (I a ) injected into this cable is: whence from eqn (4): Now, if there are N neurites connected to an isopotential compartment of area A, then instead of eqn (5) we have for the j th neurite: where and If the behaviour of the neurites is linear with V, the value of α j will be invariant with V c .Of course, this will not be true in practice; the neurites will not be semi-infinite, and they might exhibit tapering and branching, all of which will lead to departures from the predicted behaviour (Cox, 2008).Therefore, a correction algorithm is needed (described below).But we can use the present formulation of Cox's method as a starting point.Assuming i is uniform over A, substituting eqn (7) into eqn (6) (applied to the jth neurite) gives: Given that, for a particular V c , the quantity (I a,tot /i)(dI a,tot /dV c ) has a unique value, eqns ( 9) and (10) imply that: which is simply the ratio of the input conductance of the jth neurite to the total input conductance of the N neurites in parallel [with all neurites approximated as passive and uniform (Koch, 2004)].Finally, assuming that eqn (10) can be applied to t-dependent currents, we set: in eqn (10), substitute for α j from eqn (11) and solve for di/dV c to obtain: In the particular case of the mossy fibre bouton (MFB), we have N = 2, d 1 = d 2 = d (the axon diameter) and so γ = 4π 2 d 3 /R a .Because of their short length (20 μm was assumed for the compartmental model described below), the filopodial extensions attached to the MFB (Fig. 1) are assumed to be isopotential with the MFB, meaning that their area can simply be added to the MFB area to obtain A.
In order to solve eqn (13) for i(x c ; V c ,t), one needs to specify an initial condition, and there seems to be only one choice available.[To simplify notation, we sometimes omit references to x c , V c and t, and refer to V c simply as V.] By definition, i(V rest ) = 0 (where V rest is the resting membrane potential or RMP), but values of i at other V are unknown; indeed, it is these values that we are trying to determine.Unfortunately, setting V = V rest and i(V rest ) = 0 in eqn (13) leads to an indeterminate quotient because I(V rest ) is also zero (again by definition).To overcome this difficulty, we approximate i(V) and I(V) as Taylor's series expansions about V rest , retaining only the linear terms.Then, on setting V = V rest in eqn (13) and solving for di/dV, we obtain: where the negative square root is taken to ensure that di/dV → 0 as dI/dV → 0.Then, close to V rest , say |V−V rest | < |δV|, i(V) can be approximated as: For |V−V rest | ≥ |δV|, we solve eqn (13) numerically using the initial condition: δV was set to −0.1 mV for V < V rest and 0.1 mV for V > V rest (making it smaller did not affect the results).Equation ( 13) was solved for i(x c ; V c ,t) using routine DIVMRK from the IMSL Math/Library v.3.0 (Visual Numerics, Houston, TX, USA) running under Compaq Visual Fortran v.6.6 (Compaq Computer Corporation, Houston, TX, USA) or the function scipy.integrate.solve_ivpfrom the SciPy Python package (https://docs.scipy.org)running under Python 3.6 (https://www.anaconda.com).

Figure 1. Schematic representation of the dentate gyrus granule-cell model morphology
The soma was 15 μm in diameter.The first section of the main axon (160 μm long) tapered in diameter from 1 μm at the soma to 0.4 μm at 20 μm from the soma.The rest of the main axon [with sections of variable lengths in between mossy fibre boutons (MFBs); see Methods] was 0.4 μm in diameter, and the 200-μm-long collaterals were 0.2 μm in diameter.Mossy fibre boutons were 7 μm in diameter and equipped with four filopodial extensions (20 μm long and 0.1 μm in diameter).Primary dendrites were connected to the soma via an apical dendritic trunk (7 μm long, with a taper from 5 to 4 μm in diameter).Primary dendrites were 50 μm long, with a taper from 4 to 3 μm, and secondary dendrites were 350 μm long, with a taper from 3 to 1 μm.Unless stated otherwise, the axon length was 2.16 mm.

An algorithm to improve the Cox estimates of membrane current density
In practice, for reasons discussed above and in the Results section, eqn (13) yielded only approximate estimates of i, which were deemed unacceptable, especially at hyperpolarized V. Accordingly, an algorithm was devised to improve on these estimates.Specifically, although the use of the measured I in eqn (13) might produce somewhat erroneous estimates of i, one can imagine that there is a fictitious I (I * ) which, when substituted for I in eqn (13), yields the correct i.Hence, the aim is to find I * .In the present study, we used a compartmental model of a dentate gyrus granule cell implemented in NEURON v.7.2 (Carnevale & Hines, 2009) to find I * iteratively.The model is described in the next subsection.Then, given the command voltage (V c ) and the current (I) injected into a MFB, the following algorithm was used to find I * and thus i (k is the iteration index):  (Gerald & Wheatly, 1984): (viii) determine (dI * /dV c ) k and go to (ii).Both predicted and observed values of V and I were corrected for the access and seal resistances, respectively (if present in the NEURON model).The values of I * and dI * /dV needed for step (ii) were estimated from appropriate smoothing polynomials [eqns ( 18) and ( 21) below], fitted with either the IMSL routines DRCURV and DRNLIN or the Python functions numpy.polyfitand scipy.optimize.least_squares;such smoothing is especially important for the estimation of dI * /dV, which is sensitive to noise.In step (iii), i k (V c ) was passed in tabular form to the compartmental model using the FUNCTION_TABLE feature of NEURON.Note that in this case there is no need to know the channel reversal potential.The stopping criterion in step (v) is discussed below.In step (vii) we generally used the modified linear interpolation algorithm described by Gerald and Wheatly (1984) in order to improve efficiency.The combination of the analytical approximation described in the previous sub-section together with the iterative improvement described in the present subsection will be referred to as the Cox method with correction.

The dentate gyrus granule-cell model
Our model of the dentate gyrus granule cell (Fig. 1) was based on the model of Alle and Geiger (2006).A minor adjustment to the spacing of the MFBs was made in order to be consistent with the anatomical data of Acsady et al. (1998).Specifically, if x is the distance from the soma, then the inter-MFB distances were set to 160 μm for x < 800 μm, 220 μm for 800 μm ≤ x ≤ 1850 μm and 340 μm for x > 1850 μm.'Escape' of the voltage could be avoided only if the axon was ≥1.8 mm long, with MFB 3 in Fig. 1 as the recording/injection site, or 2.16 mm long for MFBs 4 and 5.This reflects the higher NaP channel densities required when less membrane area is available.In fact, mossy fibres extend to >3 mm in length (Acsady et al., 1998), but they can be severed during slicing.Unless stated otherwise, the axon length was 2.16 mm, and recordings were made at x = 664 μm, corresponding to MFB 4 in Fig. 1.
Based on the data of Schmidt-Hieber et al. ( 2007), 'collapsed' spines were added to the dendrites such that the membrane area per unit length was increased by a factor 1.5 + (x − x 0 )/(x 1 − x 0 ), where x 0 is the length of the apical trunk (7 μm) and x 1 is the distance from the soma to the tip of a dendrite (407 μm).Unless stated otherwise, the specific membrane capacitance (C m ) was assumed to be 1 μF cm −2 throughout the model neuron, and R a was set to 194 cm in the soma and dendrites (Schmidt-Hieber et al., 2007).The value of R a in the axon was adjusted to get C m close to 1 μF cm −2 (see Results).Following Schaefer et al. (2003) and Castelfranco and Hartline (2004), membrane conductances were assumed to have a uniform spatial distribution, unless stated otherwise.In addition to MFBs, mossy fibres are also equipped with small en passant varicosities (Acsady et al., 1998), but with an average diameter of <2 μm and an axonal spacing of ∼250 μm they will have only minor effects on the effective values of C m , R a and membrane specific resistance (R m ).
Experiments were conducted at 34-36°C.Whole-cell voltage-clamp recordings were obtained from four large (∼7 μm diameter) MFBs in four different hippocampal slices.The holding potential was −80 mV.Voltage ramps (from −100 to −30 mV in 0.6 or 1.2 s) were applied in groups of 10 sweeps (sweep length = 1 or 2 s, with the ramp applied from 0.2 to 0.8 s or from 0.4 to 1.6 s, intersweep interval = 2 or 5 s, respectively).After series-resistance compensation, the access resistance (R access ) was estimated to be 6-10 M .Given that I was <55 pA, the error in V introduced by R access is expected to be negligible (<0.6 mV).This was confirmed by simulations.

Preprocessing of voltage-ramp data
To ensure reasonably stationary behaviour following TTX application, a quadratic polynomial was fitted to part of the mean I(t) curve for each group of 10 sweeps, using a t interval determined by the points of inflection observed in the absence and presence of TTX (Fig. 2A).The approximate constancy of the quadratic coefficient was then used as a guide as to which 10-sweep groups to use for averaging (e.g. the filled symbols in Fig. 2B).Further analysis was then performed on the resulting grand average I(V) curves.The polynomials were fitted over a t interval determined by the points of inflection observed in the absence and presence of TTX.The inset shows the voltage-ramp protocol; the continuous portion of the V(t) plot corresponds to the I(t) plots below.B, the quadratic coefficients of fitted curves, such as those in A, were used to decide which group-mean traces to average (filled symbols).C, the resulting grand average I(V) curves (noisy traces) were fitted with eqn (18) (control) and eqn (21) (TTX) (smooth curves).The TTX-sensitive current was calculated as the difference (TTX minus control) of the fitted functions.D, the holding current (I hold ) for each sweep during the first ∼7 min.All data in A-D are from the same mossy fibre bouton.The increase in I hold during the whole-cell recording was observed in three of the four mossy fibre boutons (Table 1C).[Colour figure can be viewed at wileyonlinelibrary.com]Data from n = 4 mossy fibre boutons (MFBs; 1-4) obtained from one animal.A, parameter estimates from fits of eqn (18) to grand average I(V) data in control conditions (no TTX).B, parameter estimates from fits of eqn (21) to grand average I(V) data from the same four MFBs as in A but in the presence of 1 μM TTX.C, estimates of holding current (I hold ) at −80 mV, capacitive charging current (I C ) and apparent resting membrane potential (V rest ) in the presence and absence (control) of 1 μM TTX in the recording conditions stated in the Methods.Note that the estimate of V rest will not be negative enough if there is a significant leak of current through the seal resistance.Bootstrapped two-sided P-values for the paired differences ( = TTX − Control) are given.Note also that fitting regression lines to I hold versus t prior to the application of TTX always yielded negative slopes that were statistically significant according to an F-test: MFB 1, −0.0338 ± 0.0027 pA s −1 , P < 0.001; MFB 2, −0.0436 ± 0.0053 pA s −1 , P < 0.001; MFB 3, −0.0157 ± 0.0057 pA s −1 , P = 0.012; MFB 4, −0.0329 ± 0.0082 pA s −1 , P < 0.001.The bootstrapped 95% confidence interval for the mean slope was (−0.039, −0.020) pA s −1 .
Before TTX application (i.e. the control conditions), each grand average I(V) curve was fitted with the function: where V 0 is the ramp starting V (which was taken as −90 rather than −100 mV in order to avoid an initial artefact; see Fig. 2A) and the values of α and β coefficients are constants.The quadratic and cubic functions were constrained to be smooth and continuous at the point of inflection (V = V 1 ), i.e.: 19) Following TTX application, each grand average I(V) curve was fitted with the function: The values of V 1 , α 2 , α 3 and the β coefficients were all estimated by the least-squares fitting program.Example fits are shown in Fig. 2C, and parameter estimates are given in Table 1A (control) and B (TTX).Equations ( 18) and ( 21) were also used to approximate the fictitious current (I * ) used in the correction algorithm.

Estimation of membrane current density (i), membrane specific capacitance (C m ) and resting membrane potential (V rest )
The parameter values in Table 1A and B were used to construct mean I(V) curves for the determination of i in control conditions [eqn (18)] and TTX [eqn (21)], respectively, using the Cox method with correction (see above).Note that in control conditions, i = i NaP + i R , where i R is a residual, TTX-insensitive current density that remains when i NaP is blocked by TTX; i R implicitly includes a leak current density (i L ).If it was desired to estimate the value of C m consistent with the chosen values of R a and R seal (the seal resistance), the value of C m in the compartmental model was optimized to account for the capacitive current (I C ).The latter was estimated as: where I(V hold ) is the ramp current at V = V hold calculated from eqn ( 18) or ( 21).Specifically, starting with an initial estimate of 1 μF cm −2 , C m was updated on the k th iteration according to: where ÎC,k−1 is the NEURON estimate of I C , calculated from eqn (22) using the model estimates of I(V hold ) and I hold .Otherwise, C m was fixed at 1 μF cm −2 , and R a or R seal was adjusted in successive runs to obtain an acceptable model prediction of I C .Unless stated otherwise, the value of C m was 1 ± 0.05 μF cm −2 in the simulations reported below.Iterations were continued until the maximum errors (ε) on I and I C were both <0.1 pA [step (v) in the correction algorithm].With these tolerances, convergence was generally achieved in three to five iterations (<40 s with the Python software that accompanies this paper running on a desktop PC).The value of V rest was estimated by setting I(V) = I C and solving eqn ( 18) or ( 21) for V [a modified linear interpolation algorithm (Gerald & Wheatly, 1984) was used for this purpose].Note that the observed I(V c ) plot will not estimate V rest correctly because of I C and current flowing through R seal .Therefore, the trial value of R seal was used to correct the values of V used in eqns ( 18) and ( 21) (relative to V c ).

Modelling the effects of g NaP on EPSP conduction
An EPSP was simulated by setting the somatic V according to the following waveform: where τ 1 = 25 ms, τ 2 = 36 ms and B is related to the peak change in V ( V peak = 10 mV) according to: The time constants were chosen in order to get similar EPSP kinetics to those in Fig. S2B of Alle and Geiger (2006).A uniform leak-corrected current density was used, estimated as i(V) − i L (V), where i(V) was determined by the Cox method with correction from the pooled grand mean experimental I(V) curves (mean parameter values in Table 1A), assuming R seal = ∞ G , x = 664 μm and an axonal length of 2.16 mm.The value of i L (V) was estimated from the first 1% of the i(V) curve.In addition to i(V) − i L (V), a uniform passive conductance (g pas ) was inserted into the model axon in order to obtain R m = 60 k cm 2 and V rest = −89 mV as reported by Alle and Geiger (2006): where E Na = 50 mV, E pas is the passive reversal potential and, at the potentials used here, i NaP ≈ Figs 6 and 9).The value of R a was set to 100 cm and C m was set to 1 μF cm −2 .The effect of g NaP on EPSP conduction was assessed either by scaling i(V) − i L (V) ≈ i NaP (V) or by varying the somatic holding potential while keeping i(V) − i L (V) at its default level (i.e. the level determined by the Cox method with correction).

Statistics
Mean parameter values for experimental MFBs (n = 4, from one animal) are given with standard deviations.F tests were based on the normal distribution.Bootstrapped P-values were obtained using the percentile method.Bootstrapped confidence intervals were obtained using the BCa method.In both cases, 10,000 bootstrap samples were used.

Results
The hippocampal mossy fibre is considered to contain two uniformly distributed (specific) conductances: (i) the persistent sodium conductance (g NaP ); and (ii) a residual V-dependent conductance (g R ), which remains in the presence of the NaP-channel blocker TTX and which implicitly includes a V-independent leak conductance (g L ).Both conductances are assumed effectively to gate instantaneously compared with the time scale of the voltage ramps used here.Whole-cell voltage-clamp recordings were made on four large (∼7 μm diameter) MFBs.Specifically, V-ramps (from −100 to −30 mV in 0.6 or 1.2 s; Fig. 2A, inset) were used to estimate the stationary current densities i NaP + i R (control) and i R (in 1 μΜ TTX) using the Cox method with correction.As discussed in the Methods, this is an iterative procedure, in which estimates of current density (i) are inserted into a NEURON compartmental model of a dentate gyrus granule cell in order to generate predictions of the current (I) recorded during the V-ramp.This current was smoothed with polynomials.Unless stated otherwise, analyses were carried out on I(V) curves computed using the mean polynomial coefficients in Table 1A and B as applied to control conditions [eqn ( 18)] and TTX [eqn ( 21)], respectively.Estimates of the membrane specific capacitance (C m ) and resting membrane potential (V rest ) are also obtained.

Analysis of control voltage-ramp data
Figure 3A shows that the Cox method yields an excellent prediction for I without correction (iteration index k = 0) when the model morphology consists of a single MFB without filopodia at the midpoint of a 10-mm-long axon.
Example fits for the more realistic model of Fig. 1 are shown in Fig. 3B-D.The axon was 2.16 mm long, and simulated MFB recordings were made at 497.5, 664.5 and 891.5 μm from the soma (MFBs 3, 4 and 5 in Fig. 1).These MFBs are at distances similar to those in the study by Alle and Geiger (2006); see their Fig.4B.Despite the shorter axon, the extra membrane area of the additional MFBs means that when i is inserted into the model the resulting prediction of I is too high, especially for V < V rest (k = 0 curves in Fig. 3B-D).However, after five iterations the correction algorithm produces good agreement with Observed curves are plots of eqn (18) using the mean parameter values in Table 1A (n = 4 MFBs from one animal) and with V ≥ −90 mV in order to avoid the initial artefact (see Fig. 2A).Predicted curves were obtained from the NEURON compartmental model.Iterations (k) were continued until the maximum errors on I and I C were <0.1 pA.In all cases, C m = 1 ± 0.05 μF cm −2 ; the indicated values of R a are those required to achieve this tolerance on C m (see Fig. 4B and Table 2A).x = distance from the soma; axon length = 2.16 mm; infinite seal resistance.
the observed I (k = 4 curves in Fig. 3B-D).Plots of i = i NaP + i R versus V for k = 4 are shown in Fig. 4A.Evidently, the choice of model MFB does not greatly affect the estimate of i(V) obtained with the Cox method with correction.The form of the i(V) curves is consistent with what would be expected for an inward, regenerative current density (i NaP ).Indeed, i(V) has a similar form even without correction (k = 0); the changes in i(V) for k = 0-4 are mainly quantitative (data not shown).In contrast, the form of I(V) ∼ I(t) (Fig. 2A) gives a rather poor impression of the form of i(V).This reflects space-clamp errors.
As shown in Fig. 4B (black curves and markers), the optimal value of C m increased with the choice of R a .This reflects the fact that there is a lower axial current with higher R a values, hence more of I C must be accommodated locally (necessitating a higher C m ).Assuming that the true value of C m is 1 ± 0.05 μF cm −2 , the optimal values of R a were 85-100 cm (Fig. 4B).These estimates are consistent with those of Alle and Geiger (2006).As shown in Fig. 4C (black curves and markers), the 'passive' specific membrane resistance (R m , estimated as the chord resistance between −85 and −65 mV) decreased with R a , because more membrane current must be accommodated locally at higher R a values.Given the presence of the broad-spectrum K + channel blocker 4-aminopyridine, one would have expected R m to be higher than the value of 60 k cm 2 reported by Alle and Geiger (2006).But the plot in Fig. 4C suggests that R m = 30-50 k cm 2 if R a ≈ 100 cm.Even lower values (∼20 k cm 2 ) are obtained by using the first 1% of the i(V) curve to estimate R m (Table 3A).The reason for this unexpected result is unknown.One possibility is that the relatively depolarized V rest in control conditions (Table 1C) led to activation of i NaP (Fig. 6C and D), an influx of Na + and a consequent opening of Na + -activated K + channels, KNa (Budelli et al., 2009;Hage & Salkoff, 2012;Koh et al., 1994;Schwindt et al., 1989;Wang et al., 2003).This hypothesis The key in B applies to C and vice versa.D, a non-ideal seal resistance (in gigohms) does not have a large effect on the predicted i(V), although there is a leftward shift at more depolarized V. Progressively higher values of R a are needed as R seal is reduced (Table 2A; see also panel B).Data were derived from the analysis of experimental I(V) curves computed using the mean parameter values in Table 1A (n = 4 MFBs from one animal).
[Colour figure can be viewed at wileyonlinelibrary.com] could be tested by using a Cs-or TEA-based pipette solution to block i KNa .The sensitivity of the predicted i(V) curves to different choices of R a and various other model parameters is illustrated in the Appendix.Within 5 cm, we were unable to obtain fits for values of R a lower than those shown in Fig. 4B and C.
In the above simulations it was assumed that the seal resistance (R seal ) was infinite.For finite R seal values, the current through the gigaseal was estimated as (V − E seal )/R seal where, for want of a better alternative, the seal reversal potential (E seal ) was set to −LJP = −2 mV.A finite R seal did not have a large effect on the predicted i(V) curve, although there was a leftward shift at more depolarized V (Fig. 4D).This reflects the fact that at the observed V rest (∼ −46 mV) the inward current through R seal must be balanced by an outward membrane current.Progressively higher values of R a were needed as R seal was reduced, in order to keep C m close to 1 μF cm −2 (Table 2A; see also Fig. 4B).This is to be expected because as more current flows through the seal, less current needs to flow axially in order to account for the recorded current.But the variation of C m and R m with R a was similar to that observed with infinite R seal (Fig. 4B and C, grey curves and markers).In both cases there was a lower limit to R a below which fits were not possible, which implies that C m is >0.65 μF cm −2 (Fig. 4B).Assuming R a ≤ 120 cm 2 (Alle & Geiger, 2006), the simulations suggest that R seal was ≥15 G in control conditions in the present study (Table 2A).To ensure good coverage, we performed simulations for R seal values in the range 10 G ≤ R seal ≤ ∞ G , unless stated otherwise.In any case, the results for R seal values of 10 and 15 G are similar.

Analysis of TTX voltage-ramp data
Estimates of the holding current (I hold ) at the holding potential (V hold = −80 mV) are given in Table 1C, together with the resting potential (V rest ) and the current needed to charge the membrane capacitance (I C ).The mean I C values were very similar for the control and TTX data.The relatively depolarized control values of V rest , compared with the normal V rest of ∼−86 mV reported by Alle and Geiger (2006) (assuming an LJP of 10 mV in their study), presumably reflect blockage of K + channels by 4-aminopyridine (see Methods).The fall in V rest to more negative values after application of TTX, together with the change in sign of the quadratic coefficient (Fig. 2A  and B), are what one would expect after block of i NaP .More surprising is the fact that in three of four MFBs I hold was more negative in the presence of TTX, resulting in a statistically significant effect (Table 1C).A plot of I hold versus t for one of these MFBs is shown in Fig. 2D.Given that i NaP is a depolarizing current density, one might have expected I hold to be less negative in TTX.As discussed below, one possible explanation for this unexpected change in I hold after TTX application is a deterioration of the gigohm seal.This interpretation is supported by the change in I hold in the same direction even before TTX was applied (i.e.I hold drifted towards more negative values over time, in all experiments; see Fig. 2D and the legend to Table 1C).
Using the values of R seal and R a from Table 2A (control conditions), R seal in the presence of TTX (R TTX seal ) was varied until C m was within the range 1 ± 0.05 μF cm −2 .This resulted in the R TTX seal values shown in Table 2B.Comparing these values with those of R con seal (which are the control values from Table 2A), it seems likely that R seal was >10 G in control but fell to <5 G in TTX (assuming that R a and C m remained constant).If there was such a fall in R seal , it was probably not a direct effect of TTX (which seems unlikely) but rather a time-dependent deterioration of the gigohm seal (which is not uncommon).As in control conditions, varying R TTX seal did not have a large effect on model predictions.For example, Fig. 5A shows the fit of the model prediction to the observed I(t) for R TTX seal = 4.25 and 3.4 G (corresponding to R con seal = ∞ and 15 G , respectively; MFB 4, x = 664 μm, 2.16-mm-long axon).Figure 5B shows the corresponding predicted voltage dependence of the residual current density (i R ).The region of negative slope conductance is gone, as one would expect if i NaP were abolished.We note that fitting the model to the data for MFB 3 in Table 1C (which did not show an increase in I hold ) still necessitated a reduction of R seal in TTX (data not shown).The value of I hold hardly changed in this case because of the reduction in inward current associated with the TTX-induced change in i(V) (e.g.compare Fig. 5B and Fig. 4D).In other words, if there had been no fall in R seal , I hold would have become less negative.
If one supposes there were no change in R seal (i.e.R TTX seal = R con seal ), then with R a held constant one finds that: (i) C m approximately doubles in the presence of TTX [which might seem rather unlikely, especially as the nominal value of 1 μF cm −2 might itself be something of an overestimate (Koch, 2004)]; and (ii) the form of i R (V) is not changed; the curves in Fig. 5B are simply moved down (which is the cause of the more negative I hold in this scenario) and are somewhat steeper at high V (Fig. 5C).A similar result is obtained if one holds C m at 1 μF cm −2 and varies R a in order to fit the data (again with R TTX seal = R con seal ), but this time i R (V) is a little shallower at high V (Fig. 5D).However, by comparing corresponding R a values in Table 2A and C, it can be seen that R a must be reduced by ∼50% and to rather low values (∼50 cm if R seal is >10 G ), although still higher than the resistivity of the pipette solution (ρ p ).This solution included 155 mM KCl, which has a resistivity of ∼40 cm at 36°C (Haynes, 2013); this puts an upper limit on ρ p .In principle then, one might conjecture that gradual infiltration of the axon by pipette solution could lead to the predicted fall in R a .In practice, however, R a is found to be much larger than ρ p , presumably because of the presence of cell organelles (see appendix A of Koch, 2004) and of cytoskeletal and transport proteins.Hence, a deterioration of the gigohm seal might seem the most plausible explanation for I hold being more negative than expected in the presence of TTX compared with control conditions.In any case, as shown in Fig. 5, the basic form of i R (V) is insensitive to whether one chooses R seal , R a or C m as the variable that changes upon application of TTX (or with time).
The slope conductance at hyperpolarized V was much less (i.e.R m was much higher) in TTX than in control conditions (compare Fig. 5B with Fig. 4D).Indeed, as expected, R m (>100 k cm 2 ) was evidently higher than the value of 60 k cm 2 reported by Alle and Geiger (2006).As indicated in the previous subsection, it is the comparatively low value of R m in control conditions that is unexpected.But whatever the explanation of the increase in R m in TTX at low V, it did not manifest itself as a reduction in the initial slope of the observed I(V c ) plot, as measured by the parameter β 1 in Table 1A and B. In both TTX and control conditions, β 1 ≈ 0.6 pA mV −1 , which corresponds to an input resistance β −1 1 ≈ 1.7 G , similar to the value of 1.4 G obtained by Alle and Geiger (2006) from the V response to hyperpolarizing I pulses.At first sight, this result might seem paradoxical.However, the interpretation of input resistances in control conditions at low V is problematic, because over much of the axon i NaP + i R (V) is not even in the 'linear' hyperpolarized range but is in the region of negative conductance to the right of the peak (Fig. 4A and Fig. A3A in the Appendix).In this case, a comparison of input resistances for control and TTX treatments might seem invalid or, at best, uninformative.

Estimation of the stationary persistent sodium current density, i NaP
In the absence of space-clamp errors and after correcting for any difference in I hold , i NaP can be estimated as the TTX-sensitive current (black curve in Fig. 2C) divided by an appropriate membrane area.But in the presence of space-clamp errors, i NaP must be estimated from current densities that have been corrected for these errors.In the previous two subsections, we described how the Cox method with correction was used to estimate i NaP + i R in control conditions and i R in the presence of TTX, where i R is a residual current density that remains when i NaP is blocked.Originally, the intention was to estimate i NaP by subtracting i R (i.e.curves like those in Fig. 5B) from i NaP + i R (i.e.curves like those in Fig. 4D).But because of the increase in R m in TTX [as indicated by the smaller initial slope of the i(V) curve] this strategy had to be modified.First, a presumptive leak current density (i L , assumed linear) was estimated from the first 1% of the i NaP + i R versus V curve.The i L (V) lines were then subtracted from the i NaP (V) + i R (V) curves to yield the six i NaP (V) + i R (V) − i L (V) curves shown in Fig. 6A.These curves were constructed for various values of x and for R con seal values of 10 and ∞ G .Mean values of the leak conductance (g L = 1/R m ) and reversal potential (E L ) are given in Table 3A.Second, i R was also corrected for i L , although in some cases the initial slope of i R (V) was slightly negative, in which case the minimum value of i R was used to estimate i L and values of i R − i L to the left of this minimum were set to zero.The resulting set of six i R (V) − i L (V) curves is shown in Fig. 6B.These curves were then subtracted from the corresponding curves in Fig. 6A to yield the estimates of i NaP shown by the six curves in Fig. 6C (thin lines).This procedure eliminates any effect of the increase in R m in TTX.However, we cannot exclude the possibility of a  1B and with V ≥ −90 mV in order to avoid the initial artefact (Fig. 2A).Predicted curves were obtained using the Cox method with correction as in Fig. 3.The pair of numbers associated with each thin curve indicate seal resistance (R seal ) before (first number) and after (second number) setting g NaP = 0 in order to simulate the application of TTX.Axial resistivity (R a ) was set to its value in the absence of TTX (Table 2A), and R TTX seal was varied to obtain C m = 1 ± 0.05 μF cm −2 (Table 2B).The two predicted I(t) curves are virtually indistinguishable.Good fits were also obtained with R con seal values of 10 and 25 G (data not shown).B, residual membrane current density (i R ) as a function of V.This is the current density that remains when i NaP is blocked.In C and D, the dashed curves show the effect of holding R seal constant (∞) and either approximately doubling C m (C) or halving R a (D) in order to obtain a fit to the observed I(t) data in the presence of TTX.The fits to these data were comparable to those in A.  A, estimates of the leak specific conductance ( ḡL ) and reversal potential (E L ) obtained from the first 1% of i NaP + i R versus V curves, such as those in Fig. 4A and D. These parameters were used to generate estimates of the leak current density, i L (V) = ḡL (V − E L ), hence the curves in Fig. 6A.Grand mean values: ḡL = 47 μS cm −2 , E L = −74 mV.The value of ḡL implies Rm = 21.3 k cm 2 .Data were derived from the analysis of I(V) curves computed using the mean polynomial coefficients in Table 1A and B. B, parameter estimates obtained from the fits of eqn (28) to each of the thin curves in Fig. 6C, which were generated by subtracting the curves in Fig. 6B from those in Fig. 6A.The value of E Na (61.4 mV) was calculated from the Nernst equation.The parameter estimates obtained from the fit of eqn (28) to all six thin curves simultaneously were: ḡNaP = 69.5 μS cm −2 , V NaP = −63.0mV and k NaP = 14.9 mV.Data were derived from the same I(V) curves as in A. C, parameter estimates obtained from fits of eqn (28) to each of the thin curves in Fig. 6D, which were derived using the individual I(V) parameters for each of the experimental MFBs (1-4 in Table 1A and B).x = 664 μm and infinite R con seal were assumed when applying the Cox method with correction.Abbreviation: CV, coefficient of variation.The parameter estimates obtained from the fit of eqn (28) to all four thin curves simultaneously were: ḡNaP = 86.0μS cm −2 , V NaP = −63.8mV and k NaP = 15.8 mV.Estimates of g L and E L in control conditions are also given.D, as Table 3C, but for finite R con seal .Fits were not possible for R con seal lower than those shown, within 5 G .The parameter estimates obtained from the fit of eqn (28) to the pooled i NaP (V) data from all four MFBs were: ḡNaP = 82.3μS cm −2 , V NaP = −64.2mV and k NaP = 15.4 mV.

R. Murphy and others
J Physiol 602.8 change in i R .In particular, if there was a rundown of i R between control and TTX then i NaP will have been underestimated.
As a first attempt to quantify the voltage dependence of g NaP we suppose that, in the stationary state, g NaP (V) can be described by a generic HH model, i.e. a Boltzmann function raised to some power, p.Then, if ḡNaP is the maximum specific conductance, V NaP the V for half-maximal activation and k NaP the slope factor, we have: The thick continuous curve in Fig. 6C shows eqn (28) fitted to all six curves simultaneously with p = 3.This simple model seems to provide a reasonable description in the subthreshold V range (say, V ≤ −45 mV; Mateos-Aparicio et al., 2014), but the slope of i NaP (V) appears too shallow for V > −45 mV.Attempts to force a higher value of ḡNaP [and obtain a more negative i(V) slope for V > −45 mV] resulted in poor fits for V ≤ −45 mV, hence the fit shown in Fig. 6C is preferred.This is especially so because during an action potential the transient sodium conductance (g NaT ) will rapidly become much larger than g NaP .Hence, it is more important to model g NaP accurately in the subthreshold range.
The parameter estimates obtained from the fit of eqn ( 28 3B.It can be seen that changes in x and/or R con seal have little effect on V NaP or k NaP ; the main variation is in ḡNaP , and even this is modest.Of course, all the analyses so far have been done using the  3. A, leak-corrected current density in the absence of TTX.Leak current density (i L ) was estimated from the first 1% of i NaP + i R (V) curves like those in Fig. 4D (see Table 3A for parameter estimates).i R is the residual, TTX-insensitive current density.B, the leak-corrected i in the presence of TTX was estimated in a similar fashion from curves like those in Fig. 5B.The curves in B were then subtracted from the corresponding curves in A to yield the six thin curves in C.These curves were fitted with eqn (28) (thick continuous curve).Data in A-C were derived from the analysis of experimental I(V) curves computed using the mean parameter values in Table 1A and B (n = 4 MFBs from one animal).D, the same analysis applied to the four experimental MFBs, using the individual parameter values from Table 1A and B; x = 664 μm and infinite R con seal were assumed when applying the Cox method with correction.The thin curves give i NaP (V) for each experimental MFB; the thick curve is a fit of eqn ( 28).[Colour figure can be viewed at wileyonlinelibrary.com] mean I(V) parameter values in Table 1A and B. To obtain a sense of the variation between experimental MFBs, the analysis in Fig. 6C for x = 664 μm and infinite R con seal was repeated using the individual I(V) parameters for each of the experimental MFBs (1-4) in Table 1A and B. The resulting plots of i NaP (V) are shown in Fig. 6D (thin curves), and parameter estimates from the fits of eqn ( 28) to each of these curves are given in Table 3C.Between-experimental MFB variability appears somewhat larger than that attributable to the choice of x or R con seal in the model (cf.Fig. 6C and D).But, as in Table 3B, the main variation is in ḡNaP ; V NaP and, to a lesser extent, k NaP show lower relative variability (Table 3C).The thick curve in Fig. 6D is a fit of eqn ( 28) to the four thin curves taken together; parameter estimates were ḡNaP = 86.0μS cm −2 , V NaP = −63.8mV and k NaP = 15.8 mV, similar to the mean values in Table 3C.Fits to the individual MFB i NaP (V) curves were not possible for R con seal = 10 G , but fits were possible for R con seal = 15 G (three MFBs) and 25 G (one MFB).The resulting parameter estimates (Table 3D) are similar to those for infinite R con seal (Table 3C).We note that in all cases setting p = 1 rather than p = 3 in eqn ( 28) always resulted in a worse fit as judged by visual inspection and from the error sum of squares (SSE; data not shown).

Relating NaP to NaT
Equation ( 28) and the parameters in Table 3 provide a basis for approximating i NaP in models of the hippocampal mossy fibre (at least at subthreshold V and assuming 'instantaneous' gating).But, as discussed in the Introduction, it is also of interest to relate g NaP to g NaT .Engel and Jonas (2005) developed an HH model of g NaT in MFBs based on outside-out patch-clamp measurements.As in the classical HH model, they assumed that the value of the steady-state inactivation variable (h ∞ ) approached zero as V → ∞ (dashed curve in Fig. 7A).But as pointed out by Taddese and Bean (2002), if one allows h ∞ to approach a small positive value as V → ∞ then a persistent sodium current might arise.To test whether an HH-type model of g NaT could account for g NaP in the mossy fibre, we modified the description of h ∞ (V) in the model of Engel and Jonas (2005).Initially, we note that from Fig. 5B of Engel and Jonas (2005), a Boltzmann function with h ∞ → 0 as V → ∞ gave a better fit to the data than their HH model.Accordingly, we fitted the following Boltzmann function to their data (digitized from their Fig.5B) in order to allow h ∞ > 0 as V → ∞: Here, V h is the midpoint potential, k h is the slope factor, h 0 ∞ is a small positive offset as V → ∞, and V shift is a V offset (discussed below), which was set to zero for the fit.This curve also gave a better fit to the data than the HH model of Engel and Jonas (2005) (continuous curve in Fig. 7A).Moreover, for 0 < h 0 ∞ ≤ 0.012 (incomplete inactivation), the fit of eqn ( 29) was as good as or better than the fit with h 0 ∞ = 0 (complete inactivation) as judged by the SSE (Fig. 8A).The best fit (minimum SSE) was obtained with h 0 ∞ = 0.006 (arrow in Fig. 8A).We next fitted the modified HH model to the pooled i NaP data from the four experimental MFBs in Fig. 6D (indicated by the shaded area in Fig. 7B).Gating was assumed instantaneous, and steady-state activation (n ∞ ) was as described by Engel and Jonas (2005) but with V replaced by V − V shift .[In their model of the mossy fibre, Engel and Jonas (2005) also replaced V with V − V shift for both activation and inactivation, with V shift = 12 mV.]To fit the model, various values were chosen for h 0 ∞ , and for each of these values eqn (29) was fitted to the h ∞ (V) data with V shift = 0.Then, using the resulting estimates of V h and k h , the following function was fitted to the i NaP (V) data: with V shift and the maximum specific NaT conductance (ḡ NaT ) as adjustable parameters.Example fits are shown by the coloured curves in Fig. 7B, while Fig. 7C shows plots of the V shift and ḡNaT estimates versus h 0 ∞ .In Fig. 7D is shown g NaP (V) = i NaP (V)/(V − E Na ).As in Table 3B-D, the main uncertainty regarding g NaP is in its value (ḡ NaP ) as V becomes large.
As judged by visual inspection and the SSE (Fig. 8B), the modified HH model gave reasonable fits to the i NaP data, comparable with those of eqn (28), provided h 0 ∞ > 0.004 (compare Fig. 6D with Fig. 7B).Considering the fits of both eqns ( 29) and (30) (Fig. 8), a reasonable range of values for h 0 ∞ is 0.005 ≤ h 0 ∞ ≤ 0.012.SSE(i NaP , eqn (30)) changed by only ∼1% within this range and was within 1% of SSE(i NaP , eqn ( 28)) (Fig. 8B).The latter observation argues against overfitting, which would not be expected in any case given the form of the data, the form of eqn (30) and the fact that it has only two adjustable parameters [the others being determined by the fit of eqn ( 29)].And although the relative change in SSE(h ∞ , eqn ( 29)) was larger than that of SSE(i NaP , eqn ( 30)) (Fig. 8A), the fit of eqn (29) was still impressive (R 2 > 0.999).The best fit of eqn (30) was obtained with h 0 ∞ = 0.007 (arrow in Fig. 8B), close to the best-fitting value (h 0 ∞ = 0.006) for eqn (29); the former value was used for the fit of eqn (29) shown by the continuous curve in Fig. 7A.Within the range 0.003 ≤ h 0 ∞ ≤ 0.014, the estimated values of V h (−89.59≤ V h ≤ −89.33 mV) and k h (7.58 ≤ k h ≤ 7.78 mV) hardly changed.
One interpretation of the parameter V shift is as follows: J Physiol 602.8 (Engel & Jonas, 2005), where V LJP is a shift in the LJP and V Donnan is a shift in the Donnan potential, both calculated for the whole-cell configuration relative to the outside-out patch (i.e.whole-cell minus patch).The values of V in Fig. 7A are from Engel and Jonas (2005), who did not correct for V LJP , whereas those in Fig. 7B are from the present study and are corrected for V LJP .Hence V LJP = 0 − V patch LJP = −V patch LJP .From the solutions A, the steady-state inactivation variable (dashed curve) for the Hodgkin-Huxley g NaT model of Engel and Jonas (2005); data points are taken from their Fig.5B.The continuous curve is a fit of a Boltzmann function with an offset, h 0 ∞ [eqn (29)].Values of voltage were not corrected for the liquid junction potential (LJP).h 0 ∞ = 0.007 and V shift = 0; V h (−89.4 mV) and k h (7.7 mV) were adjustable parameters.B, fits of eqn (30) (coloured curves) to the pooled i NaP (V) data from four experimental mossy fibre boutons (MFBs; n = 4 from one animal; same data as in Fig. 6D but indicated by the shaded area to avoid having too many curves).The chosen values of h 0 ∞ are indicated.C, estimates of the maximum NaT specific conductance ( ḡNaT ) and the shift of the h ∞ -V and n ∞ -V curves (V shift ) obtained from the fits of eqn (30) and plotted against the chosen values of h 0 ∞ .D, the NaP specific conductance as a function of the LJP-corrected voltage.E and F, similar to C, but for low and high R a , respectively.Also, the predicted ḡNaT is corrected for open probability [0.529; Engel and Jonas (2005)], and i NaP + i L is higher in the model MFBs than in the axon shaft by the factors indicated in the key.Values of V shift and points for which V shift < −4.5 mV or V shift > 7.5 mV are not plotted.The short-dashed lines indicate the range and the long-dashed lines the mean reported by Engel and Jonas (2005), both corrected to 35°C assuming Q 10 = 1.2-1.5.[Colour figure can be viewed at wileyonlinelibrary.com] used by Engel and Jonas (2005), one would expect V patch LJP = 4-5 mV, hence if V Donnan = 0, V shift = V LJP ≈ −4.5 mV.This would place h 0 ∞ at the extreme right of Fig. 7C, with a rather high value of 0.014.However, V Donnan is expected to be larger for the cell than for the patch (Marty & Neher, 1995), hence V Donnan > 0, which will shift V shift and therefore h 0 ∞ to the left.According to Marty and Neher (1995), a maximum value for V Donnan is ∼12 mV, which is the value of V shift used by (Engel & Jonas, 2005) ( V LJP = 0 in their case because they did not correct for V LJP in either patch-clamp configuration).Setting V Donnan = 12 mV and V LJP = −4.5 mV then gives V shift = 7.5 mV and h 0 ∞ ≈ 0.0042 (a rather low value).Assuming the best-fitting value h 0 ∞ ≈ 0.007 gives V Donnan = 6.9 mV, well below the expected maximum of 12 mV.Hence, a value for h 0 ∞ in the required range (0.005 ≤ h 0 ∞ ≤ 0.012) might not seem implausible.As shown in Fig. 7C, the corresponding predicted values of ḡNaT are in the range 6-15 mS cm −2 , or 11-28 mS cm −2  29) with h 0 ∞ > 0 (incomplete inactivation) is as good as or better than the fit with h 0 ∞ = 0 (complete inactivation; dashed horizontal line), and the fit is always impressive (R 2 > 0.999).B, for 0.005 ≤ h 0 ∞ ≤ 0.012 (dashed vertical lines) the SSE for the fit of eqn (30) changes by <1% and is within 1% of the SSE for the fit of eqn (28) (dashed horizontal line).
if one assumes a maximum open probability of 0.529 ± 0.053 as estimated by Engel and Jonas (2005) from fluctuation analysis; 21.6 mS cm −2 using the best-fitting h 0 ∞ value (0.007).Analysis of the individual MFB curves gave similar results (Table 4), and in all cases the P-values for the estimates of V shift and ḡNaT were essentially zero, reflecting the high density of data (>10 3 points per curve) afforded by using a V ramp rather than a family of V test pulses.Clearly, however, there was a large between-MFB variation in V shift (Table 4).According to the present interpretation of V shift , this might reflect a large variability of V Donnan , because V LJP is presumably fixed by the compositions of the internal and external solutions [eqn (31)].
In contrast, Engel and Jonas (2005) estimated a value for ḡNaT of 49 mS cm −2 for the peak NaT conductance at 23°C, which suggests a value of 61-80 mS cm −2 at 35°C (long-dashed lines in Fig. 7E and F), assuming a Q 10 of 1.2-1.5 (Hille, 2001).Alle, Roth et al. (2009) reported a similar value (78 mS cm −2 when corrected for open probability) based on voltage-clamp recordings from outside-out patches and numerical simulations of action potential propagation along mossy fibres.
Perhaps the simplest explanation for this discrepancy is variability.Thus, the present study is based on four MFBs from one animal, and the range of ḡNaT values reported by Engel and Jonas (2005) is large, i.e. 11-224 mS cm −2 (adjusted to 35°C; short-dashed lines in Fig. 7E  and F).Another possibility is that only a fraction of the NaT pool is available for generating i NaP , i.e. that most of the NaT channels fully inactivate (h 0 ∞ = 0).Alternatively, activation of i KNa (see above) and/or slow inactivation of i NaP during the V ramp might have led to an underestimation of i NaP .Given the V-ramp rates used in the present study (>50 mV s −1 ), the effect of i NaP inactivation is expected to be small, based on V-ramp data on mouse layer V neocortical neurons (Fleidervish & Gutnick, 1996) and rat entorhinal cortex layer II principal neurons (Magistretti & Alonso, 1999).But the situation might be different in dentate granule cells.It is also conceivable that ḡNaT is higher in the MFB than in the axon shaft, e.g. to boost the action potential amplitude and thus enhance presynaptic Ca 2+ influx (Engel & Jonas, 2005).To assess this possibility, some further simulations were performed, this time using the grand mean I(t) curve as input (thick grey curves in Fig. 3), x = 664 μm and i NaP (V) + i L (V) in the MFBs scaled by some factor relative to its value in the axon shaft.It was not possible to treat i NaP and i L separately, but of course when the final fit was obtained, i NaP (V) could be estimated as i(V) − i L (V), with i L (V) estimated from the first 1% of the i(V) curve.Good fits were obtained, similar to those in Fig. 3. Figure 7E shows the resulting plots of ḡNaT (MFB) versus h 0 ∞ , with ḡNaT corrected for open probability, −4.5 mV ≤ V shift ≤ 7.5 mV and R a = 100 cm.Evidently, the predicted values of A, fit of the model to each of the thin curves in Fig. 6D.x = 664 μm and infinite R con seal were assumed.The parameter estimates obtained from the fit to all four thin curves simultaneously (i.e. the pooled curves) were: h 0 ∞ = 0.007, V h = −89.4mV, k h = 7.70 mV, ḡ * NaT = 21.6 mS cm −2 and V shift = 2.3 mV.B, as for A, but for finite R con seal (same values as in Table 3D).The parameter estimates obtained from the fit of the Hodgkin-Huxley model to the pooled i NaP (V) curves from all four MFBs were: h 0 ∞ = 0.007, V h = −89.4mV, k h = 7.70 mV, ḡ * NaT = 21.4 mS cm −2 and V shift = 2.2 mV.* Assuming a maximum open probability of 0.529 ± 0.053 (Engel & Jonas, 2005).
ḡNaT are still low compared with those reported by Engel and Jonas (2005) and Alle, Roth et al. (2009), even with i NaP + i L 8-fold higher in the MFBs than in the axon shaft.Better agreement is obtained with i NaP +i L 2-to 4-fold higher in the MFB than in the shaft and R a = 145-148 cm (Fig. 7F).However, the latter R a values are rather high compared with values of ∼100 cm estimated by Alle and Geiger (2006).
Our estimates of ḡNaT are based on the assumption that the observed fall in I hold in TTX (Fig. 2D and Table 1C) was the result of a fall in R seal .In principle, R seal can be held constant, but in that case we find that either C m or R a must change by a factor of two (Fig. 5C and D).This seems less plausible than a fall in R seal , especially given that it is a common observation that the gigohm seal deteriorates during a V-clamp experiment.Also, the change in I hold before application of TTX is consistent with seal deterioration over time (see Fig. 2D and the legend to Table 1C).Perhaps C m and R a could both change by a lesser amount, but in any case the basic form of the i(V) curve in TTX is not affected if C m or R a change (Fig. 5).Instead, the curve shifts down (to account for the increase in I hold ) and either increases in slope if R a is held constant (Fig. 5C) or decreases in slope if C m is held constant (Fig. 5D).Because of the resulting increase in i R − i L (TTX) [which is subtracted from i NaP + i R − i L (Control) to obtain i NaP ], the former effect would lead to ∼25% increase in our estimates of ḡNaT .This is not nearly enough to account for the discrepancy between our estimates and those reported by Engel and Jonas (2005).

A potential functional role for NaP
Table 3C and D shows the estimated variation of ḡNaP between MFBs.If such variations were under the control of the neuron, this might provide a means of modulating the analog axonal signalling described by Alle and Geiger (2006), i.e. the conduction of EPSPs by the mossy fibre axon from the soma to presynaptic terminals, such as MFBs.To assess this possibility, we investigated the conduction of simulated EPSPs along the model mossy fibre (for details, see Methods).As shown in Fig. 9A and B, omitting g NaP had almost no effect on EPSP propagation.Of course, the true effect will be larger if i NaP was underestimated during the V ramp because of i KNa activation and/or i NaP inactivation.This could be tested experimentally by injecting an EPSP waveform into the soma in the presence and absence of TTX.Increasing g NaP by a factor of four decreased EPSP attenuation with x and prolonged the EPSP waveform (Fig. 9C).Plots of the normalized EPSP height, V peak (x)/ V peak (0), versus x for various g NaP scaling factors are shown in Fig. 9E.Increasing g NaP by a factor of five resulted in a sustained depolarization of the mossy fibre after the onset of the EPSP (not shown).But for scaling factors ≤ ×4, variations in ḡNaP , e.g. by altering the fraction of the ḡNaT pool that contributes to it, could be a possible means of modulating EPSP conduction.Alternatively, if the dentate gyrus granule-cell soma is more depolarized in vivo than in vitro owing to excitatory synaptic bombardment, neuromodulation or other factors, this could boost g NaP and thus EPSP conduction.The effect of somatic depolarization is illustrated in Fig. 9D, where the soma is held at −71 mV; again V peak attenuation is reduced and the EPSP waveform prolonged.If the soma is held at −70 mV, there is a sustained depolarization of the mossy fibre after F, similar to E, but with g NaP at the default level and various somatic holding potentials (given in millivolts in the key).The default level of g NaP (g NaP × 1) was determined from the analysis of experimental I(V) curves computed using the mean parameter values in Table 1A (n = 4 MFBs from one animal).See Methods for further details.
J Physiol 602.8 the onset of the EPSP (not shown).But more modest depolarizations could provide a means of modulating EPSP conduction (Fig. 9F).In practice, the situation will be more complex than depicted here, with larger variations in g NaP possible, and perhaps necessary, because of V-dependent K + conductances, which were blocked in the present study.

Discussion
In the present study, we have estimated the stationary voltage dependence of the persistent sodium-current density (i NaP ) in hippocampal mossy fibre axons based on in vitro whole-cell patch-clamp (voltage-clamp) measurements from MFBs.This whole-cell recording configuration, with a single patch pipette sealed onto a MFB, allows only for suboptimal space clamp, because of the thin and elongated geometry of the attached axon stem.To correct for this suboptimal space clamp and estimate the voltage dependence of i NaP , Cox's 'direct' method for correcting space-clamp errors in a neurite (Cox, 2008) was extended to the case of an isopotential compartment with several attached neurites of varying diameters.In principle, the method yields estimates of the voltage dependence of the total membrane current density, i(V).Indeed, in near-ideal conditions (10-mm-long model axon with one MFB) excellent agreement was obtained between the predicted and observed currents (Fig. 3A).With a more realistic compartmental model of a dentate gyrus granule cell, however (Fig. 1), the estimates of I(V) were not acceptable at hyperpolarized membrane potentials (V), i.e. when the estimated i(V) was inserted into the compartmental model, the predicted currents (I) were significantly more negative than the currents recorded during V-ramp experiments for V < V rest (Fig. 3B-D).Nevertheless, Cox's method is still of theoretical interest, and it might be more successful in other applications.At the very least, it provides a starting estimate of i(V) that might then be improved upon by optimization.This was the approach adopted in the present study.We call the resulting procedure the Cox method with correction.Using this method, we were able to obtain good agreement between observed and predicted I when the estimates of i(V) were inserted into the compartmental model (Fig. 3B-D).Schaefer et al. (2003), Castelfranco and Hartline (2004) and Cox (2008) have described methods for analysing currents recorded during V test pulses in poorly space-clamped neurites.These methods allow the estimation of time-dependent channel gating.Our own Cox method with correction builds on this earlier work, and it can also be used to estimate time-dependent channel gating, as will be described in a subsequent paper.In the present study, however, we were concerned only with estimating the stationary voltage dependence of i NaP (with gating assumed effectively instantaneous).Voltage ramps are especially advantageous in this context, at least for non-inactivating channels, because of the very high density of data in the V domain compared with V test pulses.Yet some form of smoothing might still be helpful in reducing noise.In the present study, polynomials in V were used for this purpose [eqns ( 18) and (21); Fig. 2C].The resulting estimates of i(V) were passed to the NEURON compartmental model in tabular form.This is convenient and avoids the need to specify a particular kinetic model.Nor is it necessary to specify the reversal potential.On the contrary, if one is prepared to assume a particular kinetic model [and reversal potential; e.g. eqn (28)] then this can be fitted to the i(V) data generated by the Cox method with correction (Fig. 6C and  D).If a reasonable fit is obtained, the resulting parameter estimates provide a concise summary of the stationary voltage dependence of the conductance (Table 3B-D).
But perhaps the more interesting scenario is a poor fit, or parameter estimates that are inconsistent with those obtained by other methods (e.g.inside-out or outside-out patches).Of course, such inconsistencies must be resolved.One possibility is a problem with those other methods (see Introduction).Another is a false assumption in the Cox method with correction.As in the studies by Schaefer et al. (2003) and Castelfranco and Hartline (2004), we assumed a spatially uniform maximum specific conductance, ḡ (x).This might seem reasonable for an axon.Moreover, Schaefer et al. (2003) concluded that even if ḡ (x) is non-uniform, one can still obtain reasonable estimates of the voltage dependence, g(V), under the assumption of a uniform ḡ (x), provided that the ḡ is sufficiently high near the stimulus site and that ḡ (x) is monotonic and not too steep.Clearly, however, it is conceivable that a particular non-uniform ḡ (x) could lead to significant errors in the estimation of g(V) if ḡ (x) is assumed constant.If one is prepared to assume that a pre-established g(V) is correct, this might be used to estimate ḡ (x), i.e. given g(V), one would optimize ḡ (x) in order to obtain the observed I(V).A potential problem with this approach is that it is not clear that fixing g(V) necessarily guarantees a unique ḡ (x).This problem of non-uniqueness is also a concern in passive cable modelling (Major, 2000).
When the form of ḡ(x) is an issue, an independent estimate of ḡ (x) seems desirable.In this regard, cell-attached and/or excised patch recordings might be helpful, and they have the additional benefit of providing an estimate of g(V) for comparison with that obtained from the space-clamp error correction algorithm.We emphasize that only relative channel densities, i.e. ḡ(x)/ḡ(0), are required by iterative whole-cell methods, such as the Cox method with correction or the method of Schaefer et al. (2003).Indeed, one aim of these methods is to return the absolute densities.But even relative channel densities might be estimated poorly by cell-attached and excised patch methods in some cases (Kole et al., 2008).An alternative, perhaps more reliable way to estimate ḡ(x)/ḡ(0) is by quantitative immunocytochemistry (e.g.Kole et al., 2008;Kovacs et al., 2010;Lörincz et al., 2002;Nusser, 2007), although it must be admitted that this approach does not demonstrate channel functionality.
The reason why it is necessary to correct for space-clamp errors is that, in general, these errors cannot be measured, i.e. we cannot measure V(t) at arbitrary x throughout the neurite.If we could, such measurements might be used to constrain the fitted model and provide information on ḡ(x).Huys et al. (2006) describe a linear regression approach to the estimation of ḡ(x) for multiple channel types, in addition to synaptic weights, reversal potentials and passive electrical properties, which looks forward to the rapid optical measurement of V(x,t) (Knöpfel & Song, 2019).Such measurements might be combined with whole-cell patch-clamp recordings.Huys et al. (2006) note that their approach does not require such recordings.However, their method does require channel kinetics to be provided, and these will generally be determined by patch-clamp methods.Cell-attached and excised-patch recordings have a role to play here (although these methods can cause certain artefacts; see e.g.Kole et al., 2008;Ruppersberg et al., 1991), as do whole-cell methods using space-clamp error correction algorithms.Moreover, these whole-cell methods might provide improved patch-clamp estimates of channel densities that can be compared with those obtained using the approach of Huys et al. (2006).
Space-clamp errors are not the only source of uncertainty when estimating i(V) in neurites by V clamp.Stability of the recorded currents is always a concern.In the present study, we used a quadratic polynomial to assess the stability of I(V) and to choose which traces to analyse (Fig. 2A and B).Access resistance (R access ) is another potential source of error, but its value can be estimated and incorporated into the analysis (Montnach et al., 2021); its effect was negligible in the present study, given the small currents involved.The values of membrane specific capacitance (C m ), axial resistivity (R a ) and patch-seal resistance (R seal ) are more difficult to assess.In the present study, we assumed C m = 1 ± 0.05 μF cm −2 and then estimated the value of R a required to fit the voltage dependence of the observed current (Fig. 4B).But this estimate is also affected by the assumed value of R seal (Table 2).Moreover, we found it necessary to assume that at least one of these three variables was different in the presence of TTX than in control conditions.A reduction in R seal is considered the most likely candidate (Table 2B).Otherwise, a fall in R a to rather low values (Table 2C) or a doubling of C m to an unlikely 2 μF cm −2 is required.The putative fall in R seal might simply be a time-dependent deterioration of the gigohm seal rather than an effect of TTX, which seems very unlikely.The estimated form of i(V) was robust to the choice of R seal , although lower values of R seal resulted in an upward shift in the i(V) curve, especially at more depolarized V (Figs 4D and 5B).Likewise, i(V) was rather insensitive to different choices of R a (and their implied values of C m ; see Fig. A1 in the Appendix).The choice of axon diameter (within reasonable limits; Fig. A2B) and MFB diameter (Fig. A2C and D) also had little effect on i(V).The i(V) curve was also rather insensitive to the choice of model MFB (Fig. 4A) and the scaling of i in the soma + dendrites and the first 20 μm of the axon (Fig. A2A).The latter result reflects the clamping of V close to V rest near the soma and at the axon terminus (Fig. A3A and C).Lei et al. (2020) discuss a number of measurement errors that can arise in patch-clamp recordings and present a mathematical model for correcting them.Such errors can contribute significantly to the apparent between-cell variability of inferred kinetic parameters.
An unforeseen difficulty in estimating i NaP was an apparent increase in the membrane specific resistance (R m ) in TTX compared with control conditions, as judged from the slope of the i(V) curve at hyperpolarized V (e.g.compare Fig. 4D with Fig. 5B).This might be interpreted as a reduction in the leak conductance (g L ), but its cause is unknown.One possibility is the opening of Na + -activated K + channels (KNa) in control conditions resulting in a lower R m than expected from the study by Alle and Geiger (2006).This hypothesis could be tested by using a caesium-or TEA-based pipette solution to block i KNa .It is interesting to note that the fall in g L did not lead to an increase in apparent input resistance (β −1 1 ≈ 1.7 G ; Table 1A and B).This seemingly paradoxical result reflects the lack of space clamp (Fig. A3A) and the highly non-linear behaviour of i(V) in control conditions (Figs 4A and D and A3B).In any event, the fall in g L complicated the estimation of i NaP (V) because the latter could not be estimated simply by subtraction (control minus TTX).Instead, before subtraction, the leak current density (i L ) had to be estimated separately for control conditions and TTX in order to generate plots of i NaP + i R − i L versus V (Fig. 6A) and i R − i L versus V (Fig. 6B), where i R is the TTX-insensitive residual current (including i L ).The resulting estimates of i NaP (V) were fitted with a third-order Boltzmann function [eqn (28); Fig. 6C and D], which gave a better fit than a first-order Boltzmann function.Variation in i NaP (V), whether attributable to the choice of model MFB and R seal (Table 3B) or between-experimental MFB variation (Table 3C and D), was manifested mainly in the variation of the maximum specific conductance (ḡ NaP ).Likewise, variation in g L rather than E L was the main cause of the variability of i L (V) (Table 3C and D).Presumably, such variations in specific conductance mainly reflect biological variation rather than measurement error (Lei et al., 2020).
Figure 4B shows that the optimal value of C m is positively correlated with the choice of R a .Figure A1 shows how the predicted current densities vary with the choice of R a (assumed constant) for two choices of R con seal .The implied values of C m are also shown.The lowest R a values in Fig. A1 are the lowest possible (within 5 cm) for fitting the I(V) data; this implies C m > 0.65 μF cm −2 .Koch (2004) suggests that the usual value of 1 μF cm −2 might be something of an overestimate, hence this value might be taken as an upper bound.In Fig. A1, C m is in the approximate range 0.7-1.3μF cm −2 , with corresponding R a values of 75-140 cm.These parameter values resulted in a relative variation of ∼20% (i.e.±10%) in the predicted current densities at V = −30 mV, or 10-15% if C m ≤ 1 μF cm −2 (Fig. A1D).
In the simulations described in the main text, i NaP + i R was 10-fold higher in the first 20 μm of the axon than in the rest of the axon and 10-fold lower in the soma and dendrites.However, reducing i NaP + i R 10-fold in the initial 20 μm of the axon and/or increasing i NaP + i R 10-fold in the soma and dendrites had only minor effects on the results (Fig. A2A).This reflects the fact that V is essentially clamped to V rest in the soma and dendrites (Fig. A3A).Likewise, V approaches V rest as x → 2 mm (Fig. A3A), hence increasing the axon length from ∼2 to 3.2 mm also had little effect on the results (data not shown).In contrast to the relatively smooth decay in V with x (Fig. A3A and C), the region of negative conductance in the i(V) curve in control conditions (Fig. A1A) results in an irregular spatial variation of i NaP + i R at hyperpolarized ramp voltages (V c ; Fig. A3B).The negative conductance disappears in simulated TTX (i NaP = 0; Fig. A1B), which results in a monotonic variation of i R with x (Fig. A3D).
Figure A2B shows the effect of changing the axon diameter (d).Reducing d from 0.4 to 0.3 μm had only a minor effect on i(V).But with R a = 90 cm, a C m value of 1.72 μF cm −2 is required if d = 0.3 μm.In fact, it was not possible to obtain a fit with C m = 1 μF cm −2 ; the lowest possible value was 1.19 μF cm −2 , and this necessitated a rather low value of R a (50 cm).Increasing d to 0.5 μm required R a = 150 cm with C m = 1 μF cm −2 .However, the form of the i(V) curve for V < −50 mV was unlike that obtained for d ≤ 0.4 μm and seems inconsistent with expectation.Hence, a reasonable value for d in these particular voltage-ramp experiments is 0.4 μm, which is the value proposed by Alle and Geiger (2006).More generally, electron microscopy studies suggest that d can be smaller (Blackstad & Kjaerheim, 1961).Likewise, the MFB diameter can be smaller than the 7 μm assumed here, although 7 μm is appropriate for the large boutons selected for voltage-clamp experiments in the present study.Halving the diameter (D) of the other MFBs in the model or even eliminating them (i.e.D = d = 0.4 μm) had little effect on the estimate of i(V) (Fig. A2C).[However, as one would expect from Fig. 3A in the main text, the initial error in the predicted I(V) for k = 0 was reduced for the smaller MFB diameters.]Indeed, even reducing the assumed diameter of the recorded MFB (D 0 ) from 7 to 4 μm with D = 3.5 μm had only a modest effect on i(V) at depolarized V (Fig. A2D).2B).Negative x corresponds to the soma + dendrites, positive x to the axon.V c is the ramp voltage at the current injection site (MFB 4, 664 μm from the soma).In A, C and D, curves are plotted at 10 mV intervals; in B, curves are plotted only at the indicated values of V c for the sake of clarity.In D, the i scale is expanded for ease of comparison with the −90 mV curve in B. The scaling factors for i in the soma + dendrites and first section of the axon were 0.1 and 1, respectively (relative to the main part of the axon).Increasing these values results in V approaching V rest more closely in these regions (V rest is ∼−46 mV in A and B and ∼−64 mV in C and D).R a = 90 cm, C m = 1 μF cm −2 and axon length = 2.16 mm.Data were derived from the analysis of experimental I(V) curves computed using the mean parameter values in Table 1A and B (n = 4 MFBs from one animal).

Figure 2 .
Figure 2. Preprocessing of voltage-ramp dataA, quadratic polynomials (smooth curves) fitted to voltage-ramp current traces (noisy curves; mean of 10 sweeps).The polynomials were fitted over a t interval determined by the points of inflection observed in the absence and presence of TTX.The inset shows the voltage-ramp protocol; the continuous portion of the V(t) plot corresponds to the I(t) plots below.B, the quadratic coefficients of fitted curves, such as those in A, were used to decide which group-mean traces to average (filled symbols).C, the resulting grand average I(V) curves (noisy traces) were fitted with eqn (18) (control) and eqn (21) (TTX) (smooth curves).The TTX-sensitive current was calculated as the difference (TTX minus control) of the fitted functions.D, the holding current (I hold ) for each sweep during the first ∼7 min.All data in A-D are from the same mossy fibre bouton.The increase in I hold during the whole-cell recording was observed in three of the four mossy fibre boutons (Table1C).[Colour figure can be viewed at wileyonlinelibrary.com]

Figure 3 .
Figure 3. Analysis of control V-ramp dataA, control voltage-ramp data (thick grey curve) were fitted with a model comprising a 10-mm-long axon with a single mossy fibre bouton (MFB) at its midpoint (no soma or filopodia).The i(V) curve for implementation in NEURON was obtained directly from eqn (13) with no correction.The resulting prediction for I (black curve) has a maximum error of 21 fA for t > 0.2 s.The value of C m was set to 10 −3 μF cm −2 in order to make the capacitive current (I C ) negligible.Accordingly, the predicted I hold differs from the observed value by an amount equal to I C [∼0.8 pA; see eqn (22)].The value of R a was set arbitrarily.B-D, comparison of predicted (thin black curves) and observed (thick grey curves) injected currents for MFB 3 (B), MFB 4 (C) and MFB 5 (D) in the model of Fig. 1.Observed curves are plots of eqn (18) using the mean parameter values in Table1A(n = 4 MFBs from one animal) and with V ≥ −90 mV in order to avoid the initial artefact (see Fig.2A).Predicted curves were obtained from the NEURON compartmental model.Iterations (k) were continued until the maximum errors on I and I C were <0.1 pA.In all cases, C m = 1 ± 0.05 μF cm −2 ; the indicated values of R a are those required to achieve this tolerance on C m (see Fig.4Band Table2A).x = distance from the soma; axon length = 2.16 mm; infinite seal resistance.

Figure 4 .
Figure 4. Sensitivity of some model predictions to the choice of MFB, R a and R seal A, total membrane current density required to obtain the fits in Fig. 3 at various distances from the soma, assuming infinite seal resistance (R seal ).The choice of mossy fibre bouton (MFB) does not have a large effect on the estimate of i(V).B and C, variation of membrane capacitance (C m ) and resistance (R m ), respectively, with axial resistivity (R a ) for two values of R seal .The dashed line in B indicates the assumed true C m value of 1 μF cm −2 .The value of R m was estimated as the chord resistance between −85 and −65 mV.Fits were not possible for R a values lower than those shown (within 5 cm).The key in B applies to C and vice versa.D, a non-ideal seal resistance (in gigohms) does not have a large effect on the predicted i(V), although there is a leftward shift at more depolarized V. Progressively higher values of R a are needed as R seal is reduced (Table2A; see also panel B).Data were derived from the analysis of experimental I(V) curves computed using the mean parameter values in Table1A(n = 4 MFBs from one animal).[Colourfigure can be viewed at wileyonlinelibrary.com]

Figure 5 .
Figure 5. Analysis of TTX data Axon length = 2.16 mm, x = 664 μm.A, comparison of predicted (thin curves) and observed (thick curve) injected currents.The observed curve is a plot of eqn (21) using the mean parameter values in Table1Band with V ≥ −90 mV in order to avoid the initial artefact (Fig.2A).Predicted curves were obtained using the Cox method with correction as in Fig.3.The pair of numbers associated with each thin curve indicate seal resistance (R seal ) before (first number) and after (second number) setting g NaP = 0 in order to simulate the application of TTX.Axial resistivity (R a ) was set to its value in the absence of TTX (Table2A), and R TTX seal was varied to obtain C m = 1 ± 0.05 μF cm −2 (Table2B).The two predicted I(t) curves are virtually indistinguishable.Good fits were also obtained with R con The continuous curves in C and D are the same as that in B. Using R con seal values of 10, 15 or 25 G instead of ∞ G yields similar plots to those in C and D (data not shown).Data were derived from the analysis of experimental I(V) curves computed using the mean parameter values in Table 1B (n = 4 MFBs from one animal).[Colour figure can be viewed at wileyonlinelibrary.com] ) in Fig. 6C were ḡNaP = 69.5 μS cm −2 , V NaP = −63.0mV and k NaP = 14.9 mV.The value of E Na (61.4 mV) was calculated from the Nernst equation with [Na + ] i = 15 mM and [Na + ] o = 151.25 mM.Parameter estimates for the fit of eqn (28) to each of the six curves in Fig. 6C are given in Table

Figure 6 .
Figure 6.Estimation of the persistent sodium-current density, i NaPThe key in B applies also to A and C; resistance values are for R con seal (gigohm seal resistance in control conditions), and values in micrometres give the distance of the model mossy fibre bouton (MFB) from the soma.Further quantitative details are given in Table3.A, leak-corrected current density in the absence of TTX.Leak current density (i L ) was estimated from the first 1% of i NaP + i R (V) curves like those in Fig.4D(see Table3Afor parameter estimates).i R is the residual, TTX-insensitive current density.B, the leak-corrected i in the presence of TTX was estimated in a similar fashion from curves like those in Fig.5B.The curves in B were then subtracted from the corresponding curves in A to yield the six thin curves in C.These curves were fitted with eqn (28) (thick continuous curve).Data in A-C were derived from the analysis of experimental I(V) curves computed using the mean parameter values in Table1Aand B (n = 4 MFBs from one animal).D, the same analysis applied to the four experimental MFBs, using the individual parameter values from Table1Aand B; x = 664 μm and infinite R con seal were assumed when applying the Cox method with correction.The thin curves give i NaP (V) for each experimental MFB; the thick curve is a fit of eqn (28).[Colour figure can be viewed at wileyonlinelibrary.com]

Figure 7 .
Figure 7. Relating the persistent sodium conductance (g NaP ) to the transient sodium conductance (g NaT ) A, the steady-state inactivation variable (dashed curve) for the Hodgkin-Huxley g NaT model of Engel and Jonas (2005); data points are taken from their Fig.5B.The continuous curve is a fit of a Boltzmann function with an offset, h 0 ∞ [eqn (29)].Values of voltage were not corrected for the liquid junction potential (LJP).h 0 ∞ = 0.007 and V shift = 0; V h (−89.4 mV) and k h (7.7 mV) were adjustable parameters.B, fits of eqn (30) (coloured curves) to the pooled i NaP (V) data from four experimental mossy fibre boutons (MFBs; n = 4 from one animal; same data as in Fig. 6D but indicated by the shaded area to avoid having too many curves).The chosen values of h 0 ∞ are indicated.C, estimates of the maximum NaT specific conductance ( ḡNaT ) and the shift of the h ∞ -V and n ∞ -V curves (V shift ) obtained from the fits of eqn (30) and plotted against the chosen values of h 0 ∞ .D, the NaP specific conductance as a function of the LJP-corrected voltage.E and F, similar to C, but for low and high R a , respectively.Also, the predicted ḡNaT is corrected for open probability [0.529; Engel and Jonas (2005)], and i NaP + i L is higher in the model MFBs than in the axon shaft by the factors indicated in the key.Values of V shift and points for which V shift < −4.5 mV or V shift > 7.5 mV are not plotted.The short-dashed lines indicate the range and the long-dashed lines the mean reported by Engel and Jonas (2005), both corrected to 35°C assuming Q 10 = 1.2-1.5.[Colour figure can be viewed at wileyonlinelibrary.com]

Figure 8 .
Figure 8. Estimation of the steady-state inactivation offset (h 0 ∞ ) in a model of the transient sodium conductance Error sum of squares (SSE) as a function of the trial values of h 0 ∞ for the fits of eqn (29) (A) and eqn (30) (B); see Fig. 7A and B, respectively, for example fits.Best-fitting (minimum SSE) estimates of h 0 ∞ are indicated by the arrows.A, for h 0 ∞ ≤ 0.012 (right dashed vertical line), the fit of eqn (29) with h 0 ∞ > 0 (incomplete inactivation) is as good as or better than the fit with h 0 ∞ = 0 (complete inactivation; dashed horizontal line), and the fit is always impressive (R 2 > 0.999).B, for 0.005 ≤ h 0 ∞ ≤ 0.012 (dashed vertical lines) the SSE for the fit of eqn (30) changes by <1% and is within 1% of the SSE for the fit of eqn (28) (dashed horizontal line).

Figure 9 .
Figure 9. Simulated modulation of EPSP conduction by the persistent sodium conductance, g NaP A-D, a somatic EPSP was simulated by injecting current into the model soma (top trace).The V(x) was then recorded at various distances (x) from the soma, corresponding to the points in E and F. A, omitting g NaP has almost no effect on EPSP conduction compared with the default value of g NaP (B).However, a 4-fold increase in g NaP (C) or somatic depolarization from −89 to −71 mV (D) decreases EPSP attenuation with x and prolongs the EPSP waveform.E, peak EPSP attenuation versus x for V hold (soma) = −89 mV and various g NaP scaling factors (as indicated in the key).F, similar to E, but with g NaP at the default level and various somatic holding potentials (given in millivolts in the key).The default level of g NaP (g NaP × 1) was determined from the analysis of experimental I(V) curves computed using the mean parameter values in Table1A(n = 4 MFBs from one animal).See Methods for further details.

Figure A1 .
Figure A1.Sensitivity of the estimated membrane current density to the choice of R a ; the implied optimal values of C m are also given A, control conditions.B, simulated TTX.C, the resulting estimate of i NaP (V).i R is the residual current density that remains when i NaP is blocked.D, variability of the predicted i (∼20%) at V = −30 mV.The key in A applies to B and C; the key in B applies to all panels.Axon length = 2.16 mm and x = 664 μm.Data were derived from the analysis of experimental I(V) curves computed using the mean parameter values in Table1Aand B (n = 4 MFBs from one animal).

J
Figure A2.Sensitivity of the predicted current densities to various model parameters A, the estimate of i(V) is rather insensitive to scaling of i in the soma + dendrites and the first 20 μm of the axon.For each curve, the first value in the key gives i(V) in the soma + dendrites and the second value gives i(V) in the first 20 μm of the axon, both expressed relative to i(V) in the main part of the axon.B, effect of changing the axon diameter (d).The i(V) curve for d = 0.5 μm might seem inconsistent with expectation.The curve for d = 0.3 μm is plausible but required R a = 50 cm and C m = 1.19 μF cm −2 .C, reducing the diameter (D) of model mossy fibre boutons (MFBs) other than the recorded MFB had little effect on the estimate of i(V).D, even reducing the diameter (D 0 ) of the recorded MFB with D = 3.5 μm had only a modest effect at depolarized V.For all panels, axon length = 2.16 mm, x = 664 μm and R seal = ∞ G .Data were derived from the analysis of experimental I(V) curves computed using the mean parameter values in Table 1A (n = 4 MFBs from one animal).[Colour figure can be viewed at wileyonlinelibrary.com]

Table 2 . Analysis of voltage-ramp experiments
a ) and gigohm seal resistance (R seal ) required to fit the experimental I(V) curves assuming C m = 1 ± 0.05 μF cm −2 .The experimental I(V) curves were computed using the mean polynomial coefficients in Table1A andB as applied to control conditions [eqn (18)] and TTX [eqn (21)], respectively; hence the experimental I(V) curves are based on n = 4 mossy fibre boutons (MFBs) obtained from one animal.x is the distance of the voltage-clamped model MFB from the soma.A, required values of R a (in ohms centimetres) in the absence of TTX (control) for given values of R seal .B, required values of R seal in the presence of TTX (R TTX seal ; in gigohms) assuming no change in R a and for given values of R seal in the absence of TTX (R con seal ); the value of R a was set to the appropriate values in A (with R seal = R con seal ).C, required values of R a (in ohms centimetres) in the presence of TTX assuming no change in R seal or C m .