Excitation properties of computational models of unmyelinated peripheral axons

Abstract

Biophysically based computational models of nerve fibers are important tools for designing electrical stimulation therapies, investigating drugs that affect ion channels, and studying diseases that affect neurons. Although peripheral nerves are primarily composed of unmyelinated axons (i.e., C-fibers), most modeling efforts focused on myelinated axons. We implemented the single-compartment model of vagal afferents from Schild et al. (1994) (Schild JH, Clark JW, Hay M, Mendelowitz D, Andresen MC, Kunze DL. J Neurophysiol 71: 2338–2358, 1994) and extended the model into a multicompartment axon, presenting the first cable model of a C-fiber vagal afferent. We also implemented the updated parameters from the Schild and Kunze (1997) model (Schild JH, Kunze DL. J Neurophysiol 78: 3198–3209, 1997). We compared the responses of these novel models with those of three published models of unmyelinated axons (Rattay F, Aberham M. IEEE Trans Biomed Eng 40: 1201–1209, 1993; Sundt D, Gamper N, Jaffe DB. J Neurophysiol 114: 3140–3153, 2015; Tigerholm J, Petersson ME, Obreja O, Lampert A, Carr R, Schmelz M, Fransén E. J Neurophysiol 111: 1721–1735, 2014) and with experimental data from single-fiber recordings. Comparing the two models by Schild et al. (1994, 1997) revealed that differences in rest potential and action potential shape were driven by changes in maximum conductances rather than changes in sodium channel dynamics. Comparing the five model axons, the conduction speeds and strength-duration responses were largely within expected ranges, but none of the models captured the experimental threshold recovery cycle—including a complete absence of late subnormality in the models—and their action potential shapes varied dramatically. The Tigerholm et al. (2014) model best reproduced the experimental data, but these modeling efforts make clear that additional data are needed to parameterize and validate future models of autonomic C-fibers.

NEW & NOTEWORTHY Peripheral nerves are primarily composed of unmyelinated axons, and there is growing interest in electrical stimulation of the autonomic nervous system to treat various diseases. We present the first cable model of an unmyelinated vagal nerve fiber and compare its ion channel isoforms and conduction responses with other published models of unmyelinated axons, establishing important tools for advancing modeling of autonomic nerves.

INTRODUCTION

Biophysical computational models of nerve fibers are important tools for analyzing and designing therapeutic electrical stimulation and block, for investigating drugs that affect ion channels, and for studying diseases that affect axon ultrastructure or ion channels. Although autonomic and somatic nerves are composed primarily of unmyelinated axons (Fig. 1), most modeling efforts focused on representing myelinated axons. Most published models of unmyelinated axons (i.e., C-type axons) are based on nonmammalian experimental data (e.g., Rattay and Aberham 1993), and recent models of peripheral unmyelinated axons target the somatic nervous system (Sundt et al. 2015; Tigerholm et al. 2014). Thus, to address the growing interest in pathologies of the autonomic nervous system and associated treatments (Birmingham et al. 2014; Famm et al. 2013; Fox 2017; Waltz 2016), we implemented the single-compartment model of vagal afferents from the study by Schild et al. (1994); conducted a rigorous examination of the implementation details and comparisons with published figures; extended the single-compartment model into a multicompartment cable model axon, presenting the first C-fiber model of a vagal afferent; and compared the responses of the novel model with those of three published models of unmyelinated C-type axons (Rattay and Aberham 1993; Sundt et al. 2015; Tigerholm et al. 2014). These models and simulations establish important tools for advancing modeling of autonomic nerves.

Figure 1.

Percentage of axons that are unmyelinated in peripheral nerves.

METHODS

All model code is available on ModelDB (https://senselab.med.yale.edu/modeldb/, Accession No. 266498; https://modeldb.yale.edu/266498) (McDougal et al. 2017). The supplements, the source data, and the plotting code for all figures are available on the SPARC portal (RRID:SCR_017041) (Pelot et al. 2020). Symbols and abbreviations are defined the GLOSSARY.

We implemented the Schild et al. (1994) single-compartment models for A- and C-type vagal sensory neurons using NEURON v7.4 (Hines and Carnevale 1997), including eight nonlinear ionic conductances (two Na⁺ currents, four K⁺ currents, and two Ca²⁺ currents), three ion transporters (Na⁺-Ca²⁺ exchanger, Ca²⁺ pump, and Na⁺-K⁺ pump), two background currents (Na⁺ and Ca²⁺), and dynamics of Ca²⁺ concentrations in the intracellular and extracellular spaces. To validate our implementation further, we also coded the model in MATLAB R2017b (MathWorks, Inc., Natick, MA) [both numerical (backward Euler) and analytical solutions of individual ion channel dynamics to generate voltage clamp data] and in Brian 2 (Goodman and Brette 2009). One author implemented the full (Schild et al. 1994) model in NEURON as well as the individual nonlinear ion channels in MATLAB. Another author implemented the full (Schild et al. 1994) model in Brian. The numerical and analytical voltage clamp data generated in MATLAB were consistent in all cases and are therefore simply referenced as the “MATLAB” data in the results.

Although the original model defined a spherical compartment (30 µm diameter), NEURON operates with cylinders; therefore, we used a cylinder that is 20 µm in diameter and 45 µm in length to match the volume and surface area from the original publication, noting that the circular end-caps of a cylindrical compartment in NEURON are not included in the surface area. The neural membrane is surrounded by a perineural cylindrical shell, 0.5 μm in thickness, outside of which was the extracellular bath. The model assumes constant concentrations for Na⁺ and K⁺ in the intracellular and perineural spaces, resulting in constant reversal potentials; conversely, the model includes Ca²⁺ accumulation dynamics for the intracellular and perineural compartments with a constant extracellular concentration, as well as intracellular Ca²⁺ buffering and diffusion of Ca²⁺ between the perineural space and the extracellular bath.

We simulated the model at room and body temperatures. The text in the original publication indicates that simulations were run at 22°C and 37°C (Schild et al. 1994), but in the table of constants (their Table 4), the temperature is listed as 296 K, which is 22.85°C. Therefore, we ran the room temperature simulations at 22.85°C and used 22.85°C for the Q₁₀ reference temperature, although the 0.85°C difference in room temperature values did not noticeably affect the results. Additional implementation notes are provided in the supplemental data, including a note on the nomenclature of the K⁺ currents (Supplement 1), an error in the published Ca²⁺ diffusion equation (Supplement 2), a note on the Q₁₀ temperature scaling factors (Supplement 4), a note on the values for Faraday’s constant and the gas constant (Supplement 5), and an inconsistency in the value for the perineural space thickness (Supplement S6). We used a time step of 5 µs with backward Euler integration, unless otherwise indicated. All stimulation currents in the single-compartment models were delivered intracellularly.

We also implemented an updated version of the model using values and equations from the study by Schild and Kunze (1997). Specifically, we used the modified maximum conductances for all eight nonlinear ion channels and two ionic background currents (Fig. 13), as well as the modified voltage-dependent Na⁺ channel equations (Table 5); the revised conductances were taken from the caption of Fig. 9 in the publication, except for the fast and slow Na⁺ conductances, which were from the caption of Fig. 11. For the Schild and Kunze (1997) Na⁺ channels, we used a Q₁₀ reference temperature of 22°C, taking the mean of their stated range from 21°C to 23°C. Supplement 3 notes an error in the signs of the S_1/2 values in the Schild and Kunze (1997) model.

Figure 13.

Comparison of ion channels in seven cable models of peripheral axons. We aligned similar channel models and isoforms in rows. The seven models compared here are McIntyre et al. (2002), Rattay and Aberham (1993), Schild et al. (1994), Schild and Kunze (1997), Sundt et al. (2015), and Tigerholm et al. (2014), where Schild modeled nodose ganglion soma, MRG modeled myelinated axons of the somatic peripheral nervous system, and Tigerholm, Sundt, and Rattay modeled unmyelinated axons of the somatic peripheral nervous system. We consulted many publications to map the isoforms across the models (Alexander et al. 2019; BBP/EPFL 2020; Benarroch 2015; Catterall et al. 2005; Debanne et al. 2011; Delmas 2008; Ertel et al. 2000; Fux et al. 2018; Goldin et al. 2000; Ranjan et al. 2019; Rasband 2010; Rush et al. 2007; Schild and Kunze 2012; Tsantoulas and McMahon 2014); note that many studies focus on cell bodies (ganglia) rather than axons. HCN channel, hyperpolarization-activated cyclic nucleotide-gated channel. Note 1: the Schild 1997 conductances for the non-Na⁺ channels were taken from the caption of Fig. 9 of the publication. Note 2: for myelinated axons, different voltage-dependent K⁺ channel isoforms are present at the juxtaparanode (Kv1.1, 1.2, 1.4, and 3.4) than at the nodes of Ranvier (Rasband 2010; Tsantoulas and McMahon 2014). A recent study also indicated that thermo- and mechano-sensitive K⁺ channels are important at the nodes of Ranvier (Kanda et al. 2019). Note 3: only added to soma, stem axon, and proximal axon in the original model, not the distal axon. Note 4: only added to soma in the original model. Note 5: only added to soma in the original model; conductance value from cal.mod on ModelDB.

Table 5.

Comparison of voltage-dependent equations for the Na_f and Na_s ion channels of the C-type models in Schild et al. (1994) and Schild and Kunze (1997)

	Schild 1994	Schild 1997
Naf	${\tau }_m=0.75\times \exp \left(-{\left(0.0635\right)}^2{\left(V_m+40.35\right)}^2\right)+0.12$ ${\tau }_h=6.5\times \exp \left(-{\left(0.0295\right)}^2{\left(V_m+75\right)}^2\right)+0.55$ ${\tau }_j=\frac{25}{1+\exp \left(\frac{V_m-20}{4.5}\right)}+0.01$ $m_{\inf }=\frac{1}{1+\exp \left(\frac{\left(V_m+41.35\right)}{-4.75}\right)}$ $h_{\inf }=\frac{1}{1+\exp \left(\frac{\left(V_m+62\right)}{4.5}\right)}$ $j_{\inf }=\frac{1}{1+\exp \left(\frac{\left(V_m+40\right)}{1.5}\right)}$	${\tau }_m=1.15\times \exp \left(-{\left(0.06\right)}^2{\left(V_m+40\right)}^2\right)+0.21$ ${\tau }_h=18\times \exp \left(-{\left(0.043\right)}^2{\left(V_m+62.5\right)}^2\right)+1.35$ $m_{\inf }=\frac{1}{1+\exp \left(\frac{\left(V_m+31.62\right)}{-6.98}\right)}$ $h_{\inf }=\frac{1}{1+\exp \left(\frac{\left(V_m+65.99\right)}{5.97}\right)}$
Nas	${\tau }_m=1.5\times \exp \left(-{\left(0.0595\right)}^2{\left(V_m+20.35\right)}^2\right)+0.15$ ${\tau }_h=4.95\times \exp \left(-{\left(0.0335\right)}^2{\left(V_m+20\right)}^2\right)+0.75$ $m_{\inf }=\frac{1}{1+\exp \left(\frac{\left(V_m+20.35\right)}{-4.45}\right)}$ $h_{\inf }=\frac{1}{1+\exp \left(\frac{\left(V_m+18\right)}{4.5}\right)}$	${\tau }_m=1.45\times \exp \left(-{\left(0.058\right)}^2{\left(V_m+14.5\right)}^2\right)+0.26$ ${\tau }_h=10.75\times \exp \left(-{\left(0.067\right)}^2{\left(V_m+13.5\right)}^2\right)+3.15$ $m_{\inf }=\frac{1}{1+\exp \left(\frac{\left(V_m+11.29\right)}{-5.54}\right)}$ $h_{\inf }=\frac{1}{1+\exp \left(\frac{\left(V_m+31\right)}{5.2}\right)}$

Note that although the form of the steady-state equation is correct in Schild and Kunze (1997), the signs of the S_1/2 values (e.g., in their Table 1 and in the caption for their Fig. 8) are incorrect (Supplement 3). The dynamic responses of the gating parameters for activation (m), inactivation (h), and slow inactivation (j) of the voltage-gated sodium channels are characterized by their steady-state values (m_inf, h_inf, j_inf) and their time constants (τ_m, τ_h, τ_j).

We extended the Schild et al. (1994) and Schild and Kunze (1997) single-compartment models into multicompartment cable models of unmyelinated axons. In our single-compartment implementation, the volumes and surface areas—used to compute many other values—were hard-coded; in the cable model implementation, we instead calculated these parameters during initialization based on the specified geometrical dimensions. We eliminated the intracellular volume adjustment made in the Schild et al. (1994) model—where the intracellular volume was computed as the volume of the spherical compartment minus 10% to account for cell organelle volumes—since the axon has fewer organelles than the soma; the threshold to generate an action potential with an extracellular cathodic pulse was unaffected. We defined the perineural space—used for the Ca²⁺ accumulation mechanism—by assuming a cylindrical shell with a constant thickness around each compartment that formed a concentration gradient with the bulk extracellular bath. We assigned this thickness to be 0.5 µm to match the thickness of the single compartment in the Schild et al. (1994) model (Supplement 6). We used an intracellular resistivity of 100 Ω-cm (Table 1).

Table 1.

Intracellular resistivity values used in published peripheral axon models

Peripheral Axon Model	Intracellular Resistivity, Ω-cm	References
McIntyre and Grill (2002)	70	No reference
Tigerholm et al. (2014)	35.4	(Hodgkin and Huxley 1952)
Sundt et al. (2015)	100	(Choi and Waxman 2011; Lüscher et al. 1994)
Rattay and Aberham (1993)	100	No reference

We compared the conduction responses of the cable models (Schild et al. 1994; Schild and Kunze 1997) with those of three published unmyelinated axon models (Rattay and Aberham 1993; Sundt et al. 2015; Tigerholm et al. 2014); the ion channels and other mechanisms for each model are provided in Fig. 13. The model by Rattay and Aberham (1993) was originally implemented to compare responses of the original Hodgkin–Huxley model adjusted to 37°C (as used herein) with those of three other axon models (FH, CRRSS, and SE). The model by Sundt et al. (2015) was originally implemented to investigate action potential propagation past the T-junction of the dorsal root ganglion. The model by Tigerholm et al. (2014) was originally developed to reproduce activity-dependent changes in action potential conduction in unmyelinated somatic afferents.

We implemented the model by Rattay and Aberham (1993) using the equations and parameters in the publication. We used the code for the ion channels from the Sundt et al. (2015) model provided on ModelDB (https://senselab.med.yale.edu/modeldb/, Accession No. 187473) (McDougal et al. 2017). We implemented the unmyelinated axon model from the study by Tigerholm et al. (2014) using the code for the ion channel mechanisms generously provided by the senior author Dr. Erik Fransén. Table 2 summarizes differences identified between the code for the ion channel dynamics and the equations provided in the publication by Tigerholm et al. (2014). We also traced back several of the citations used to develop the ion channel equations mentioned by Tigerholm et al. (2014):

Table 2.

Summary of differences between code and publication for ion channel equations in Tigerholm et al. (2014)

Ion Channel	Publication	Code
Nav1.7	βs = 132.05 − (132.05)/(1 + exp((Vm + 384.9)/28.5))	βs = 132.05 − (132.05)/(1 + exp((Vm − 384.9)/28.5))
Nav1.9	αm = 1.032/(1 − exp((Vm + 6.99)/14.87115))	αm = 1.032/(1 + exp((Vm + 6.99)/−14.87115))
Nav1.9	βh = 0.13496/(1 + exp((Vm + 10.27853)/29.09334))	βh = 0.13496/(1 + exp((Vm + 10.27853)/−9.09334))
KDR	ninf = 1/(1 + exp(−(Vm + 45)/15.4))	ninf = 1/(1 + exp(−(Vm + 25)/15.4))
KDR	N/A	“k” is used as a variable in NEURON, defining a constant in millivolts, although “k” is an NMODL keyword for the potassium ion
KM	If Vm ≥ −60 mV, τns = 13 × Vm + 1000 × Q10K	If Vm ≥ −60 mV, τns = (13 × Vm + 1000) × Q10K
KA	ninf = 1/[1 + exp(−(Vm + 5.4 + 15)/16.4)]4	ninf = 1/[1 + exp(−(Vm + 5.4 − 15)/16.4)]4
KA	hinf = 1/[1 + exp((Vm + 49.9 + 15)/4.6)]	hinf = 1/[1 + exp((Vm + 49.9 − 15)/4.6)]
h	If Vm ≥ −70 mV, τn,s = 300 + 542 × exp((Vm + 25)/20)	If Vm ≥ −70 mV, τn,s = 300 + 542 × exp((Vm + 25)/−20)
KNa	w = 1/[1 + (38.7/Nain)3.5]	w = 0.37/[1 + (38.7/Nain)3.5]
Nav1.8 (τh), KDR (τn if Vm > −31 mV), KA (τn, τh), h (τn,s, τn,f)	Q10 should be a multiplicative factor applied to entire time constant equations rather than only one term (i.e., missing parentheses)	Q10 is a multiplicative factor for τ
Na+-K+ pump	Methods: ikpump = gbar/((1 + 1/ksp)2) − (1.62/(1 + (6.7(mM)/(nai + 8(mM)))3) + 1.0/(1 + (67.6(mM)/(nai + 8(mM)))3)) Appendix: ikpump = gbar/((1 + 1/ksp)2)× (1.62/(1 + (6.7(mM)/(nai + 8(mM)))3) + 1.0/(1 + (67.6(mM)/(nai + 8(mM)))3))	ikpump = gbar/((1+1/ksp)2) × (1.62/(1 + (6.7(mM)/(nai + 8(mM)))3) + 1.0/(1 + (67.6(mM)/(nai + 8(mM)))3))

The boldface symbols and values indicate the differences between the publication and the code.

• Na_v1.7 code is consistent with that reported by Sheets et al. (2007), whereas the publication is not.

• Na_v1.8 equations match those reported by Sheets et al. (2007), with the addition of Q₁₀ for the m and h gates.

• Na_v1.9 ${\overline{g}}_{Na}$ differs from that reported by Herzog et al. (2001), where the model has a maximum conductance of 0.0948 mS/cm² in the publication and code, while the maximum conductance reported by Herzog et al. (2001) is 6.9005 mS/cm². However, the model is consistent with the equations in Herzog et al. (2001).

• K_dr equation for n_inf was shifted by 10 mV in the publication and code compared with that reported by Sheets et al. (2007) to “better fit the experimental data.” However, no experimental data were cited in Tigerholm et al. (2014).

The models used different approaches for their passive leak currents, where in some cases, the leak current was used to attain a target resting potential. The original reversal potentials for Na⁺, K⁺, and leak provided in the Rattay and Aberham (1993) model resulted in a rest potential of 0 mV; however, the publication specified a rest potential of −70 mV, which we achieved by shifting all three reversal potential values by −70 mV. Sundt et al. (2015) used a nonspecific leak (passive) channel with a maximum conductance from the literature, but the reversal potential of the leak channel was calculated using the target rest potential. Tigerholm et al. (2014) calculated the maximum conductances of the sodium and potassium leak (balancing) currents to achieve their target rest potential. Schild et al. (1994) included sodium and calcium leak (background) currents without justifying their maximum conductance values, but they were not calculated based on a target resting potential.

Using our implementation of each published model, we first reproduced figures from the original publications. When reproducing published results, we used the temperature and geometry of the original models. Subsequently, we standardized the geometries of the unmyelinated axons before quantifying the conduction responses (Table 3) and simulated all axon models at 37°C. The diameters of mammalian unmyelinated axons typically range from ∼0.3 to 2 µm (Fazan et al. 2001; Illanes et al. 1990; Mei et al. 1980; Nanobashvili et al. 1994; Ochoa and Mair 1969; Schmalbruch 1986; Sharma and Thomas 1975; Soltanpour and Santer 1996). Therefore, unless otherwise indicated, we modeled axons 1 µm in diameter and 5 mm in length, with 600 segments that were each 8.33 µm in length. We selected the segment length by doubling the number of segments for each model until the threshold for intracellular stimulation changed by <1%, and we used the smallest converged segment length for all model axons. Except for the Tigerholm et al. (2014) model, we used linear (passive) end segments for each axon model with 100 µS/cm² and 1 µF/cm² and a reversal potential equal to the rest potential of the other (nonlinear) axon segments. We used a time step of 5 µs with backward Euler integration, except for the results with PW = {10, 20} µs, for which we used a time step of 1 µs. We ensured initial steady state at rest before starting the target simulation at t = 0 by initializing all parameters using the model’s V_rest value and by stepping from t = −1,000 ms to t = 0 in 1-ms time steps.

Table 3.

Summary of geometrical changes made to published unmyelinated models to quantify and compare their responses after reproducing published figures

Tigerholm et al. (2014)	We removed the “branch” and “cone” sections and only used the “parent” axon.
Sundt et al. (2015)	We omitted the T-junction and soma and instead only used a linear axon. We only used the Nav1.7, Nav1.8, delayed rectifier potassium, and leak channels. In the original paper, other mechanisms (KCNQ, KCa, Cav, and intracellular Ca2+ dynamics) were only inserted into the soma (while we are focused on the axon) and the Na+-K+ pump was only used in a sensitivity analysis.
Rattay and Aberham (1993)	We extended the single-compartment model into a multicompartment axon.

The model parameters from the original publications (including geometry and temperature) were used for the data reproductions in Supplement 9. Our new data in Figs. 14 and 15 instead used standardized geometries, where all axons were 5 mm long with 600 segments, each 8.33 µm in length, and simulated at 37°C.

For each model, we quantified conduction speed, action potential shape, threshold recovery cycle, and strength-duration properties. Unless otherwise indicated, we quantified conduction speed and action potential shape using intracellular stimuli delivered at the first active compartment, whereas threshold recovery cycle and strength-duration properties were quantified using intracellular stimuli delivered at the center compartment. We used five different axon diameters for quantifying the conduction speed, from 0.5 to 1.5 µm in 0.25-µm increments, and 1-µm fibers for the other responses. For the threshold recovery cycle, we delivered monophasic anodic paired pulses with different interstimulus intervals (ISI = {2, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, and 500} ms). The first pulse was delivered at 2× threshold and we used a binary search algorithm to find the threshold for the second pulse, capped at 1,000× threshold. We recorded at the last active node to check for two propagated action potentials. For the strength-duration curve, we used a binary search algorithm to find the activation threshold of anodic pulses with different pulse widths (PW = {0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, and 100} ms), recording at the last active node to check for a propagated action potential.

RESULTS

In the first section of the results, we implemented the single-compartment model from the study by Schild et al. (1994), reproducing the voltage clamp currents and spiking data from the original publication. We then implemented the Schild and Kunze (1997) model, with updated maximum conductances and Na⁺ channel dynamics, and we compared the thresholds and action potentials of the two single-compartment models. In the second section of the results, we extended these implementations into the first multicompartment cable models of unmyelinated vagal afferents and compared their conduction responses with those of three published models of unmyelinated peripheral axons as well as with experimental data from the literature.

Single-compartment vagal afferent model.

Although our focus is on C-type models—corresponding to unmyelinated axons—Schild et al. (1994) first defined an A-type model, which was modified to shift the responses to fit C-type biophysics. Therefore, we first reproduced their A-type responses. Specifically, we reproduced Figs. 3 and 4 from the study by Schild et al. (1994) in our three implementations (NEURON, Brian, and MATLAB), showing currents under a voltage clamp for the six primary ion channels (Na_f, Na_s, Ca_n, Ca_t, K_d, and K_trans) for the A-type model at 22.85°C (Fig. 2). These traces were generated by simulating a single ion channel (except for the transient K⁺ current, where its two component current mechanisms were inserted) and using the modified conductances and ion concentrations indicated for each case in the captions and text of the original publication (Table 4); each voltage clamp simulation was initialized at −80 mV. We concluded that the Ca_n and Ca_t voltage clamp traces seen in the work by Schild et al. (1994) used constant Ca²⁺ concentrations, although the full model includes Ca²⁺ dynamics. The current traces from all three implementations (NEURON, Brian, and MATLAB) matched for all six nonlinear ionic currents at all voltage clamp levels. Our data also match the Schild et al. (1994) results, except for K_d (Fig. 2D) and Ca_n (Fig. 2E). Specifically, the published and simulated traces are substantially different at the lower voltage clamp levels for both currents, but they are approximately consistent at more depolarized levels. Given that the NEURON, Brian, and MATLAB implementations were all consistent, we concluded that there was an error in the parameters or equations of the Schild et al. (1994) publication for these particular currents.

Figure 2.

Currents under voltage clamp in a single-compartment model of the A-type nonlinear ionic conductances at 22.85°C quantified for vagal sensory neurons (Schild et al. 1994), comparing digitized data from the original publication (red dots) and our implementations in NEURON, Brian, and MATLAB (solid lines). Note the different axis bounds for each ion channel. Fast (A) and slow (B) Na⁺ currents; transient (C) and delayed rectifier (D) K⁺ currents; and N-type (E) and T-type (F) Ca²⁺ currents using constant Ca²⁺ concentrations.

Table 4.

Ion channel conductances and ion concentrations for default A-type model and for A-type voltage clamp data reproductions for Schild et al. (1994) at room temperature

Channel	Default Values for A-Type Neurons			Values for A-Type Vclamp Data
	Max Conductance, nS	Ion Concentration, mM		Max Conductance, nS	Ion Concentration, mM
		Intracellular	Extracellular		Intracellular	Extracellular
Naf	2,050	8.9	154.0	29.5 [Fig. 3 caption]	8.9	50 [Fig. 3 caption]
Nas	0.01			25.5 [Fig. 3 caption]	8.9	154.0 [implied by text on p. 2341]
Ktrans (note 1)	35 for IA and 10 for ID	145.0	5.4	28.0 for IA and 11.5 for ID [Fig. 4 caption]	145 [Fig. 4 caption]	5.4 [Fig. 4 caption]
Kd (note 1)	10			30.0 [Fig. 4 caption]	145 [Fig. 4 caption]	5.4 [Fig. 4 caption]
Can	1	94e-6 to 354e-6 (note 2)	2	22.5 [Fig. 3 caption]	1 (note 3)	2 (note 3)
Cat	0.35			2.75 [Fig. 3 caption]	1 (note 3)	2 (note 3)

The values in the left half of the table are for the default A-type model from Table 4 of Schild et al. (1994), except for the Ca²⁺ concentrations, which were determined from our model implementation. The values in the right half of the table were used to reproduce the voltage clamp (Vclamp) data, as shown in Fig. 2, where we simulated a single ion channel at a time (except for the transient K⁺ current, K_trans, where its two component current mechanisms were used). The references in square brackets refer to the origin in Schild et al. (1994). We used the “pipette solution” values from Schild et al. (1994) for the intracellular concentrations and the “bath solution” values for the extracellular concentrations. Note 1: see Supplement 1 for the K⁺ current nomenclature. Note 2: see Supplement 7 for the resting Ca²⁺ concentrations depending on neuron type (A vs. C), temperature (room vs. body), as well as maximum conductances and Na⁺ channel dynamics [Schild et al. (1994) vs. Schild and Kunze (1997)]. Note 3: the Ca²⁺ concentrations for the voltage clamp studies are from Mendelowitz and Kunze (1992), as referenced in Schild et al. (1994).

Schild et al. (1994) included modifications to model C-type vagal sensory neurons as well as Q₁₀ factors to model responses at 37°C. Specifically, the C-type model has a different membrane capacitance, different maximum ionic channel conductances, and shifts in the voltage-dependencies of the steady-state gating parameters as compared to the A-type model. However, the publication did not provide voltage clamp data for the C-type model. We compared voltage clamp currents for the same six ion channels for our three implementations of the C-type model at 37°C (NEURON, Brian, and MATLAB) with the default model conductances and ion concentrations but initiated each simulation at −80 mV, as done for the A-type voltage clamp simulations. As for the A-type model, the data from our three implementations were consistent (Fig. 3).

Figure 3.

Currents under voltage clamp in a single-compartment model of the C-type nonlinear ionic conductances at 37°C quantified for vagal sensory neurons (Schild et al. 1994) for our implementations in NEURON, Brian, and MATLAB. Note the different axis bounds for each ion channel. Fast (A) and slow (B) Na⁺ currents; transient (C) and delayed rectifier (D) K⁺ currents; and N-type (E) and T-type (F) Ca²⁺ currents with full Ca²⁺ dynamics.

Following these voltage clamp simulations, we quantified spiking behavior, seeking to reproduce Figs. 5 and 6 from the study by Schild et al. (1994), which show action potentials from the A-type and C-type models at 22.85°C and 37°C in response to short- and long-duration pulses, respectively (Fig. 4). Supplement 7 provides the resting potential and steady-state Ca²⁺ concentrations of our single-compartment model in different configurations (A-type/C-type, 22.85°C and 37°C), as compared with that reported by Schild et al. (1994). The responses in our NEURON and Brian implementations were consistent but differed from responses shown in the original publication (Schild et al. 1994). Specifically, using the input currents specified in the publication, the action potentials for the NEURON and Brian implementations had shorter rise times, without the slow depolarization shown in the Schild et al. (1994) model. Furthermore, as discussed below for Fig. 6, even when stimulating with the threshold current determined for our implementation—rather than the stimulation amplitudes defined in the original publication—our rise times were still faster.

Figure 4.

Action potentials for C-type (top) and A-type (bottom) implementations of Schild et al. (1994) at 22.85°C (left) and 37°C (right) using short-duration (2 ms; first and third rows) and long-duration (200 ms; second and fourth rows) pulses, comparing with Figs. 5 and 6 in the original Schild et al. (1994) publication. Stimulation parameters are listed in the titles of each panel. The values of V_rest are provided in Supplement 7. For the shorter stimulation pulses (2 ms), the voltage was held near rest before stimulation (−48 mV for the C-type and −64 mV for the A-type). For the longer stimulation pulses (200 ms), the voltage was held at hyperpolarized voltages before stimulation (−80 mV for the C-type and −73 mV for the A-type). Note that the NEURON (blue) and Brian (green) traces are overlaid.

Figure 6.

Threshold currents and threshold responses (defined as the rising edge of V_m past +40 mV) for our NEURON implementation of the Schild et al. (1994) C-type neuron at 37°C for 2-ms (top) and 200-ms (bottom) pulses. The left-hand NEURON simulations used the initial holding potentials as in the original publication (top, −48 mV; bottom, −80 mV). The right-hand NEURON simulations began at rest (−46.5 mV).

We also sought to reproduce Figs. 7 and 8 from the study by Schild et al. (1994), which show the component currents during action potentials for the A-type and C-type models, respectively, in response to a 200-ms pulse. The NEURON and Brian models were consistent but did not elicit an action potential in the C-type model at the 40 pA amplitude used in the original publication; thus, we increased the magnitude to 80 pA for the C-type model. The C-type plots (Fig. 5, top) show similar magnitudes and shapes of the component currents between the NEURON/Brian models and the Schild model, but the NEURON/Brian models spike more rapidly, as shown in Fig. 4. The A-type results (Fig. 5, bottom) are well matched across NEURON, Brian, and Schild, including magnitude, shape, and timing, despite the mismatches in the K_d and Ca_n voltage clamp currents (Fig. 2).

Figure 5.

Component currents during an action potential in response to a 200-ms pulse for both C-type (top) and A-type (bottom) models at 22.85°C. These plots recreate the simulation conditions of Figs. 7 and 8 in the study by Schild et al. (1994). We stimulated at 80 pA for the C-type NEURON and Brian models, since they did not elicit an action potential at the 40 pA stimulation level given in the Schild et al. (1994) model; 40 pA was used for the A-type simulations. The models were stimulated from the resting membrane potential (−47 mV for the C-type and −59 mV for the A-type). In the first column (inward currents), Na_s is not plotted for the A-type (bottom) because its magnitude was much smaller than the other currents and was not provided in Fig. 7 in the study by Schild et al. (1994). In the second column (outward currents), the “K” current corresponds to the delayed rectifier K⁺ current; the “A” and “D” currents combine to form the transient K⁺ current (Supplement 1). Note the different x-axes, spanning either 50 or 200 ms, and the different y-axes for the current traces between the top and bottom rows. The NEURON and Brian traces are overlaid.

To investigate the source of the discrepancy in spike timing for the C-type model between the NEURON/Brian and Schild et al. (1994) models (Figs. 4 and 5), we considered the possibility that the current amplitudes provided in the Schild et al. (1994) model may reflect the experimental stimulation amplitudes rather than the simulation amplitudes. The published figures seemed to show near-threshold responses, given the slow subthreshold rise before the spike. Thus, using a binary search algorithm, we determined thresholds for the C-type NEURON model at 37°C to achieve a rising edge of V_m past +40 mV in response to 2- and 200-ms pulses, starting either from the resting or from the initial holding potential used in the Schild et al. (1994) model (Fig. 6). Although our thresholds were ∼0.5–2× the stimulus amplitudes used by Schild et al. (1994), the spike times were all earlier than those shown in the original publication, with a shorter period of slow depolarization before the upstroke. Using the 2-ms pulse, we further decreased the stimulation amplitude, from our threshold of 286 pA to 270 pA, which delayed the peak of the transmembrane potential to ∼7.5 ms, matching the latency of the published action potential, but peaked at only 24 mV (data not shown).

Finally, we reproduced the insets in Fig. 5B from the study by Schild et al. (1994), showing repetitive firing of the A-type model in response to a 1,000-ms pulse at 22.85°C and 37°C (Fig. 7); note that the C-type model does not fire repetitively [see the insets in Fig. 6B from the study by Schild et al. (1994)]. It was found that the NEURON and Brian implementations fired more rapidly when compared with the data in the original publication. We observed a slight difference in the NEURON and Brian spike times, accumulated by the end of the 1,000-ms pulse. Therefore, we compared different numerical integration methods. The default integration methods in NEURON and Brian are backward Euler and forward Euler, respectively, both first-order methods. The integration method used by Schild et al. (1994) was a fifth-order Runge–Kutta–Merson algorithm with a variable time step. Thus, we evaluated higher-order integration methods: second-order Crank–Nicholson in NEURON and fourth-order Runge–Kutta in Brian. The differences in the spiking frequencies across the integration methods were small and could only be observed during a very long stimulation pulse. Zooming in on the last spike during the 1,000-ms pulse revealed slight differences in spike times between the four simulations (NEURON and Brian implementations, each with two integration methods) (Fig. 8). In NEURON, for simulations at both 22.85°C and 37°C, the Crank–Nicholson integration decreased the spiking frequency as compared with backward Euler, and in BRIAN, the Runge–Kutta integration increased the spiking frequency as compared with forward Euler. Furthermore, at 37°C, the NEURON data had a slower firing rate than Brian with the Euler methods but a higher firing rate with the higher-order integration methods. Thus, the slight differences in firing rate between the NEURON and Brian implementations are consistent with differences due to the methods of numerical integration.

Figure 7.

Response of the A-type model at 22.85°C (top) and 37°C (bottom) with a 1,000-ms stimulation pulse (40 pA) for NEURON and Brian implementations, with spike times from Fig. 5B in the original publication (Schild et al. 1994) indicated with black dots. The frequencies in the legends provide the mean firing rates.

Figure 8.

Last action potential in response to a 1,000-ms stimulation pulse (40 pA) for the A-type model (left, 22.85°C; right, 37°C) using two different numerical integration methods in each of the NEURON and Brian implementations, all using a time step of 5 μs. Note that at 37°C (right), the integration method can move the Brian spike time from being the earliest to being the latest, as compared with the NEURON responses. The Euler methods are first order, the Crank–Nicholson method is second order, and the Runge–Kutta method is fourth order. The frequencies in the legend provide the mean firing rate for each method at room temperature (left) and body temperature (right).

We considered the updated maximum ion channel conductances and nonlinear Na⁺ equations published in the article by Schild and Kunze (1997) that were intended to explain the heterogeneous firing behavior across a population of vagal C-type afferents. Based on voltage clamp data to quantify the Na⁺ channel characteristics, they modified the Schild et al. (1994) equations for Na_f and Na_s (Table 5) and maximum conductances for all ion channels (Fig. 13). To validate our implementation of the Schild and Kunze (1997) model, we first reproduced the steady-state gating parameter values and time constant values for each of the Na_f and Na_s activation and inactivation gating parameters (Fig. 9). Schild and Kunze (1997) also provided voltage clamp data (and associated steady-state and time constant parameters) for one specific cell studied in vitro, which we reproduced in Fig. 10. The shape and time courses of the currents under voltage clamp match well; differences in peak amplitudes are likely due to the experimental traces being representative examples, rather than mean traces to which the model parameters were fit.

Figure 9.

Steady-state (left) and time constant (middle and right) values for the activation (blue) and inactivation (red) gating parameters for Na_f (top) and Na_s (bottom) in the Schild and Kunze (1997) model, which we reproduced in MATLAB, without any Q₁₀ correction (therefore simulating the default room temperature parameters). The equations represent the mean values across 20 cells tested in vitro. The dynamic responses of the gating parameters for activation (m) and inactivation (h) of the voltage-gated sodium channels are characterized by their steady-state values (m_inf and h_inf) and their time constants (T_m and T_h).

Figure 10.

Currents under voltage clamp for the Na_f (A) and Na_s (B) currents at six different voltages for a single vagal C-type cell, showing the in vitro data in red dots (Schild and Kunze 1997) and our data simulated in NEURON in solid blue lines, without any Q₁₀ correction (therefore simulating the default room temperature parameters).

After this verification of our implementation of the Schild and Kunze (1997) Na⁺ channels, we inserted the updated maximum conductances and Na⁺ channel mechanisms into our implementation of the Schild et al. (1994) model to compare the firing properties of the models. Schild and Kunze (1997) did not include a rationale for the updated conductance values, and thus, it was unclear which set of maximum conductances would be suitable. Therefore, we evaluated three equation-conductance combinations (Table 6): the 1994 model, the 1997 model, and the hybrid model (1997 Na⁺ channel equations with 1994 maximum conductances). The models differed with respect to excitability, resting membrane potential, and action potential shape. The threshold of the 1997 model was ∼2× higher than the 1994 model for a 2-ms pulse (Table 6). This can be attributed to several differences. First, the resting membrane potential was more hyperpolarized for the 1997 model (−69 vs. −47 mV; Supplement 7), thus requiring more current to depolarize to spike threshold. In addition, the maximum conductance for the Na_f channel (TTX-S) was lower in the 1997 model (22 vs. 69 mS/cm²), and the smaller conductance resulted in lower excitability (i.e., higher thresholds). The action potential shape was also different between the 1994 and 1997 models, with higher peak voltage and a “shoulder” in the downstroke of the 1997 model (Fig. 11). The higher peak voltage may result from increased Na⁺ conductance; despite reducing the Na_f peak conductance threefold, the 1997 model increased the Na_s peak conductance ∼20× (22 vs. 1.04 mS/cm²). The “shoulder” in the 1997 model downstroke is also caused by the increased level of Na_s current. The 1994 and 1997 models have the same Na⁺ concentrations, and the Na⁺ concentration was constant in each compartment; therefore, the difference in the action potential peak was not driven by different Na⁺ reversal potentials.

Table 6.

Three single-compartment C-type models using equation–conductance combinations from Schild et al. (1994) and Schild and Kunze (1997), with the stimulus thresholds determined in our NEURON implementations to achieve a rising edge in V_m past +30 mV

Temp., °C	Model	Threshold for	Threshold for
		0.2-ms Pulse, nA	2-ms Pulse, nA
22.85	Schild 1994	2.54	0.267
	Hybrid	N/A; spontaneously active
	Schild 1997	5.04	0.562
37	Schild 1994	2.36	0.258
	Hybrid	N/A; spontaneously active
	Schild 1997	5.11	0.598

The hybrid model is the same as the Schild 1994 model, but with the Na⁺ channel dynamics from Schild 1997. Schild 1997 has different Na⁺ channel dynamics and different maximum conductances for all channels as compared to Schild 1994.

Figure 11.

Single-compartment action potentials at 37°C for the C-type Schild et al. (1994) model (blue), the Schild and Kunze (1997) model (yellow), and the hybrid model with Schild 1994 maximum conductances and Schild 1997 voltage-dependent Na⁺ channel equations (red). The stimulus was a 0.2-ms pulse delivered at t = 5 ms. We used the threshold amplitudes (Table 6) for the Schild 1994 (2.36 nA) and the Schild 1997 (5.11 nA) models. We also delivered 5.11 nA to the hybrid model, evoking the first action potential, even though it is spontaneously active (i.e., fires without any stimulus; this is suggested by the hyperpolarizing drift of V_m before the stimulation is delivered at t = 5 ms and by the second action potential).

With no stimulus, the single-compartment hybrid model was spontaneously active and continuously fired action potentials. This was consistent with Fig. 9 from the study by Schild and Kunze (1997), where they provided a plot with the firing characteristics of their model with different proportions of the two types of Na⁺ channels, i.e., TTX-R Na_s channels and TTX-S Na_f channels. The Schild et al. (1994) Na⁺ conductances put the model close to 0% TTX-R Na⁺ conductance (for the C-type neuron, ${\overline{g}}_{Nas}/\left({\overline{g}}_{Naf}+{\overline{g}}_{Nas}\right)=0.0295/\left(1.95+0.0295\right)*100\%=1.5\%$), resulting in spontaneous firing. Conversely, the conductances in the Schild and Kunze (1997) model put the TTX-R proportion at 50%.

Overall, our single-compartment model comparisons revealed that changing the Na⁺ channel equations from the Schild et al. (1994) model to the hybrid model affected excitability, causing spontaneous activity, whereas the action potential widths, peaks, and afterhyperpolarizations were similar. Conversely, when changing the maximum conductances of all channels from the hybrid model to the Schild and Kunze (1997) model, the excitability decreased [loss of spontaneous activity from the hybrid model and higher thresholds than the original Schild et al. (1994) model], the resting potential was more hyperpolarized, and the action potential shape changed substantially, with a higher peak, a narrower width, and a slight shoulder on the repolarizing phase. These differences in action potential peak and shape were maintained in the multicompartment cable model implementations (Fig. 12).

Extension into multicompartment cable models and comparison with published models.

We extended the single-compartment Schild et al. (1994) and Schild and Kunze (1997) models into multicompartment cable models with the full complement of membrane ionic mechanisms, maintaining the conductance densities. All models successfully propagated an action potential, although the hybrid model was spontaneously active, as observed in the single-compartment implementation. We applied an intracellular stimulus at 25% of the axon length and recorded the transmembrane potential at 50% and 75% of the axon length. The resulting intracellular thresholds and conduction speeds are provided in Table 7 at 22.85°C and 37°C; plots of the transmembrane potential are provided in Fig. 12. As expected, the thresholds are lower and the conduction speeds are faster at the warmer temperature. The conduction speed for the Schild et al. (1994) model at 37°C is at the lower bound of the expected range (∼0.5 to 1.5 m/s) (Andrews et al. 1980; Sato et al. 1985), whereas the Schild and Kunze (1997) model conducts more slowly, possibly due to its reduced Na_f maximum conductance, since the two models have the same intracellular resistivity. The rest potential, the action potential amplitude and shape, and the duration of the afterhyperpolarization are noticeably different between the Schild et al. (1994) and Schild and Kunze (1997) models (Fig. 12).

Table 7.

Intracellular thresholds and conduction speeds for the two 1-μm axon models (Schild et al. 1994; Schild and Kunze 1997) at room and body temperatures

Temperature, °C	Model	Threshold for 0.2-ms Pulse, nA	Threshold for2-ms Pulse,nA	Conduction Speed, m/s
22.85	Schild 1994	0.816	0.161	0.35
	Schild 1997	1.81	0.320	0.19
37	Schild 1994	0.648	0.143	0.51
	Schild 1997	1.48	0.302	0.29

Figure 12.

Transmembrane potential for the multicompartment cable model axons Schild et al. (1994) (A) and Schild and Kunze (1997) (B) models at 37°C recorded (Rec) at 50% (blue) and 75% (red) of the axon length in response to an intracellular stimulus (Stim) at 25% of the axon length (starting at t = 5 ms, dt = 5 µs, PW = 0.2 ms, Istim = threshold from Table 7).

We compared our cable model implementations of the Schild et al. (1994) and Schild and Kunze (1997) models with those of three published cable models of unmyelinated axons (Rattay and Aberham 1993; Sundt et al. 2015; Tigerholm et al. 2014). Figure 13 presents a comparison of ion channels represented in seven peripheral nerve fiber models, including three models of peripheral somatic unmyelinated axons (Rattay and Aberham 1993; Sundt et al. 2015; Tigerholm et al. 2014), A-type and C-type vagal afferent neurons (Schild et al. 1994; Schild and Kunze 1997), and the MRG model of peripheral mammalian myelinated axons (McIntyre et al. 2002). In addition to comparing the ion channels across fiber models, Fig. 13 provides maximum conductance values, ion channel isoforms, and notes on the qualitative behavior of the channels.

We reproduced figures from each of the publications to validate our implementations (Supplement 9). Using model axons with consistent geometries (Table 3), we quantified conduction responses for the five unmyelinated peripheral axon models at 37°C, including the conduction speed (CV), action potential shape, threshold recovery cycle (RC), and strength-duration properties (Figs. 14 and 15). We also compared the responses of the models with experimental data at 37°C from the literature; see Supplement 8 for details on the experimental data.

Figure 14.

Conduction responses from intracellular stimulation of five unmyelinated axon models at 37°C with standardized geometry (Table 3) overlaid with published experimental data. See Supplement 8 for notes on the literature review. The plots of the transmembrane potential (V_m) as a function of time were shifted to V_rest = 0 to allow comparison of the action potential shapes. The original V_rest values were −60 mV for Sundt, −70 mV for Rattay, −55 mV for Tigerholm, −47 mV for Schild 1994, and −69 mV for Schild 1997. We quantified the action potential duration by measuring the spike duration from the transmembrane potential departure from rest to return to rest. The inset for the recovery cycle has the same axis bounds as the main panel, but has a logarithmic x-axis. [1] Andrews et al. (1980), [2] Sato et al. (1985), [3] Grundfest and Gasser (1938), Fig. 10, [4] Grundfest and Gasser (1938), Fig. 11, [5] Grundfest and Gasser (1938), [6] Iggo (1958), and [7] Paintal (1967).

Figure 15.

Strength-duration relationships for five 1-µm unmyelinated axon models at 37°C with standardized geometry (Table 3) using intracellular stimulation overlaid with published C-fiber strength-duration data (Koslow et al. 1973; Woodbury and Woodbury 1990). See Supplement 8 for notes on the literature review. Panel D shows the raw data while panels A to C show the thresholds normalized by threshold at 1 ms (A), except for the Woodbury 1990 data in anesthetized rats, where the threshold at the longest PW (600 µs) was used for normalization, threshold at 100 ms (B), or rheobase estimated by fitting the log-transformed Weiss equation, log₁₀(I_th) = log₁₀(I_rh * (1 + T_ch/PW)) to the data for thresholds up to PW = 1 ms (C). Panel E shows the different rheobase estimates.

Conduction speeds were comparable across the five models, with faster conduction speeds for the larger axon diameters. The conduction speeds overlapped with the lower values found in the literature, except for the Schild and Kunze (1997) model, which conducted more slowly for all fiber diameters (Fig. 14, top left).

The action potentials for the five models exhibited remarkably different shapes and durations (Fig. 14, top right and bottom right). The Rattay and Aberham (1993) and Sundt et al. (2015) models had the shortest spike durations (0.51 and 1.15 ms, respectively), whereas the Schild et al. (1994) and Schild and Kunze (1997) models had the longest spike durations (7.41 ms and 5.49 ms, respectively). The Tigerholm et al. (2014) model had an intermediate spike duration of 2.12 ms. Experimental measurements of extracellular voltage changes yield spike durations of ∼1.5–2.5 ms (Grundfest and Gasser 1938; Iggo 1958; Paintal 1967). The Schild and Kunze (1997) model had the largest peak voltage (132 mV above rest), whereas the Rattay and Aberham (1993) model had the smallest peak (65 mV above rest). The Schild and Kunze (1997) and Tigerholm et al. (2014) models showed a prominent “shoulder” during the repolarization phase, which has been seen experimentally in vagal fiber recordings from unmyelinated fibers and certain myelinated fibers (Li and Schild 2007).

Threshold recovery cycles from the models and experiments all exhibited a large increase in thresholds at short interstimulus intervals (early subnormal period) due to the refractory period (Fig. 14, bottom left). This early subnormal period generally extended to longer ISIs for the models with wider action potentials. Although the experimental data showed a late subnormal period (increase in thresholds for ISIs greater than ∼60 ms), with a slow recovery to baseline (>500 ms), none of the models exhibited a late subnormal period. At intermediate ISIs, there is a phase with reduced thresholds (supernormal period) in the experimental data and for three of the models (Rattay and Aberham 1993; Schild et al. 1994; Tigerholm et al. 2014), although the experimental peak was larger (∼−30% change in thresholds vs. ∼−1%, −8%, and −5% for the models, respectively). Furthermore, a larger supernormal period in the models was associated with peak supernormality at a larger ISI (5, 85, and 30 ms, respectively). Conversely, the other two models did not have a supernormal period (Schild and Kunze 1997; Sundt et al. 2015).

The modeled strength-duration curves with intracellular stimulation matched well with the experimental data, whether normalized to the threshold at 1 ms (Fig. 15A) or normalized to rheobase as estimated by fitting the Weiss equation to the data for PW ≤ 1 ms (Fig. 15C). Normalizing by rheobase allows comparison with experimental measurements where the absolute thresholds are variable. However, the normalization can be quite sensitive; the raw thresholds for the Rattay and Aberham (1993), Schild et al. (1994), and Sundt et al. (2015) models were very similar (Fig. 15D), but the rheobases were slightly different, causing them to have substantially different chronaxies, as shown by the normalized data (Figs. 15, A–C). Furthermore, the differences between the curves depend on the method used to estimate the rheobase (Fig. 15E), and depending on the axon’s biophysics, longer pulse widths may be required to capture the rheobase (Pelot and Grill 2020). This demonstrates that it is important that all strength-duration curves be analyzed with the same methods for comparisons across datasets.

DISCUSSION

Developing therapies that use electrical stimulation of the nervous system to treat disease requires engineering design within a very large parameter space. Computational models provide important tools for quantifying neural responses across parameter dimensions and values. However, although the majority of axons in peripheral nerves are unmyelinated, the utility of existing models in representing the excitability of autonomic C-fibers is not clear. We implemented models of peripheral C-type axons incorporating ionic mechanisms from vagal sensory neurons and compared their conduction responses with those of three existing models of unmyelinated axons. The Tigerholm model best reproduced the experimental data, although its membrane properties are based on cell bodies of somatic peripheral afferents, and none of the axon models captured the experimental recovery cycle. The original Tigerholm model reproduced the experimental conduction speed recovery cycle following a long train of prepulses (Tigerholm et al. 2014), but we did not simulate a train of prepulses and did not include the model’s superficial branch axon and cone attachment to the parent axon in our standardized geometry. The other models varied in their responses, particularly in the action potential shapes. These studies also emphasize the importance of having a second party verify code and its consistency with the publication, publishing code, providing complete and detailed methods in publications, and reproducing past work before building upon it.

We implemented single-compartment models of A-type and C-type vagal afferent neurons from the Schild et al. (1994) model using NEURON, Brian, and MATLAB. By using three separate implementations, as well as different initial code authors, we ensured that discrepancies between the published results and our simulations were not due to implementation errors but rather due to incorrect or incomplete information in the publication. The primary discrepancies were in the Ca_n and K_d voltage clamp data for the A-type model at 22.85°C (Fig. 2), faster action potential upstroke and higher thresholds (Figs. 4–6), and a faster firing rate for the A-type model (Fig. 7). Publishing and comparing simulated currents under voltage clamp provided important verifications because they allow examination of the equations and parameters of a specific ionic mechanism, whereas the sources of differences in complete model responses, such as an action potential, are difficult to trace, given the interdependence across mechanisms. The discrepancy in the Ca_n currents under voltage clamp likely had little effect on the complete model response, given the low amplitude of the Ca_n current relative to the other inward currents (Fig. 5); conversely, K_d is one of the largest outward currents (Fig. 5, labeled “K”), and errors in its implementation could have important effects on the model. The comparisons across the complete A-type model responses were inconsistent: the firing behavior matched in some cases (Fig. 5, showing the transmembrane potential and component currents) but not others (Figs. 4 and 7), which suggests incorrect information in the stimulation protocol rather than in the base mechanisms. Although the Schild et al. (1994) model did not include C-type voltage clamp data for validation, the complete C-type model responses consistently showed faster action potential rise in our implementation across stimulation protocols (Figs. 4–6), suggesting errors in model components that impacted the rising phase of the action potential, such as the Na_f equations and parameters or the membrane capacitance. Overall, the C-type model differed from the A-type model in that it had a more depolarized rest potential (Supplement 7), greater intracellular Ca²⁺ concentration at rest (Supplement 7), more depolarized Na_f and Na_s activation (Fig. 2 vs. Fig. 3; these figures are also at different temperatures), higher activation threshold for spiking (Fig. 5), wider action potential (Fig. 5; note different x-axes), smaller Na_f and K_trans current magnitudes during an action potential (Figs. 5 and 13), and lack of repetitive firing during a long stimulation pulse (Fig. 6 vs. Fig. 7).

We simulated the models at both room temperature and body temperature. We used the Q₁₀ scaling factors provided in the Schild et al. (1994) model to scale the time constant equations, although maximum conductance may also change with temperature (Collins and Rojas 1982; Fitzhugh 1966; Hodgkin and Huxley 1952; Moore 1958). At higher temperatures, the thresholds were lower and the conduction speeds in the multicompartment model were faster. In modifying the model to use the maximum conductances and Na⁺ ion channel dynamics from the Schild and Kunze (1997) model, the rest potential was more hyperpolarized (Figs. 11 and 12 and Supplement 7), the activation thresholds were ∼2× higher (Tables 6 and 7), and the action potential had a higher peak with a more rectangular (less triangular) profile (Figs. 11, 12, and 14); these three differences occurred in both the single- and multicompartment models.

The five models of unmyelinated axons had conduction speeds in the lower end of the range of experimental measurements, although the modeled conduction speeds may be shifted into the experimental range by adjustments to the maximum Na⁺ channel conductance and/or the intracellular resistivity (unpublished data). The action potential shapes differed substantially across the models (Fig. 14); the action potentials had more rapid downstrokes for the models with fewer channels (Rattay and Aberham 1993; Sundt et al. 2015) and were much wider and somewhat taller for the models with more ionic mechanisms, such as Ca²⁺ currents, ion accumulation, and Na⁺-K⁺-ATPase pump (Schild et al. 1994; Schild and Kunze 1997; Tigerholm et al. 2014) (Fig. 13); these additional mechanisms have slower responses than the fundamental Na⁺ and K⁺ channels. There were no changes in rest potential, action potential peak, action potential shape, or afterpotential shape between the single-compartment (Fig. 11) and multicompartment (Fig. 12) Schild models. The action potentials for the Schild and Kunze (1997) and Tigerholm et al. (2014) models exhibited a prominent “hump” or “shoulder” on the downstroke, although Schild et al. (1994) also described a hump for their C-type spike, which they attributed to the slow inactivation of Na_s in the model and also to the slow inactivation of Ca_n in vivo; it has been shown that action potentials are broader for the cell bodies of unmyelinated vagal fibers than for the cell bodies of myelinated fibers (Li et al. 2007; Li and Schild 2007). As seen in Fig. 11, the more prominent hump in the Schild and Kunze (1997) model occurred specifically due to the changes in maximum conductances from the Schild et al. (1994) model, not the changes in Na⁺ channel dynamics. In the Tigerholm et al. (2014) model, Na_v1.8 and K_dr are the dominant currents during the hump. The hump is distinct from the depolarizing afterpotential described in the study by McIntyre et al. (2002), given the substantially smaller amplitude and longer time course of the afterpotential. The action potential durations for the two Schild models were much longer than those reported for single fibers in the literature, whereas that of the Rattay model was slightly shorter; action potential durations of the Tigerholm and Sundt models were within the published range.

In addition to different action potential shapes, the models had different threshold recovery cycles (Fig. 14). Three of the models [i.e., the Rattay, Tigerholm, and Schild (1994) models] exhibited a small supernormal period, with larger peak supernormality when the peak of the supernormal period occurred at a longer ISI. However, none of the models reproduced the large supernormality at relatively short ISIs seen in vivo. Interestingly, although the Schild and Kunze (1997) model had a very wide action potential with a prominent hump on the downstroke, its recovery cycle lacked a supernormal period, perhaps due to the rectangular shape of the action potential, lacking a gradual decrease in V_m that would allow a second stimulus to depolarize the axon back above threshold with a smaller current than when stimulating from rest. Prior modeling papers posited mechanisms of action for the supernormal period of the recovery cycle. For myelinated fibers, McIntyre et al. (2002) demonstrated that the supernormal period is due to persistent Na⁺ channel activation and discharge of the internodal axolemma. For unmyelinated fibers, Tigerholm et al. (2015) investigated the conduction speed recovery cycle rather than the threshold recovery cycle, and they suggested three contributing factors for the supernormal period in the conduction speed recovery cycle: 1) perineural K⁺ accumulation causing reduced K_dr current, 2) intracellular Na⁺ accumulation, and 3) reduction in Na_v1.7 current relative to Na_v1.8 current. Both the McIntyre paper and the Tigerholm paper relate the supernormal period in the recovery cycle to the depolarizing afterpotential. None of the models exhibited a late subnormal period. This finding was interesting for the Tigerholm model, given that the published conduction speed recovery cycle does show a subnormal period from ∼100 to 300 ms (Tigerholm et al. 2014), albeit when applying paired pulses after a train of low-frequency prepulses between 0.5 and 2 Hz to stabilize the conduction speed and including the tapered axon geometry at the distal end; the prepulse conditioning—which would affect ion concentrations—may be required to capture the effects at longer ISIs. Although we did compare the models’ recovery cycles with limited data from the literature [n = 2 from Grundfest and Gasser (1938)], Gasser’s A-fiber thresholds recovery cycle data compare well with more recent A-fiber in vivo recordings (Supplement 8).

The strength-duration responses for all models compared well with the literature (Fig. 15), although normalization of the strength-duration curve and reporting of chronaxie and rheobase values must be done using consistent methods across datasets (Pelot and Grill 2020), and the chosen normalization method can affect comparisons between datasets (Fig. 15). Furthermore, all published in vivo data available for the conduction responses used extracellular stimulation and recordings (Supplement 8), which may affect the action potential duration and strength-duration responses.

We considered incorporating the additional K⁺ current, I_dtx [sensitive to α-dendrotoxin (α-DTX)], from the Glazebrook et al. (2002) model, which provides another update to the original Schild et al. (1994) model. However, we concluded that I_dtx is effectively already included in the Schild et al. (1994) model. The Glazebrook et al. (2002) model shows the experimental K⁺ current sensitivities to DTX alone, 4-AP alone, and DTX + 4-AP, as well as the sensitivities to DTX, TEA alone, and DTX + TEA. They found the same results with 4-AP alone and with DTX + 4-AP; similarly, they found the same results with TEA alone and with DTX + TEA. This indicates that the DTX current is a subset of the 4-AP- and TEA-sensitive currents, which are already included in the Schild et al. (1994) model: I_K,Trans = I_K,A + I_K,D, the transient K⁺ current, and I_K, the delayed rectifier current, are 4-AP- and TEA-sensitive, respectively. Thus, although the I_dtx current could be added to the model with accompanying changes in the conductances for I_K,Trans and I_K to provide more detailed parameterization, it would not change the firing properties.

The width of the perineural space in the Schild et al. (1994) model was 0.5 µm (Supplement 6), which we also used in our cable model extension. However, we also simulated the Schild unmyelinated cable models at 37°C with 0.03 µm perineural thickness (data not shown) to be consistent with other published models and with EM analyses of fiber bundles [stated as 29 nm in Tigerholm et al. (2014); 10 to 50 nm (nominal 30 nm) in Meffin et al. (2012); and 20 to 30 nm in Waxman (1978, p. 11)]. Both the Schild (1994 and 1997) cable models successfully propagated an action potential when using a perineural space thickness of 0.03 µm, and the action potential shapes and conduction speeds were unaffected.

These modeling efforts make clear the additional data that are needed to parameterize and validate future models of autonomic C-fibers. As aforementioned, in vivo validation data for small autonomic axons are sparse. For the ion channels themselves, consideration is required for potential differences in ion channel isoforms and densities in the cell body versus the axon (Benarroch 2015; Debanne et al. 2011; Sundt et al. 2015; Tsantoulas and McMahon 2014), where McIntyre et al. (2002) used data from axonal patch clamps, but Tigerholm et al. (2014) used data from somas, in autonomic versus somatic axons, in afferent versus efferent axons, and potentially in other functional groupings. Furthermore, although the action potential alone is driven by only a few ion channels, the responses to other stimulus protocols, especially repetitive activation as might be used for functional stimulation, may be more protracted and require additional mechanisms to reproduce the correct biophysics. For example, many of the published (Tigerholm et al. 2014) responses involve long stimulus trains. Furthermore, Schild and Kunze (1997) eliminated the j gate in the Na_f equation set (Table 5), possibly due to its very slow time constant, and this might contribute to determining the response to long epochs of continuous firing. Our compilations and simulations herein provide an important overview of the current state of unmyelinated axon models and the effects of different model channel sets on key conduction responses.

GRANTS

This work was supported by the National Institutes of Health (NIH) SPARC program (OT2 OD025340) and Duke University (University Scholars Program, Myra & William Waldo Boone Fellowship, and Pratt School of Engineering Faculty Discretionary Fund).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

N.A.P., D.C.C., N.D.T., and W.M.G. conceived and designed research; N.A.P., D.C.C., B.J.T., and E.D.L. performed experiments; N.A.P., D.C.C., B.J.T., and C.S.H. analyzed data; N.A.P. and D.C.C. interpreted results of experiments; N.A.P., D.C.C., and B.J.T. prepared figures; N.A.P. and D.C.C. drafted manuscript; N.A.P., D.C.C., B.J.T., N.D.T., C.S.H., and W.M.G. edited and revised manuscript; N.A.P., D.C.C., B.J.T., N.D.T., E.D.L., C.S.H., and W.M.G. approved final version of manuscript.

ENDNOTE

At the request of the authors, readers are herein alerted to the fact that additional materials related to this manuscript may be found at https://doi.org/10.26275/IIWV-K07F (Pelot et al. 2020). These materials are not a part of this manuscript and have not undergone peer review by the American Physiological Society (APS). APS and the journal editors take no responsibility for these materials, for the website address, or for any links to or from it.

GLOSSARY

AP

action potential

4-AP

4-aminopyridine

Ca_n

N-type voltage-gated calcium ion channel

Ca_t

T-type voltage-gated calcium ion channel

Ca_v

voltage-gated calcium ion channel

CRRSS

Chiu-Ritchie-Rogart-Stagg-Sweeney (model)

CV

conduction velocity (i.e., conduction speed)

dt

time step

FH

Frankenhaeuser-Huxley (model)

Istim

stimulation current amplitude

ISI

interstimulus interval

Irh

rheobase current; threshold amplitude for a pulse of infinite duration

Ith

threshold current

K_d

delayed rectifier voltage-gated potassium ion channel

K_trans

transient voltage-gated potassium ion channel

K_v

voltage-gated potassium ion channel

Max

maximum

Min

minimum

MRG

McIntyre-Richardson-Grill (model)

N/A

not applicable

Na_f

fast voltage-gated sodium ion channel

Na_s

slow voltage-gated sodium ion channel

Na_v

voltage-gated sodium ion channel

PW

pulse width (duration of one phase)

Q₁₀

multiplicative scaling factor for every 10 ^oC change in temperature

RC

recovery cycle

S_1/2

one quarter of the reciprocal of the slope at V_1/2 along the curve of the gating parameter curve versus V_m, where V_1/2 is the transmembrane potential at half-activation (V_m at gating parameter = 0.5); see Supplement 3

SE

Schwarz-Eikhof (model)

SPARC

Stimulating Peripheral Activity to Relieve Conditions (program of the National Institutes of Health)

Tch

chronaxie; pulse width with a threshold amplitude of two times the rheobase (Irh)

TEA

tetraethylammonium

TTX

tetrodotoxin

TTX-R

tetrodotoxin-resistant

TTX-S

tetrodotoxin-sensitive

V_m

transmembrane potential

V_rest

rest potential