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REVIEW 4 major objections 7 minor 2 references

Improved Hodgkin & Huxley-type model for action potentials in squid

T0 review · 4 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Electrodiffusion model predicts squid rebound spikes at 20 °C

desk verdict A genuine GHK-based reformulation of HH with a clear head, but the central no-spike-train claim lacks the required bifurcation analysis. read the letter →

arxiv 1908.05086 v2 pith:FVBAP2KC submitted 2019-08-14 q-bio.NC

P. J. Stiles , C. G. Gray This is my paper · ORCID
classification q-bio.NC
keywords actionpotentialsquidgiantaxonGoldman-Hodgkin-KatzequationelectrodiffusionofmeanforcegatingkineticsreboundspikingHodgkin-Huxleymodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the classic Hodgkin-Huxley (HH) model's empirical conductance rules can be replaced by a simpler four-variable model built from the Goldman-Hodgkin-Katz (GHK) electrodiffusion current equation. It claims this model reproduces the squid giant axon's main behaviors: action potentials, propagation speed around 22 m/s, rebound spiking at 18.5 to 20 °C where the HH model fails, and repetitive firing after external calcium is lowered. It also predicts that perfused axons do not fire spike trains under sustained constant-current stimulation, in line with experiment but contrary to HH. The result matters because it ties action-potential generation to electrodiffusion quantities, such as permeabilities, free-energy barriers, and gating relaxation times, rather than to fitted gating powers, and it removes several known HH discrepancies.

What carries the argument

The central object is the generalized GHK current equation, Eq. (3), in which the resting-state Goldman-Hodgkin-Katz formula is extended by replacing the constant membrane potential and permeabilities with time-dependent ones: $i = -e^2 P(t) V_m(t)\,[c^{\mathrm{int}}-c^{\mathrm{ext}}\exp(e\beta V_m(t))]/[1-\exp(e\beta V_m(t))]$. The permeabilities are exponential functions of potentials of mean force, $P(t)=(fD/L)\exp(-\beta w(t))$, and the potentials of mean force for sodium and potassium channels depend linearly on the gating variables $m$, $h$, and $n$. The dynamical system closes with three first-order relaxation equations using constant time constants $\tau_m$, $\tau_h$, $\tau_n$ and sigmoidal steady-state gating functions. This machinery carries the argument by turning gating from empirical conductance multipliers into modulated free-energy barriers, so that the same electrodiffusion formalism produces both the resting potential and the action potential.

What would settle it

Measure the spatial profile of the potential of mean force for sodium and potassium ions inside squid voltage-gated channels under voltage-clamp conditions; if the profile is not approximately flat or its gating dependence cannot be captured by the three PMF barriers, Eq. (3) fails. A simpler check: voltage-clamp the fully open sodium conductance and test whether the current follows the GHK form across the full range of membrane potentials; a systematic deviation would break the predicted I-V curve and threshold.

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Extended reading notes

Core claim

The central claim is that a four-variable dynamical system, consisting of membrane depolarization $V$ plus gating variables $m$, $h$, and $n$, with ionic currents from the GHK constant-field equation, reproduces squid axon firing without HH's $m^3 h$ and $n^4$ dependencies. Gating enters through time-dependent permeabilities $P(t)=(fD/L)\exp(-\beta w(t))$, where the potential of mean force $w(t)$ for a sodium or potassium channel is a linear combination of open- and closed-gate free-energy barriers weighted by the gating variables; chloride is ungated and static. The model is a resonator with a single stable resting fixed point, and it predicts stationary action potentials peaking near 120 mV, a propagating speed of 22.3 m/s at 20 °C versus the experimental 21.2 m/s at 18.5 °C, rebound spiking after prolonged hyperpolarization at both 6.3 °C and 20 °C, and repetitive firing when the sodium activation barrier is lowered from $\beta w_\mathrm{Na}^O=3.0$ to $1.48$ to mimic low external calcium. Unlike HH, it does not produce spike trains under prolonged constant current in perfused, pumps-off axons.

Load-bearing premise

The whole current-voltage relation rests on the Goldman constant-field approximation together with the assumption that each channel's potential of mean force is spatially constant inside the pore; if the real free-energy profile is strongly structured or changes during fast transients, the predicted currents, resting potential, and action potentials would shift materially.

Editorial extensions

If this is right

  • HH's $m^3 h$ and $n^4$ gating terms are not necessary to reproduce squid action potentials; gating can be represented as exponential changes in channel free-energy barriers.
  • Rebound spiking after prolonged hyperpolarization is available near physiological temperature, so inhibitory rebound excitation may be common in squid axons, not limited to the cold temperatures at which HH can produce it.
  • The low-calcium repetitive firing regime is tied to a single control parameter, the sodium activation barrier $\beta w_\mathrm{Na}^O$, lowered from 3.0 to 1.48, giving a concrete bifurcation that could be checked against calcium-binding measurements.
  • Perfused squid axons with pumps off should not sustain spike trains under constant current, matching Clay's experiments and distinguishing perfused from freshly dissected fibers.
  • The model is a resonator with one stable fixed point, so subthreshold oscillations and resonate-and-fire behavior arise naturally from gating delays rather than from any explicit inductive element.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the effective free-energy barrier interpretation holds, the same equations could connect extracellular calcium concentration quantitatively to sodium-channel permeability, yielding a testable prediction for how threshold current shifts with $[\mathrm{Ca}^{2+}]_\mathrm{out}$.
  • The model's success would suggest that atomic-level free-energy profiles from molecular dynamics could be inserted into Eq. (3) in place of the single constant PMF, producing a more mechanistic bridge from channel structure to whole-axon firing.
  • Because the chloride concentration choice is load-bearing for the resting potential, the model could be tested by switching internal chloride to the higher values reported by later experiments and checking whether the predicted action-potential shape survives.
  • A related testable extension would be to apply the same GHK-based gating to other axon types, such as myelinated fibers, replacing the empirical gating powers still used in Frankenhaeuser-Huxley-type models.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The paper proposes a four-variable model for squid giant axon action potentials, replacing the HH phenomenological gating expressions with a dynamic GHK electrodiffusion formulation. The model assumes spatially constant potentials of mean force inside channels, first-order gating kinetics with constant time constants, and sigmoidal steady-state gating functions. It reports that the model reproduces the stationary action potential waveform, propagates a pulse at 22.3 m/s at 20C (vs HH's 21.2 m/s at 18.5C), predicts rebound spiking at 20C (where HH fails), predicts repetitive firing under low external calcium after brief depolarizing current, and predicts no spike trains under prolonged constant current in perfused axons. The paper also discusses temperature dependencies qualitatively. Parameter values are explicitly tabulated.

Significance. The model is a serious attempt to simplify the HH framework while grounding it in electrodiffusion. It avoids the arbitrary m^3h and n^4 powers, uses only four variables, and produces several falsifiable predictions, including rebound spiking at 20C and low-calcium repetitive firing. The manuscript is commendably transparent: it explicitly states that the seven PMF barriers, three relaxation times, steepness parameters, and area fractions were selected to reproduce the HH action potential shape and voltage-clamp transients, so those agreements are not independent validations. The genuinely predictive elements are the propagation speed, rebound spiking, and low-calcium firing. The paper also acknowledges discrepancies with experimental permeability ratios. However, the universal negative claim about constant-current spike trains is not justified by the local stability analysis, and the speed result lacks any sensitivity or error assessment. If the missing global bifurcation analysis and sensitivity studies are added, the model would be a valuable contribution.

major comments (4)
  1. [Section 3; Section 2.1] The concluding claim that the model 'does not predict spike trains of repetitive action potentials during prolonged constant-current stimulation of any size' is not supported by the analysis. Section 2.1 establishes only that the zero-stimulus steady state is the unique fixed point and is linearly stable (Fig. 3). A dynamical system can have a unique, linearly stable fixed point and still possess stable limit cycles for nonzero parameter values; in the HH model, tonic firing arises through a Hopf bifurcation of a single fixed point. Equations (9b) and (11a-c) are nonlinear, so the existence of oscillations for I_stim ≠ 0 is a global question. A universal negative over all stimulus amplitudes requires either a bifurcation diagram in I_stim, a Lyapunov argument, or direct long-time simulations for a range of amplitudes. Because this claim is one of the four headline agreements, the missing analysis is load-bearing.
  2. [Section 2.2 (paragraph before Table 1)] The similarity of the model's stationary action potential to the HH recordings is partly built in by construction. The authors state: 'Other parameters were selected to ensure that stationary membrane action potentials (see Fig. 4) have temporal profiles similar to those recorded by HH,' and that the seven PMF barriers, three gating relaxation times, and channel area fractions were chosen with this criterion and with the additional requirement that voltage-clamp transients 'at least qualitatively' resemble those of HH. Consequently, the agreement in Fig. 4 should not be presented as an independent success. The independent tests are the propagation speed (Section 2.4), rebound spiking at 20C (Figs. 1B and 7), and low-calcium repetitive firing (Section 2.3); these should be foregrounded.
  3. [Section 2.4 (Fig. 6)] The propagation speed of 22.3 m/s at 20C is compared with the HH experimental estimate of 21.2 m/s at 18.5C, but the result is a single deterministic simulation with no error bars, no sensitivity analysis, and no discussion of how the speed depends on key parameters such as axon radius a, internal resistivity R, gating time constants, and PMF barriers. Given the 1.5C temperature difference and the acknowledged fitting of the parameters to stationary AP properties, the agreement could be fortuitous. A parameter scan or at least a statement of the expected sensitivity of v to the main parameters is needed to support the 'good agreement' claim.
  4. [Section 1, Eq. (2)] The derivation of the current-voltage relation assumes the potential of mean force w(x,t) inside a channel is spatially constant, so the only x-dependence in the flux integral is the linear electrostatic term. The paper gives no evidence that this approximation is adequate during fast transients, when the gating variables are changing rapidly. If the true PMF is spatially structured, the predicted currents, resting potential, and action potential shape could shift materially. The authors should either justify this approximation quantitatively (e.g., by comparing with a spatially varying PMF or with known free-energy profiles from molecular dynamics) or state clearly that the model is a coarse-graining whose quantitative predictions are approximate.
minor comments (7)
  1. [Eq. (7)] The GHK equation is garbled in the typeset; the subscripts and superscripts for internal and external concentrations are misaligned, obscuring the numerator and denominator structure. Please re-set the equation carefully.
  2. [Table 1] The steepness parameter s_h is listed as '11' without units; since h_ss depends on the dimensionless variable m, s_h is dimensionless, but this should be stated explicitly. The notation for τ_m, τ_h and s_m, s_h is also cramped and should be clarified.
  3. [Section 2.3] The stimulus current amplitude switches from -65 and -69 μA cm^-2 in the first part of the paragraph to -269 μA cm^-2 later; the relationship between these values and the bifurcation description should be clarified.
  4. [Fig. 1 caption] The labels 'A action potential V(t) (mV)' and 'B action potential V(t) (mV)' are malformed; they should read 'A. Action potential...' and 'B. Action potential...'.
  5. [References] The reference to 'Stiles and Gray, 2019' is the manuscript under review itself (arXiv:1908.05086); citing the submitted manuscript as a reference is inappropriate and should be removed.
  6. [Figures 6 and 7] The full text contains only placeholder text for Fig. 6 and Fig. 7 with no graphical content; the final version must include these plots so the propagation and rebound waveforms can be inspected.
  7. [Reference list entry for Perram and Stiles (2010)] The reference list contains an editorial note appended to the Perram and Stiles (2010) entry ('This model incorrectly describes...'); such commentary belongs in the main text, not in the reference list.

Circularity Check

1 steps flagged · score 6.0 of 10

Stationary action-potential waveform is fitted to HH data and then reported as a prediction; propagation speed, rebound spiking, and low-calcium firing remain independent checks, so circularity is partial.

  1. fitted input called prediction [Section 2.2, after Eqs. (12); Table 1 caption; Section 2.2 'Stationary membrane action potentials']
    "Other parameters were selected to ensure that stationary membrane action potentials ()Vt (see Fig. 4) have temporal profiles similar to those recorded by HH. Our selected values for the seven PMF barriers w, three cation gating relaxations times τ, and channel membrane area-fractions f used this criterion. We also used the additional requirement that voltage-clamp electric current transients and steady states should, at least qualitatively, resemble those measured by HH."

    The model's AP waveform is not predicted independently: it is the fitting target. Equations (3)-(6), (9b), (11) and (12) contain many free parameters (seven PMF barriers, three relaxation times, three area fractions, steepness parameters). The paper says these values were 'selected to ensure that stationary membrane action potentials ... have temporal profiles similar to those recorded by HH' and to match voltage-clamp transients/steady states. Section 2.2 then reports the resulting AP (Fig. 4) with peak height and latency as a 'prediction'. Reproducing the HH waveform under those conditions is thus by construction, the pattern 'fitted input called prediction'.

full rationale

Partial circularity score 6. The derivation of the GHK-type current (Eq. (3)) from Nernst-Planck-Kramers with the constant-field and constant-PMF ansatz is explicit and self-contained; self-citations (Perram & Stiles 2010, Stiles & Gray 2019) supply prior derivation but the equations are re-stated in this paper. The fitted-input step is the calibration of the model parameters to HH's stationary AP and voltage-clamp data, after which the stationary AP is presented as a prediction. The paper's other headline results (22.3 m/s propagation vs 21.2 m/s experimental; rebound spiking at 20 C; repetitive firing on reducing the sodium-activation PMF barrier) were not used in that calibration, so they are genuine model outputs. However, the Section 3 claim that prolonged constant-current stimulation 'of any size' produces no spike trains is not supported by the uniqueness/stability analysis in Section 2.1; that is a correctness/evidence gap rather than a circularity. The reference-note self-correcting Perram & Stiles (2010) is not load-bearing circularity, and the no-spike-train assertion is unsupported rather than definitionally forced.

Assumptions & free parameters 15 free parameters · 8 assumptions · 0 invented entities

The electrodiffusion current law and cable equation are standard, but the channel-gating kinetics, sigmoid shapes, and all PMF barrier values are imposed or fitted, introducing many degrees of freedom not independently constrained by first principles.

free parameters (15)
  • Na activation open PMF barrier beta*w_Na^O = 3.0
    Chosen to make model action potentials resemble HH recordings.
  • Na activation closed PMF barrier beta*w_Na^C = 12.8
    Chosen with other barriers to fit action potential shape and voltage-clamp transients.
  • Na inactivation open PMF barrier beta*w_Na^daggerO = 1.7
    Chosen by authors as part of the seven-barrier fit.
  • Na inactivation closed PMF barrier beta*w_Na^daggerC = 8
    Chosen by authors as part of the seven-barrier fit.
  • K open PMF barrier beta*w_K^O = 3.0
    Chosen to reproduce HH action potential shape.
  • K closed PMF barrier beta*w_K^C = 10.9
    Chosen with other parameters to match action potential shape.
  • Cl PMF barrier beta*w_Cl = 6.9
    Chosen to set chloride leak current.
  • Na activation relaxation time tau_m = 0.12 ms
    Constant relaxation time selected to match HH-like kinetics.
  • Na inactivation relaxation time tau_h = 2.5 ms
    Constant relaxation time selected to match HH-like kinetics.
  • K activation relaxation time tau_n = 2 ms
    Constant relaxation time selected to match HH-like kinetics.
  • Na activation steepness s_m = 0.16 per mV
    Chosen to set resting m near 0.021 and threshold near 6.55 mV.
  • Na inactivation steepness s_h = 11
    Chosen to make resting h near 0.995.
  • K activation steepness s_n = 0.15 per mV
    Chosen so resting n is 0.5 in the perfused state.
  • Steady-state gating offsets m_T, V_T = 0.26 and 12 mV
    Temperature-dependent parameters in the sigmoidal steady-state gating curves.
  • Channel area fractions f_Na, f_K, f_Cl = 1.0e-5, 3.5e-6, 5.0e-6 respectively
    Chosen by authors; these interact with PMF barriers, so there is parameter degeneracy.
assumptions (8)
  • standard math Nernst-Planck-Kramers equation governs electrodiffusive ion flux in channels (Eq. 1).
    Background electrodiffusion theory assumed without derivation.
  • domain assumption Goldman constant-field approximation: the electric field inside the membrane is spatially constant during time-dependent transients.
    Used to derive Eq. (2) and Eq. (3); explicit in Section 1.
  • ad hoc to paper The potential of mean force w(t) inside each channel is spatially constant.
    Stated as 'We adopt the simplest possible model for the PMF and assume w(x,t) inside a channel is spatially constant.' This is a tractability assumption, not derived.
  • ad hoc to paper Gating variables obey first-order relaxation with constant time constants, and h depends on m rather than V.
    Equations (11a-c) replace HH voltage-dependent relaxation times with constants, motivated by the authors' prior work.
  • ad hoc to paper Steady-state gating functions have the sigmoidal forms in Eq. (12).
    The tanh forms and their parameters are imposed, not derived from electrodiffusion or molecular kinetics.
  • domain assumption The perfused squid axon has pumps switched off, with internal chloride 40 mM and no active transport.
    Used to set the resting state; discussed in Section 3 with reference to Baker et al.
  • domain assumption Surface potentials, electric double layers, and gating currents are neglected.
    Explicitly stated in Section 1; the authors note these would alter quantitative predictions.
  • standard math Standard cable equation assumptions for action potential propagation: negligible external resistance, quasi-static magnetic field, and given parameter inequalities.
    Used to derive and solve Eq. (13) in Section 2.4.

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Pith. "Pith review of Improved Hodgkin & Huxley-type model for action potentials in squid." pith.science (2026). https://pith.science/paper/FVBAP2KC

@misc{pith2026190805086,
  author       = {Pith},
  title        = {Pith review of: Improved Hodgkin & Huxley-type model for action potentials in squid},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FVBAP2KC}},
  note         = {Machine review of arXiv:1908.05086}
}
read the original abstract

By extending the crude Goldman-Hodgkin-Katz electrodiffusion model for resting-state membrane potentials in perfused giant axons of squid, we reformulate the Hodgkin-Huxley (HH) phenomenological quantitative model to create a new model which is simpler and based more fundamentally on electrodiffusion principles. Our dynamical system, like that of HH, behaves as a 4-dimensional resonator exhibiting subthreshold oscillations. The predicted speed of propagating action potentials at 20 degrees Celsius is in good agreement with the HH experimental value at 18.5 degrees Celsius. After the external concentration of calcium ions is reduced, the generation of repetitive rebound action potentials is predicted by our model, in agreement with experiment, when the membrane is stimulated by a brief (0.1 ms) depolarizing current. Unlike the HH model, our model predicts, in agreement with experiment, that prolonged constant-current stimulation does not generate spike trains in perfused axons. Our resonator model predicts rebound spiking following prolonged hyperpolarizing stimulation, observed at 18.5 degrees Celsius by HH but not predicted at this temperature by their quantitative model. Spiking promoted by brief hyperpolarization is also predicted, at room temperature, by our electrodiffusion model, but only at much lower temperatures (ca. 6 degrees Celsius) by the HH model. We discuss qualitatively, more completely than do HH, temperature dependences of the various physical effects which determine resting and action potentials.

Figures

Figures reproduced from arXiv: 1908.05086 by the authors.

Figure 5
Figure 5. Persistent spike train of membrane action potentials, triggered by a 0.1 ms pulse of transverse depolarizing membrane current-density -2 stim i = −69μA cm when low external calcium ion concentrations in the bulk electrolyte reduce the free energy barrier O β wNa of sodium ions from 3.0 to 1.48 inside open sodium channels of the perfused squid axon at 20 C° . Amplitudes and frequencies of action potentials in this ef… view at source ↗

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [374]

    Null space in the Hodgkin-Huxley equations

    https://doi.org/10.1113/jphysiol.1962.sp007026 24 Best, E.N., 1979. Null space in the Hodgkin-Huxley equations. A critical test. Biophys. J. 27 (1), 87-104. https://doi.org/10.1016/S0006-3495(79)85204-2 Bezanilla, F., 2018. Gating currents. J. Gen. Physiol. 150 (7), 911-932. https://doi.org/10.1085/jgp.201812090 Bezanilla, F., Armstrong, C.M., 1977. Inact...

    work page doi:10.1113/jphysiol.1962.sp007026 1962

  2. [913]

    Rectification and inductance in the squid giant axon

    https://doi.org/10.1152/jn.1998.80.2.903 Cole, K.S., 1941. Rectification and inductance in the squid giant axon. J. Gen. Physiol. 25 (1), 29-51. https://doi.org/10.1085/jgp.25.1.29 Cole, K.S., Guttman, R., Bezanilla, F., 1970. Nerve membrane excitation without threshold. Proc. Natl. Acad. Sci. USA 65 (4), 884-891. https://doi.org/10.1073/pnas.65.4.884 Del...

    work page arXiv 2001

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