From modelling to understanding: the signals in nerves

J\"uri Engelbrecht; Kert Tamm; Tanel Peets

REVIEW 4 major objections 5 minor 84 references

From modelling to understanding: the signals in nerves

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Nerve signals are a coupled wave ensemble, not just electricity.

desk verdict A candid review of the authors' own coupled-model program: useful synthesis, but the qualitative-match evidence is too thin to validate the assumed coupling structure. read the letter →

arxiv 2412.17413 v1 pith:MLII6GF2 submitted 2024-12-23 physics.bio-ph

Jüri Engelbrecht , Kert Tamm , Tanel Peets This is my paper · ORCID

classification physics.bio-ph MSC 92C2035Q9274J30

keywords actionpotentialwaveensemblenervesignalpropagationmechanoelectricalcouplingtemperaturemathematicalmodellingmyelinatedaxoncoupledequations

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

This review paper argues that the signal moving along a nerve axon is not the electrical action potential alone but an ensemble of coupled waves: a pressure wave in the inner fluid, longitudinal and transverse deformations of the membrane wall, and a temperature change. The authors assemble this ensemble into one system of coupled equations, with the electrical signal acting as the trigger and the other waves generated by its changes. Numerical solutions in dimensionless form reproduce the measured shapes of these accompanying waves, and a version in physical units that includes myelin geometry predicts conduction velocities up to about 68 m/s, inside the measured range for myelinated fibres. The paper's aim is to provide a modular, physics-first scaffold on which more detailed models of nerve signalling can be built.

What carries the argument

The carrying object is the coupled 'wave ensemble' model: an action-potential block (either the simplified two-variable model or the full ion-current model), a damped wave equation for axoplasmic pressure, an improved density-wave equation for the longitudinal membrane wave, the transverse displacement taken proportional to the longitudinal gradient as in rod theory, and a heat equation with source terms for temperature. Coupling forces F1, F2, F3 enter as linear combinations of derivatives such as ZX, JT, ZT, PT, and UT, with free coefficients; temperature also uses internal variables for exo- and endothermic reactions. For myelinated axons, the machinery is a transmission-line pair derived from electromagnetic equations with inductance retained, and myelination enters through a thickness ratio and a length ratio between myelin segments and nodes of Ranvier.

What would settle it

Record the action potential, membrane displacement, and temperature simultaneously in a single unmyelinated axon and check whether the displacement and temperature waveforms are locked to the derivatives of the electrical signal. If the mechanical peak precedes the electrical peak, or if no choice of the free coupling coefficients reproduces the measured amplitude (about 1 nm transverse displacement) and temperature time course, the central assumption is refuted.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the propagation of a nerve signal can be described as a wave ensemble whose components obey coupled continuum equations: the action potential, a pressure wave in the axoplasm, a longitudinal density wave and a transverse displacement in the biomembrane, and a temperature field. The electrical variables are the cause; mechanical and thermal responses are driven by coupling forces that are written as linear combinations of space and time derivatives of the field variables, in line with the observation that the rate of change of stimulation, not its absolute value, excites the nerve. The dimensionless proof-of-concept model yields profiles that match experiments qualitatively, and the myelinated-axon extension in physical units, which adds the myelin geometry through length and thickness ratios, raises the predicted action-potential velocity into the 67.7 m/s range. The model is deliberately modular: any block, including the action-potential generator, can be replaced by a more accurate or even experimentally measured description.

Load-bearing premise

The load-bearing premise is that all mechanical and thermal waves are generated by changes in the electrical variables, so the coupling forces can be written as linear combinations of derivatives of the field variables with free coefficients; if the real coupling is not of this form, the qualitative match is a fitting artifact rather than a test of the mechanism.

Editorial extensions

If this is right

If the ensemble model is correct, mechanical and thermal recordings alongside the electrical one are not side effects but complementary views of the same propagating event, so optical and thermal measurements can be used to constrain the electrical model.
The modular structure implies that replacing the action-potential block with a measured signal still yields the accompanying waves, at the cost of losing feedback from mechanics and temperature onto the electrical signal.
Including myelin geometry through length and thickness ratios predicts conduction velocities from about 0.5 m/s for unmyelinated axons up to 67.7 m/s for myelinated ones, consistent with the observed range of 10 to 120 m/s and supporting saltatory conduction as a geometric effect.
The same building-block strategy can be extended to other excitable tissues once their structural parameters are known.

Reading between the lines

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

A reader should treat the qualitative match as a proof of concept, not a quantitative validation: the coupling coefficients in F1, F2, F3 are free parameters, so a direct quantitative comparison of predicted transverse displacement amplitude (about 1 nm) and temperature transient against simultaneous recordings would be the real test.
The derivative-coupling hypothesis predicts specific phase relationships: the pressure and membrane waves should be locked to the time derivative of the action potential, so simultaneous AP and displacement measurements could distinguish this mechanism from one in which a mechanical wave drives the electrical signal.
The model suggests a testable extension: if myelination enters through geometry ratios, then varying node length or myelin thickness in computational experiments should change velocity in a predictable way, and the predicted ceiling near 68 m/s could be checked against systematic measurements across axon diameters.
If the modular framework is portable, it could be applied to cardiac or muscle fibres, where electromechanical and thermal coupling are also observed.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Take a look at this one if you want the Tallinn group's coupled-model framework in one place. It's not a new result — the abstract says it is a review of their own studies. But as a synthesis it does a real service: the 'building block' model from their monograph is written out explicitly (Appendix A), the myelinated extension with the Lieberstein-inspired Maxwell equations is in Appendix B, and the seven guidelines at the end are clear. They also give a fair critical pass over the alternatives — Heimburg-Jackson solitons, El Hady-Machta, Chen et al., Rvachev's pressure-wave-driven AP — and they are candid that much of the coupling is phenomenological.

What I'd flag is exactly the stress-test point. The coupling forces F1-F3 are asserted as linear combinations of derivatives of Z, J, P, U. That form is motivated by the Du Bois-Reymond 'changes matter' idea, but it is not derived from the conservation laws, and it rules out the opposite causal direction (mechanical wave driving the AP) that they themselves mention with Rvachev. So the statement that the model matches experiments 'rather well qualitatively' does not actually validate that specific causal structure; it shows that a flexible family with free coefficients can be made to look like the data. Without parameter values, error bars, or a direct comparison to the cited experimental profiles, that claim is not independently checkable from this preprint.

The myelinated velocity result (up to 67.7 m/s) also rests on two phenomenological parameters, gamma and mu, so it's a demonstration that the model can produce plausible speeds, not a prediction. To the authors' credit, they say the dimensionless model lacks physical parameters for relaxation times and that more experiments are needed. So the paper is honest about its status.

My bottom line: it's a solid review/position paper for people entering this niche or who want a single reference for the program. It deserves peer review as a review article, but the referee should ask them to temper the validation claim and point to the specific papers where the numerical comparisons live, ideally with parameter tables. I wouldn't cite it in my own work — I'd cite the underlying papers — but I might put it on the reading list for a student wanting the framework.

Referee Report

4 major / 5 minor

Summary. This review paper consolidates the authors' decade of work on nerve-signal modelling. It argues that a propagating action potential is accompanied by a pressure wave, longitudinal and transverse membrane waves, and temperature changes, and it presents a coupled system of equations in Appendix A (FHN for the AP, wave equation with dissipation for pressure, improved Heimburg-Jackson equation for density, and heat equation for temperature, with coupling forces F1-F3), claiming 'rather good qualitative match' with experimental profiles. It then summarizes a myelinated-axon extension in Appendix B in which inserting inductance and phenomenological myelination parameters gamma and mu into a Lieberstein-type cable model yields conduction velocities up to 67.7 m/s, and it closes with five modelling guidelines.

Significance. If the proposed framework is taken as a modular scaffold, it has value: it places electrical, mechanical, and thermal effects in one explicit set of partial differential equations, it emphasizes physical conservation laws in Section 3, and it is self-consciously modular ('building blocks'). The guidelines in Section 7 are sensible and the review of existing models is useful. However, the significance as a validated model is currently limited: the central comparison to experiments is qualitative, the coupling structure is assumed rather than derived, and most quantitative results are delegated to earlier papers, including an unreviewed preprint. The paper is more a programmatic hypothesis than a validated quantitative model at this stage.

major comments (4)

[Appendix A (after eq. (d))] The coupling forces F1 = eta1 Z_X + eta2 J_T + eta3 Z_T, F2 = gamma1 P_T + gamma2 J_T - gamma3 Z_T, and F3 = tau1 Z^2 + tau2(P_T + phi2(P)) + tau3(U_T + phi3(U)) - tau4 Omega are asserted as the mathematical implementation of the authors' hypotheses, not derived from the conservation laws discussed in Section 3. Because the coefficients are free and no parameter values are tabulated, the qualitative agreement in Fig. 3 cannot establish that this particular derivative-linear form is the correct coupling structure; a different causal arrangement with similar flexibility could plausibly be tuned to match the same profiles. The manuscript itself lists mechanically sensitive ion channels in Section 5 and cites Rvachev [68] for pressure-driven AP, so the restriction to electrically driven derivative terms is an additional assumption that must either be defended from the physics or explicitly presented as a testable hypothesis.
[Section 5, Fig. 3] The statement that 'the computational results ... demonstrate a good qualitative match with experimentally measured profiles' is not substantiated in this manuscript. No experimental curves are overlaid, no error bars or discrepancy metrics are given, and no list of the dimensionless parameter values used in the simulation is provided. Since the 'proof of concept' claim rests on this match, the paper should report at least one concrete comparison with a published measurement (e.g., Tasaki [76,77] or Terakawa [78]) and specify the parameter set used, including the coupling coefficients.
[Appendix B / Section 6] The claim that including myelination geometry in physical units yields AP velocities up to 67.7 m/s is presented on the basis of the companion paper [75], which is an arXiv preprint, and the description here leaves several load-bearing choices underspecified. In particular, the physical origin and fitted value of gamma in Eq. (3) are not given, the range of mu-ratio is stated but not connected to the resulting velocities, and the statement in Section 6 that 'a closer match to measurements' is achieved is not quantified. The authors should either report the full parameter set and matching statistics in this paper or restrict the claim to a summary of a peer-reviewed, published result.
[Section 6, temperature relaxation paragraph] The manuscript itself concedes that 'the lack of physical parameters does not permit to calculate the relaxation time in physical units.' This admission limits the thermal component of the model, which is one of the five components claimed to match experiments. The authors should state whether the dimensionless temperature profile in Fig. 3 is predictive or merely illustrative, and they should indicate which experimental data are needed to close the model.

minor comments (5)

[Title / running header] The title contains a typo: 'NER VES' should be 'NERVES'.
[Section 4, bullet list] In the bullet list on axon scales, 'typical neutron' should read 'typical neuron'.
[Section 1, paragraph citing Hodgkin] The sentence ending with 'and thei [33] mentioned' is incomplete or contains a typo; the citation should be integrated grammatically.
[Figure 3 caption] The caption should identify which curve corresponds to which component (AP, PW, LW, Theta, TW) and should state the parameter values used; as printed, the two panels are difficult to interpret.
[Appendix B, Eq. (1)] The capacitance combination (Ca*pi*a^2 + Cm*2*pi*a) mixes per-unit-length and per-area quantities; a sentence explaining the resulting units would prevent confusion.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: the myelinated-velocity result is produced by a velocity-determining phenomenological parameter and is self-cited from the authors' own preprint, though the unmyelinated proof-of-concept model is explicit and not circular.

fitted input called prediction [Appendix B, around Eqs. (3)-(7), text after Eq. (7)]
"Let us take Lieberstein eqs. (1) and (2), introducing parameters µ and γ characterizing the AP propagation velocity increase from saltatory conduction [5] and other relevant mechanisms. ... parameter γ is a phenomenological coefficient which determines conduction velocity between adjacent nodes of Ranvier. ... Under the considered parameter combinations we can observe the AP propagation velocities up to 67.7 [m/s] [75]"

Equation (3) inserts the factor (1+γ·µ) multiplying the axial-current gradient ∂i_a/∂x, so the propagation velocity is scaled by the very parameters introduced for that purpose, and γ is explicitly described as the phenomenological coefficient 'which determines conduction velocity.' The reported maximum velocity, 67.7 m/s, is therefore not a first-principles prediction from the Maxwell/Lieberstein base alone; it is the output of a parameter introduced to control that output. The numerical evidence is also imported from the authors' own arXiv preprint [75], closing the validation loop inside the same group's modelling chain. This is a fitted input presented as a result, rather than an independent derivation of myelinated conduction velocity.

full rationale

The paper is mostly a transparent review: the unmyelinated model is assembled from explicitly stated hypotheses and building blocks, and the qualitative agreement with experiments is offered as a proof of concept rather than as a parameter-free prediction. The derivative-linear coupling ansatz is underdetermined by a qualitative match, but the paper labels it as a hypothesis rather than deriving it, so that concern is a validation weakness, not circularity. The one place where a claimed output is effectively built into an input is Appendix B: the myelinated velocity increase is inserted through the phenomenological γ and μ parameters, and the resulting velocity range is cited from the authors' own prior preprint [75]. Since Equation (3) scales the propagation term by (1+γμ), the up-to-67.7-m/s velocity is a consequence of the very parameter introduced to produce it, making this step partially circular. The rest of the framework, including the unmyelinated proof-of-concept equations and the discussion of alternative mechanisms such as Rvachev's pressure-wave trigger, retains independent content, so the overall circularity is moderate rather than total.

Assumptions & free parameters 8 free parameters · 7 assumptions · 1 invented entities

The central model in Appendix A has a large number of coefficients introduced by hand and not fixed by independent data in this paper; the myelinated model adds gamma and mu. The main axioms are the HH-trigger assumption, continuum representations of axon components, the derivative-coupling hypothesis, and the rod-theory relation W proportional to U_X.

free parameters (8)

FHN parameters a1, b1, epsilon = not specified
Control AP shape, threshold, and time scale in the dimensionless proof-of-concept model; no values are given in the preprint.
mechanical activation coefficient beta_i (b_i = -beta_i U) = not specified
Couples AP to density change in the membrane; chosen by hand.
pressure coupling coefficients eta_1, eta_2, eta_3 = not specified
Set the strength of F1 in the pressure wave equation; no values provided.
membrane coupling coefficients gamma_1, gamma_2, gamma_3 = not specified
Set the strength of F2 in the improved Heimburg-Jackson equation; no values provided.
thermal coupling coefficients tau_1..tau_4, plus phi_2, phi_3 and Omega = not specified
Set the source and sink terms in the temperature equation, including the abstracted endothermic reaction; no values provided.
iHJ coefficients c0, N, M, H1, H2, mu_2 = not specified
Dispersion, nonlinearity, and damping constants for membrane density waves; no numerical values are given.
myelination parameter gamma = not specified
Introduced in Eq. (3) as a phenomenological coefficient determining conduction velocity between nodes; adjusted to reach observed speeds.
mu-ratio L2/L1 = range 0 to 325 used
Geometrical ratio but also treated as an adjustable parameter controlling the AP velocity increase in the myelinated model; velocity up to 67.7 m/s is reported for chosen values.

assumptions (7)

standard math Conservation laws (Maxwell equations, Newton's second law, Fourier and Joule laws) govern the electrical, mechanical, and thermal components.
Invoked in Section 3 and Guideline 1 as the physical basis for all governing equations.
domain assumption Hodgkin-Huxley paradigm: electrical signals are the primary carriers and trigger all other processes.
Section 5 assumption list; the model is built with AP as driver, and the paper notes alternative mechanisms are not incorporated.
domain assumption The axon can be represented as a tube of viscous fluid with a thin elastic/viscoelastic bilayer wall.
Section 4; this continuum idealization underlies the wave equations for PW, LW, and TW.
domain assumption Du Bois-Reymond law: changes, not absolute values, are the stimuli, so coupling forces involve derivatives.
Section 5 and Guideline 2; used to choose the form of F1, F2, F3.
domain assumption Transverse displacement of the biomembrane is proportional to the gradient of longitudinal displacement, W proportional to U_X.
Section 5 and Appendix A; taken from the theory of rods, but not directly verified for biomembranes in this paper.
ad hoc to paper Coupling forces F1, F2, F3 can be represented as linear combinations of derivative terms with free coefficients.
Appendix A; this is a modeling choice, not a derivation, and is load-bearing for the qualitative match.
domain assumption Internal variables can describe exo- and endothermic reactions and myelin-sheath microstructure.
Section 6 and Appendix A; used for temperature relaxation and taken from previous work [19, 72, 74].

invented entities (1)

abstracted endothermic reaction variable Omega
purpose: Models a slow temperature-recovery process after the nerve pulse by acting as a sink in the heat equation.
Phenomenological internal variable introduced without an independent experimental handle; its relaxation time cannot be computed in physical units, as the paper acknowledges.

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Cite this review

Pith. "Pith review of From modelling to understanding: the signals in nerves." pith.science (2026). https://pith.science/paper/MLII6GF2

@misc{pith2026241217413,
  author       = {Pith},
  title        = {Pith review of: From modelling to understanding: the signals in nerves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLII6GF2}},
  note         = {Machine review of arXiv:2412.17413}
}

read the original abstract

This paper attempts to review our studies on the propagation of signals in nerves over the past decade. The need for interdisciplinary studies is stressed that helps to understand the physical mechanisms of coupling the electrical, mechanical, and thermal effects in nerves. Based on the analysis of structural properties of axons and possible mechanisms of interaction between different physical phenomena, a set of assumptions and hypotheses is formulated. As a proof of concept, a rather general mathematical model is presented for describing a wave ensemble in unmyelinated axons. This model is composed of several governing equations ("building blocks") which are coupled by forces describing the interaction between the effects. The numerical simulation using the dimensionless variables demonstrated a rather good qualitative match with experiments. The further generalisation of this model in physical units for the processes in myelinated axons permits a closer match to measurements. Based on modelling and in silico experiments, the guidelines for modelling such a complex electrophysiological process are formulated. These guidelines reflect the importance of following the physical principles in modelling together with interdisciplinary knowledge from continuum mechanics and mathematics.

Figures

Figures reproduced from arXiv: 2412.17413 by the authors.

**Figure 2.** Sketch of a wave ensemble in an axon. Based on assumptions and hypotheses, it is possible to build up a mathematical model that describes all the elements of the ensemble. In the first stage, for the sake of generality, we use dimensionless variables and start with modelling the processes in an unmyelinated axon. It permits us to pay attention to coupling effects. The model is a system of coupled differential equati… view at source ↗

**Figure 3.** The wave ensemble (from Engelbrecht et al 2021b) [2 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The scheme of the myelinated axon. It is proposed [28] that under the myelin sheath, the passive cable equation (the diffusion-type equation) governs the process and in Ranvier nodes, the usual HH equation works. Tamm et al. [75] have specified 10 [PITH_FULL_IMAGE:fig…

Discussion (0). Continue with ORCID to comment.

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