The Lie Group Basis of Neuronal Membrane Architecture: Why the Hodgkin–Huxley Equations Take Their Form^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript aims to demonstrate that the HH model can be explained by three fundamental symmetries: conformational state compactness, conductance scaling invariance, and temporal translation invariance. Although it is interesting to explore more theoretical foundations for the HH model—since some parts of the original model were established by fitting experimental data—the present study does not provide sufficiently convincing or substantive evidence. My main concerns are as follows.

(1) The HH model is not totally empirical. It has certain theoretical bases, such as equations (1) to (3) (page 2) derived from Ohm’s law, Kirchhoff's current law, and the first-order kinetics of reversible reactions. I don’t think using a so-called Lie group structure to explain the model is a better choice than relying on these original theoretical foundations. 

(2) I don’t agree with the authors that the six questions listed in page 3 are all empirical choices. For example, the value range of gating variables (m, h, n) must be from 0 to 1 because these variables are probability values. At lease, I think this probability theory is easier to understand than “Compactness of Conformational States” provided by this study.

(3) For the power integers of the gating variables, such as the 4th power of gating variable n, are not fixed numbers. They depend on the structures, functions and mechanisms of the described ion channels that have been gradually revealed and confirmed by the development of channel theory. In addition, the original HH model included only the fast sodium channel and the delayed rectifier potassium channel. Many more ion channels have been discovered that can be modeled with similar biophysical bases for various activation and inactivation gate variables derived from the HH model but with distinct power integers. I don’t see that the theory provide by this study has such inclusiveness.

Similarly, the rate constants (termed rate functions in this manuscript) for gate variables of various ion channels do not always have “exponential voltage dependencies” as claimed in this manuscript.

(4) Resulting from “Multiplicative Scaling of Conductances”, Equation (23) in section 2.2 shows that the conductance ratio between Na+ and K+ remains constant. This is not true in the HH model!  The “Temporal Transition Invariance” in section 2.3 is also incomprehensible, since the action potential waveform described by the HH model depends on time.

Author Response

Please see the uploaded PDF reviewer1_response.pdf. This is a detailed point-by-point response addressing the comments of Reviewer #1.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript presents a fascinating and novel theoretical framework that derives the Hodgkin-Huxley (HH) equations from fundamental symmetry principles (Lie groups). The authors successfully address "why" the equations take their specific form (bounded variables, integer exponents, exponential voltage dependence), moving the field from phenomenological curve-fitting to first-principles derivation. This is a significant contribution to theoretical neuroscience. However, there are some incongruences in the bibliography and some physical justifications that need refinement before publication.

Major Points:

1. Citation Mismatches (Crucial Correction Needed):

   There appears to be a significant error in the citation mapping, particularly regarding biological claims supported by physics textbooks.

   • Line 149: The text states: "Structural biology has revealed that these channels possess discrete conformational states [6, 7]". Reference [6] is listed as "Weinberg, S. (1995). The quantum theory of fields". Reference [7] is "Drzewiecki... Modeling strategies in physiology".

   • Steven Weinberg's book on Quantum Field Theory is not a valid reference for the structural biology of ion channels. The authors likely meant to cite Hille (currently Ref [8]) or seminal structural biology papers (e.g., MacKinnon, Doyle). Please audit all references to ensure they match the text claims.

2. Physical Justification of Conductance Scaling (Symmetry 2):

   While the mathematical derivation of the scale invariance leading to exponential functions is elegant, the physical justification needs strengthening.

   • In Section 2.2.1, you argue that multiplying all conductances by λ leaves relative dynamics unchanged. While true for the relative contributions, in a real biological system, scaling conductances ($g$) without scaling capacitance ($C_m$) or current ($I$) changes the time constant ($\tau = C/g$).

   • Please clarify if this symmetry assumes a concomitant scaling of current, or if it strictly refers to the form of the steady-state equations independent of timescale. The link between this abstract scaling symmetry and the emergence of the Boltzmann-type voltage dependence (e^{E/E_0}) is the core of your paper and deserves a more rigorous physical (thermodynamic) argument, not just a mathematical one.

3. Integer Exponents and Biology:

   Theorem 5 argues that integer exponents (m^3, n^4$) emerge from irreducible representations (winding numbers).

   • While this fits the original HH data perfectly, modern fitting often finds non-integer exponents (e.g., fractal kinetics or complex cooperativity) provide better fits for some channels.

   • Comment: It would be beneficial to add a sentence in the Discussion acknowledging that while the ideal symmetric system yields integers, biological deviations (symmetry breaking) could explain experimentally observed fractional exponents. This would make the theory more robust against criticisms from experimentalists.

4. The Semidirect Product Structure:

   The derivation of SO(2) \ltimes \mathbb{R}^2 is central.

   • Could the authors briefly explain (perhaps in the discussion or a simplified appendix) why the product is semidirect and not a direct product in intuitive terms? It implies the time-evolution/scaling acts on the conformational state. Making this interaction explicit for a biophysics audience (who may not be versed in Lie algebra) would strictly increase the paper's impact.

Minor Points:

1. Check the formatting of the equation (45) approximation.

2. The list of predictions matches the findings well, but ensure the citation for "confirmed by voltage-clamp data" is appropriate (Ref [1] and [12] are fine, but Hille [8] is the standard bible for this).

Author Response

Please see the attachment reviewr2_response.pdf. This document is a point-by-point response to all of the comments raised by Reviewer #2.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript attempts to provide a theoretical interpretation of the Hodgkin–Huxley equations based on Lie group theory and symmetry principles. The authors claim that key structural features of the Hodgkin–Huxley model, such as bounded gating variables, exponential voltage dependencies, integer exponents, and first-order kinetics, can be derived from an underlying symmetry group. However, despite the ambitious scope, the presentation is unclear and the logical connection between the proposed symmetry framework and the established biophysical model is not sufficiently demonstrated.

• The abstract contains extensive mathematical symbols, which is not appropriate for the journal format. In addition, the abstract is overly complex and does not clearly communicate the main idea, objectives, or concrete contributions of the study.

 

• The introduction section is very brief and does not adequately contextualize the problem within existing literature, nor does it clearly motivate the necessity of the proposed symmetry-based approach.

• The methodology is written in an unclear and confusing manner. The derivation steps and assumptions are not presented systematically, and the overall structure of the manuscript lacks coherence.

• Throughout the manuscript, it remains unclear what is genuinely derived as a new theoretical result and what is merely a reinterpretation of already known properties of the Hodgkin–Huxley model. Both the aim of the study and the actual achievements are ambiguous.

• For these reasons, I recommend that the manuscript be rejected.

Author Response

Please see the attachment reviewr3_response.pdf. This document is a point-by-point response to all of the comments raised by Reviewer #3.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript has been improved.

Reviewer 3 Report

Comments and Suggestions for Authors

I do not recommend accepting the study in its current form.