{"id":"e16bda63-00ad-446b-aac7-1329f9a11ad3","arxiv_id":"2506.11776","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For FitzHugh-Nagumo, Morris-Lecar, Hodgkin-Huxley, and Hindmarsh-Rose neurons, periodic traveling-wave signals propagate stably along feedforward chains because transverse Floquet multipliers stay inside the unit circle.","lead":"This paper tests a mathematical framework for propagating periodic signals along feedforward chains of model neurons, using four standard neuron models. It finds that the signals remain stable in all tested cases, which matters for understanding locomotion, peristalsis, and robot design.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central existence claim rests on unreproduced Floquet-multiplier computations; with no error estimates and an apparent table inconsistency, the numerical evidence is the load-bearing weak point.","rationale":"Stress-test logic: the abstract's central claim is an existence claim ('can be transversely Floquet stable') across four neuron models. For an existence claim, a handful of parameter sets is logically sufficient; the weakness is not paucity alone but that the 'can' is established only by the numbers in Tables 1-4. The theory (Theorem 2.3) is published and appears correctly applied: the lifted orbit's Floquet multipliers decompose into CPG and transverse multipliers, and the feedforward block-triangular structure makes the transverse multipliers depend only on the CPG orbit. The simulations in Sections 4-7 plausibly realize the intended phase-synchronous orbits, and the figures qualitatively support the existence of the periodic traveling waves. Thus the mathematical architecture is sound. The vulnerable step is the quantitative claim that all transverse multipliers have modulus below 1 for the stated cases. Nothing in the manuscript lets a reader check that claim: no data files, no code, no tolerances, no convergence study. The apparent inconsistency in Table 4 for (7.23) is a concrete red flag that the tables cannot be taken at face value. If an independent recomputation confirms every value, the paper's claim stands and the CONDITIONAL verdict could be upgraded to ACCEPT. If one multiplier is actually above 1 for a parameter set, the claim as stated fails for that model, and Section 9's universal or broad-range language would need to be withdrawn. This is exactly what a single, well-specified numerical test can decide, which is why the concern is load-bearing but not fatal.","tokens_in":25871,"tokens_out":14312,"duration_ms":135160,"concrete_test":"Recompute all transverse Floquet multipliers in Tables 1-4 with an independent, publicly available high-precision solver (e.g., scipy.integrate.solve_ivp with rtol=atol=1e-12), determining the period T by a zero crossing of a Poincare section, integrating the full linearized system over exactly one period, and computing the monodromy matrix eigenvalues. Require that every maximum absolute multiplier matches the published value to two significant digits and lies below 1 by at least the recomputed uncertainty. Pay special attention to the Hindmarsh-Rose (7.23) row: if the true multiplier is 0.887 (not 0.0887), repeat with a fine parameter scan around g = -2 to see whether the multiplier crosses 1 before the coupling leaves the physically plausible range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's new contribution is numerical: it applies the previously proved Theorem 2.3 and reports that, for all four neuron models, the maximum absolute transverse Floquet multiplier is below 1 for the tested parameter sets (Tables 1-4). The abstract's 'can be' claim would be established by a single accurate example per model, but the tables are the only evidence that the examples are transversely Floquet stable. There are no error estimates, no convergence checks with respect to integration tolerance or settling time, no statement of the ODE solver or tolerances for the Mathematica runs, and no code or data files. One row compounds this: Table 4, parameters (7.23), lists a transverse multiplier 0.887 with absolute value 0.0887; at face value 0.887 is much closer to (but still below) 1, so the margin of stability is 0.113, not 0.9113. If this is a transcription error, it shows the tables need verification; if it is a genuine multiplier, the claimed margin shrinks by an order of magnitude for that model. Section 9 then generalizes to 'universally or over broad parameter ranges', which is not supported by two to four fixed parameter sets per model. The load-bearing assumption is therefore that every tabulated absolute multiplier is accurate to well below its distance from 1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper applies the feedforward-lift framework developed by the authors in [81] to four standard neuron models: FitzHugh–Nagumo, Morris–Lecar, Hodgkin–Huxley, and Hindmarsh–Rose. For a 7-node feedforward chain built on a 3-node Z3-symmetric CPG, it reports numerical Floquet multipliers and transverse Jacobian eigenvalues for a small number of parameter sets per model, claiming that the lifted periodic orbits are transversely Floquet stable. It also compares two transverse stability notions, discusses an informal 'transverse stability on average' condition, and presents simulations of robustness to synchrony-breaking perturbations.","tokens_in":26171,"tokens_out":17187,"duration_ms":141308,"significance":"The paper's theoretical backbone is rigorous and imported from [81]: stability of a feedforward lift reduces to Floquet stability of the CPG orbit plus the transverse multipliers of a single module, independent of chain length. Demonstrating this condition in four biophysically standard models is a useful check of the theory's applicability, and the specific parameter sets constitute falsifiable predictions. However, the new evidence is numerical and currently compromised by table inconsistencies, missing error estimates and solver details, and conclusions in Section 9 that go beyond the tested parameter range. If these issues are resolved, the paper would be a solid application-oriented contribution.","major_comments":[{"comment":"The tables contain internal arithmetic inconsistencies that affect the reported stability margins. In Table 2 (Section 5.3), the transverse Floquet multiplier for parameter set (5.16) is listed as -0.609 ± 0.0575i with absolute value 0.0838, but its modulus is sqrt(0.609^2 + 0.0575^2) ≈ 0.612. In Table 4 (Section 7.3), the multiplier 0.887 for (7.23) is listed with absolute value 0.0887, whereas the absolute value is 0.887. Consequently, the 'largest transverse eigenvalue' values quoted in Section 8.1 are inconsistent with the tables as printed: for FitzHugh–Nagumo the maximum is 0.715 (parameter set (4.10)), not 0.435; for Morris–Lecar it is approximately 0.612, not 0.0986; and for Hindmarsh–Rose it is 0.887 (if the entry is taken literally), not 0.820. Please recheck all numerical results and correct the tables, since the stability margins change by up to an order of magnitude.","section":"Tables 2 and 4; Section 8.1"},{"comment":"The Floquet multipliers are reported to three decimal places without any error estimates, convergence checks with respect to integration tolerance or settling time t0, or a statement of the numerical integrator and tolerances used for the Mathematica runs. Section 3.1's method requires an accurate period T and a reliable fundamental matrix at time t0; without evidence that the computed multipliers are accurate to well below their distance from 1, the central claim rests on unverified numerics. The authors should provide reproducible details (solver, tolerances, t0, residual errors) or make code and data available.","section":"Section 3.1; Tables 1–4"},{"comment":"The conclusion that the four models satisfy transverse stability conditions 'either universally or over broad parameter ranges' is not supported by the evidence: Sections 4.1, 5.1, 6.1, and 7.1 test only 4, 2, 2, and 3 parameter sets per model, respectively. These examples establish the existence claim ('can be') but not a universal or broad-range claim. Please limit the conclusions to the tested parameter sets or add systematic parameter scans with evidence of coverage.","section":"Section 9"}],"minor_comments":[{"comment":"The acronym 'GPG' appears in Section 1.4 ('a very simple GPG') and in Section 8.2 ('one of the simplest GPGs'); these should be 'CPG'.","section":"Sections 1.4 and 8.2"},{"comment":"In the derivation of the Floquet matrix E, the line 'Y(t0 + T) = (P(t0)e^{BT})((t0)^{-1}Y(t0))' is missing the factor P(t0)^{-1}; the correct expression is (P(t0)e^{BT}P(t0)^{-1})Y(t0).","section":"Section 3.1"},{"comment":"In item (a), the reference 'Equation (2.2)' should be 'Equation (2.3)', since the displayed transverse Floquet equation is labelled (2.3).","section":"Definition 2.2"},{"comment":"The layout of Table 1 is ambiguous because the header 'abs.' appears twice and it is unclear which entries correspond to CPG multipliers and which to transverse multipliers; please reformat with separate subheadings.","section":"Table 1"},{"comment":"The references to 'Table 4.3', 'Table 5.3', 'Table 6.3', and 'Table 7.3' should be cross-referenced to the actual table numbers, since the tables are numbered 1–4.","section":"Section 8.1"},{"comment":"The paper invokes 'transverse stability on average' to explain stability when transverse eigenvalues are sometimes positive, but it also states in Section 1.4 that this concept has not been made rigorous; these remarks should be labelled as heuristic.","section":"Sections 4.3, 5.2, 7.2"},{"comment":"Minor typos include 'the the' in Remark 3.2, 'correponding' in Section 8.1, 'most noticable' in Section 8.2, and 'it therefore make sense' in Section 8.1.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a numerical application of the authors' own previously published theorem; its novelty is limited to the examples. The table inconsistencies are likely transcription errors, but they undermine confidence in the empirical claims, and the lack of reproducibility details is a serious issue for a computational paper. I recommend requiring code/data and a thorough numerical verification before publication, and checking the journal's data-sharing policy. The heavy reliance on [81] is appropriate given the derivation, but the paper should state clearly which statements are new."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe takeaway: this is a competent numerical companion to the authors' earlier theoretical paper [81]. It doesn't prove new theorems, but it does something useful: it checks the transverse Floquet stability condition from that theory against four standard neuron models and shows that the intuitively appealing 'transverse eigenvalues negative everywhere' criterion is too strong—short intervals of positive real part don't necessarily destroy Floquet stability. The biological framing (leech, locomotion, C. elegans) is clear and the exposition is readable.\n\nWhat's new: explicit computation of transverse Floquet multipliers for FitzHugh-Nagumo, Morris-Lecar, Hodgkin-Huxley, and Hindmarsh-Rose networks, and the comparison of two stability notions. The observation that Floquet stability can survive short intervals of positive transverse eigenvalues is worth having.\n\nWhere it's soft: the numerical evidence is thin and unreproduced. Each model gets one to three parameter sets. There are no error estimates, no convergence checks, no solver details for the Mathematica runs, and no code or data files. The abstract's 'for all these neuron models ... can be transversely Floquet stable' is supported by the examples, but Section 9 pushes further to 'universally or over broad parameter ranges,' which the evidence doesn't justify. There's also an apparent typo in Table 4: for parameters (7.23), the multiplier 0.887 is listed with absolute value 0.0887. At face value, that's inconsistent; either the multiplier or the norm is off by a factor of ten. That kind of slip makes you wonder about the rest of the tables. The dependence on [81] is heavy but legitimate—that theorem is published and rigorous.\n\nNet: the theoretical foundation is sound, the new work is a reasonable first numerical pass, and the paper deserves a serious referee. The referee should ask for the code, error bounds, and a rewritten Section 9 that matches the evidence. I'd accept it for review with revisions expected.","headline":"Solid numerical companion to the authors' theory paper, but the evidence is thin and Section 9 overreaches; worth reviewing with revisions.","tokens_in":26682,"tokens_out":3049,"would_cite":false,"duration_ms":29537,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","37C75","92B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that periodic signals from a central pattern generator propagate stably along feedforward chains of four standard neuron models, with numerical transverse Floquet multipliers below 1 in every tested case.","keywords":["feedforward lift","central pattern generator","Floquet stability","phase synchrony","FitzHugh-Nagumo","Morris-Lecar","Hodgkin-Huxley","Hindmarsh-Rose"],"falsifier":"Scan a fine grid of coupling strengths and input currents for any of the four models and recompute the transverse Floquet multipliers with a checked integration tolerance: finding a parameter set where the CPG orbit remains Floquet stable but some transverse multiplier has absolute value $\\geq 1$ would refute the claim that propagated signals are generically transversely stable. The claim would also fail if a perturbation localized at one node grew along an extended chain instead of decaying back to the phase-locked pattern.","tokens_in":25651,"feed_emoji":"🧠","tokens_out":10470,"duration_ms":93705,"temperature":0.7,"pith_summary":"This paper asks whether a periodic rhythm, once generated by a small central pattern generator (CPG) subnetwork, can travel stably along a feedforward chain of neurons and keep its phase pattern. Building on a companion theory paper, the authors take a periodic orbit of the CPG and 'lift' it to the full chain; stability of the lifted signal then reduces to the CPG orbit being Floquet stable and every transverse Floquet multiplier having absolute value below 1. Numerical tests on four standard neuron models—FitzHugh-Nagumo, Morris-Lecar, Hodgkin-Huxley, and Hindmarsh-Rose—report that all computed transverse multipliers lie inside the unit circle, so the propagating signal is transversely Floquet stable in every tested case. The paper also contrasts this Floquet condition with the simpler criterion of transverse stability of the synchrony subspace, finding parameter sets where Jacobian eigenvalues briefly have positive real part while Floquet multipliers remain stable, a 'stability on average' effect. If the claim holds, a CPG rhythm can be copied into arbitrarily long chains at no extra stability cost, which bears on biological locomotion, peristalsis, and continuum robots.","feed_headline":"Stable rhythm propagation verified in four neuron models","feed_subtitle":"FitzHugh-Nagumo, Morris-Lecar, Hodgkin-Huxley, and Hindmarsh-Rose chains all keep the CPG beat without losing phase.","key_machinery":"The central object is the feedforward lift: a network in which a CPG subnetwork is both a subnetwork and a quotient network of a larger network, with every added arrow pointing forward in an ordering of the non-CPG nodes. The identity that carries the argument is Theorem 2.3 of [81]: because the Jacobian is block-triangular in feedforward order, the Floquet multipliers of the lifted periodic orbit are exactly the Floquet multipliers of the CPG orbit together with the transverse Floquet multipliers, which depend only on the CPG orbit and not on chain length. Thus stability one step along the chain implies stability for a chain of any length. Numerically, the paper computes the fundamental matrix over one period $T$ and forms $E = VU^{-1}$, whose eigenvalues are the Floquet multipliers; the phase-synchrony variant (Theorem 3.1) reduces the computation further to a single module of orbit representatives.","core_discovery":"The central claim, stated in the abstract as 'for all these neuron models the propagating signal can be transversely Floquet stable', rests on Theorem 2.3 of [81]: a periodic orbit of a CPG lifts to a Floquet-stable periodic orbit of any feedforward lift if and only if the CPG orbit is Floquet stable and, for every chain node, the transverse Floquet multipliers have absolute value less than 1. The numerical work computes these multipliers for the four neuron models on a 7-node feedforward example and finds all multipliers inside the unit circle for the chosen parameter sets: four parameter sets for FitzHugh-Nagumo, two for Morris-Lecar, two for Hodgkin-Huxley, and three for Hindmarsh-Rose. The paper further shows that transverse stability of the synchrony subspace—negative real parts of the transverse eigenvalues at every point of the orbit—is sufficient but not necessary, and exhibits parameter sets where those eigenvalues have small positive real parts on short intervals even though the Floquet multipliers remain below 1.","pith_inferences":["The same feedforward-lift mechanism likely transfers beyond neurons: any oscillator with a stable periodic orbit could drive a chain, so the results plausibly apply to gene-regulatory delay lines and to continuum robots whose 'neurons' are mechanical segments.","The observed 'stability on average' suggests a rigorous probabilistic criterion: weighting the transverse eigenvalues by the invariant measure on the periodic orbit, a positive average contraction rate might imply a measure-theoretic attractor for the lifted orbit.","A direct experimental test would perturb one segment's coupling strength in a leech or nematode preparation and measure whether downstream phase differences stay within a few percent while amplitudes recover.","The eigenvalue-versus-multiplier gap warns that instantaneous Jacobian eigenvalues alone can mislead for higher-dimensional node spaces, so master-stability-function-style analyses of periodic orbits without Floquet multipliers may yield wrong stability conclusions."],"forward_implications":["Stability of a propagated signal is independent of chain length: if the first module of a feedforward chain is transversely stable, every additional copied module is automatically stable.","Transverse Floquet multipliers need to be computed only for the CPG nodes, so the stability check does not grow with the number of chain nodes.","A CPG with cyclic symmetry $\\mathbb{Z}_k$ creates phase patterns whose phase shifts are integer multiples of $T/k$, and these propagate down the chain as apparent traveling waves.","Small, independent perturbations to node dynamics or couplings in the chain preserve the existence, phase pattern, and approximate synchrony of the signal, with phase relations more robust than amplitudes.","The largest transverse Floquet multiplier can serve as a rule-of-thumb stability index, indicating which neuron models propagate signals with the largest margin of safety (Morris-Lecar above Hindmarsh-Rose in the tested cases)."],"supporting_citations":[{"why":"Supplies the feedforward-lift construction and Theorem 2.3, the necessary and sufficient Floquet-stability condition at the core of this paper.","marker":"[81]"},{"why":"Provides the Floquet theory and the numerical method the paper uses to compute multipliers from the fundamental matrix over one period.","marker":"[39]"},{"why":"Gives the network-dynamics formalism of balanced colorings, synchrony subspaces, and quotient networks that defines what a feedforward lift is.","marker":"[26]"},{"why":"Defines the FitzHugh-Nagumo equations used in Section 4's simulations and stability computations.","marker":"[18]"},{"why":"Defines the Morris-Lecar model used in Section 5's simulations and stability computations.","marker":"[60]"},{"why":"Defines the Hodgkin-Huxley model used in Section 6's simulations and stability computations.","marker":"[42]"},{"why":"Defines the Hindmarsh-Rose model used in Section 7's simulations and stability computations.","marker":"[40]"}],"fun_headline_variants":["Four neuron models keep the beat in feedforward chains","Rhythm propagation proven stable in four neuron types","Floquet-stable signals in standard neuron chain networks","Synchronous periodic signals survive four neuron models","CPG rhythm lifts to stable chains in all tested models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the one to three parameter sets tested per neuron model are representative enough to conclude that propagating signals are transversely Floquet stable 'universally or over broad parameter ranges', while the reported Floquet multipliers carry no error estimates or convergence checks.","fun_headline_variants_meta":{"raw":{"variants":["Four neuron models keep the beat in feedforward chains","Rhythm propagation proven stable in four neuron types","Floquet-stable signals in standard neuron chain networks","Synchronous periodic signals survive four neuron models","CPG rhythm lifts to stable chains in all tested models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000476,"raw_usage":{"total_tokens":2361,"prompt_tokens":943,"completion_tokens":1418,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":1343}},"tokens_in":559,"tokens_out":1418,"duration_ms":11200,"temperature":1.0,"reasoning_tokens":1343,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:03:22.619338+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Scan a fine grid of coupling strengths and input currents for any of the four models and recompute the transverse Floquet multipliers with a checked integration tolerance: finding a parameter set where the CPG orbit remains Floquet stable but some transverse multiplier has absolute value $\\geq 1$ would refute the claim that propagated signals are generically transversely stable. The claim would also fail if a perturbation localized at one node grew along an extended chain instead of decaying back to the phase-locked pattern.","supporting_citations":[{"cited_title":"Stewart and D","cited_arxiv_id":null,"evidence_quote":"Supplies the feedforward-lift construction and Theorem 2.3, the necessary and sufficient Floquet-stability condition at the core of this paper."},{"cited_title":"Hassard, N.D","cited_arxiv_id":null,"evidence_quote":"Provides the Floquet theory and the numerical method the paper uses to compute multipliers from the fundamental matrix over one period."},{"cited_title":"Golubitsky and I","cited_arxiv_id":null,"evidence_quote":"Gives the network-dynamics formalism of balanced colorings, synchrony subspaces, and quotient networks that defines what a feedforward lift is."},{"cited_title":"FitzHugh","cited_arxiv_id":null,"evidence_quote":"Defines the FitzHugh-Nagumo equations used in Section 4's simulations and stability computations."},{"cited_title":"Morris and H","cited_arxiv_id":null,"evidence_quote":"Defines the Morris-Lecar model used in Section 5's simulations and stability computations."},{"cited_title":"Hodgkin and A.F","cited_arxiv_id":null,"evidence_quote":"Defines the Hodgkin-Huxley model used in Section 6's simulations and stability computations."},{"cited_title":"Hindmarsh and R.M","cited_arxiv_id":null,"evidence_quote":"Defines the Hindmarsh-Rose model used in Section 7's simulations and stability computations."}],"review_version":1}