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Synchronous Propagation of Periodic Signals in Feedforward Networks of Standard Model Neurons

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Solid numerical companion to the authors' theory paper, but the evidence is thin and Section 9 overreaches; worth reviewing with revisions. read the letter →

arxiv 2506.11776 v1 pith:Q6OTWMG6 submitted 2025-06-13 nlin.CD math.DSq-bio.NC

Ian Stewart , David Wood This is my paper · ORCID
classification nlin.CDmath.DSq-bio.NC MSC 34C1537C7592B20
keywords feedforwardliftcentralpatterngeneratorFloquetstabilityphasesynchronyFitzHugh-NagumoMorris-LecarHodgkin-HuxleyHindmarsh-Rose
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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).

Reading between the lines

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

  • 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.
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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

3 major / 7 minor

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.

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 (3)
  1. [Tables 2 and 4; Section 8.1] 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.
  2. [Section 3.1; Tables 1–4] 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.
  3. [Section 9] 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.
minor comments (7)
  1. [Sections 1.4 and 8.2] 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'.
  2. [Section 3.1] 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).
  3. [Definition 2.2] In item (a), the reference 'Equation (2.2)' should be 'Equation (2.3)', since the displayed transverse Floquet equation is labelled (2.3).
  4. [Table 1] 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.
  5. [Section 8.1] 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.
  6. [Sections 4.3, 5.2, 7.2] 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.
  7. [Various] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: previously proved parameter-free theorem plus direct Floquet-multiplier computations carry the argument.

full rationale

The paper's central claim is an existence/stability assertion backed by explicit computation. The load-bearing theoretical result, Theorem 2.3, is quoted from the authors' earlier paper [81]: the Floquet multipliers of a feedforward lift are the CPG multipliers plus the transverse multipliers, and the lift is stable iff the CPG orbit is Floquet stable and every transverse multiplier has modulus < 1. This is a parameter-free theorem whose stated assumptions concern the feedforward-lift construction, not the particular neuron models or the desired conclusion, so citing it is independent support rather than circularity. The paper then supplies the missing evidence for the present claim: for each of the four models it fixes explicit parameter values, simulates the 7-node feedforward chain, computes the monodromy matrix by the method of Section 3.1, and tabulates CPG and transverse Floquet multipliers (Tables 1-4). All tabulated transverse moduli are below 1, from which the 'stable' verdict follows by direct application of the criterion. No parameter is fitted to force the outcome, and no quantity called a prediction is a renamed input. The remaining weaknesses are evidentiary rather than circular: Section 1.4 admits that 'transverse stability on average' 'has not been made rigorous'; Section 3 notes the rigidity converse was proved only after repairing an overlooked technical condition; and Table 4 contains an apparent inconsistency (0.887 vs. absolute value 0.0887) that calls for verification, as do the absent integration tolerances and error estimates. These matters affect confidence in the numerical evidence, not the logical structure of the derivation.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on an established mathematical framework, a previously proved stability theorem, and hand-chosen parameter sets for four neuron models. No new physical entities are postulated. The main load-bearing ingredients beyond the imported theorems are the specific parameter choices and the accuracy of the numerical Floquet multiplier computations.

free parameters (4)
  • FitzHugh-Nagumo parameters and coupling (4.9)-(4.12) = a=0.05, b=2.5, gamma=0.3, c=0.4/-0.6/-0.8/-0.4, I=0/2
    Chosen by hand to produce synchronous or Z3 phase-shifted periodic states; no systematic parameter scan is reported.
  • Morris-Lecar parameters (5.16)-(5.17) = gCa=5, gK=8, gL=3, VCa=7, VK=-70, VL=50, v1=1, v2=1, v3=-10, v4=14.5, C=20 or 1, Iapp=295 or 300, a=-0.9, T0=5
    Parameter values are inherited from standard barnacle muscle examples and chosen to give oscillatory states with the intended phase pattern.
  • Hodgkin-Huxley parameters (6.18)-(6.19) = gK=10 or 100, gNa=20 or 50, gl=20, VK=-150 or 50, VNa=100 or 10, Vl=20, Cm=40, I=0 or -50, a=-1 or -1.3
    Chosen by hand to produce stable periodic states with the 1/3-period phase pattern; only two parameter sets are shown.
  • Hindmarsh-Rose parameters (7.21)-(7.23) = a=1, b=3 or 4, c=1 or 2, d=5, r=0.1 or 0.05, s=0 or 2, xR=-1.6, I=10 or 5, g=-1 or -2
    Standard spiking/bursting parameters adjusted by hand; three parameter sets are shown, one of which has a possible table transcription issue.
assumptions (4)
  • domain assumption The network dynamics formalism of admissible ODEs, balanced colorings, and quotient networks is valid.
    Invoked throughout Section 2, following [26, 79, 30]; it is the standard framework the paper builds on.
  • domain assumption Theorem 2.3 and Theorem 3.1 from [81] correctly characterize Floquet stability of feedforward lifts.
    The stability decomposition into CPG multipliers and transverse multipliers is imported from the authors' prior published work; the current paper does not reprove it.
  • standard math Floquet theory and the numerical computation of fundamental matrices via integration give accurate multipliers.
    Section 3.1 describes the method but supplies no error estimates or convergence checks for the reported multipliers.
  • standard math Hyperbolic periodic orbits persist under sufficiently small C1 perturbations, and this underpins the robustness discussion.
    Used in Section 8.1 to argue that forced symmetry-breaking leads to approximate synchrony; the quantitative robustness is only tested by simulation.

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Pith. "Pith review of Synchronous Propagation of Periodic Signals in Feedforward Networks of Standard Model Neurons." pith.science (2026). https://pith.science/paper/Q6OTWMG6

@misc{pith2026250611776,
  author       = {Pith},
  title        = {Pith review of: Synchronous Propagation of Periodic Signals in Feedforward Networks of Standard Model Neurons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q6OTWMG6}},
  note         = {Machine review of arXiv:2506.11776}
}
read the original abstract

Periodic signals propagating along chains are common in biology, for example in locomotion and peristalsis, and are also of interest for continuum robots. In previous work we constructed such networks as 'feedforward lifts' of a central pattern generator (CPG). When the CPG undergoes periodic oscillations, created by Hopf bifurcation or other mechanisms, it can then transmit periodic signals along one or more feedforward chains in a synchronous or phase-synchronous manner. We proved necessary and sufficient conditions for the stability of these lifted periodic orbits, in several senses. Here we examine the implications of the resulting theory for chains of neurons, using several standard neuron models: FitzHugh-Nagumo, Morris-Lecar, Hindmarsh-Rose, and Hodgkin-Huxley. We compare different notions of transverse stability, and summarize some numerical simulations showing that for all these neuron models the propagating signal can be transversely Floquet stable. Finally we discuss implications for less idealized models.

Figures

Figures reproduced from arXiv: 2506.11776 by the authors.

Figure 1
Figure 1. A 7-node network with one node-type and one-arrow type. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Left: Medicinal leech showing two linear series of heart chambers, one on each side, redrawn from [10]. Right: timing signal CPG for the leech heartbeat (shaded box) and propagating chains of neurons, redrawn from [48, Figure 4A,B], HE = heart excitatory motor neuron. HN = heart interneuron. All connections shown are inhibitory. Example 1.1. The leech Hirudo medicinalis has two series of heart chambers, one on each … view at source ↗
Figure 3
Figure 3. Z3 traveling wave periodic states in a 3-node ring, for four standard model neu￾rons. Each subfigure shows three superposed time series of corresponding node variables. Top left: FitzHugh–Nagumo. Top right: Morris–Lecar. Bottom left: Hodgkin–Huxley. Bottom right: Hindmarsh–Rose. For example, the CPG of [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (17 more)
Figure 1
Figure 1. Figure 1: This 7-node network is a feedforward chain with a single feedback connection [PITH_FULL_IMAGE:figures/full_fig_p015_1.png]
Figure 4
Figure 4. Figure 4: Synchronous periodic state. Time range = [50 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: shows a Z3 traveling wave periodic state. Parameters are: I = 0 a = 0.05 b = 2.5 γ = 0.3 c = −0.6 (4.10) 55 60 65 70 -0.2 -0.1 0.1 0.2 55 60 65 70 -0.2 -0.1 0.1 0.2 55 60 65 70 -0.2 -0.1 0.1 0.2 55 60 65 70 -0.2 -0.1 0.1 0.2 55 60 65 70 -0.2 -0.1 0.1 0.2 55 60 65 70 -0…
Figure 6
Figure 6. Figure 6: Cases where real parts of eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: For Floquet stable periodic states, real parts of eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Morris–Lecar model. Left: Superposed time series for Vc where 1 ≤ c ≤ 7. Right: Superposed time series for Wc where 1 ≤ c ≤ 7. Parameter values are (5.16). 55 60 65 70 75 -1 1 2 3 50 55 60 65 70 75 0.75 0.80 0.85 0.90 [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Morris–Lecar model. Left: Superposed time series for Vc where 1 ≤ c ≤ 7. Right: Superposed time series for Wc where 1 ≤ c ≤ 7. Parameter values are (5.17). In contrast, [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Morris–Lecar model. Left: Real parts of eigenvalues of Dx1 f on the periodic orbit x(t). Right: Imaginary parts of eigenvalues of Dx1 f on the periodic orbit x(t). Parameter values are (5.16). 55 60 65 70 75 -10 -8 -6 -4 -2 0 2 4 55 60 65 70 75 -4 -2 2 4 [PITH_FULL_I…
Figure 11
Figure 11. Figure 11: Morris–Lecar model. Left: Real parts of eigenvalues of Dx1 f on the periodic orbit x(t). Right: Imaginary parts of eigenvalues of Dx1 f on the periodic orbit x(t). Parameter values are (5.17). It is also possible to find parameter values giving approximately sinusoida…
Figure 12
Figure 12. Figure 12: Hodgkin–Huxley. Superposed time series for [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Hodgkin–Huxley. Real parts of eigenvalues of D [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Hodgkin–Huxley. Parameter values as in (6.19). [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: Three simulations of the 7-node feedforward chain of Figure 1 with [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: Transverse eigenvalues for the 7-node feedforward chain of Figure 1 with [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: Traveling wave periodic states in the 7-node FitzHugh–Nagumo feedforward [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]
Figure 18
Figure 18. Figure 18: Graphs showing (top left) the effect of altering the parameter b in node 4 on the amplitude of node 4, and (top right) phase difference between nodes 4 and 1. Bottom: the feedforward network extended to 30 nodes, whose time series are superposed, when parameter b is m…
Figure 19
Figure 19. Figure 19: Graph showing the effect on the amplitude of node 4 ( [PITH_FULL_IMAGE:figures/full_fig_p032_19.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.