arxiv: 2605.08901 · v1 · submitted 2026-05-09 · 🌊 nlin.AO

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Extreme forcing and wave dynamics in weakly nonlocally coupled excitable FitzHugh-Nagumo systems

A.V. Bukh, N.I. Semenova, V.V. Semenov

Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:54 UTC · model grok-4.3

classification 🌊 nlin.AO

keywords FitzHugh-Nagumotraveling wavesLévy noiseperiodic forcingsynchronizationcollective dynamicsexcitable systemscounter-propagating waves

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The pith

Periodic forcing creates regular synchronized traveling waves in coupled excitable systems, while Lévy noise produces counter-propagating waves that become random walks as intensity increases.

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

The paper examines how three kinds of extreme external forcing affect the collective wave behavior in a network of weakly coupled excitable oscillators modeled by FitzHugh-Nagumo equations. Periodic pulses lead to regions of synchronization in parameter space where the system shows orderly dynamics, including multiple waves or mixed states of partial sync and wave motion. Lévy noise, however, disrupts this order and favors waves moving in opposite directions that evolve into erratic random motion with stronger noise. This classification helps map how external extremes can steer collective patterns in such systems. A sympathetic reader would care because these findings clarify control mechanisms for wave propagation in excitable media like biological tissues or chemical reactions.

Core claim

Under periodic Gaussian forcing the system develops synchronization tongues featuring highly regular collective dynamics that can support multiple traveling waves or the coexistence of partial synchronization with ongoing wave propagation. Lévy noise instead eliminates regular patterns and establishes counter-propagating waves whose behavior shifts toward random walking as the noise intensity grows. The work maps these regimes across parameter space for each forcing type.

What carries the argument

Weakly nonlocally coupled FitzHugh-Nagumo oscillators driven by extreme external forcing, whose collective traveling-wave states are classified by forcing type and parameters.

If this is right

Periodic forcing produces synchronization tongues with highly regular collective dynamics.
Periodic forcing may induce multiple traveling waves or coexistence of partial synchronization with wave propagation.
Lévy noise suppresses regular behavior.
Lévy noise generates a regime of counter-propagating waves.
Increasing Lévy noise intensity transitions the system to random walking dynamics.

Where Pith is reading between the lines

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

Controlling the type and intensity of extreme forcing could steer wave directionality and coherence in real excitable systems such as neural or cardiac tissues.
Similar transitions might appear in other oscillator networks when subjected to heavy-tailed noise instead of Gaussian perturbations.
Experiments could test whether adjusting noise parameters alone switches between ordered and disordered wave regimes without changing coupling.
These results imply that noise statistics act as a tunable parameter for pattern selection in extended excitable media.

Load-bearing premise

Numerical integration of the coupled equations faithfully reproduces the dynamical regimes without artifacts from time-stepping, spatial discretization, or boundary conditions.

What would settle it

Simulations with high-intensity Lévy noise showing sustained regular waves or synchronized behavior instead of counter-propagating waves and random walks would contradict the central claim.

Figures

Figures reproduced from arXiv: 2605.08901 by A.V. Bukh, N.I. Semenova, V.V. Semenov.

**Figure 2.** Figure 2: FIG. 2. Regime diagram on the parameter plane [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Regime diagram on the parameter plane [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Regime diagram on the parameter plane [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

read the original abstract

The influence of extreme external forcing on traveling-wave dynamics in an ensemble of weakly nonlocally coupled excitable FitzHugh--Nagumo systems is studied. Three types of external exposure are considered: periodic Gaussian pulses, periodic pulses modulated by Gaussian white noise, and L\'evy noise with tunable distribution parameters. Periodic forcing produces synchronization tongues with highly regular collective dynamics and may induce multiple traveling waves or coexistence of partial synchronization with wave propagation. In contrast, L\'evy noise suppresses regular behavior and generates a regime of counter-propagating waves, which with increasing intensity transitions to random walking dynamics. The study provides a comprehensive classification of the observed dynamical regimes and presents their organization in parameter space for different types of extreme external forcing.

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.

Desk Editor's Note private letter to a colleague

The paper maps synchronization tongues and wave regimes in weakly nonlocal FHN systems under Gaussian pulses and Lévy noise, but the numerics lack reported convergence checks.

read the letter

This work classifies how extreme forcings reshape collective behavior in an ensemble of weakly nonlocally coupled FitzHugh-Nagumo oscillators. Periodic Gaussian pulses produce synchronization tongues with regular dynamics and can support multiple traveling waves or partial sync coexisting with propagation. Lévy noise, by contrast, disrupts regularity and drives a transition from counter-propagating waves to random-walk-like motion as intensity rises. The main contribution is the organized regime diagram in parameter space for these three forcing types, which extends earlier studies on coupled excitable media by adding tunable Lévy statistics and noise-modulated pulses. The authors do a service by laying out the observed transitions clearly for this specific coupling strength and model. That said, the claims rest entirely on numerical integration of the nonlocal integral term. Weak nonlocality requires discretizing the coupling kernel over a finite range, and Lévy jumps introduce large excursions that can interact badly with time-step size or boundary conditions. No convergence tests, dt-refinement studies, or spatial-resolution checks are described in the abstract or framing, so it is hard to rule out discretization artifacts mimicking the reported wave speeds or randomness. The work is a solid numerical survey for specialists in nonlinear dynamics of excitable media who already work with FHN or similar oscillator lattices. It does not claim broad theoretical advance, so it will mainly interest readers already running similar simulations. I would send it to peer review rather than desk-reject, provided the methods section supplies the missing integration details and at least basic convergence evidence; the regime map itself is useful reference material once the numerics are secured.

Referee Report

2 major / 2 minor

Summary. The manuscript numerically examines the influence of three extreme external forcings—periodic Gaussian pulses, periodic pulses modulated by Gaussian white noise, and Lévy noise with tunable parameters—on traveling-wave dynamics in an ensemble of weakly nonlocally coupled excitable FitzHugh-Nagumo oscillators. It reports that periodic forcing generates synchronization tongues featuring highly regular collective behavior and can produce multiple traveling waves or coexistence of partial synchronization with wave propagation. In contrast, Lévy noise suppresses regularity, inducing a regime of counter-propagating waves that transitions to random-walking dynamics with increasing noise intensity. The work provides a classification of these regimes and their organization in parameter space.

Significance. If the observed regimes prove robust under numerical refinement, the classification distinguishes the effects of periodic versus heavy-tailed forcings on coherence and wave propagation in coupled excitable media. This could inform models of biological systems (e.g., neural or cardiac tissue) subject to extreme perturbations and supply concrete parameter-space maps for further analytic or experimental work.

major comments (2)

[Numerical Methods] Numerical Methods section: No convergence tests are reported for the discretization of the weakly nonlocal coupling integral, the choice of time step, spatial grid resolution, or truncation of Lévy jumps. Because the central claims rest on specific wave speeds, counter-propagation, and transitions to random walking, it is essential to demonstrate that these features persist under dt-refinement and increased spatial points; otherwise discretization or boundary artifacts could artifactually produce the reported regimes.
[Results] Results and classification (e.g., around the parameter-space diagrams): Regime boundaries (synchronization tongues, counter-propagating waves, random walking) appear defined by visual inspection of space-time plots or order-parameter thresholds. Quantitative, reproducible diagnostics—such as averaged wave-speed distributions, two-point correlation functions, or Lyapunov exponents—should be supplied to make the classification objective and to allow readers to assess sensitivity to the precise thresholds used.

minor comments (2)

[Figures] Figure captions should explicitly state the exact parameter values (coupling strength, forcing amplitude, Lévy index, etc.) and integration details used for each panel to facilitate reproducibility.
[Abstract and Introduction] The abstract and introduction could more clearly distinguish the three forcing types by their mathematical definitions (e.g., explicit forms of the Lévy stable distribution) rather than descriptive labels alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive overall assessment of our work. We address the two major points below and will incorporate revisions to strengthen the numerical validation and objectivity of the regime classification.

read point-by-point responses

Referee: [Numerical Methods] Numerical Methods section: No convergence tests are reported for the discretization of the weakly nonlocal coupling integral, the choice of time step, spatial grid resolution, or truncation of Lévy jumps. Because the central claims rest on specific wave speeds, counter-propagation, and transitions to random walking, it is essential to demonstrate that these features persist under dt-refinement and increased spatial points; otherwise discretization or boundary artifacts could artifactually produce the reported regimes.

Authors: We agree that explicit convergence tests are necessary to substantiate the robustness of the reported wave dynamics. In the revised manuscript we will add a dedicated subsection (or appendix) presenting results from systematic refinement of the time step dt, spatial grid size, and the cutoff for Lévy jumps. These tests will compare key observables—wave speeds, directionality of propagation, and the transition to random-walk behavior—across the refined discretizations to confirm that the identified regimes are not discretization artifacts. revision: yes
Referee: [Results] Results and classification (e.g., around the parameter-space diagrams): Regime boundaries (synchronization tongues, counter-propagating waves, random walking) appear defined by visual inspection of space-time plots or order-parameter thresholds. Quantitative, reproducible diagnostics—such as averaged wave-speed distributions, two-point correlation functions, or Lyapunov exponents—should be supplied to make the classification objective and to allow readers to assess sensitivity to the precise thresholds used.

Authors: We accept that the current classification relies heavily on visual inspection supplemented by order-parameter thresholds. To improve objectivity and reproducibility we will augment the revised manuscript with quantitative diagnostics, specifically averaged wave-speed distributions and two-point spatial correlation functions, computed across multiple realizations. These metrics will be used to delineate regime boundaries more rigorously and to quantify their sensitivity to threshold choices. While full Lyapunov spectra are computationally demanding for the system sizes considered, we will explore adding representative values where feasible. revision: yes

Circularity Check

0 steps flagged

No circularity: purely numerical classification of observed regimes

full rationale

The paper performs direct numerical integration of the weakly nonlocally coupled FitzHugh-Nagumo system under three classes of external forcing and reports the resulting dynamical regimes (synchronization tongues, traveling waves, counter-propagating waves, random walking) as observed outcomes. No analytic derivation, parameter fitting, or first-principles prediction is claimed; the classification is extracted from simulation trajectories. No self-citations, ansatzes, or uniqueness theorems appear in the load-bearing steps, and the model equations are integrated as given without any reduction of outputs back to fitted inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5431 in / 1102 out tokens · 43504 ms · 2026-05-12T00:54:28.476883+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
The model was integrated using the Heun’s method ... Lévy noise ... Janicki–Weron algorithm ... regime diagrams on (A,T), (D,T), (s,α) planes
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
weak nonlocal coupling ... r=0.006 ... σ=0.2 ... no mention of cost functions or golden-ratio identities

Reference graph

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