Emergence of long-term rhythmicity within a frustrated triangle oscillator-network

Hiroshi Kawakami; Hiroshi Ueno; Masatomo Matsushima; Yoshiki Kamiya

arxiv: 1907.05487 · v1 · pith:5YD3CWVJnew · submitted 2019-07-04 · 🌊 nlin.AO

Emergence of long-term rhythmicity within a frustrated triangle oscillator-network

Masatomo Matsushima , Hiroshi Ueno , Yoshiki Kamiya , Hiroshi Kawakami This is my paper

Pith reviewed 2026-05-25 02:26 UTC · model grok-4.3

classification 🌊 nlin.AO

keywords oscillator networkmultistabilityelectronic fireflyfrustrated trianglerhythmicitynonlinear dynamicscoupled oscillators

0 comments

The pith

A triangle network of electronic firefly oscillators produces long-term rhythmicity and multiple stable states.

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

The paper builds a physical network of three coupled electronic firefly oscillators arranged in a triangle to model interactions among brain cells. In this frustrated configuration the circuits generate persistent rhythmic patterns that continue over long times. The observed behavior is reproduced by simple mathematical models that display multiple coexisting stable states. The work therefore links network geometry and frustration directly to the appearance of sustained rhythms.

Core claim

Within a frustrated triangle of electronic firefly oscillators, long-term rhythmicity emerges and the system exhibits multiple stability that is captured by elementary mathematical models.

What carries the argument

The frustrated triangle oscillator-network realized with electronic firefly circuits, which produces multistability and sustained rhythms.

If this is right

Frustration arising from the closed triangle topology is sufficient to generate persistent rhythmic activity.
Simple differential-equation models already contain the multistability seen in the hardware.
The same circuit platform can be used to explore how network geometry shapes collective rhythms.
Multiple stable states imply that the system can switch between different rhythmic patterns depending on initial conditions.

Where Pith is reading between the lines

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

Similar frustration effects may contribute to rhythm generation in real neural circuits when three or more cells form closed loops.
Scaling the triangle to larger networks could reveal whether multistability persists or gives way to more complex collective states.
The hardware model offers a low-cost testbed for checking whether specific coupling strengths or delays destroy the long-term rhythms.

Load-bearing premise

The electronic firefly circuit and its mathematical model accurately capture the essential dynamics of a brain cell network.

What would settle it

Recordings from the physical circuit that fail to show multiple coexisting long-term rhythms, or a mathematical model that cannot reproduce the observed states, would falsify the central claim.

Figures

Figures reproduced from arXiv: 1907.05487 by Hiroshi Kawakami, Hiroshi Ueno, Masatomo Matsushima, Yoshiki Kamiya.

**Figure 1.** Figure 1: Circuit arrangement example These oscillators change the behavior of vibration by receiving a stimulus. Therefore, due to the arrangement of the oscillators, the strength of the stimulus received by the oscillators changes, and it is possible to see many synchronized patterns of light. In the next section, we explain the circuit used for the oscillator. 3 Experimental system We introduce the electric circu… view at source ↗

**Figure 2.** Figure 2: Actual oscillator (left), Circuit Diagram (right) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Calculation of phase difference θ = Y T ∗ 2π, φ = R T ∗ 2π. (1) These show the synchronization by drawing the phase plane of the phase difference θ and φ. We now show the experimental results. First, Fig.4 shows the phase difference of automatically [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: The phase difference of natural vibrations [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: A:12cm, B:12cm, C:12cm (Regular Triangle) [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 7.** Figure 7: λ1,2,3=1 (The distribution of energy U and Vector) [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 9.** Figure 9: λ1,2 = 1, λ3 = 0.5 (The distribution of energy U and Vector) [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗

read the original abstract

This study tries to simulate a brain cell network using an electric circuit oscillator called electronic firefly. Multiple stability was observed in the electric circuit oscillator which is expressed by simple mathematical models.

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 reports a hardware observation of multiple stable rhythms in a three-node electronic firefly circuit, but the abstract supplies almost no evidence that this differs from prior coupled-oscillator work.

read the letter

The central observation is that a small triangle of electronic firefly oscillators shows multiple long-term rhythms under frustration. The authors model the circuit with simple equations and present it as a brain-network analog. That hardware demonstration is the concrete piece of work here; everything else follows from it. The circuit itself appears to be a known platform, so the contribution rests on whether the triangle configuration and the reported stability patterns are new or just a straightforward extension. The abstract gives no equations, no parameter values, and no comparison to earlier studies on the same or similar oscillators, which makes it impossible to judge the size of the step. The brain-cell framing is mentioned but not developed; it functions more as motivation than as a tested claim. On the positive side, a physical realization of even a small frustrated network can be useful for people who want to check numerical predictions against real components without building large simulations. The soft spots are the lack of visible literature grounding and the absence of any quantitative link between the circuit data and the claimed brain relevance. If the full paper contains clear time-series data, parameter values, and a direct comparison to existing models, those gaps could be filled. If not, the result stays at the level of an incremental circuit observation. This is the kind of paper that might interest a narrow group working on small-scale analog oscillator networks or hardware implementations of coupled systems. A reader already familiar with firefly circuits or frustrated oscillator triangles would get the most out of the methods section. It is coherent enough on its own terms to deserve referee time rather than an immediate desk reject, mainly because hardware results are still relatively rare and can be checked independently once the details are supplied.

Referee Report

1 major / 0 minor

Summary. The manuscript describes an experimental setup using an electronic firefly circuit oscillator arranged as a frustrated triangle network to simulate aspects of a brain cell network. It reports the observation of multiple stability and the emergence of long-term rhythmicity, which the authors state can be expressed using simple mathematical models.

Significance. An experimental observation of multiple stability in a simple frustrated oscillator circuit could provide a useful physical analog for studying multistable dynamics in networks. However, with no equations, data, circuit parameters, or model derivations provided, it is impossible to determine whether the claimed observations are supported or reproducible, limiting any assessment of significance.

major comments (1)

[Abstract] Abstract: The central claim that 'multiple stability was observed in the electric circuit oscillator which is expressed by simple mathematical models' is stated without any supporting equations, data, circuit description, or model details. This absence makes it impossible to verify whether the models support the observations or to assess the soundness of the experimental result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and comments on our manuscript. We address the major comment point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that 'multiple stability was observed in the electric circuit oscillator which is expressed by simple mathematical models' is stated without any supporting equations, data, circuit description, or model details. This absence makes it impossible to verify whether the models support the observations or to assess the soundness of the experimental result.

Authors: We agree that the provided abstract is brief and does not include equations, data, circuit parameters, or model derivations. The manuscript text as presented is limited to a short description without these supporting elements, which prevents verification of the claims. We will revise the manuscript to add the circuit description, mathematical models, and relevant experimental details or data. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical observation with simple models

full rationale

The paper reports an experimental observation of multiple stability in an electronic firefly circuit oscillator, expressed via simple mathematical models, framed as a brain-network simulation. No derivation chain, fitted parameters renamed as predictions, self-citations as load-bearing premises, or ansatz smuggling is described or visible in the abstract or provided context. The central claim rests on circuit behavior data rather than any self-referential reduction of a result to its inputs. This is the expected non-finding for an observational study without visible theoretical derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone.

pith-pipeline@v0.9.0 · 5547 in / 935 out tokens · 19514 ms · 2026-05-25T02:26:26.671198+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

U = λ1 cos(θ) + λ2 cos(φ) + λ3 cos(φ-θ) ... stable at the lowest energy U
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

three electric circuit oscillators ... anti-phase synchronization ... multistability

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

33 extracted references

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[1] [1]

Maruoka, K.Kubota, R

H. Maruoka, K.Kubota, R. Kurokawa, et al. Periodic organization of a major subtype of pyramidal neurons in neocortical layer v. J. Neurosci, 3117-11(31(50)):18522–42, 2011

2011

[2] [2]

Lattice system of functionally distinct cell types in the neocortex

Hisato Maruoka, Nao Nakagawa, Shun Tsuruno, Seiichiro Sakai, Taisuke Yoneda, and Toshihiko Hosoya. Lattice system of functionally distinct cell types in the neocortex. Science, (358(6363)):610–615, 2017

2017

[3] [3]

Sebastian, M

A. Sebastian, M. L. Gallo, G. W. Burr, S. Kim, M. BrightSky, and E. Eleftheriou. Tutorial: Brain-inspired computing using phase-change memory devices. Journal of Applied Physics, 124(111101), 2018

2018

[4] [4]

G. Dion, S. Mejaouri, and J. Sylvestre. Reservoir computing with a single delay- coupled non-linear mechanical oscillator. Journal of Applied Physics , 124(152132), 2018

2018

[5] [5]

Vodenicarevic, N

D. Vodenicarevic, N. Locatelli, J. Grollier, and D. Querlioz. Nano-oscillator-based classiﬁcation with a machine learning-compatible architecture. Journal of Applied Physics, 124(152117), 2018

2018

[6] [6]

Popescu, G

B. Popescu, G. Csaba, D. Popescu, A. H. Fallahpour, P. Lugli, W. Porod, and M. Becherer. Simulation of coupled spin torque oscillators for pattern recognition. Journal of Applied Physics , 124(152128), 2018

2018

[7] [7]

D. E. Nikonov, G. Csaba, W. Porod, T. Shibata, D. Voils, D. Hammerstrom, I. A. Young, and G. I. Bourianoﬀ. Coupled-oscillator associative memory array operation for pattern recognition. IEEE Journal on Exploratory Solid-State Computational Devices and Circuits, 1(15675486):85–93, 2015

2015

[8] [8]

Csaba, A

G. Csaba, A. Raychowdhury, S. Datta, and W. Porod. Computing with coupled oscillators: Theory, devices, and applications. 2018 IEEE ISCAS , (18228738), 2018. 7

2018

[9] [9]

Ziegler, C

M. Ziegler, C. Wenger, E. Chicca, and H. Kohlstedt. Tutorial: Concepts for closely mimicking biological learning with memristive devices: Principles to emulate cellular forms of learning. Journal of Applied Physics , 124(152003), 2018

2018

[10] [10]

Brunner, B

D. Brunner, B. Penkovsky, B. A. Marquez, M. Jacquot, I. Fischer, and L. Larger. Tutorial: Photonic neural networks in delay systems. Journal of Applied Physics , 124(152004), 2018

2018

[11] [11]

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2018

[12] [12]

J. C. Coulombe, M. C. A. York, and J. Sylvestre. Computing with networks of nonlinear mechanical oscillators. PLOS ONE, 12(6)(e0178663), 2017

2017

[13] [13]

Prokopenko, I

O.Sulymenko, O. Prokopenko, I. Lisenkov, J. ˚Akerman, V. Tyberkevych, A. N. Slavin, and R. Khymyn. Ultra-fast logic devices using artiﬁcial “neurons” based on antiferromagnetic pulse generators. Journal of Applied Physics , 124(152115), 2018

2018

[14] [14]

A. A. Oumate, S. Vaidyanathan, K. Zourmba, B. Gambo, and A. Mohamadou. Analysis, circuit design and synchronization of a new hyperchaotic system with three quadratic nonlinearities. Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors, pages 251–270, 2018

2018

[15] [15]

Synchronization of electric ﬁreﬂies by us- ing square wave generators

T.Kousaka, H.Kawakami, and T.Ueta. Synchronization of electric ﬁreﬂies by us- ing square wave generators. IEICE TRANS. FUNDAMENTALS, E81-4(4):656–663, 1998

1998

[16] [16]

Wu amd N

Y. Wu amd N. Wang, L. Li, and J. Xiao. Anti-phase synchronization of two coupled mechanical metronomes. Chaos, 22(023146), 2012

2012

[17] [17]

Astakhov, A

S. Astakhov, A. Gulai, N. Fujiwara, and J. Kurths. The role of asymmetrical and repulsive coupling in the dynamics of two coupled van der pol oscillators. Chaos, 26(023102), 2016

2016

[18] [18]

J. Jia, Z. Shangguan, H. Li, Y. Wu, W. Liu, J. Xiao, and J. Kurths. Experimental and modeling analysis of asymmetrical on-oﬀ oscillation in coupled non- identical inverted bottle oscillators. Chaos, 26(116301), 2016

2016

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Aonishi, K

T. Aonishi, K. Kurata, and M. Okada. Statistical mechanics of an oscillator associa- tive memory with scattered natural frequencies. Phys. Rev. Let., 82(13), 1999

1999

[20] [20]

Giver, Z

M. Giver, Z. Jabeen, and B. Chakraborty. Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions. Phys. Rev., E 83(046206), 2011

2011

[21] [21]

Levnajic

Z. Levnajic. Emergent multistability and frustration in phase-repulsive networks of oscillators. Phys. Rev., E 84(016231), 2011

2011

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Kralemann, A

B. Kralemann, A. Pikovsky, and M. Rosenblum. Reconstructing phase dynamics of oscillator networks. Chaos, 21(025104), 2011

2011

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cognitive

V. K. Vanag. “cognitive” modes in small networks of almost identical chemical oscillators with pulsatile inhibitory coupling. Chaos, 29(033106), 2019. 8

2019

[24] [24]

Z. G. Esfahani, L. L. Gollo, and A. Valizadeh. Stimulus-dependent synchronization in delayed-coupled neuronal networks. Scientiﬁc Reports, 6(23471), 2016

2016

[25] [25]

Y. Liu, M. Sebek, F. Mori, and I. Z. Kiss. Synchronization of three electrochemical oscillators: From local to global coupling. Chaos, 28(045104), 2018

2018

[26] [26]

Farcot, S

E. Farcot, S. Best, R. Edwards, I. Belgacem, X. Xu, and P. Gill. Chaos in a ring circuit. Chaos, 29(043103), 2019

2019

[27] [27]

A. B. C. Buskermolen, Hamsini Suresh, S. S. Shishvan, A. Vigliotti, A. DeSimone, N. A. Kurniawan, C. V. C. Bouten, and V. S. Deshpande. Entropic forces drive cellular contact guidance. Biophys. Journal, 116:1994–2008, 2019

1994

[28] [28]

Goldstein, M

D. Goldstein, M. Giver, and B. Chakraborty. Synchronization patterns in geometri- cally frustrated rings of relaxation oscillators. Chaos, 25(123109), 2015

2015

[29] [29]

M. H. Matheny, J. Emenheiser, W. Fon, A. Chapman, A. Salova, M. Rohden, J. Li, M. H. de Badyn, M. P´ osfai, L. D.-Osorio, M. Mesbahi, J. P. Crutchﬁeld, M. C. Cross, R. M. D’Souza, and M. L. Roukes. Exotic states in a simple network of nanoelectromechanical oscillators. Science, 363(6431), 2019

2019

[30] [30]

Self-entrainment of a population of coupled non-linear oscillators

Yoshiki Kuramoto. Self-entrainment of a population of coupled non-linear oscillators. International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics , 39:420–422, 1975

1975

[31] [31]

Chandra, M

S. Chandra, M. Girvan, and E. Ott. Continuous versus discontinuous transitions in the d -dimensional generalized kuramoto model: Odd d is diﬀerent. Phys. Rev., X 9(011002), 2019

2019

[32] [32]

Superslow relaxation in identical phase oscillators with random and frus- trated interactions

H.Daido. Superslow relaxation in identical phase oscillators with random and frus- trated interactions. Chaos, 28(045102), 2018

2018

[33] [33]

Barr´ e and D

J. Barr´ e and D. M´ etivier. Bifurcations and singularities for coupled oscillators with inertia and frustration. Phys. Rev. Let., 117(214102), 2016. 9

2016