Pith. sign in

REVIEW 1 major objections 51 references

n:m Phase-Locking of Coupled Oscillators with Nonlinearities in Coupling Strength and Heterogeneity

T0 review · 1 major / 0 minor · reviewed 2026-05-23 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read A scalar reduction method predicts n:m phase-locking when both coupling strength and oscillator heterogeneity are nonlinear.

desk verdict The paper gives a scalar reduction for n:m locking under nonlinear heterogeneity and coupling, but the weak-perturbation premise undercuts the claim that small heterogeneity produces large shifts. read the letter →

arxiv 2409.14566 v4 submitted 2024-09-22 q-bio.NC

Youngmin Park This is my paper · ORCID
classification q-bio.NC
keywords phase-lockingcoupledoscillatorsheterogeneitynonlinearcouplingscalarreductionbiologicalneuralmodelsphasedifference
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

The paper introduces a reduction that collapses high-dimensional coupled oscillator systems to a single scalar equation for the phase difference. This equation tracks the birth and death of n:m locked states as nonlinear coupling strength and nonlinear heterogeneity vary. The approach applies to concrete biological models such as the Van der Pol oscillator and thalamic neurons, and it shows that even modest heterogeneity produces locked-state changes invisible to identical-oscillator approximations. The reduction therefore lets researchers analyze synchronization in more realistic, heterogeneous settings without solving the full system.

What carries the argument

The scalar reduction to an equation governing the phase difference, which folds nonlinear heterogeneity directly into the reduced dynamics.

What would settle it

Numerical integration of the full thalamic or Van der Pol system at a chosen nonlinear coupling strength and heterogeneity level, followed by checking whether the observed n:m locked intervals match the intervals predicted by the scalar equation.

Watch

Extended reading notes

Core claim

The scalar reduction determines the existence and stability of n:m phase-locked states by incorporating nonlinearities in both coupling strength and heterogeneity, which is treated as a weak nonlinear perturbation of the vector field, and it correctly reproduces the emergence and disappearance of these states across several biological oscillator models.

Load-bearing premise

Heterogeneity is formulated as a relatively weak but nonlinear alteration of the vector field(s).

Editorial extensions

If this is right

  • The method applies directly to the nonradial isochron clock, thalamic neural oscillator, and Van der Pol oscillator.
  • Even small nonlinear heterogeneity produces phase-locked states absent from identical-oscillator models.
  • High-dimensional systems of coupled biological oscillators become amenable to stability analysis in more realistic parameter regimes.
  • Locked-state boundaries can be traced explicitly as functions of both nonlinear coupling strength and nonlinear heterogeneity.

Reading between the lines

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

  • The same reduction could be tested on small networks of three or more oscillators provided the heterogeneity remains weak and localized.
  • Models of neural population rhythms that currently assume identical cells may need re-examination once nonlinear cell-to-cell differences are included.
  • The technique might extend to forced systems with time-varying inputs if the forcing can be absorbed into the effective coupling term.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The paper introduces a scalar reduction method for forced or coupled oscillator systems that incorporates nonlinearities in both coupling strength and heterogeneity, where heterogeneity is modeled as a relatively weak nonlinear alteration of the vector field. The method is applied to three models—the nonradial isochron clock, a thalamic neural oscillator, and the Van der Pol oscillator—to determine existence and stability of n:m phase-locked states. The central claim is that the reduction captures the emergence and disappearance of these states as functions of the nonlinear parameters and that even small heterogeneity produces significant alterations to phase-locked states that cannot be captured by assuming identical oscillators.

Significance. If the reduction is valid in the claimed regimes, the approach would provide a practical tool for analyzing phase-locking in high-dimensional biological oscillator networks under more realistic heterogeneous conditions, extending beyond the identical-oscillator limit that is common in the literature.

major comments (1)
  1. [scalar reduction derivation and application to biological models] The derivation of the scalar reduction (described in the abstract as relying on relatively weak nonlinear heterogeneity) is load-bearing for the central claim that the method accurately predicts both the n:m locking boundaries and the qualitative difference from the identical-oscillator case. The abstract provides no indication that higher-order terms were retained or that non-perturbative numerical validation was performed in regimes where heterogeneity strength produces O(1) shifts in locking boundaries; this creates a potential inconsistency between the perturbative premise and the reported biological examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comment regarding the scalar reduction derivation and its application below.

read point-by-point responses

  1. Referee: [scalar reduction derivation and application to biological models] The derivation of the scalar reduction (described in the abstract as relying on relatively weak nonlinear heterogeneity) is load-bearing for the central claim that the method accurately predicts both the n:m locking boundaries and the qualitative difference from the identical-oscillator case. The abstract provides no indication that higher-order terms were retained or that non-perturbative numerical validation was performed in regimes where heterogeneity strength produces O(1) shifts in locking boundaries; this creates a potential inconsistency between the perturbative premise and the reported biological examples.

    Authors: The scalar reduction is derived perturbatively to leading order in the heterogeneity strength, but incorporates the full nonlinear dependence on the heterogeneity and coupling parameters within that order. This allows small heterogeneity amplitudes to induce O(1) changes in the locking boundaries due to the nonlinear terms. In the manuscript, we validate the reduction by direct numerical comparison of the phase-locking predictions from the scalar equation against simulations of the full high-dimensional models for each of the three biological examples. These comparisons confirm the accuracy of the boundaries even in regimes with significant shifts. We agree that the abstract could more clearly indicate the presence of this numerical validation and will revise the abstract to include a brief mention of the validation performed. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: scalar reduction derived from stated assumptions without reduction to inputs

full rationale

The paper introduces a scalar reduction for forced/coupled oscillators under the explicit premise of relatively weak nonlinear heterogeneity. The abstract and description present this as a derived method applied to concrete models (nonradial isochron clock, thalamic oscillator, Van der Pol), with claims about phase-locked states following from that application. No equations or steps are shown to be self-definitional, fitted parameters renamed as predictions, or dependent on self-citations whose content reduces to the present result. The derivation chain remains independent of the reported outcomes.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Review limited to abstract; no explicit free parameters, invented entities, or additional axioms beyond the stated formulation of heterogeneity could be identified.

assumptions (1)
  • domain assumption Heterogeneity can be formulated as a relatively weak but nonlinear alteration of the vector field(s)
    Directly stated in the abstract as the basis for the reduction method.

how reviews work

0 comments
Cite this review

Pith. "Pith review of n:m Phase-Locking of Coupled Oscillators with Nonlinearities in Coupling Strength and Heterogeneity." pith.science (2026). https://pith.science/paper/2409.14566

@misc{pith2026240914566,
  author       = {Pith},
  title        = {Pith review of: n:m Phase-Locking of Coupled Oscillators with Nonlinearities in Coupling Strength and Heterogeneity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2409.14566}},
  note         = {Machine review of arXiv:2409.14566}
}
abstract

We introduce a scalar reduction method for forced or coupled systems with nonlinearities in both heterogeneity and coupling strength. Heterogeneity is formulated as a relatively weak but nonlinear alteration of the vector field(s). The method can be used to determine the existence and stability of $n{:}m$ phase-locked states in a variety of forced or coupled biological oscillator models, including the nonradial isochron clock, a thalamic neural oscillator, and the Van der Pol oscillator. The proposed scalar reduction successfully captures the emergence and disappearance of phase-locked states as a function of nonlinear coupling strength and nonlinear heterogeneity. We find that even small amounts of heterogeneity can significantly alter phase-locked states in ways that cannot be captured by assuming identical oscillators. The proposed method enables a reduction and analysis of high-dimensional systems of coupled oscillators in more biologically realistic settings.

Figures

Figures reproduced from arXiv: 2409.14566 by the authors.

Figure 1
Figure 1. The forcing function (29) as a function of [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Frequency-locking and drift in a forced nonradial isochron clock. All panels use the same horizontal axis, [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. One-parameter bifurcation diagrams of the forced nonradial isochron clock as a function of coupling strength [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Two-parameter bifurcation diagrams (Arnold tongues) of the forced nonradial isochron clock. The vertical [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Frequency-locking and drift in a forced thalamic neuron. All panels use the same horizontal axis, [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: One-parameter bifurcation diagrams of a forced thalamic neuron as a function of coupling strength [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: The diagrams were computed numerically using XPPAUTO [8] using Fourier [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 7
Figure 7. Figure 7: Two-parameter bifurcation diagrams (Arnold tongues) of a forced thalamic neuron. The vertical axis [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Representative trajectories of the scalar reduction (red dashed for order [PITH_FULL_IMAGE:figures/full_fig_p041_8.png]
Figure 9
Figure 9. Figure 9: One-parameter bifurcation diagrams for 1:1 phase-locking in a pair of thalamic neurons given varying levels [PITH_FULL_IMAGE:figures/full_fig_p042_9.png]
Figure 10
Figure 10. Figure 10: One-parameter bifurcation diagrams for n:m phase-locking in a pair of thalamic neurons given varying levels of heterogeneity. All panels show the existence of fixed points in the O(ε) reduction (red), O(ε 2 ) reduction (blue), and full model (black). Solid (dashed) cu…
Figure 11
Figure 11. Figure 11: One-parameter bifurcation diagrams for n:m phase-locking in a pair of thalamic neurons given varying levels of heterogeneity. All panels show the existence of fixed points in the O(ε) reduction (red), O(ε 2 ) reduction (blue), and full model (black). Solid (dashed) cu…
Figure 12
Figure 12. Figure 12: Two-parameter bifurcation diagrams of the reduced coupled thalamic model for [PITH_FULL_IMAGE:figures/full_fig_p044_12.png]
Figure 13
Figure 13. Figure 13: Original H- and J -functions (black) vs the corresponding Fourier approximation (blue) for 1:1 phase￾locking. 0 φ 2π/n 0.22 0.24 0.26 H X A 1:2, O(ε) 0 φ 2π/n −6.25 −6.00 −5.75 bJX B 1:2, O(ε) 0 φ 2π/n 0.00 0.02 H Y C1:2, O(ε 2) 0 φ 2π/n −0.5 0.0 bJX D1:2, O(ε 2) 0 φ …
Figure 14
Figure 14. Figure 14: Original H- and J -functions (black) vs the corresponding Fourier approximation (blue) for 1:2 phase￾locking. 0 φ 2π/n 0.2 0.4 H X A 2:1, O(ε) 0 φ 2π/n −6.25 −6.00 −5.75 bJX B 2:1, O(ε) 0 φ 2π/n 0.0 0.1 H Y C2:1, O(ε 2) 0 φ 2π/n −1 0 1 bJX D2:1, O(ε 2) 0 φ 2π/n −625 −…
Figure 15
Figure 15. Figure 15: Original H- and J -functions (black) vs the corresponding Fourier approximation (blue) for 2:1 phase￾locking. 45 [PITH_FULL_IMAGE:figures/full_fig_p045_15.png]
Figure 16
Figure 16. Figure 16: Original H- and J -functions (black) vs the corresponding Fourier approximation (blue) for 1:3 phase￾locking. 0 φ 2π/n 0.1 0.2 0.3 H X A 3:1, O(ε) 0 φ 2π/n −6.25 −6.00 −5.75 bJX B 3:1, O(ε) 0 φ 2π/n −0.05 0.00 0.05 H Y C3:1, O(ε 2) 0 φ 2π/n −1 0 bJX D3:1, O(ε 2) 0 φ 2…
Figure 17
Figure 17. Figure 17: Original H- and J -functions (black) vs the corresponding Fourier approximation (blue) for 3:1 phase￾locking. 0 φ 2π/n 0.22 0.24 H X A 2:3, O(ε) 0 φ 2π/n −6.25 −6.00 −5.75 bJX B 2:3, O(ε) 0 φ 2π/n 0.00 0.01 H Y C2:3, O(ε 2) 0 φ 2π/n −0.50 −0.25 0.00 bJX D2:3, O(ε 2) 0…
Figure 18
Figure 18. Figure 18: Original H- and J -functions (black) vs the corresponding Fourier approximation (blue) for 2:3 phase￾locking. 46 [PITH_FULL_IMAGE:figures/full_fig_p046_18.png]
Figure 19
Figure 19. Figure 19: Original H- and J -functions (black) vs the corresponding Fourier approximation (blue) for 3:2 phase￾locking. 0 φ 2π/n 0.23 0.24 H X A 3:4, O(ε) 0 φ 2π/n −6.25 −6.00 −5.75 bJX B 3:4, O(ε) 0 φ 2π/n −0.005 0.000 0.005 H Y C3:4, O(ε 2) 0 φ 2π/n −0.2 0.0 bJX D3:4, O(ε 2) …
Figure 20
Figure 20. Figure 20: Original H- and J -functions (black) vs the corresponding Fourier approximation (blue) for 3:4 phase￾locking. 0 φ 2π/n 0.23 0.24 H X A 4:3, O(ε) 0 φ 2π/n −6.25 −6.00 −5.75 bJX B 4:3, O(ε) 0 φ 2π/n 0.0025 0.0050 0.0075 H Y C4:3, O(ε 2) 0 φ 2π/n −0.2 0.0 bJX D4:3, O(ε 2…
Figure 21
Figure 21. Figure 21: Original H- and J -functions (black) vs the corresponding Fourier approximation (blue) for 4:3 phase￾locking. 47 [PITH_FULL_IMAGE:figures/full_fig_p047_21.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    Adhikari, J

    S. Adhikari, J. G. Restrepo, and P. S. Skardal , Synchronization of phase oscillators on complex hypergraphs , Chaos: An Interdisciplinary Journal of Non- linear Science, 33 (2023)

    work page 2023

  2. [2]

    Bhaskar, B

    A. Bhaskar, B. Shayak, A. T. Zehnder, and R. H. Rand , Integer ra- tio self-synchronization in pairs of limit cycle oscillators , in International De- sign Engineering Technical Conferences and Computers and Information in En- gineering Conference, vol. 85482, American Society of Mechanical Engineers, 2021, p. V011T11A010

    work page 2021

  3. [3]

    R. J. Butera Jr, J. Rinzel, and J. C. Smith , Models of respiratory rhythm generation in the pre-B¨ otzinger complex. ii. populations of coupled pacemaker neu- rons, Journal of Neurophysiology, 82 (1999), pp. 398–415

    work page 1999

  4. [4]

    C. C. Canavier, F. G. Kazanci, and A. A. Prinz , Phase resetting curves allow for simple and accurate prediction of robust N:1 phase locking for strongly coupled neural oscillators, Biophysical Journal, 97 (2009), pp. 59–73

    work page 2009

  5. [5]

    J. Chen, K. Wong, H. Zheng, and J. Shuai , Phase signal coupling induced n: m phase synchronization in drive-response oscillators , Physical Review E, 63 (2001), p. 036214

    work page 2001

  6. [6]

    I. R. Epstein and J. A. Pojman, An introduction to nonlinear chemical dynam- ics: oscillations, waves, patterns, and chaos , Oxford University Press, 1998

    work page 1998

  7. [7]

    G. B. Ermentrout , n:m phase-locking of weakly coupled oscillators , J. Math. Biol., 12 (1981), pp. 327–342, https://doi.org/10.1007/BF00276920

    work page doi:10.1007/bf00276920 1981

  8. [8]

    G. B. Ermentrout, Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students , vol. 14, SIAM, 2002

    work page 2002

Show all 51 references
  1. [9]

    G. B. Ermentrout and D. H. Terman , Mathematical foundations of neuro- science, vol. 35 of Interdisciplinary Applied Mathematics, Springer, New York, 2010, https://doi.org/10.1007/978-0-387-87708-2

    2010 doi

  2. [10]

    Glass and R

    L. Glass and R. Perez, Fine structure of phase locking, Physical Review Letters, 48 (1982), p. 1772. 37

    1982

  3. [11]

    E. E. Goldwyn and A. Hastings, Small heterogeneity has large effects on syn- chronization of ecological oscillators, Bulletin of Mathematical Biology, 71 (2009), pp. 130–144

    2009

  4. [12]

    Golubitsky and I

    M. Golubitsky and I. Stewart , The symmetry perspective: from equilibrium to chaos in phase space and physical space , vol. 200, Springer Science & Business Media, 2003

    2003

  5. [13]

    Horv ´ath, D

    V. Horv ´ath, D. J. Kutner, M. D. Zeng, and I. R. Epstein , Phase- frequency model of strongly pulse-coupled Belousov-Zhabotinsky oscillators , Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), p. 023128, https://doi.org/10.1063/1.5082161, https://doi.org/10...

    2019 doi

  6. [14]

    Hunter, M

    I. Hunter, M. M. Norton, B. Chen, C. Simonetti, M. E. Mous- taka, J. Touboul, and S. Fraden , Pattern formation in a four-ring reaction-diffusion network with heterogeneity , Physical Review E, 105 (2022), p. 024310, https://doi.org/10.1103/PhysRevE.105.024310, https://link. a...

    2022 doi

  7. [15]

    E. M. Izhikevich, Dynamical systems in neuroscience: the geometry of excitabil- ity and bursting , Computational Neuroscience, MIT Press, Cambridge, MA, 2007

    2007

  8. [16]

    Kuramoto, Chemical oscillations, waves, and turbulence , vol

    Y. Kuramoto, Chemical oscillations, waves, and turbulence , vol. 19 of Springer Series in Synergetics, Springer-Verlag, Berlin, 1984, https://doi.org/10.1007/ 978-3-642-69689-3

    1984

  9. [17]

    Lengler, F

    J. Lengler, F. Jug, and A. Steger, Reliable neuronal systems: the importance of heterogeneity, PloS one, 8 (2013), p. e80694

    2013

  10. [18]

    Llibre, D

    J. Llibre, D. D. Novaes, and M. A. Teixeira , Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), pp. 563– 583, https://doi.org/10.1088/0951-7715/27/3/563

    2014 doi

  11. [19]

    Maggia, S

    M. Maggia, S. A. Eisa, and H. E. Taha , On higher-order averaging of time- periodic systems: reconciliation of two averaging techniques , Nonlinear Dynamics, 99 (2020), pp. 813–836

    2020

  12. [20]

    J. R. Magnus and H. Neudecker, Matrix differential calculus with applications in statistics and econometrics , John Wiley & Sons, 2019

    2019

  13. [21]

    Marder, S

    E. Marder, S. Kedia, and E. O. Morozova, New insights from small rhythmic circuits, Current Opinion in Neurobiology, 76 (2022), p. 102610

    2022

  14. [22]

    E. T. Mau, M. Rosenblum, and A. Pikovsky , High-order phase reduction for coupled 2d oscillators , Chaos: An Interdisciplinary Journal of Nonlinear Science, 33 (2023)

    2023

  15. [23]

    Mauroy, I

    A. Mauroy, I. Mezi ´c, and J. Moehlis , Isostables, isochrons, and Koopman spectrum for the action–angle representation of stable fixed point dynamics , Phys. D, 261 (2013), pp. 19–30

    2013

  16. [24]

    McKinley, M

    J. McKinley, M. Zhang, A. Wead, C. Williams, E. Tognoli, and C. Bee- tle, Third party stabilization of unstable coordination in systems of coupled oscil- lators, in Journal of Physics: Conference Series, vol. 2090, IOP Publishing, 2021, p. 012167

  17. [25]

    Mezi ´c, Spectrum of the koopman operator, spectral expansions in functional spaces, and state-space geometry, J

    I. Mezi ´c, Spectrum of the koopman operator, spectral expansions in functional spaces, and state-space geometry, J. Nonlinear. Sci., 30 (2020), pp. 2091–2145. 38

    2020

  18. [26]

    Milne and F

    R. Milne and F. Guichard, Coupled phase-amplitude dynamics in heterogeneous metacommunities, Journal of Theoretical Biology, 523 (2021), p. 110676

    2021

  19. [27]

    R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), pp. 1645–1662, https://doi.org/ 10.1137/0150098

    1990 doi

  20. [28]

    Nicks, R

    R. Nicks, R. Allen, and S. Coombes , Insights into oscillator network dy- namics using a phase-isostable framework , Chaos: An Interdisciplinary Journal of Nonlinear Science, 34 (2024)

    2024

  21. [29]

    M. M. Norton, N. Tompkins, B. Blanc, M. C. Cambria, J. Held, and S. Fraden, Dynamics of reaction-diffusion oscillators in star and other networks with cyclic symmetries exhibiting multiple clusters , Physical review letters, 123 (2019), p. 148301

    2019

  22. [30]

    Ott and T

    E. Ott and T. M. Antonsen , Low dimensional behavior of large systems of globally coupled oscillators , Chaos, 18 (2008), p. 037113

    2008

  23. [31]

    Park and G

    Y. Park and G. B. Ermentrout, Weakly coupled oscillators in a slowly varying world, J. Journal of Computational Neuroscience, 40 (2016), pp. 269–281, https: //doi.org/10.1007/s10827-016-0596-6

    2016 doi

  24. [32]

    Y. Park, S. Heitmann, and G. B. Ermentrout, The utility of phase models in studying neural synchronization, in Computational Models of Brain and Behavior, A. A. Moustafa, ed., Wiley Online Library, 2017, ch. 36, pp. 493–504

    2017

  25. [33]

    Park and D

    Y. Park and D. Wilson, High-order accuracy computation of coupling functions for strongly coupled oscillators , SIAM Journal on Applied Dynamical Systems, 20 (2021), pp. 1464–1484

    2021

  26. [34]

    Park and D

    Y. Park and D. Wilson, N-body oscillator interactions of higher-order coupling functions, SIAM Journal on Applied Dynamical Systems, 23 (2024), pp. 1471–1503

    2024

  27. [35]

    P´erez-Cervera, T

    A. P´erez-Cervera, T. M. Seara, and G. Huguet , Global phase-amplitude description of oscillatory dynamics via the parameterization method, arXiv preprint arXiv:2004.03647, (2020)

    2004

  28. [36]

    Rosenblum and A

    M. Rosenblum and A. Pikovsky , Numerical phase reduction beyond the first order approximation, Chaos, 29 (2019), p. 011105

    2019

  29. [37]

    J. E. Rubin and D. Terman , High frequency stimulation of the subthalamic nu- cleus eliminates pathological thalamic rhythmicity in a computational model , Jour- nal of Computational Neuroscience, 16 (2004), pp. 211–235

    2004

  30. [38]

    M. A. Schwemmer and T. J. Lewis, The theory of weakly coupled oscillators, in Phase response curves in neuroscience: theory, experiment, and analysis, Springer, 2011, pp. 3–31

    2011

  31. [39]

    Shirasaka, W

    S. Shirasaka, W. Kurebayashi, and H. Nakao, Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems , Chaos, 27 (2017), p. 023119

    2017

  32. [40]

    R. B. Stein, E. R. Gossen, and K. E. Jones , Neuronal variability: noise or part of the signal? , Nature Reviews Neuroscience, 6 (2005), pp. 389–397

    2005

  33. [41]

    S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and E. Ott , Crowd synchrony on the millennium bridge , Nature, 438 (2005), pp. 43–44

    2005

  34. [42]

    Urban and S

    N. Urban and S. Tripathy , Circuits drive cell diversity , Nature, 488 (2012), pp. 289–290. 39

    2012

  35. [43]

    K. C. A. Wedgwood, K. K. Lin, R. Thul, and S. Coombes, Phase-amplitude descriptions of neural oscillator models , J. Math. Neurosci., 3 (2013), p. 2

    2013

  36. [44]

    D. Wilson. Personal communication

  37. [45]

    Wilson, Phase-amplitude reduction far beyond the weakly perturbed paradigm , Physical Review E, 101 (2020), p

    D. Wilson, Phase-amplitude reduction far beyond the weakly perturbed paradigm , Physical Review E, 101 (2020), p. 022220

    2020

  38. [46]

    Wilson and G

    D. Wilson and G. B. Ermentrout , Greater accuracy and broadened appli- cability of phase reduction using isostable coordinates , J. Math. Biol., 76 (2018), pp. 37–66, https://doi.org/10.1007/s00285-017-1141-6

    2018 doi

  39. [47]

    Wilson and G

    D. Wilson and G. B. Ermentrout , An operational definition of phase char- acterizes the transient response of perturbed limit cycle oscillators , SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 2516–2543

    2018

  40. [48]

    Wilson and G

    D. Wilson and G. B. Ermentrout, Phase models beyond weak coupling, Phys. Rev. Lett., 123 (2019), p. 164101

    2019

  41. [49]

    Wilson and J

    D. Wilson and J. Moehlis, Isostable reduction of periodic orbits, Phys. Rev. E, 94 (2016), p. 052213

    2016

  42. [50]

    A. T. Winfree , The geometry of biological time , vol. 12, Springer Science & Business Media, 2001

    2001

  43. [51]

    Zang and E

    Y. Zang and E. Marder , Neuronal morphology enhances robustness to pertur- bations of channel densities, Proceedings of the National Academy of Sciences, 120 (2023), p. e2219049120. 40 □0.02 0.00 H 1:1, b = 0.025A O(ε) O(ε2) 0 2πφ 0 500 1000 1500 2000 t 0.000 0.025 H 1:2, b = ...

    2023

Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.