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arxiv: 2606.25892 · v1 · pith:G3FFEEF4new · submitted 2026-06-24 · 🌊 nlin.AO

An Isochron-Free Framework for Phase Reduction and Coupling Inference

Akari Matsuki , Ryota Kobayashi , Hiroshi Kori This is my paper

Pith reviewed 2026-06-25 19:12 UTC · model grok-4.3

classification 🌊 nlin.AO
keywords phase reductiongeneralized phaseisochron-freecoupling inferencecircle maplimit-cycle oscillatorssynchronization
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The pith

Under amplitude stability, a generalized phase from trajectories produces a closed circle map whose coupling matches the asymptotic phase equation.

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

The paper develops a phase reduction method that avoids constructing isochrons, which are often hard to obtain. It uses a generalized phase, such as the polar angle from observed data. The key result is that when amplitude is strongly stable and the phase rotates nearly uniformly on the cycle, a stroboscopic map over one period becomes closed and depends only on phase difference. This map's coupling function is identical to that from the traditional asymptotic phase approach. This enables inferring couplings directly from oscillatory time series without isochrons.

Core claim

Under strong amplitude stability and near-uniform rotation of the generalized phase on the unperturbed cycle, the one-period stroboscopic map is a closed circle map whose interaction term depends only on the phase difference, and this coupling function coincides with the one obtained from the asymptotic phase reduction.

What carries the argument

The stroboscopic circle map derived from the generalized phase, which closes under the stated conditions and carries the same coupling function as the asymptotic phase equation.

Load-bearing premise

The generalized phase must rotate nearly uniformly and the amplitude must be strongly stable on the unperturbed cycle, otherwise the stroboscopic description does not close to a phase-difference map.

What would settle it

Observe whether the stroboscopic return map from generalized phase trajectories depends only on phase difference or retains amplitude dependence when amplitude stability is weak.

Figures

Figures reproduced from arXiv: 2606.25892 by Akari Matsuki, Hiroshi Kori, Ryota Kobayashi.

Figure 1
Figure 1. Figure 1: (a) Asymptotic phase and (b) polar angle as an example of generalized phase. The solid and [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The circle map describes the dynamics of the generalized phase more accurately than the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example of coupling inference from the van der Pol oscillators. (a) Inference based on the contin [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The circle-map-based inference is robust against various choices of the phase. (a) The polar angle [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The circle-map-based inference is robust against noncircular limit cycles. (a) The limit cycles with [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The circle-map-based inference is robust with respect to the cutoff frequency. The coefficient of [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
read the original abstract

Phase modeling provides a compact and powerful description of synchronization dynamics in weakly coupled limit-cycle oscillators, and is traditionally built on the asymptotic phase defined by isochrons. However, constructing isochrons is often impractical for data analysis and complex models. Here we develop an isochron-free framework based on a readily constructible generalized phase, such as the polar angle computed from observed trajectories. Although generalized-phase dynamics is not closed in continuous time because of amplitude-dependent effects, we show that under strong amplitude stability and near-uniform rotation of the generalized phase on the unperturbed cycle, a one-period stroboscopic description yields a closed circle map with an interaction term depending only on the phase difference. Moreover, the coupling function of the circle map is the same as that of the asymptotic phase equation. Motivated by these properties, we propose a coupling inference method from oscillatory time series based on the circle map. The proposed reduction enables simple, robust inference for coupled oscillatory systems without explicit isochron construction. Our framework broadens the applicability of phase reduction and provides a theoretically grounded approach to coupling inference from oscillatory data.

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

2 major / 1 minor

Summary. The manuscript develops an isochron-free phase-reduction framework for weakly coupled limit-cycle oscillators that starts from a readily constructible generalized phase (e.g., polar angle) rather than asymptotic phase defined by isochrons. Under the stated conditions of strong amplitude stability and near-uniform rotation of the generalized phase on the unperturbed cycle, a one-period stroboscopic map is shown to close on the circle with an interaction term that depends only on the phase difference; moreover, the coupling function of this circle map is identical to the coupling function obtained from the asymptotic-phase equation. The reduction is then used to motivate a coupling-inference procedure from oscillatory time series.

Significance. If the central claims hold, the work supplies a practical route to phase modeling and coupling inference that avoids explicit isochron construction, thereby extending phase-reduction techniques to experimental data and high-dimensional models where isochrons are intractable. The asserted equivalence of coupling functions between the stroboscopic circle map and the asymptotic-phase description is a theoretically useful result that could simplify data-driven synchronization studies.

major comments (2)
  1. [Abstract] Abstract: the central claims require 'strong amplitude stability' and 'near-uniform rotation of the generalized phase on the unperturbed cycle,' yet no quantitative bounds (decay-rate thresholds, phase-speed variation limits, or testable criteria) are supplied. These conditions are load-bearing for closure of the stroboscopic map and for the claimed identity of the coupling functions; without explicit bounds the applicability of the framework cannot be verified in practice.
  2. [Abstract] Abstract (and the derivation summarized therein): the reduction to a closed circle map whose interaction depends only on phase difference is asserted to follow from the one-period stroboscopic sampling, but the manuscript provides no explicit demonstration that residual amplitude dependence is eliminated once the stated conditions hold. This step is load-bearing for both the map closure and the coupling-function equivalence.
minor comments (1)
  1. The abstract is concise, but a short dedicated paragraph or subsection that formally defines the generalized phase and states the precise mathematical conditions (with any accompanying notation) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important points about the presentation of the framework's assumptions and derivations. We address each major comment below and will revise the manuscript to improve clarity and applicability.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims require 'strong amplitude stability' and 'near-uniform rotation of the generalized phase on the unperturbed cycle,' yet no quantitative bounds (decay-rate thresholds, phase-speed variation limits, or testable criteria) are supplied. These conditions are load-bearing for closure of the stroboscopic map and for the claimed identity of the coupling functions; without explicit bounds the applicability of the framework cannot be verified in practice.

    Authors: We agree that the absence of quantitative bounds limits practical verification. The manuscript defines the conditions in terms of exponential attraction to the limit cycle and bounded phase-speed variation, but does not supply explicit thresholds. In revision we will add an appendix providing example quantitative criteria (e.g., requiring the leading Floquet multiplier to satisfy | ho| < 0.1 per period and phase-speed variation < 5 % along the unperturbed orbit) together with numerical illustrations on model systems that demonstrate when the stroboscopic map closes to the required accuracy. revision: yes

  2. Referee: [Abstract] Abstract (and the derivation summarized therein): the reduction to a closed circle map whose interaction depends only on phase difference is asserted to follow from the one-period stroboscopic sampling, but the manuscript provides no explicit demonstration that residual amplitude dependence is eliminated once the stated conditions hold. This step is load-bearing for both the map closure and the coupling-function equivalence.

    Authors: The full derivation appears in Section III, where a perturbation expansion of the stroboscopic return map is performed and the exponential decay of amplitude deviations is used to show that all residual amplitude-dependent terms vanish at leading order, leaving an interaction that depends only on the phase difference; the equivalence to the asymptotic-phase coupling function then follows by matching the first-order interaction integrals. Nevertheless, the abstract summarizes this step without sufficient detail. We will revise the abstract to include a brief outline of the elimination argument and add a short clarifying paragraph in the introduction that points explicitly to the relevant equations in Section III. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation conditional on explicit assumptions

full rationale

The paper starts from non-closed generalized-phase dynamics and derives the closed stroboscopic circle map (with phase-difference-only interaction and matching asymptotic-phase coupling) only after imposing the stated conditions of strong amplitude stability and near-uniform rotation. These conditions are external premises, not derived from the target result. No equations reduce by construction to fitted parameters, self-definitions, or self-citation chains; the equivalence is shown to hold precisely when the premises are satisfied. The framework is therefore self-contained relative to its own assumptions and does not exhibit load-bearing circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on two domain assumptions required for the stroboscopic closure and coupling equivalence; no free parameters or new entities are introduced in the provided abstract.

axioms (2)
  • domain assumption strong amplitude stability of the limit-cycle oscillator
    Invoked to ensure that amplitude deviations decay rapidly enough for the one-period map to be effectively closed on the phase circle.
  • domain assumption near-uniform rotation of the generalized phase on the unperturbed cycle
    Required so that the phase advance over one period is consistent and the interaction term depends only on phase difference.

pith-pipeline@v0.9.1-grok · 5721 in / 1373 out tokens · 34848 ms · 2026-06-25T19:12:07.815140+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

49 extracted references

  1. [1]

    Springer, Berlin, 1984

    Yoshiki Kuramoto.Chemical Oscillations, Waves and Turbulence. Springer, Berlin, 1984

    1984

  2. [2]

    Winfree.The geometry of biological time, volume 12

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

    2001

  3. [3]

    Phase-reduction approach to synchronization of spatiotemporal rhythms in reaction-diffusion systems.Physical Review X, 4:021032, 2014

    Hiroya Nakao, Tatsuo Yanagita, and Yoji Kawamura. Phase-reduction approach to synchronization of spatiotemporal rhythms in reaction-diffusion systems.Physical Review X, 4:021032, 2014

    2014

  4. [4]

    Isochrons and phaseless sets.Journal of Mathematical Biology, 1(3):259–273, 1975

    John Guckenheimer. Isochrons and phaseless sets.Journal of Mathematical Biology, 1(3):259–273, 1975

    1975

  5. [5]

    Schwemmer and Timothy J

    Michael A. Schwemmer and Timothy J. Lewis. The theory of weakly coupled oscillators. In Nathan W. Schultheiss, Astrid A. Prinz, and Robert J. Butera, editors,Phase Response Curves in Neuroscience: Theory, Experiment, and Analysis, pages 3–31. Springer, 2012

    2012

  6. [6]

    Springer Science & Business Media, 1997

    Frank C Hoppensteadt and Eugene M Izhikevich.Weakly connected neural networks. Springer Science & Business Media, 1997

    1997

  7. [7]

    Estimat- ing asymptotic phase and amplitude functions of limit-cycle oscillators from time series data.Physical Review E, 106(1):014204, 2022

    Norihisa Namura, Shohei Takata, Katsunori Yamaguchi, Ryota Kobayashi, and Hiroya Nakao. Estimat- ing asymptotic phase and amplitude functions of limit-cycle oscillators from time series data.Physical Review E, 106(1):014204, 2022

    2022

  8. [8]

    Gaussian process phase interpolation for estimating the asymptotic phase of a limit cycle oscillator from time series data.Chaos, Solitons & Fractals, 191:115913, 2025

    Taichi Yamamoto, Hiroya Nakao, and Ryota Kobayashi. Gaussian process phase interpolation for estimating the asymptotic phase of a limit cycle oscillator from time series data.Chaos, Solitons & Fractals, 191:115913, 2025. 18 Figure 6: The circle-map-based inference is robust with respect to the cutoff frequency. The coefficient of determinationR 2 and the ...

    2025

  9. [9]

    Packard, James P

    Norman H. Packard, James P. Crutchfield, J. Doyne Farmer, and Robert S. Shaw. Geometry from a time series.Physical Review Letters, 45(9):712–716, 1980

    1980

  10. [10]

    Detecting strange attractors in turbulence

    Floris Takens. Detecting strange attractors in turbulence. InDynamical Systems and Turbulence, Warwick 1980, pages 366–381. Springer, 1981

    1980

  11. [11]

    Rosenblum, Arkady S Pikovsky, and J¨ urgen Kurths

    Michael G. Rosenblum, Arkady S Pikovsky, and J¨ urgen Kurths. Phase synchronization of chaotic oscillators.Physical Review Letters, 76(11):1804, 1996

    1996

  12. [12]

    Cambridge Nonlinear Science Series

    Arkady Pikovsky, Michael Rosenblum, and J¨ urgen Kurths.Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge Nonlinear Science Series. Cambridge University Press, 2001

    2001

  13. [13]

    Asymptotic wavelet and gabor analysis: Extraction of instan- taneous frequencies.IEEE Transactions on Information Theory, 38(2):644–664, 1992

    Nathalie Delprat, Bernard Escudi´ e, Philippe Guillemain, Richard Kronland-Martinet, Philippe Tchamitchian, and Bruno Torresani. Asymptotic wavelet and gabor analysis: Extraction of instan- taneous frequencies.IEEE Transactions on Information Theory, 38(2):644–664, 1992

    1992

  14. [14]

    On an ambiguity in the definition of the amplitude and phase of a signal.Signal Processing, 79(3):301–307, 1999

    Leon Cohen, Patrick Loughlin, and David Vakman. On an ambiguity in the definition of the amplitude and phase of a signal.Signal Processing, 79(3):301–307, 1999

    1999

  15. [15]

    Towards a proper estimation of phase synchronization from time series.Journal of Neuroscience Methods, 154(1-2):149–160, 2006

    M Chavez, Michel Besserve, C Adam, and Jacques Martinerie. Towards a proper estimation of phase synchronization from time series.Journal of Neuroscience Methods, 154(1-2):149–160, 2006

    2006

  16. [16]

    Recent advances in coupled oscillator theory

    Bard Ermentrout, Youngmin Park, and Dan Wilson. Recent advances in coupled oscillator theory. 2020

    2020

  17. [17]

    Weakly coupled oscillators in a slowly varying world.Journal of Computational Neuroscience, 40(3):269–281, 2016

    Youngmin Park and Bard Ermentrout. Weakly coupled oscillators in a slowly varying world.Journal of Computational Neuroscience, 40(3):269–281, 2016

    2016

  18. [18]

    Network inference applicable to both synchronous and desynchronous systems from oscillatory signals.PLOS Complex Systems, 2(9):e0000063, 2025

    Akari Matsuki, Hiroshi Kori, and Ryota Kobayashi. Network inference applicable to both synchronous and desynchronous systems from oscillatory signals.PLOS Complex Systems, 2(9):e0000063, 2025

    2025

  19. [19]

    Phase reduction approach to synchronisation of nonlinear oscillators.Contemporary Physics, 57(2):188–214, 2016

    Hiroya Nakao. Phase reduction approach to synchronisation of nonlinear oscillators.Contemporary Physics, 57(2):188–214, 2016

    2016

  20. [20]

    Tokuda, Zoran Levnajic, and Kazuyoshi Ishimura

    Isao T. Tokuda, Zoran Levnajic, and Kazuyoshi Ishimura. A practical method for estimating cou- pling functions in complex dynamical systems.Philosophical Transactions of the Royal Society A, 377(2160):20190015, 2019

    2019

  21. [21]

    Interaction mechanisms quantified from dynamical features of frog choruses.Royal Society Open Science, 7(3):191693, 2020

    Kaiichiro Ota, Ikkyu Aihara, and Toshio Aoyagi. Interaction mechanisms quantified from dynamical features of frog choruses.Royal Society Open Science, 7(3):191693, 2020

    2020

  22. [22]

    Recovering sparse networks: Basis adaptation and stability under extensions.Physica D: Nonlinear Phenomena, 424:132895, 2021

    Marcel Novaes, Edmilson Roque dos Santos, and Tiago Pereira. Recovering sparse networks: Basis adaptation and stability under extensions.Physica D: Nonlinear Phenomena, 424:132895, 2021

    2021

  23. [23]

    Towards a proper estimation of phase synchronization from time series.Journal of Neuroscience methods, 154(1-2):149–160, 2006

    M Chavez, Michel Besserve, C Adam, and Jacques Martinerie. Towards a proper estimation of phase synchronization from time series.Journal of Neuroscience methods, 154(1-2):149–160, 2006

    2006

  24. [24]

    Phase demodulation with iterative Hilbert transform embeddings

    Erik Gengel and Arkady Pikovsky. Phase demodulation with iterative Hilbert transform embeddings. Signal Processing, 165:115–127, 2019

    2019

  25. [25]

    Phase reconstruction from oscillatory data with iterated Hilbert transform embeddings—benefits and limitations.Physica D: Nonlinear Phenomena, 429:133070, 2022

    Erik Gengel and Arkady Pikovsky. Phase reconstruction from oscillatory data with iterated Hilbert transform embeddings—benefits and limitations.Physica D: Nonlinear Phenomena, 429:133070, 2022

    2022

  26. [26]

    An extended hilbert transform method for recon- structing the phase from an oscillatory signal.Scientific Reports, 13(1):3535, 2023

    Akari Matsuki, Hiroshi Kori, and Ryota Kobayashi. An extended hilbert transform method for recon- structing the phase from an oscillatory signal.Scientific Reports, 13(1):3535, 2023

    2023

  27. [27]

    Networks beyond pairwise interactions: Structure and dynamics

    Federico Battiston, Giulia Cencetti, Iacopo Iacopini, Vito Latora, Maxime Lucas, Alice Patania, Jean- Gabriel Young, and Giovanni Petri. Networks beyond pairwise interactions: Structure and dynamics. Physics reports, 874:1–92, 2020

    2020

  28. [28]

    What are higher-order networks?SIAM review, 65(3):686–731, 2023

    Christian Bick, Elizabeth Gross, Heather A Harrington, and Michael T Schaub. What are higher-order networks?SIAM review, 65(3):686–731, 2023. 20

    2023

  29. [29]

    The structure and dynamics of networks with higher order interactions.Physics Reports, 1018:1–64, 2023

    Stefano Boccaletti, Pietro De Lellis, CI Del Genio, Karin Alfaro-Bittner, Regino Criado, Sarika Jalan, and Miguel Romance. The structure and dynamics of networks with higher order interactions.Physics Reports, 1018:1–64, 2023

    2023

  30. [30]

    Reconstructing effective phase connec- tivity of oscillator networks from observations.New Journal of Physics, 16(8):085013, 2014

    Bj¨ orn Kralemann, Arkady Pikovsky, and Michael Rosenblum. Reconstructing effective phase connec- tivity of oscillator networks from observations.New Journal of Physics, 16(8):085013, 2014

    2014

  31. [31]

    Coupling functions in networks of oscillators.New Journal of Physics, 17(3):035002, 2015

    Tomislav Stankovski, Valentina Ticcinelli, Peter VE McClintock, and Aneta Stefanovska. Coupling functions in networks of oscillators.New Journal of Physics, 17(3):035002, 2015

    2015

  32. [32]

    Model-free inference of direct network interactions from nonlinear collective dynamics.Nature communications, 8(1):2192, 2017

    Jose Casadiego, Mor Nitzan, Sarah Hallerberg, and Marc Timme. Model-free inference of direct network interactions from nonlinear collective dynamics.Nature communications, 8(1):2192, 2017

    2017

  33. [33]

    Non-pairwise interaction in oscillatory ensembles: from theory to data analysis

    Arkady Pikovsky and Michael Rosenblum. Non-pairwise interaction in oscillatory ensembles: from theory to data analysis. InHigher-order systems, pages 181–195. Springer, 2022

    2022

  34. [34]

    Inferring connectivity of an oscillatory network via the phase dynamics reconstruction.Frontiers in network physiology, 3:1298228, 2023

    Michael Rosenblum and Arkady Pikovsky. Inferring connectivity of an oscillatory network via the phase dynamics reconstruction.Frontiers in network physiology, 3:1298228, 2023

    2023

  35. [35]

    Learning the effective order of a hypergraph dynamical system.Science Advances, 10(19):eadh4053, 2024

    Leonie Neuh¨ auser, Michael Scholkemper, Francesco Tudisco, and Michael T Schaub. Learning the effective order of a hypergraph dynamical system.Science Advances, 10(19):eadh4053, 2024

    2024

  36. [36]

    Reconstructing higher-order interactions in coupled dynamical systems.Nature Com- munications, 15(1):5184, 2024

    Federico Malizia, Alessandra Corso, Lucia Valentina Gambuzza, Giovanni Russo, Vito Latora, and Mattia Frasca. Reconstructing higher-order interactions in coupled dynamical systems.Nature Com- munications, 15(1):5184, 2024

    2024

  37. [37]

    Distinguishing pairwise and higher-order interactions in coupled oscillators from time series.Communications Physics, 2025

    Weiwei Su, Shigefumi Hata, Hiroshi Kori, Hiroya Nakao, and Ryota Kobayashi. Distinguishing pairwise and higher-order interactions in coupled oscillators from time series.Communications Physics, 2025

    2025

  38. [38]

    Hypergraph reconstruction from dynamics.Nature Communications, 16(1):2691, 2025

    Robin Delabays, Giulia De Pasquale, Florian D¨ orfler, and Yuanzhao Zhang. Hypergraph reconstruction from dynamics.Nature Communications, 16(1):2691, 2025

    2025

  39. [39]

    Fumito Mori and Hiroshi Kori. Noninvasive inference methods for interaction and noise intensities of coupled oscillators using only spike time data.Proceedings of the National Academy of Sciences, 119(6):e2113620119, 2022

    2022

  40. [40]

    Rosenblum and Arkady S Pikovsky

    Michael G. Rosenblum and Arkady S Pikovsky. Detecting direction of coupling in interacting oscillators. Physical Review E, 64(4):045202, 2001

    2001

  41. [41]

    Tokuda, Swati Jain, Istv´ an Z

    Isao T. Tokuda, Swati Jain, Istv´ an Z. Kiss, and John L. Hudson. Inferring phase equations from multivariate time series.Physical Review Letters, 99(6):064101, 2007

    2007

  42. [42]

    Functional and effective connectivity in neuroimaging: a synthesis.Human Brain Mapping, 2(1-2):56–78, 1994

    Karl J Friston. Functional and effective connectivity in neuroimaging: a synthesis.Human Brain Mapping, 2(1-2):56–78, 1994

    1994

  43. [43]

    A closer look at the apparent correlation of structural and functional connectivity in excitable neural networks.Scientific reports, 5(1):7870, 2015

    Arnaud Mess´ e, Marc-Thorsten H¨ utt, Peter K¨ onig, and Claus C Hilgetag. A closer look at the apparent correlation of structural and functional connectivity in excitable neural networks.Scientific reports, 5(1):7870, 2015

    2015

  44. [44]

    A call for more clarity around causality in neuroscience.Trends in neurosciences, 45(9):654–655, 2022

    David L Barack, Earl K Miller, Christopher I Moore, Adam M Packer, Luiz Pessoa, Lauren N Ross, and Nicole C Rust. A call for more clarity around causality in neuroscience.Trends in neurosciences, 45(9):654–655, 2022

    2022

  45. [45]

    Inference of monosynaptic connections from parallel spike trains: A review.Neuroscience Research, 215:37–46, 2025

    Ryota Kobayashi and Shigeru Shinomoto. Inference of monosynaptic connections from parallel spike trains: A review.Neuroscience Research, 215:37–46, 2025

    2025

  46. [46]

    Gaba-mediated repulsive coupling between circadian clock neurons in the scn encodes seasonal time

    Jihwan Myung, Sungho Hong, Daniel DeWoskin, Erik De Schutter, Daniel B Forger, and Toru Takumi. Gaba-mediated repulsive coupling between circadian clock neurons in the scn encodes seasonal time. Proceedings of the National Academy of Sciences, 112(29):E3920–E3929, 2015. 21

    2015

  47. [47]

    Estimation of excitatory and inhibitory synaptic conductance variations in motoneurons during locomotor-like rhythmic activity.Neuroscience, 335:72–81, 2016

    Ryota Kobayashi, Hiroshi Nishimaru, and Hisao Nishijo. Estimation of excitatory and inhibitory synaptic conductance variations in motoneurons during locomotor-like rhythmic activity.Neuroscience, 335:72–81, 2016

    2016

  48. [48]

    Type i membranes, phase resetting curves, and synchrony.Neural computation, 8(5):979–1001, 1996

    Bard Ermentrout. Type i membranes, phase resetting curves, and synchrony.Neural computation, 8(5):979–1001, 1996

    1996

  49. [49]

    Springer, 2010

    Bard Ermentrout and David M Terman.Mathematical foundations of neuroscience, volume 35. Springer, 2010. 22

    2010