We present a phase autoencoder that encodes the asymptotic phase of a limit-cycle oscillator, a fundamental quantity characterizing its synchronization dynamics. This autoencoder is trained in such a way that its latent variables directly represent the asymptotic phase of the oscillator. The trained autoencoder can perform two functions without relying on the mathematical model of the oscillator: first, it can evaluate the asymptotic phase and the phase sensitivity function of the oscillator; second, it can reconstruct the oscillator state on the limit cycle in the original space from the phase value as an input. Using several examples of limit-cycle oscillators, we demonstrate that the asymptotic phase and the phase sensitivity function can be estimated only from time-series data by the trained autoencoder. We also present a simple method for globally synchronizing two oscillators as an application of the trained autoencoder.

1.
A. T.
Winfree
,
The Geometry of Biological Time
(
Springer
,
1980
), Vol. 2.
2.
A.
Pikovsky
,
M.
Rosenblum
, and
J.
Kurths
,
Synchronization: A Universal Concept in Nonlinear Science
(Cambridge University Press, Cambridge, 2001).
3.
Y.
Kuramoto
,
Chemical Oscillations, Waves and Turbulence
(Dover Publications, Mineola, NY, 2003), https://www.bibsonomy.org/bibtex/28ce3643f7eaa20e80268987ef83a1dd9/rincedd.
4.
T.
Stankovski
,
V.
Ticcinelli
,
P. V.
McClintock
, and
A.
Stefanovska
, “
Coupling functions in networks of oscillators
,”
New J. Phys.
17
,
035002
(
2015
).
5.
T.
Stankovski
,
V.
Ticcinelli
,
P. V.
McClintock
, and
A.
Stefanovska
, “
Neural cross-frequency coupling functions
,”
Front. Syst. Neurosci.
11
,
33
(
2017
).
6.
L.
Borgius
,
H.
Nishimaru
,
V.
Caldeira
,
Y.
Kunugise
,
P.
Löw
,
R.
Reig
,
S.
Itohara
,
T.
Iwasato
, and
O.
Kiehn
, “
Spinal glutamatergic neurons defined by EphA4 signaling are essential components of normal locomotor circuits
,”
J. Neurosci.
34
,
11
(
2014
).
7.
J. J.
Collins
and
I. N.
Stewart
, “
Coupled nonlinear oscillators and the symmetries of animal gaits
,”
J. Nonlinear Sci.
3
,
349
(
1993
).
8.
R.
Kobayashi
,
H.
Nishimaru
, and
H.
Nishijo
, “
Estimation of excitatory and inhibitory synaptic conductance variations in motoneurons during locomotor-like rhythmic activity
,”
Neuroscience
335
,
08027
(
2016
).
9.
T.
Funato
,
Y.
Yamamoto
,
S.
Aoi
,
T.
Imai
,
T.
Aoyagi
,
N.
Tomita
, and
K.
Tsuchiya
, “
Evaluation of the phase-dependent rhythm control of human walking using phase response curves
,”
PLoS Comput. Biol.
12
,
1004950
(
2016
).
10.
B.
Kralemann
,
M.
Frühwirth
,
A.
Pikovsky
,
M.
Rosenblum
,
T.
Kenner
,
J.
Schaefer
, and
M.
Moser
, “
In vivo cardiac phase response curve elucidates human respiratory heart rate variability
,”
Nat. Commun.
4
,
3418
(
2013
).
11.
M.
Garcia
,
A.
Chatterjee
,
A.
Ruina
, and
M.
Coleman
, “
The simplest walking model: Stability, complexity, and scaling
,”
J. Biomech. Eng.
120
,
281–288
(
1998
).
12.
M.
Rohden
,
A.
Sorge
,
M.
Timme
, and
D.
Witthaut
, “
Self-organized synchronization in decentralized power grids
,”
Phys. Rev. Lett.
109
,
064101
(
2012
).
13.
H.
Nakao
, “
Phase reduction approach to synchronisation of nonlinear oscillators
,”
Contemp. Phys.
57
,
188
214
(
2016
).
14.
G. B.
Ermentrout
and
D. H.
Terman
,
Mathematical Foundations of Neuroscience
(Springer-Verlag, New York, 2010), https://link.springer.com/book/10.1007/978-0-387-87708-2.
15.
B.
Monga
,
D.
Wilson
,
T.
Matchen
, and
J.
Moehlis
, “
Phase reduction and phase-based optimal control for biological systems: A tutorial
,”
Biolog. Cyber.
113
,
11
46
(
2019
).
16.
Y.
Kuramoto
and
H.
Nakao
, “
On the concept of dynamical reduction: The case of coupled oscillators
,”
Philos. Trans. R. Soc. A
377
,
20190041
(
2019
).
17.
B.
Ermentrout
,
Y.
Park
, and
D.
Wilson
, “
Recent advances in coupled oscillator theory
,”
Philos. Trans. R. Soc. A
377
,
20190092
(
2019
).
18.
L.
Glass
and
M. C.
Mackey
,
From Clocks to Chaos: The Rhythms of Life
(
Princeton University Press
,
1988
).
19.
S.
Shirasaka
,
W.
Kurebayashi
, and
H.
Nakao
, “
Phase-amplitude reduction of limit cycling systems
,” in
The Koopman Operator in Systems and Control
, Lecture Notes in Control and Information Sciences Vol. 484, edited by A. Mauroy, I. Mezić, and Y. Susuki (Springer, Cham, Switzerland, 2020).
20.
D.
Wilson
, “
An adaptive phase-amplitude reduction framework without O ( ϵ ) constraints on inputs
,”
SIAM J. Appl. Dyn. Syst.
21
,
204
230
(
2022
).
21.
K.
Taira
and
H.
Nakao
, “
Phase-response analysis of synchronization for periodic flows
,”
J. Fluid Mech.
846
,
R2
(
2018
).
22.
F. C.
Hoppensteadt
and
E. M.
Izhikevich
,
Weakly Connected Neural Networks
(
Springer Science & Business Media
,
1997
), Vol.
126
.
23.
K. C.
Wedgwood
,
K. K.
Lin
,
R.
Thul
, and
S.
Coombes
, “
Phase-amplitude descriptions of neural oscillator models
,”
J. Math. Neurosci.
3
,
2
(
2013
).
24.
D.
Wilson
and
J.
Moehlis
, “
Isostable reduction of periodic orbits
,”
Phys. Rev. E
94
,
052213
(
2016
).
25.
A.
Mauroy
and
I.
Mezić
, “
Global stability analysis using the eigenfunctions of the Koopman operator
,”
IEEE Trans. Autom. Control
61
,
3356
3369
(
2016
).
26.
S.
Shirasaka
,
W.
Kurebayashi
, and
H.
Nakao
, “
Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems
,”
Chaos
27
,
023119
(
2017
).
27.
A.
Mauroy
and
I.
Mezić
, “
Global computation of phase-amplitude reduction for limit-cycle dynamics
,”
Chaos
28
,
073108
(
2018
).
28.
K.
Kotani
,
Y.
Ogawa
,
S.
Shirasaka
,
A.
Akao
,
Y.
Jimbo
, and
H.
Nakao
, “
Nonlinear phase-amplitude reduction of delay-induced oscillations
,”
Phys. Rev. Res.
2
,
033106
(
2020
).
29.
H.
Nakao
, “Phase and amplitude description of complex oscillatory patterns in reaction-diffusion systems,” in Physics of Biological Oscillators: New Insights into Non-Equilibrium and Non-Autonomous Systems (Springer, 2021), pp. 11–27.
30.
S.
Takata
,
Y.
Kato
, and
H.
Nakao
, “
Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and floquet theory
,”
Chaos
31
,
093124
(
2021
).
31.
B.
Ermentrout
, “
Type I membranes, phase resetting curves, and synchrony
,”
Neural Comput.
8
,
979
1001
(
1996
).
32.
K.
Ota
,
M.
Nomura
, and
T.
Aoyagi
, “
Weighted spike-triggered average of a fluctuating stimulus yielding the phase response curve
,”
Phys. Rev. Lett.
103
,
024101
(
2009
).
33.
T.
Netoff
,
M. A.
Schwemmer
, and
T. J.
Lewis
, “
Experimentally estimating phase response curves of neurons: Theoretical and practical issues
,” in
Phase Response Curves in Neuroscience
, Springer Series in Computational Neuroscience Vol. 6, edited by N. Schultheiss, A. Prinz, and R. Butera (Springer, New York, 2012), pp. 95–129.
34.
T.
Imai
,
K.
Ota
, and
T.
Aoyagi
, “
Robust measurements of phase response curves realized via multicycle weighted spike-triggered averages
,”
J. Phys. Soc. Jpn.
86
,
024009
(
2017
).
35.
P. J.
Schmid
, “
Dynamic mode decomposition of numerical and experimental data
,”
J. Fluid Mech.
656
,
5
28
(
2010
).
36.
J. N.
Kutz
,
S. L.
Brunton
,
B. W.
Brunton
, and
J. L.
Proctor
,
Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems
(
SIAM
,
2016
).
37.
M. O.
Williams
,
I. G.
Kevrekidis
, and
C. W.
Rowley
, “
A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition
,”
J. Nonlinear Sci.
25
,
1307
1346
(
2015
).
38.
B.
Lusch
,
J. N.
Kutz
, and
S. L.
Brunton
, “
Deep learning for universal linear embeddings of nonlinear dynamics
,”
Nat. Commun.
9
,
4950
(
2018
).
39.
N.
Namura
,
S.
Takata
,
K.
Yamaguchi
,
R.
Kobayashi
, and
H.
Nakao
, “
Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data
,”
Phys. Rev. E
106
,
014204
(
2022
).
40.
B.
Kralemann
,
L.
Cimponeriu
,
M.
Rosenblum
,
A.
Pikovsky
, and
R.
Mrowka
, “
Phase dynamics of coupled oscillators reconstructed from data
,”
Phys. Rev. E
77
,
066205
(
2008
).
41.
A.
Mauroy
,
I.
Mezić
, and
J.
Moehlis
, “
Isostables, isochrons, and Koopman spectrum for the action–angle representation of stable fixed point dynamics
,”
Phys. D
261
,
19
30
(
2013
).
42.
K.
Fukami
,
K.
Taira
, and
H.
Nakao
, “Data-driven transient lift attenuation for extreme vortex gust-airfoil interactions,” arXiv:2403.00263 (2024).
43.
A.
Mardt
,
L.
Pasquali
,
H.
Wu
, and
F.
Noé
, “
VAMPnets for deep learning of molecular kinetics
,”
Nat. Commun.
9
,
5
(
2018
).
44.
C.
Wehmeyer
and
F.
Noé
, “
Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics
,”
J. Chem. Phys.
148
,
241703
(
2018
).
45.
S. E.
Otto
and
C. W.
Rowley
, “
Linearly recurrent autoencoder networks for learning dynamics
,”
SIAM J. Appl. Dyn. Syst.
18
,
558
593
(
2019
).
46.
N.
Takeishi
,
Y.
Kawahara
, and
T.
Yairi
, “
Learning Koopman invariant subspaces for dynamic mode decomposition
,” in
NeurIPS
(
2017
).
47.
E.
Yeung
,
S.
Kundu
, and
N.
Hodas
, “Learning deep neural network representations for Koopman operators of nonlinear dynamical systems,” in
American Control Conference
(2019), pp. 4832–4839.
48.
K.
Champion
,
B.
Lusch
,
J. N.
Kutz
, and
S. L.
Brunton
, “
Data-driven discovery of coordinates and governing equations
,”
Proc. Natl. Acad. Sci. U.S.A.
116
,
22445
22451
(
2019
).
49.
O.
Azencot
,
N. B.
Erichson
,
V.
Lin
, and
M.
Mahoney
, “Forecasting sequential data using consistent Koopman autoencoders,” in International Conference on Machine Learning (2020), pp. 475–485.
50.
Y.
Li
,
H.
He
,
J.
Wu
,
D.
Katabi
, and
A.
Torralba
, “Learning compositional Koopman operators for model-based control,” arXiv:1910.08264 (2019).
51.
N.
Berman
,
I.
Naiman
, and
O.
Azencot
, “Multifactor sequential disentanglement via structured Koopman autoencoders,” arXiv:2303.17264 (2023).
52.
M.
Han
,
J.
Euler-Rolle
, and
R. K.
Katzschmann
, “Desko: Stability-assured robust control with a deep stochastic Koopman operator,” in International Conference on Learning Representations (2021).
53.
G. E.
Hinton
and
R. R.
Salakhutdinov
, “
Reducing the dimensionality of data with neural networks
,”
Science
313
,
504
507
(
2006
).
54.
K.
Fukami
and
K.
Taira
, “Grasping extreme aerodynamics on a low-dimensional manifold,”
Nat. Commun.
14
, 6480 (2023).
55.
A.
Paszke
,
S.
Gross
,
S.
Chintala
,
G.
Chanan
,
E.
Yang
,
Z.
DeVito
,
Z.
Lin
,
A.
Desmaison
,
L.
Antiga
, and
A.
Lerer
, “Automatic differentiation in PyTorch,” in NIPS Autodiff Workshop (2017).
56.
V.
Nair
and
G. E.
Hinton
, “Rectified linear units improve restricted Boltzmann machines,” in Proceedings of the 27th international conference on machine learning (ICML-10) (Omnipress, 2010), pp. 807–814.
57.
S.
Ioffe
and
C.
Szegedy
, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in International Conference on Machine Learning (PMLR, 2015), pp. 448–456.
58.
A.
Paszke
,
S.
Gross
,
F.
Massa
,
A.
Lerer
,
J.
Bradbury
,
G.
Chanan
,
T.
Killeen
,
Z.
Lin
,
N.
Gimelshein
,
L.
Antiga
, et al., “
PyTorch: An imperative style, high-performance deep learning library
,” in
NeurIPS
(
2019
).
59.
J. T.
Stuart
, “
On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane poiseuille flow
,”
J. Fluid Mech.
9
,
353
370
(
1960
).
60.
L.
Landau
, “Comptes rendues DOKL,”
Acad. Sci. USSR
44
, 311 (1944).
61.
A. L.
Hodgkin
and
A. F.
Huxley
, “
A quantitative description of membrane current and its application to conduction and excitation in nerve
,”
J. Physiol.
117
,
500
(
1952
).
62.
H.
Nakao
,
S.
Yasui
,
M.
Ota
,
K.
Arai
, and
Y.
Kawamura
, “
Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations
,”
Chaos
28
,
045103
(
2018
).
63.
P.
Mircheski
,
J.
Zhu
, and
H.
Nakao
, “
Phase-amplitude reduction and optimal phase locking of collectively oscillating networks
,”
Chaos
33
,
103111
(
2023
).
64.
A.
Stefanovska
and
M.
Brai
, “
Physics of the human cardiovascular system
,”
Contemp. Phys.
40
,
31
(
1999
).
65.
S. H.
Strogatz
,
Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering
(
CRC Press
,
2018
).
66.
W.
Kurebayashi
,
S.
Shirasaka
, and
H.
Nakao
, “
Phase reduction method for strongly perturbed limit cycle oscillators
,”
Phys. Rev. Lett.
111
,
214101
(
2013
).
67.
W.
Kurebayashi
,
T.
Yamamoto
,
S.
Shirasaka
, and
H.
Nakao
, “
Phase reduction of strongly coupled limit-cycle oscillators
,”
Phys. Rev. Res.
4
,
043176
(
2022
).
68.
J.
Newman
,
J. P.
Scott
,
J. R.
Adams
, and
A.
Stefanovska
, “
Intermittent phase dynamics of non-autonomous oscillators through time-varying phase
,”
Phys. D
461
,
134108
(
2024
).
69.
P. T.
Clemson
and
A.
Stefanovska
, “
Discerning non-autonomous dynamics
,”
Phys. Rep.
542
,
297
368
(
2014
).
70.
M.
Raissi
,
P.
Perdikaris
, and
G. E.
Karniadakis
, “
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
,”
J. Comput. Phys.
378
,
686
707
(
2019
).
71.
G. E.
Karniadakis
,
I. G.
Kevrekidis
,
L.
Lu
,
P.
Perdikaris
,
S.
Wang
, and
L.
Yang
, “
Physics-informed machine learning
,”
Nat. Rev. Phys.
3
,
422
440
(
2021
).
72.
S.
Greydanus
,
M.
Dzamba
, and
J.
Yosinski
, “
Hamiltonian neural networks
,” in
NeurIPS
(
2019
).
73.
Z.
Chen
,
M.
Feng
,
J.
Yan
, and
H.
Zha
, “Learning neural hamiltonian dynamics: a methodological overview,” arXiv:2203.00128 (2022).
74.
P.
Toth
,
D. J.
Rezende
,
A.
Jaegle
,
S.
Racanière
,
A.
Botev
, and
I.
Higgins
, “Hamiltonian generative networks,” arXiv:1909.13789 (2019).
75.
A.
Daigavane
,
A.
Kosmala
,
M.
Cranmer
,
T.
Smidt
, and
S.
Ho
, “Learning integrable dynamics with action-angle networks,” arXiv:2211.15338 (2022).
76.
K.
Yawata
(2024). “Phase autoencoder for limit-cycle oscillators,” Github. https://github.com/kyoukuntaro/PhaseAutoencoder.
You do not currently have access to this content.