206
E. Gengel and A. Pikovsky
based on some assumption about the dynamics, which typically are only approximately fulfilled. This topic certainly deserves further studies.
Biological systems are noisy. We have given an example here, where IHTE also
improves the reconstruction of the phase in presence of fluctuations (see Fig. 12.9).
However, the very concept of a monotonously growing phase should be reconsidered
for noisy signals. Here we largely avoided problems by smoothing the observed
signal, but in this approach some features of the modulation might be lost.
Acknowledgements Both authors thank Aneta Stefanovska and Peter McClintock for the kind invitation to contribute to this interdisciplinary work. Erik Gengel thanks the Friedrich-Ebert-Stiftung
for financial support. This paper was supported by the RSF grant 17-12-01534. The analysis in Sec.
3.3 was supported by the Laboratory of Dynamical Systems and Applications NRU HSE, of the
Russian Ministry of Science and Higher Education (Grant No. 075-15-2019-1931).
References
1. A. Bahraminasab, F. Ghasemi, A. Stefanovska, P.V.E. McClintock, H. Kantz, Direction of
coupling from phases of interacting oscillators: a permutation information approach. Phys.
Rev. Lett. 100, 084101 (2008)
2. H. Balzter, N.J. Tate, J. Kaduk, D. Harper, S. Page, R. Morrison, M. Muskulus, P. Jones,
Multi-scale entropy analysis as a method for time-series analysis of climate data. Climate 3(1),
227–240 (2015)
3. D. Benitez, P. Gaydecki, A. Zaidi, A. Fitzpatrick, The use of the Hilbert transform in ECG
signal analysis. Comput. Biol. Med. 31(5), 399–406 (2001)
4. K.A. Blaha, A. Pikovsky, M. Rosenblum, M.T. Clark, C.G. Rusin, J.L. Hudson, Reconstruction
of two-dimensional phase dynamics from experiments on coupled oscillators. Phys. Rev. E
84(4), 046201 (2011)
5. E. Brown, J. Moehlis, P. Holmes, On the phase reduction and response dynamics of neural
oscillator populations. Neural Comput. 16(4), 673–715 (2004)
6. S.L. Brunton, J.N. Kutz, Data-Driven Science and Engineering: Machine Learning, Dynamical
Systems, and Control (Cambridge University Press, 2019)
7. R. Cestnik, M. Rosenblum, Reconstructing networks of pulse-coupled oscillators from spike
trains. Phys. Rev. E 96(1), 012209 (2017)
8. R. Chestnik, M. Rosenblum, Inferring the phase response curve from observation of a continuously perturbed oscillator. Sci. Rep. 8(1), 13606 (2018)
9. L. Cimponeriu, M. Rosenblum, A. Pikovsky, Estimation of delay in coupling from time series.
Phys. Rev. E 70, 046213 (2004)
10. L. Cohen, P. Loughlin, D. Vakman, On an ambiguity in the definition of the amplitude and
phase of a signal. Sig. Process. 79(3), 301–307 (1999)
11. M. Feldman, Hilbert Transform Applications in Mechanical Vibration (Wiley, 2011)
12. E. Gengel, A. Pikovsky, Phase demodulation with iterative Hilbert transform embeddings. Sig.
Process. 165, 115–127 (2019)
13. R. Guevara Erra, J.L. Perez Velazquez, M. Rosenblum, Neural synchronization from the perspective of non-linear dynamics. Front. Comput. Neurosci. 11, 98 (2017)
14. N.-H. Holstein-Rathlou, K.-P. Yip, O.V. Sosnovtseva, E. Mosekilde, Synchronization phenomena in nephron–nephron interaction. Chaos: Interdiscip. J. Nonlinear Sci. 11(2), 417–426
(2001)
15. N. Jajcay, S. Kravtsov, G. Sugihara, A.A. Tsonis, M. Paluš, Synchronization and causality
across time scales in El Niño Southern Oscillation. NPJ Clim. Atmos. Sci. 1(1), 1–8 (2018)
E. Gengel and A. Pikovsky
based on some assumption about the dynamics, which typically are only approximately fulfilled. This topic certainly deserves further studies.
Biological systems are noisy. We have given an example here, where IHTE also
improves the reconstruction of the phase in presence of fluctuations (see Fig. 12.9).
However, the very concept of a monotonously growing phase should be reconsidered
for noisy signals. Here we largely avoided problems by smoothing the observed
signal, but in this approach some features of the modulation might be lost.
Acknowledgements Both authors thank Aneta Stefanovska and Peter McClintock for the kind invitation to contribute to this interdisciplinary work. Erik Gengel thanks the Friedrich-Ebert-Stiftung
for financial support. This paper was supported by the RSF grant 17-12-01534. The analysis in Sec.
3.3 was supported by the Laboratory of Dynamical Systems and Applications NRU HSE, of the
Russian Ministry of Science and Higher Education (Grant No. 075-15-2019-1931).
References
1. A. Bahraminasab, F. Ghasemi, A. Stefanovska, P.V.E. McClintock, H. Kantz, Direction of
coupling from phases of interacting oscillators: a permutation information approach. Phys.
Rev. Lett. 100, 084101 (2008)
2. H. Balzter, N.J. Tate, J. Kaduk, D. Harper, S. Page, R. Morrison, M. Muskulus, P. Jones,
Multi-scale entropy analysis as a method for time-series analysis of climate data. Climate 3(1),
227–240 (2015)
3. D. Benitez, P. Gaydecki, A. Zaidi, A. Fitzpatrick, The use of the Hilbert transform in ECG
signal analysis. Comput. Biol. Med. 31(5), 399–406 (2001)
4. K.A. Blaha, A. Pikovsky, M. Rosenblum, M.T. Clark, C.G. Rusin, J.L. Hudson, Reconstruction
of two-dimensional phase dynamics from experiments on coupled oscillators. Phys. Rev. E
84(4), 046201 (2011)
5. E. Brown, J. Moehlis, P. Holmes, On the phase reduction and response dynamics of neural
oscillator populations. Neural Comput. 16(4), 673–715 (2004)
6. S.L. Brunton, J.N. Kutz, Data-Driven Science and Engineering: Machine Learning, Dynamical
Systems, and Control (Cambridge University Press, 2019)
7. R. Cestnik, M. Rosenblum, Reconstructing networks of pulse-coupled oscillators from spike
trains. Phys. Rev. E 96(1), 012209 (2017)
8. R. Chestnik, M. Rosenblum, Inferring the phase response curve from observation of a continuously perturbed oscillator. Sci. Rep. 8(1), 13606 (2018)
9. L. Cimponeriu, M. Rosenblum, A. Pikovsky, Estimation of delay in coupling from time series.
Phys. Rev. E 70, 046213 (2004)
10. L. Cohen, P. Loughlin, D. Vakman, On an ambiguity in the definition of the amplitude and
phase of a signal. Sig. Process. 79(3), 301–307 (1999)
11. M. Feldman, Hilbert Transform Applications in Mechanical Vibration (Wiley, 2011)
12. E. Gengel, A. Pikovsky, Phase demodulation with iterative Hilbert transform embeddings. Sig.
Process. 165, 115–127 (2019)
13. R. Guevara Erra, J.L. Perez Velazquez, M. Rosenblum, Neural synchronization from the perspective of non-linear dynamics. Front. Comput. Neurosci. 11, 98 (2017)
14. N.-H. Holstein-Rathlou, K.-P. Yip, O.V. Sosnovtseva, E. Mosekilde, Synchronization phenomena in nephron–nephron interaction. Chaos: Interdiscip. J. Nonlinear Sci. 11(2), 417–426
(2001)
15. N. Jajcay, S. Kravtsov, G. Sugihara, A.A. Tsonis, M. Paluš, Synchronization and causality
across time scales in El Niño Southern Oscillation. NPJ Clim. Atmos. Sci. 1(1), 1–8 (2018)