Phase synchronization

For other types of Synchronization, see Synchronization (disambiguation).

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Phase synchronization is the process by which two or more cyclic signals tend to oscillate with a repeating sequence of relative phase angles.

Phase synchronisation is usually applied to two waveforms of the same frequency with identical phase angles with each cycle. However it can be applied if there is an integer relationship of frequency, such that the cyclic signals share a repeating sequence of phase angles over consecutive cycles. These integer relationships are called Arnold tongues which follow from bifurcation of the circle map.

One example of phase synchronization of multiple oscillators can be seen in the behavior of Southeast Asian fireflies [1][2]. At dusk, the flies begin to flash periodically with random phases and a gaussian distribution of native frequencies. As night falls, the flies, sensitive to one another's behavior, begin to synchronize their flashing. After some time all the fireflies within a given tree (or even larger area) will begin to flash simultaneously in a burst.

Thinking of the fireflies as biological oscillators, we can define the phase to be 0° during the flash and +-180° exactly halfway until the next flash. Thus, when they begin to flash in unison, they synchronize in phase.

Phase synchronization patterns (n:m synchronization) have been also observed in the interaction of physiological and organ systems, for example in maternal-fetal cardiac phase-synchronization [3][4], brain blood flow velocity vs. peripheral blood pressure in stroke [5], cortico-muscular [6] and cardio-respiratory [7][8] coupling with the degree of synchronization changing across physiological states (sleep/wake, sleep stages) and age groups [8][9].

One way to keep a local oscillator "phase synchronized" with a remote transmitter uses a phase-locked loop.

## See also

* Algebraic connectivity
* Coherence
* Kuramoto model
* Synchronization (alternating current)

## References

1. ↑ Strogatz, Steven (2003). Sync: The Emerging Science of Spontaneous Order. Grand Central Publishing. ISBN 9780786868445.
2. ↑ Pikovsky, Arkady; Rosenblum, Michael; Kurths, Jürgen (2001). Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge Nonlinear Science Series. Cambridge: Cambridge University Press. ISBN 978-0-521-53352-2. https://www.cambridge.org/core/books/synchronization/E46C1FC3ADC82EEA75AE6F5B9B74E28C.
3. ↑ Van Leeuwen, P.; Geue, D.; Thiel, M.; Cysarz, D.; Lange, S.; Romano, M. C.; Wessel, N.; Kurths, J. et al. (2009-08-18). "Influence of paced maternal breathing on fetal-maternal heart rate coordination". Proceedings of the National Academy of Sciences of the United States of America 106 (33): 13661–13666. doi:10.1073/pnas.0901049106. ISSN 1091-6490. PMID 19597150.
4. ↑ Ivanov, Plamen Ch.; Ma, Qianli D. Y.; Bartsch, Ronny P. (2009). "Maternal–fetal heartbeat phase synchronization". Proceedings of the National Academy of Sciences 106 (33): 13641–13642. doi:10.1073/pnas.0906987106. PMID 19706494. Bibcode: 2009PNAS..10613641I.
5. ↑ Chen, Zhi; Hu, Kun; Stanley, H. Eugene; Novak, Vera; Ivanov, Plamen Ch (March 2006). "Cross-correlation of instantaneous phase increments in pressure-flow fluctuations: applications to cerebral autoregulation". Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics 73 (3 Pt 1): 031915. doi:10.1103/PhysRevE.73.031915. ISSN 1539-3755. PMID 16605566. PMC 2140229. https://pmc.ncbi.nlm.nih.gov/articles/PMC2140229/.
6. ↑ Tass, P.; Rosenblum, M. G.; Weule, J.; Kurths, J.; Pikovsky, A.; Volkmann, J.; Schnitzler, A.; Freund, H.-J. (1998-10-12). "Detection of n : m Phase Locking from Noisy Data: Application to Magnetoencephalography" (in en). Physical Review Letters 81 (15): 3291–3294. doi:10.1103/PhysRevLett.81.3291. ISSN 0031-9007. https://link.aps.org/doi/10.1103/PhysRevLett.81.3291.
7. ↑ Schäfer, Carsten; Rosenblum, Michael G.; Kurths, Jürgen; Abel, Hans-Henning (1998). "Heartbeat synchronized with ventilation" (in en). Nature 392 (6673): 239–240. doi:10.1038/32567. ISSN 1476-4687. https://www.nature.com/articles/32567.
8. ↑ 8.0 8.1 Bartsch, Ronny P.; Schumann, Aicko Y.; Kantelhardt, Jan W.; Penzel, Thomas; Ivanov, Plamen Ch. (2012-06-26). "Phase transitions in physiologic coupling". Proceedings of the National Academy of Sciences 109 (26): 10181–10186. doi:10.1073/pnas.1204568109. PMID 22691492. PMC 3387128. https://www.pnas.org/doi/abs/10.1073/pnas.1204568109.
9. ↑ Bartsch, Ronny P.; Liu, Kang Kl; Ma, Qianli Dy; Ivanov, Plamen Ch (2014). "Three Independent Forms of Cardio-Respiratory Coupling: Transitions across Sleep Stages". Computing in Cardiology 41: 781–784. ISSN 2325-8861. PMID 25664348. PMC 4319215. https://pmc.ncbi.nlm.nih.gov/articles/PMC4319215/.

## External links

A tutorial on calculating Phase locking and Phase synchronization in Matlab.

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