Synchronization of Kuramoto Oscillators on Knots

Synchronization of Kuramoto Oscillators on Knots

Loading document ...
Loading page ...


Author(s): Amelia Carolina Sparavigna

Download Full PDF Read Complete Article

DOI: 10.18483/ijSci.217 628 1162 85-88 Volume 2 - Jul 2013


A knot is a circle embedded in the space. Projecting a knot on a plane, we obtain a diagram which is known as the knot diagram. The vertices of the diagram, where the curved lines are crossed, can be considered as sites occupied by oscillators. The synchronization of these oscillators can be studied by means of a Kuramoto model. Here we propose to define some order parameters, of the complete knot diagram and of its regions, to study the synchronization of the system with regard to the different parts of it.


Knots, Synchronization, Kuramoto model


  1. D.J. Watts, S.H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature, 393 (1998) 440-442
  2. A.T. Winfree, Biological rhythms and the behavious of populations of coupled oscillators, J. Theor. Biol. 16 (1967) 15
  3. S.H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D 143 (2000) 1–20
  4. Y. Kuramoto, H. Arakai, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, 1984
  5. D. Cumin, C.P. Unsworth, Generalising the Kuramoto model for the study of neuronal synchronisation in the brain, Physica D 226 (2007) 181–196
  6. M.V.L. Bennet, R.S. Zukin, Electrical coupling and neuronal synchronisation in the mammalian brain, Neuron 41 (2004) 495–511
  7. H. Sompolinsky, D. Golomb, D. Kleinfeld, Cooperative dynamics in visual processing, Phys. Rev. A 43 (1991) 6990–7011.
  8. M. Cassidy, P. Mazzone, A. Oliviero, et al., Movement-related changes in synchronisation in the human basal ganglia, Brain 125 (2002) 1235–1246
  9. W. Klimesch, Memory processes, brain oscillations and EEG synchronisation, Int. J. Psychophysiol. 24 (1996) 61–100.
  10. R.D. Traub, R.K. Wong, Cellular mechanisms of neural synchronisation is epilepsy, Science 216 (1982) 745–747.
  11. F. Mormann, K. Lehnertz, P. David, C.E. Elger, Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients, Physica D 144 (2000) 358–369.
  12. F.H. Lopes das Silva, P. Suffczynski, J. Parra, et al., Epilepsies as dynamic diseases of brain systems: Basic models and the search for EEG/Magnetoencephalography (MEG) signals of impending seizures, Epilepsia 42 (Suppl. 7) (2001) 3
  13. F.J. Varela, The preictal state: Dynamic neural changes preceding seizure onset, Epilepsia 42 (Suppl. 7) (2001) 3
  14. H. Hong and M. Y. Choi, Beom Jun Kim, Synchronization on small-world networks, Phys. Rev. E, 65 (2002) 026139-1-5
  15. Wikipedia, Knot Theory,
  16. L.H. Kauffman, On knots, Annals of Mathematical Studies, Princeton University Press, 1987
  17. L.H. Kauffman, Knots and Physics, World Scientific, 2001
  18. A. Kawauchi, A survey of knot theory, Birkhäuser, 1996
  19. P.R. Cromwell, Knots and links, Cambridge University Press, 2004

Cite this Article:

International Journal of Sciences is Open Access Journal.
This article is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License.
Author(s) retain the copyrights of this article, though, publication rights are with Alkhaer Publications.

Search Articles

Issue June 2023

Volume 12, June 2023

Table of Contents

World-wide Delivery is FREE

Share this Issue with Friends:

Submit your Paper