ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Pankratova E. V., Belykh V. N. The peculiarities of transition to complete synchronization in networks of Hodgkin– Huxley elements. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 2, pp. 3-17. DOI: 10.18500/0869-6632-2008-16-2-3-17

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2008no2p003.pdf
(downloads: 146)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
519.179.2
DOI: 
10.18500/0869-6632-2008-16-2-3-17

The peculiarities of transition to complete synchronization in networks of Hodgkin– Huxley elements

Autors: 
Pankratova Evgenija Valerevna, Volga State Academy of Water Transport (VGAVT)
Belykh Vladimir Nikolaevich, Volga State Academy of Water Transport (VGAVT)
Abstract: 

In this paper we consider various networks of mutually coupled identical Hodgkin– Huxley systems. The peculiarities of transition to complete synchronization in networks subjected to suprathreshold periodic driving and common random forcing are examined both theoretically and through numerical simulation. The conditions for global stability of complete synchronization in networks of two «star»-coupled structures are obtained within the framework of connection graph stability method. Various scenarios determining the increase of the number of elements in such ensembles are considered. The behavior of the coupling strength necessary to achieve complete synchronization in the presence of random forcing is examined.

Key words: 
-
Reference: 
  1. Anischenko VS, Vadivasova TE, Postnov DE, and Safonova MA. Synchronization of chaos. Int. J. Bifurcat. Chaos. 1992;2(3):633–644. DOI: 10.1142/S0218127492000756.
  2. Parmananda P. Generalized synchronization of spatiotemporal chemical chaos. Phys. Rev. E. 1997;56(2):1595–1598. DOI: 10.1103/PhysRevE.56.1595.
  3. Abarbanel HD, Rabinovich MI, Selverston A, Bazhenov MV, Huerta R, Sushchik MM, Rubchinskii LL. Synchronisation in neural networks. Phys. Usp. 1996;39(4):337–362. DOI: 10.1070/PU1996v039n04ABEH000141.
  4. Belykh I, Lange E, and Hasler M. Synchronization of bursting neurons: what matters in the network topology. Phys. Rev. Lett. 2005;94(18):188101. DOI: 10.1103/physrevlett.94.188101.
  5. Zhou C and Kurths J. Noise-induced synchronization and coherence resonance of a Hodgkin–Huxley model of thermally sensitive neurons. Chaos. 2003;13(1):401–409. DOI: 10.1063/1.1493096.
  6. Gray CM, Konig P, Engel AK, and Singer W. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature. 1989;338(6213):334–337. DOI: 10.1038/338334a0.
  7. Stopfer M, Bhagavan S, Smith BH, and Laurent G. Impaired odour discrimination on desynchronization of odour-encoding neural assemblies. Nature. 1997;390(6655):70–74. DOI: 10.1038/36335.
  8. Steinmetz PN, Roy A, Fitzgerald PJ, Hsiao SS, Johnson KO, and Niebur E. Attention modulates synchronized neuronal firing in primate somatosensory cortex. Nature. 2000;404(6774):187–190. DOI: 10.1038/35004588.
  9. Hodgkin AL and Huxley AF. A quantitative description of membrane current and its application conduction and excitation in nerve. J. Physiol. 1952;117(4):500–544. DOI: 10.1113/jphysiol.1952.sp004764.
  10. Keener J and Sneyd J. Mathematical Physiology. Berlin: Springer Verlag; 1998. 767 p. DOI: 10.1007/b98841.
  11. Belykh VN, Belykh IV, Hasler M. Connection graph stability method for synchronized coupled chaotic systems. Physica D. 2004;195(1–2):159–187. DOI: 10.1016/j.physd.2004.03.012.
  12. Bloom F, Lazerson A, Hofstadter L. Brain, Mind and Behavior. W.H. Freeman; 1988. 394 p.
  13. Belykh IV, Hasler M, Lauret M, Nijmeijer H. Synchronization and graph topology. Int. J. Bifurcat. Chaos. 2005;15(11):3423–3433. DOI: 10.1142/S0218127405014143.
  14. Belykh VN, Pankratova EV. Chaotic synchronization in ensembles of coupled neurons modeled by the FitzHugh-Rinzel system. Radiophys. Quantum Electron. 2006;49(11):910–921. DOI: 10.1007/s11141-006-0124-z.
  15. Pankratova EV, Belykh VN, and Mosekilde E. Role of the driving frequency in a randomly perturbed Hodgkin–Huxley neuron with suprathreshold forcing. Eur. Phys. J. B. 2006;53(4):529–536. DOI: 10.1140/epjb/e2006-00401-9.
  16. Rosenblum MG, Pikovsky AS, and Kurths J. From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 1997;78(22):4193–4196. DOI: 10.1103/PhysRevLett.78.4193.
  17. Toral R, Mirasso CR, Hernandez-Garcia E, and Piro O. Analytical and numerical studies of noise-induced synchronization of chaotic systems. Chaos. 2001;11(3):665–673. DOI: 10.1063/1.1386397.
  18. Jensen RV. Synchronization of randomly driven nonlinear oscillators. Phys. Rev. E. 1998;58(6):R6907–R6910. DOI: 10.1103/PhysRevE.58.R6907.
  19. Pikovsky A, Rosenblum M, and Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge, UK: Cambridge University Press; 2001. 411 p. DOI: 10.1017/CBO9780511755743.
  20. Belykh VN and Pankratova EV. Synchronization and control in ensembles of periodic and chaotic neuronal elements with time dependent coupling. IFAC Proceedings Volumes. 2007;40(14):120–125. DOI: 10.3182/20070829-3-RU-4912.00020.
  21. Belykh VN, Pankratova EV, and Mosekilde E. Dynamics and synchronization of noise perturbed ensembles of periodically activated neuron cells. Int. J. Bifurcat. Chaos. 2008;18(9):2807–2815. DOI: 10.1142/S0218127408022044.
  22. Belykh VN, Belykh IV, Hasler M. Blinking model and synchronization in small-world networks with a time-varying coupling. Physica D. 2004;195(1–2):188–206. DOI: 10.1016/j.physd.2004.03.013.
Received: 
29.10.2007
Accepted: 
12.02.2008
Published: 
30.04.2008
Short text (in English):
PDF icon a2008no2p003.doc_.pdf
(downloads: 107)
  • 1924 reads