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.

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1. Anischenko V.S., Vadivasova T.E., Postnov D.E., and Safonova M.A. Synchronization of chaos // Int. J. Bifurcation Chaos Appl. Sci. Eng. 1992. Vol. 2. P. 633.

2. Parmananda P. Generalized synchronization of spatiotemporal chemical chaos // Phys. Rev. E. 1997. Vol. 56. P. 1595.

3. Абарбанель Г.Д.И., Рабинович М.И., Селверстон А., Баженов М.В., Хуэрта Р., Сущик М.М., Рубчинский Л.Л. Синхронизация в нейронных ансамблях // Успехи физических наук. 1996. Т. 166, No 4. С. 363.

4. Belykh I., Lange E., and Hasler M. Synchronization of bursting neurons: what matters in the network topology // Phys. Rev. Lett. 2005. Vol. 94. P. 188101.

5. Zhou C. and Kurths J. Noise-induced synchronization and coherence resonance of a Hodgkin–Huxley model of thermally sensitive neurons // Chaos. 2003. Vol. 13. P. 401.

6. Gray C.M., Ko ̈nig P., Engel A.K., and Singer W. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties // Nature. 1989. Vol. 338. P. 334.

7. Stopfer M., Bhagavan S., Smith B.H., and Laurent G. Impaired odour descrimination on desynchronization of odour-encoding neural assemblies // Nature. 1997. Vol. 390. P. 70.

8. Steinmetz P.N., Roy A., Fitzgerald P.J., Hsiao S.S, Johnson K.O., and Niebur E. Attention modulates synchronized neuronal firing in primate somatosensory cortex // Nature. 2000. Vol. 404. P. 187.

9. Hodgkin A.L. and Huxley A.F. A quantative description of membrane current and its application conduction and excitation in nerve // J. Physiology. 1952. Vol. 117. P. 500.

10. Keener J. and Sneyd J. Mathematical Physiology. Berlin: Springer Verlag, 1998.

11. Belykh V.N., Belykh I.V., Hasler M. Connection graph stability method for synchronized coupled chaotic systems // Physica D. 2004. Vol. 195. P. 159.

12. Блум Ф., Лейзерсон А., Хофстедтер Л. Мозг, разум и поведение Москва: Мир, 1988.

13. Belykh I.V., Hasler M., Lauret M., Nijmeijer H. Synchronization and graph topology // Int. J. Bif. and Chaos. 2005. Vol. 15. P. 3423.

14. Белых В.Н., Панкратова Е.В. Хаотическая синхронизация в ансамблях связанных нейронов, моделируемых системой ФитцХью–Ринцеля // Изв. вузов. Радиофизика. 2006. Т. 49. С. 910.

15. Pankratova E.V., Belykh V.N., and Mosekilde E. Role of the driving frequency in a randomly perturbed Hodgkin–Huxley neuron with suprathreshold forcing // Europ. Phys. J. B. 2006. P. 00401.

16. Rosenblum M.G., Pikovsky A.S., and Kurths J. From phase to lag synchronization in coupled chaotic oscillators // Phys. Rev. Lett. 1997. Vol.78. P. 4193.

17. Toral R., Mirasso C.R., Hernandez-Garcia E., and Piro O. Analytical and numerical studies of noise-induced synchronization of chaotic systems // Chaos. 2001. Vol.11. P. 665.

18. Jensen R.V. Synchronization of randomly driven nonlinear oscillators // Phys. Rev. E. 1998. Vol. 58. P. 6907.

19. Пиковский А., Розенблюм М., Куртс Ю. Синхронизация. Фундаментальное нелинейное явление. Москва: Техносфера, 2003.

20. Belykh V.N. and Pankratova E.V. Synchronization and control in ensembles of periodic and chaotic neuronal elements with time dependent coupling // Proceedings of 3rd IFAC Workshop «Periodic Control Systems» PSYCO-07, 2007.

21. Belykh V.N., Pankratova E.V., and Mosekilde E. Dynamics and synchronization of noise perturbed ensembles of periodically activated neuron cells // Int. J. Bif. Chaos. 2008 (in press).

22. Belykh V.N., Belykh I.V., Hasler M. Blinking model and synchronization in small-world networks with a time-varying coupling // Physica D. 2004. Vol. 195. P. 188.

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