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

Cite this article as:

Dmitrichev A. S., Nekorkin V. I. Stationary localized activity structures in two-dimensional ensemble of fitzhugh–nagumo neurons with oscillatory threshold. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 3, pp. 71-87. DOI:


Stationary localized activity structures in two-dimensional ensemble of fitzhugh–nagumo neurons with oscillatory threshold

Dmitrichev Aleksej Sergeevich, Institute of Applied Physics of the Russian Academy of Sciences
Nekorkin Vladimir Isaakovich, Federal state budgetary educational institution of higher professional education "Nizhny Novgorod state University named N. And.Lobachevsky"

We present the analysis of spatiotemporal dynamics of two-dimensional ensemble of electrically coupled FitzHugh–Nagumo neurons with oscillatory threshold. We show that in this system spatially localized activity structures can be formed. Such structures propagate through the system without changing their shape and velocity. We demonstrate that there exist two types of the structures: single and bound states. General characteristics of localized structures such as regions of existence, geometrical sizes and velocity are investigated. We also study structures interaction and give explanation for their existence and stability in terms of trajectories in associating with the ensemble multidimensional phase space.

Key words: 

1. Leznik E., Makarenko V., Llinas R. Electrotonically mediated oscillatory patterns in neuronal ensembles: An in vitro voltage-dependent dye-imaging study in the inferior olive //J. Neurosci. 2002. Vol. 20, No 7. P. 2804. 2. Wang X.-J. Synaptic reverberation underlying mnemonic persistent activity //Trends Neurosci. 2001. Vol. 24. P. 455. 3. Wu J.-Y., Guan Li, Tsau Yang. Propagating activation during oscillations and evoked responses in neocortical slices // J. Neurosci. 1999. Vol. 19, No 12. P. 5005. 4. Peinado A. Traveling slow waves of neural activity: a novel form of network activity in developing neocortex //J. Neurosci. 2000. Vol. 20. P. RC54. 5. Jung P., Milton J. Epilepsy as a dynamical disease. Springer, New York, 2003. 6. Dahlem M.A., et al. Control of sub-excitable waves in neural networks by nonlocal coupling //New trends and tools in complex networks / Eds. R. Criado, J. Pello, M. Romance. Spain: Universidad Rey Juan Carlos, 2007. 7. Kaminaga A., Vanag V.K., Epstein I.R. «Black spots» in a surfactant-rich Belousov–Zhabotinsky reaction dispersed in a water-in-oil microemulsion system //J. Chem. Phys. 2005. Vol. 122. P. 174706. 8. Vanag V.K., Epstein I.R. Stationary and oscillatory localized patterns, and subcritical bifurcations //Phys. Rev. Lett. 2004. Vol. 92. P. 128301. 9. Sakurai T., Mihaliuk E., Chirila F., Showalter K. Design and control of wave propagation patterns in excitable media //Science. 2002. Vol. 296. P. 2009. 10. Mihaliuk E., Sakurai T., Chirila F., Showalter K. Experimental and theoretical studies of feedback stabilization of propagating wave segment //Faraday Discuss. 2001. Vol. 120. P. 283. 11. Astrov Y.A., Ammelt E., Purwins H.G. Experimental evidence for zigzag instability of solitary stripes in a gas-discharge system //Phys. Rev. Lett. 1997. Vol. 78. P. 3129. 12. Muller I., Ammelt E., Purwins H.G. Self-organized quasiparticles: breathing filaments in a gas discharge system //Phys. Rev. Lett. 1999. Vol. 82. P. 3428. 13. Astrov Y.A., Purwins H.G. Plasma spots in a gas discharge system: birth, scattering and formation of molecules //Physics Letters A. 2001. Vol. 283. P. 349. 14. Розанов Н.Н. Асимметричные движущиеся локализованные структуры в широкоапертурном нелинейном интерферометре //Оптика и спектроскопия. 2007. Т. 102, No 2. С. 292. 15. Taranenko V.B., Slekys G., Weiss C.O. Spatial resonator solitons //CHAOS. 2003. Vol. 13, No 2. P. 777. 16. Umbanhowar P.B., Melo F., Swinney H.L. Localized excitations in a vertically vibrated granular layer //Nature. 1996. Vol. 382. P. 793. 17. Попцова М.С. Трансформация автоволн в локально неоднородных активных средах: Автореф. дис... канд. физ.-мат. наук. М., 2004. 18. Дудченко О.А., Гурия Г.Т. Резонансный характер долгоживущих возбуждений в слабовозбудимых активных средах //Труды LVIII научной конференции МФТИ «Современные проблемы фундаментальных и прикладных наук». 2005. P. 4. 19. Krischer K., Mikhailov A.S. Bifurcation to traveling spots in reaction-diffusion systems //Phys. Rev. Lett. 1994. Vol. 73. P. 3165.  20. Sendina-Nadal I., et al. Wave propagation in subexcitable media with periodically modulated excitability //Phys. Rev. Lett. 2001. Vol. 86, No 8. P. 1646. 21. Заикин А.Н. Формирование, распространение и взаимодействие экситонов (автоволн-квазичастиц) в активной среде //Физическая мысль России. 1995. No 1. С. 54. 22. Bode M., Liehr A.W., Schenk C.P., Purwins H.G. Interaction of dissipative solitons: particle-like behaviour of localized structures in a three-component reaction-diffusion system //Physica D. 2002. Vol. 161. P. 45. 23. Nishiura Y. Scattering of traveling spots in dissipative systems //CHAOS. 2005. Vol. 15. P. 047509. 24. Hughes S.W., et al. All thalamocortical neurones possess a T-type Ca2+ ’window’ current that enables the expression of bistability-mediated activities //J. Physiol. 1999. Vol. 517. P. 805. 25. Некоркин В.И., Дмитричев А.С., Щапин Д.С., Казанцев В.Б. Динамика модели нейрона со сложнопороговым возбуждением //Математическое моделирование. 2005. Т. 17, No 6. C. 75. 26. Kazantsev V.B., Nekorkin V.I. Spiking patterns emerging from wave instabilities in a one-dimensional neural lattice //Phys. Rev. E. 2003. Vol. 68. P. 017201. 27. Некоркин В.И., Щапин Д.С., Дмитричев А.С. Сложная волновая динмика ансамбля нейроноподобных элементов со сложнопороговым возбуждением //Изв. вузов. Прикладная нелинейная динамика. 2007. Т. 15, No 1. C. 3. 28. Nekorkin V.I., et al. Heteroclinic contours and self-replicated solitary waves in a reaction-diffusion lattice with complex threshold excitation// Phisyca D. 2008 (принята к печати). 29. Nekorkin V.I., Velarde M.G. Sinergetic phenomena in active lattices. Springer-Verlag, 2002. 357 p.

Short text (in English): 
Full text: