Chaotic oscillations in a map-based model of neural activity

Chaos

Volumn 17, Issue 4, 2007, Pages

  Courbage, M ^a   Nekorkin, V I ^b   Vdovin, L V ^b  

^a UNIVERSITÉ PARIS DIDEROT   (France)

^b LOBACHEVSKY STATE UNIVERSITY OF NIZHNY NOVGOROD   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85000257165     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2795435     Document Type: Article

References (28)

1. E. R. Kandel, J. H. Schwartz, and T. M. Jessell, Principles of Neural Science (Prentice-Hall, Englewood Cliffs, NJ, 1991).
2. M. I. Rabinovich, P. Varona, A. I. Selverston, and H. D. I. Abarbanel, " Dynamical principles in neuroscience., " Rev. Mod. Phys. 78, 1213 (2006).
3. H. R. Wilson and J. D. Cowan, " Excitatory and inhibitory interaction in localized population of model neurons., " Biophys. J. 12, 1 (1972).
4. R. FitzHugh, " Mathematical models of the threshold phenomena in the nerve membrane., " Bull. Math. Biophys. 17, 257 (1955).
5. J. L. Hindmarsh and R. M. Rose, " A model of neuronal bursting using three coupled first order differential equations., " Philos. Trans. R. Soc. London, Ser. B 221, 87 (1984).
6. C. Morris and H. Lecar, " Voltage oscillations in the barnacle giant muscle fiber., " Biophys. J. 25, 87 (1981).
7. D. R. Chialvo, " Generic excitable dynamics on the two-dimensional map., " Chaos, Solitons Fractals 5, 461 (1995).
8. O. Kinouchi and M. Tragtenberg, " Modeling neurons by simple maps., " Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 2343 (1996).
9. S. Kuva, G. Lima, O. Kinouchi, M. Tragtenberg, and A. Roque, " A minimal model for excitable and bursting element., " Neurocomputing 38-40, 255 (2001).
10. G. de Vries, " Bursting as an emergent phenomenon in coupled chaotic maps., " Phys. Rev. E 64, 051914 (2001).
11. N. F. Rulkov, I. Timofeev, and M. Bazhenov, " Oscillations in large-scale cortical networks: map-based model., " J. Comput. Neurosci. 17, 203 (2004).
12. R. D. Traub, J. G. R. Jefferys, and M. A. Whittington, Fast Oscillations in Cortical Circuits (MIT Press, Cambridge, 1999).
13. R. Llinas and Y. Yarom, " Oscillatory properties of guinea-pig inferior olivary neurones and their pharmacological modulation: An in vitro study., " J. Physiol. (London) 315, 569 (1986).
14. R. Llinas, I of Vortex. From Neurones to Self (MIT Press, Cambridge, 2002).
15. B. Cazelles, M. Courbage, and M. Rabinovich, " Anti-phase regularization of coupled chaotic maps modelling bursting neurons., " Europhys. Lett. 56, 504 (2001).
16. N. F. Rulkov, " Regularization of synchronized chaotic bursts., " Phys. Rev. Lett. 86, 183 (2001).
17. N. F. Rulkov, " Modeling of spiking-bursting neural behavior using two-dimensional map., " Phys. Rev. E 65, 041922 (2002).
18. A. L. Shilnikov and N. F. Rulkov, " Origin of chaos in a two dimensional map modeling spiking-bursting neural activity., " Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 3325 (2003).
19. A. L. Shilnikov and N. F. Rulkov, " Subthreshold oscillations in a map-based neuron model., " Phys. Lett. A 328, 177 (2004).
20. H. Tanaka, " Design of bursting in two-dimensional discrete-time neuron model., " Phys. Lett. A 350, 228 (2006).
21. M. Courbage, V. B. Kazantsev, V. I. Nekorkin, and V. Senneret, " Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons., " Chaos 14, 1148 (2004).
22. E. M. Izhikevich and F. Hoppensteadt, " Classification of bursting mappings., " Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 3847 (2004).
23. V. S. Afraimovich and S.-B. Hsu, Lectures on Chaotic Dynamical Systems (American Mathematical Society, Providence, 2003), p. 354.
24. V. S. Afraimovich and L. P. Shilnikov, " Strange attractors and quasiattractors., " in Nonlinear Dynamics and Turbulence, edited by, G. I. Barenblatt,,, G. Iooss,, and, D. D. Joseph, (Pitam, Boston, 1983), pp. 1-34.
25. V. I. Arnold, V. S. Afraimovich, Yu. S. Ilyashenko, L. P. Shilnikov, Bifurcation Theory, Dyn. Sys. V. Encyclopaedia Mathematics Sciences (Springer, Berlin, 1994).
26. L. S. Bernardo and R. P. Foster, " Oscillatory behaviour in the inferiorolive neurons: mechanism, modulation, cell agregates., " Brain Res. Bull. 17, 773 (1986).
27. R. S. K. Wang and D. A. Prince, " After potential generation in hippocampal pyramidal cells., " J. Neurophysiol. 45, 86 (1981).
28. M. Deschenes, J. P. Roy, and M. Steriade, " Thalamic bursting mechanism: An inward slow current revealed by membrane hyperpolarization., " Brain Res. 239, 289 (1982).