Bifurcations due to small time-lag in coupled excitable systems

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 15, Issue 5, 2005, Pages 1775-1785

  Burić, Nikola ^a   Todorovic, Dragana ^a  

^a UNIVERSITY OF BELGRADE   (Serbia)

Author keywords

FitzHugh Nagumo excitable systems; ODE; Time lag

Indexed keywords

APPROXIMATION THEORY; DIFFERENTIAL EQUATIONS; DYNAMICS; MATHEMATICAL MODELS; NUMERICAL METHODS; RELAXATION OSCILLATORS;

FITZHUGH-NAGUMO EXCITABLE SYSTEMS; ODE; OSCILLATORY DYNAMICS; TIME LAG;

DELAY CIRCUITS;

EID: 22144487735     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127405012831     Document Type: Article

References (31)

1. Arrowsimth, D. [1990] An Introduction to Dynamical Systems (Cambridge University Press, Cambridge).
2. Burić, N. & Todorovic, D. [2003] "Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling," Phys. Rev. E67, 066222.
3. Campbell, S. R. & Wang, D. [1996] "Relaxation oscillators with time delay coupling," The Ohio State University CIS-Technical Report No. 47.
4. Clay, J. R. & Shrier, A. [1999] "On the role of sub-threshold dynamics in neuronal signaling," J. Theor. Biol. 197, 207-216.
5. Coullet, P. & Spiegel, E. [1983] "Amplitude equations for systems with competing instabilities," SIAM J. Appl. Math. 776-821.
6. Ermentrout, G. B. & Kopell, N. [1990] "Oscillator death in systems of coupled neural oscillators," SIAM J. App. Math. 50, 25-46.
7. Faria, T. & Magalhaes, L. T. [1995] "Normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation," J. Diff. Eqs. 122, 181-200.
8. Faro, J. & Velasco, S. [1997] "An approximation for prey-predator models with time delay," Physica D110, 313-322.
9. FitzhHugh, R. [1955] "Mathematical models of threshold phenomena in the nerve membrane," Bull. Math. Biophys. 17, 257-278.
10. Fox, J. J., Jayaprakash, C., Wang, D. & Campbell, S. R. [2001] "Synchronization in relaxation oscillator networks with conduction delays," Neural Comput. 13, 1003-1021.
11. Gopalsamy, K. & Leung, I. [1996] "Delay induced periodicity in a neural network of excitation and inhibition," Physica D89, p. 395.
12. Grossberg, S. [1967] "Nonlinear difference-differential equations in prediction and learning theory," Proc. Natl. Acad. Sci. USA 58, 1329-1334.
13. Hoppensteadt, F. C. & Izhikevich, E. M. [1997] Weakly Connected Neural Networks (Springer-Verlag, NY).
14. Izhikevich, E. M. [1998] "Phase models with explicit time delay," Phys. Rev. E58, 905-908.
15. Izhikevich, E. M. [2000] "Neural excitability, spiking and bursting," Int. J. Bifurcation and Chaos 10, 1171-1266.
16. Kim, S., Park, S. H. & Ryu, C. S. [1997] "Multystability in coupled oscillator systems with time delay," Phys. Rev. Lett. 79, p. 2911.
17. Kuznetsov, Y. [1995] Elements of Applied Bifurcation Theory (Springer-Verlag, NY).
18. Kuznetsov, Y. [1997] "Explicit normal form coefficients for all codim 2 bifurcations of equilibria in ODEs," Report MAS-R9730.
19. Meinardus, G. & Nuernberg, G. (eds.) [1985] Delay Equations, Approximation and Application (Birkhauser Verlag, Basel-Boston-Studgart).
20. Murray, J. D. [1990] Mathematical Biology (Springer, NY).
21. Ncube, I., Campbell, S. A. & Wu, J. [2002] "Change in criticality of synchronous Hopf bifurcation in a multiply delayed neural system," Fields Inst. Commun. 36, 1-15.
22. Nagumo, J., Arimoto, S. & Yoshizava, S. [1962] "An active pulse transmission line simulating nerve axon," Proc IREE 50, p. 2061.
23. Nakamura, Y., Tominaga, F. & Munakata, T. [1994] "Clustering behaviour of time-delayed nearest-neighbor coupled oscillators," Phys. Rev. E49, p. 4849.
24. Olier, J. & Belair, J. [1997] "Bifurcation, stability and monotonicity properties of a delayed neural network model," Physica D102, 349-363.
25. Ramana Reddy, D. V., Sen, A. & Johnston, G. L. [1998] "Time delay induced death in coupled limit cycle oscillators," Phys. Rev. Lett. 80, 5109-5112.
26. Ramana Reddy, D. V., Sen, A. & Johnston, G. L. [1999] "Time delay effects on coupled limit cycle oscillators at Hopf bifurcation," Physica D129, p. 15.
27. Schuster, H. G. & Wagner, P. [1989] "Mutual entrainment of two limit cycle oscillators with time delayed coupling," Prog. Theor. Phys. 81, 939-945.
28. Shayer, L. P. & Campbell, S. A. [2000] "Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays," SIAM J. Appl. Math. 61, 673-700.
29. Strogatz, S. H. [1998] "Death by delay," Nature 394, 316-317.
30. Ye, H., Michel, A. & Wang, K. [1995] "Qualitative analysis of Cohen-Grossberg neural network with multiple delays," Phys. Rev. E51, p. 2611.
31. Yeung, M. K. S. & Strogatz, S. H. [1999] "Time delay in the Kuramoto model of coupled oscillators," Phys. Rev. Lett. 82, 648-651.