Stability and direction of Hopf bifurcations in a pair of identical tri-neuron network loops

Nonlinear Dynamics

Volumn 61, Issue 3, 2010, Pages 569-578

  Yuan, Sanling ^a   Li, Pan ^a  

^a UNIVERSITY OF SHANGHAI FOR SCIENCE AND TECHNOLOGY   (China)

Author keywords

Bifurcation direction; Delay; Neural networks; Periodic orbits; Stability

Indexed keywords

BIFURCATION DIRECTION; CENTER MANIFOLD REDUCTIONS; DELAY; NEURON NETWORKS; NUMERICAL SIMULATION; PERIODIC ORBITS; RETARDED DIFFERENTIAL EQUATIONS; THEORETICAL RESULT;

DIFFERENTIAL EQUATIONS; NEURAL NETWORKS; STABILITY;

HOPF BIFURCATION;

EID: 77956614897     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-010-9671-0     Document Type: Article

References (20)

1. Bergman, H., Feingold, A., Nini, A., Raz, A., Slovin, H., Abeles, M., Vaadia, E.: Physiological aspects of information processing in the basal ganglia of normal and Parkinsonian primates. Trends Neurosci. 21, 32-38 (1998)
2. Boccaletti, S., Pecora, L. M., Pelaez, A.: Unifying framework for synchronization of coupled dynamical systems. Phys. Rev. E 63 (3), 066219 (2001)
3. Campbell, S. A., Edwards, R., Van den Driessche, P.: Delayed coupling between two neural network loops. SIAM J. Appl. Math. 65 (1), 316-335 (2004)
4. Chow, S. N., Hale, J. K.: Methods of Bifurcation Theory. Springer, New York (1982)
5. Collins, J. J., Stewart, I.: Hexapodal gaits and coupled nonlinear oscillator models. Biol. Cybern. 68, 287-298 (1993)
6. Collins, J. J., Stewart, I.: A group-theoretic approach to rings of coupled biological oscillators. Biol. Cybern. 71, 95-103 (1994)
7. Faria, T.: On a planar system modelling a neuron network with memory. J. Differ. Equ. 168, 129-149 (2000)
8. Faria, T., Magalháes, L. T.: Normal form for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differ. Equ. 122, 181-200 (1995)
9. Faria, T., Magalháes, L. T.: Normal form for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Differ. Equ. 122, 201-224 (1995)
10. Golubitsky, M., Stewart, I.: Nonlinear dynamics of networks: the groupoid formalism. Bull. Am. Math. Soc. 43, 305-364 (2006)
11. Gray, C. M.: Synchronous oscillations in neuronal systems: Mechanism and functions. J. Comput. Neurosci. 1, 11-38 (1994)
12. Grosse, P., Cassidy, M. J., Brown, P.: EEG-EMG, MEG-EMG and EMG-EMG frequency analysis: Physiological principles and clinical applications. Clin. Neurophysiol. 113, 1523-1531 (2002)
13. Hale, J. K., Verduyn Lunel, S. M.: Introduction to Functional Differential Equations. Springer, New York (1993)
14. Hsu, C. H., Yang, T. S.: Periodic oscillations arising and death in delay-coupled neural loops. Int. J. Bifur. Chaos 17, 4015-4032 (2007)
15. Kandel, E. R., Schwartz, J. H., Jessell, T. M.: Principles of Neural Science. McGraw-Hill, New York (2000)
16. Peng, Y., Song, Y.: Stability switches and Hopf bifurcations in a pair of identical tri-neuron network loops. Phys. Lett. A 373, 1744-1749 (2009)
17. Dahlem, M. A., Hiller, G., Panchuk, A., Schöll, E.: Dynamics of delay-coupled excitable neural systems. Int. J. Bifur. Chaos 19, 745-752 (2009)
18. Song, Y., Tade, M. O., Zhang, T.: Bifurcation analysis and spatiotemporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling. Nonlinearity 22, 975-1001 (2009)
19. Song, Y., Makarov, V. A., Velarde, M. G.: Stability switches, oscillatory multistability, and spatiotemporal patterns of nonlinear oscillations in recurrently delay coupled neural networks. Biol. Cybern. 101, 147-167 (2009)
20. Tass, P. A.: Phase Resetting in Medicine and Biology: Stochastic Modeling and Data Analysis. Springer, Berlin (1999)