Synchronization for two coupled oscillators with inhibitory connection

Mathematical Methods in the Applied Sciences

Volumn 33, Issue 7, 2010, Pages 892-903

  Xiao, Ke ^a   Guo, Shangjiang ^a  

^a HUNAN UNIVERSITY   (China)

Author keywords

Center manifold; Delay; Hopf bifurcation; Normal form; Oscillators; Stability; Synchronous

Indexed keywords

BIFURCATION PARAMETER; CENTER MANIFOLD; CENTER MANIFOLD REDUCTIONS; COUPLED OSCILLATORS; NEURAL OSCILLATOR; NORMAL FORM; NORMAL FORM THEORY; NUMERICAL SIMULATION; OSCILLATORY NEURAL NETWORKS; PERIODIC SOLUTION; THEORETICAL RESULT;

COMPUTER SIMULATION; NEURAL NETWORKS; OSCILLATORS (ELECTRONIC);

HOPF BIFURCATION;

EID: 77951643724     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.1225     Document Type: Article

References (25)

1. Barahona M, Pecora LM. Synchronization in small-world systems. Physical Review Letters 2002; 89:054101. (Pubitemid 34916880)
2. Luo AC. A theory for synchronization of dynamical systems. Communications in Nonlinear Science and Numerical Simulation 2009; 14:1901-1951.
3. Wang XF, Chen GR. Synchronous in small-world dynamical networks. International Journal of Bifurcation and Chaos 2002; 12:187-192.
4. Wu CW. Synchronization in Coupled Chaotic Circuits and Systems. World Scientific: Singapore, 2002.
5. Li CG, Chen GR. Synchronization in general complex dynamical networks with coupling delays. Physica A 2004; 343:263-278.
6. Guo S, Huang L, Wang L. Linear stability and hopf bifurcation in a two-neuron network with three delays. International Journal of Bifurcation and Chaos 2004; 14:2799-2810. (Pubitemid 39326275)
7. Guo S, Chen Y, Wu J. Two-parameter bifurcations in a network of two neurons with multiple delays. Journal of Differential Equations 2008; 244:444-486.
8. Aronson D, Ermentrout GB, Kopell N. Amplitude response of coupled oscillators. Physica D 1990; 41:403-449.
9. Ermentrout GB, Kopell N. Multiple pulse interactions and averaging in systems of coupled neural oscillators. Journal of Mathematical Biology 1991; 29:195-217.
10. Kawato M, Sokabe M, Suzuki R. Synergism and antagonism of neurons caused by an electrical synapse. Biological Cybernetics 1979; 34:81-89. (Pubitemid 10245394)
11. Hindmarsh JL, Rose RM. A model of the nerve impulse using two first-order differential equations. Nature 1982; 296:162-164.
12. Hansel D, Mato G, Meunier C. Phase dynamics for weakly coupled Hodgkin-Huxley neurons. Europhysics Letters 1993; 23:367-372.
13. Wilson HR, Cowan JD. Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal 1972; 12:1-24.
14. Kryukov VI, Borisyuk GN, Borisyuk RM, Kirillov AB, Kovalenko EL. Metastable and unstable states in the brain. In Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, Dobrushin RL, Kryukov VI, Toom AL (eds). Manchester University Press: Manchester, 1990; 225-357.
15. MacGregor RJ. Neural and Brain Modelling. Academic Press: New York, 1987.
16. Wang X, Rinzel J. Spindle rhythmicity in the reticularis thalami nucleus: synchronization among mutually inhibitory neurons. Neuroscience 1993; 53:899-904. (Pubitemid 23126086)
17. Cymbalyuk GS, Nikolaev EV, Borisyuk RM. In-phase and antiphase self-oscillations in a model of two electrically coupled pacemakers. Biological Cybernetics 1994; 71:153-160.
18. Marcus CM, Waughand FR, Westervelt RM. Nonlinear dynamics and stability of analog neural networks. Physica D 1991; 51:234-247.
19. Babcock KL, Westervelt RM. Complex dynamics in simple neural circuits. In Neural Networks for Computing, Denker JS (ed.). American Institute of Physics: New York, 1986; 288-293.
20. Babcock KL, Westervelt RM. Dynamics of simple electronic neural networks with added inertia. Physica D 1987; 23:464-469.
21. Gopalsamy K, Leung I. Delay induced periodicity in a neural netlet of excitation and inhibition. Physica D 1996; 89:395-426. (Pubitemid 126687262)
22. Guo S, Lamb JSW. Equivariant Hopf bifurcation for neutral functional differential equations. Proceedings of the American Mathematical Society 2008; 136:2031-2041.
23. Wu J. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 1998; 350:4799-4838. (Pubitemid 128166065)
24. Hassard B, Kazarinoff N, Wan YH. Theory of Application of Hopf Bifurcation. London Mathematical Society Lecture Notes Series, vol.41. Cambridge University Press: Cambridge, 1981.
25. Golubitsky M, Stewart IN. Hopf bifurcation in the presence of symmetry. Archive for Rational Mechanical Analysis 1985; 87:107-165.