Simple zero singularity analysis in a coupled FitzHugh-Nagumo neural system with delay

Neurocomputing

Volumn 73, Issue 4-6, 2010, Pages 874-882

  Zhen, Bin ^a   Xu, Jian ^a  

^a TONGJI UNIVERSITY   (China)

Author keywords

Delay; FHN model; Pitchfork bifurcation; Simple zero singularity; Transcritical bifurcation

Indexed keywords

CENTER MANIFOLD REDUCTIONS; COUPLED NEURONS; COUPLING STRENGTHS; NEURAL SYSTEMS; NORMAL FORM; PITCHFORK BIFURCATION; PITCHFORK BIFURCATIONS; SIGNAL TRANSMISSION; SIGNIFICANT IMPACTS; SINGULARITY ANALYSIS; STEADY-STATE BIFURCATION; TRANSCRITICAL BIFURCATION;

BIFURCATION (MATHEMATICS);

ANALYTIC METHOD; ARTICLE; ARTIFICIAL NEURAL NETWORK; COMPUTER SIMULATION; FITZHUGH NAGUMO NEURAL SYSTEM; LINEAR SYSTEM; MATHEMATICAL ANALYSIS; MATHEMATICAL COMPUTING; MATHEMATICAL PARAMETERS; NONLINEAR SYSTEM; PRIORITY JOURNAL; QUANTITATIVE STUDY; SIMPLE ZERO SINGULARITY ANALYSIS; TIME DELAY;

EID: 75749148282     PISSN: 09252312     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.neucom.2009.09.015     Document Type: Article

References (27)

1. FitzHugh R. Impulses and physillogical states in theoretical models of nerve membrane. Biophys. J. 1 (1961) 445-466
2. Nagumo J., Arimoto S., and Yoshizawa S. An active pulse transmission line simulating nerve axon. Proc. IRE 50 (1962) 2061-2070
3. Hodgkin A.L., and Huxley A.F. A quantitative description of membrane and its application to conduction and excitation in nerve. J. Physiol. 117 (1952) 500-544
4. Bautin A.N. Qualitative investigation of a particular nonlinear system. J. Appl. Math. Mech. 39 (1975) 606-615
5. Tetsushi U., Hisayo M., Takuji K., and Hiroshi K. Bifurcation and chaos in coupled BVP oscillators. Int. J. Bifur. Chaos 14 (2004) 1305-1324
6. Tetsushi U., and Hiroshi K. Bifurcation in asymmetrically coupled BVP oscillators. Int. J. Bifur. Chaos 13 (2003) 1319-1327
7. Kunichika T., Kazuyuki A., and Hiroshi K. Bifurcations in synaptically coupled BVP neurons. Int. J. Bifur. Chaos 11 (2001) 1053-1064
8. Nikola B., and Dragana T. Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling. Phys. Rev. E 67 (2003) 0662215-21
9. Nikola B., Ines G., and Nebojsa V. Type I vs. type II excitable systems with delayed coupling. Chaos, Solitons Fractals 2 (2005) 1221-1233
10. Wang Q.Y., Lu Q.S., and Chen G.R. Bifurcation and synchronization of synaptically coupled FHN models with time delay. Chaos, Solitons Fractals 39 (2009) 918-925
11. Shepherd G.M. Neurobiology (1983), Oxford University, New York
12. Murray J.D. Mathematical Biology (1990), Springer, New York
13. Faria T., and Magalhaes L.T. Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Diff. Eq. 122 (1995) 201-224
14. Chow S.N., Li C., and Wang D. Normal Forms and Bifurcation of Planar Vector Fields (1994), Cambridge University, New York
15. Chua L.O., and Kokubu H. Normal forms for nonlinear vector fields-part I: theory and algorithm. IEEE Trans. Circuits Syst. 35 (1998) 863-880
16. Chua L.O., and Kokubu H. Normal forms for nonlinear vector fields-part II: applications. IEEE Trans. Circuits Syst. 35 (1998) 51-70
17. Hale J. Theory of Functional Differential Equations (1977), Springer, New York
18. Wang H.B., and Jiang W.H. Multiple stabilities analysis in a magnetic bearing system with time delays. Chaos, Solitons Fractals 27 (2006) 789-799
19. Jiang W.H., and Wei J.J. Bifurcation analysis in Van der Pol's oscillator with delayed feedback. J. Comput. Appl. Math. 213 (2008) 604-615
20. Jiang W.H., and Yuan Y. Bogdanov-Takens singularity in Van der Pol's oscillator with delayed feedback. Physica D 227 (2007) 149-161
21. Kakiuchi N., and Tchizawa K. On an explicit duck solution and delay in the FitzHugh-Nagumo equation. J. Diff. Eq. 141 (1997) 327-339
22. Yang K. Delay Differential Equation with Applications in Population Dynamics (1993), Academic Press, London
23. Wang Z.D., Liu Y.R., and Liu X.H. State estimation for jumping recurrent neural networks with discrete and distributed delays. Neural Networks 1 (2009) 41-48
24. Liu Y.R., Wang Z.D., and Liu X.H. Asymptotic stability for neural networks with mixed time-delays: the discrete-time case. Neural Networks 1 (2009) 67-74
25. Wang Y., Wang Z.D., and Liang J.L. Global synchronization for delayed complex networks with randomly occurred nonlinearities and multiple stochastic disturbances. J. Phys. A: Math. Theor. 13 (2009) 135101
26. Liang J.L., Wang Z.D., and Liu X.H. State estimation for coupled uncertain stochastic networks with missing measurements and time-varying delays: the discrete-time case. IEEE Trans. Neural Networks 20 (2009) 781-793
27. Liu Y.R., Wang Z.D., Liang J.L., and Liu X.H. Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time-delays. IEEE Trans. Neural Networks 20 (2009) 1102-1116