Two-parameter bifurcations in a network of two neurons with multiple delays

Journal of Differential Equations

Volumn 244, Issue 2, 2008, Pages 444-486

  Guo, Shangjiang ^a,b   Chen, Yuming ^b   Wu, Jianhong ^c  

^a HUNAN UNIVERSITY   (China)

^b WILFRID LAURIER UNIVERSITY   (Canada)

^c YORK UNIVERSITY   (Canada)

Author keywords

Bifurcation; Center manifold; Delay; Neural network; Normal form; Stability

Indexed keywords

EID: 36849021007     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2007.09.008     Document Type: Article

References (41)

1. Algaba A., Merino M., Freire E., Gamero E., and Rodriguez-Luis A.J. Some results on Chua's equation near a triple-zero linear degeneracy. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003) 58-608
2. Buono P.L., and Bélair J. Restrictions and unfolding of double Hopf bifurcation in functional differential equations. J. Differential Equations 189 (2003) 234-266
3. Bélair J., and Campbell S.A. Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM J. Appl. Math. 54 (1994) 1402-1424
4. Bélair J., Campbell S.A., and van den Driessche P. Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math. 56 (1996) 245-255
5. Broer H.W., and Vegter G. Subordinate Šil'nikov bifurcations near some singularities of vector fields having low codimension. Ergodic Theory Dynam. Systems 4 (1984) 509-525
6. Chen Y., and Wu J. Existence and attraction of a phase-locked oscillation in a delayed network of two neurons. Differential Integral Equations 14 (2001) 1181-1236
7. Diekmann O., van Gils S.A., Verduyn Lunel S.M., and Walther H.-O. Delay Equations, Functional-, Complex-, and Nonlinear Analysis (1995), Springer-Verlag, New York
8. 3. Nonlinearity 11 (1998) 1037-1047
9. Elphick C., Tirapegui E., Brachet M.E., Coullet P., and Iooss G. A simple global characterization for normal forms of singular vector fields. Phys. D 29 (1987) 95-127
10. Faria T. On a planar system modelling a neuron network with memory. J. Differential Equations 168 (2000) 129-149
11. Faria T., and Magalhães L.T. Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities. J. Dynam. Differential Equations 8 (1996) 35-70
12. Freire E., Gamero E., Rodríguez-Luis A.J., and Algaba A. A note on the triplezero linear degeneracy: Normal forms, dynamical and bifurcation behaviors of an unfolding. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 12 (2002) 2799-2820
13. Gavrilov N. On some bifurcations of an equilibrium with one zero and a pair of pure imaginary roots. Methods of Qualitative Theory of Differential Equations (1978), GGU, Gorkii
14. Guckenheimer J. On a codimension two bifurcation. Dynamical Systems and Turbulence. Warwick 1980 (Coventry, 1979/1980). Lecture Notes in Math. vol. 898 (1981), Springer-Verlag, Berlin 99-142
15. Guckenheimer J. Multiple bifurcation problems of codimension two. SIAM J. Math. Anal. 15 (1984) 1-49
16. Guckenheimer J., and Holmes P.J. Nonlinear Oscillations: Dynamical System and Bifurcations of Vector Fields (1983), Springer-Verlag, New York
17. Guo S., and Huang L. Periodic solutions in an inhibitory two-neuron network. J. Comput. Appl. Math. 161 (2003) 217-229
18. Guo S., Huang L., and Wang L. Linear stability and Hopf bifurcation in a two-neuron network with three delays. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004) 2799-2810
19. Guo S., Huang L., and Wu J. Global attractivity of a synchronized periodic orbit in a delayed network. J. Math. Anal. Appl. 281 (2003) 633-646
20. Guo S., Huang L., and Wu J. Convergence and periodicity in a delayed network of neurons with threshold nonlinearity. Electron. J. Differential Equations 2003 61 (2003) 1-14
21. Guo S., Huang L., and Wu J. Regular dynamics in a delayed network of two neurons with all-or-none activation functions. Phys. D 206 (2005) 32-48
22. Hale J.K., and Verduyn Lunel S.M. Introduction to Functional Differential Equations (1993), Springer-Verlag, New York
23. Holmes P.J. Unfolding a degenerate nonlinear oscillators: A codimension two bifurcation. In: Helleman R.H.G. (Ed). Nonlinear Dynamics (1980), New York Academy of Science, New York 473-488
24. Hopfield J.J. Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81 (1984) 3088-3092
25. Iooss G., and Adelmeyer M. Topics in Bifurcation Theory and Applications (1992), World Scientific, Singapore
26. Iooss G., and Langford W.F. Conjectures on the routes to turbulence via bifurcation. In: Helleman R.H.G. (Ed). Nonlinear Dynamics (1980), New York Academy of Science, New York 489-505
27. Kuznetsov Y.A. Elements of Applied Bifurcation Theory. second ed. (1998), Springer-Verlag, New York
28. Li S., Liao X., Li C., and Wong K.-W. Hopf bifurcation of a two-neuron network with different discrete time delays. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005) 1589-1601
29. Milton J. Dynamics of Small Neural Populations (1996), Amer. Math. Soc., Providence, RI
30. Olien L., and Bélair J. Bifurcations, stability, and monotonicity properties of a delayed neural network model. Phys. D 102 (1997) 349-363
31. Ruan S., and Wei J. Periodic solutions of planar systems with two delays. Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 1017-1032
32. Shayer L.P., and Campbell S.A. Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays. SIAM J. Appl. Math. 61 (2000) 673-700
33. Skinner F.K., Bazzazi H., and Campbell S.A. Two-cell to N-cell heterogeneous, inhibitory networks: Precise linking of multistable and coherent properties. J. Comput. Neurosci. 18 (2005) 343-352
34. Takens F. A nonstabilizable jet of a singularity of a vector field. Dynamical Systems, Proc. Sympos. Univ. Bahia, Salvador, 1971 (1973), Academic Press, New York 583-597
35. Takens F. Normal forms for certain singularities of vector fields. Ann. Inst. Fourier (Grenoble) 23 (1973) 163-195
36. Takens F. Singularities of vector fields. Publ. Math. Inst. Hautes Études Sci. 43 (1974) 47-100
37. Tu F., Liao X., and Zhang W. Delay-dependent asymptotic stability of a two-neuron system with different time delays. Chaos Solitons Fractals 28 (2006) 437-447
38. Wei J., and Ruan S. Stability and bifurcation in a neural network model with two delays. Phys. D 130 (1999) 255-272
39. Wei J.J., and Velarde M.G. Bifurcation analysis and existence of periodic solutions in a simple neural network with delays. Chaos 14 (2004) 940-953
40. Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. second ed. (2003), Springer-Verlag, New York
41. Wu J. Introduction to Neural Dynamics and Signal Transmission Delay (2001), Walter de Gruyter, New York