Analyzing oscillations for an N-node recurrent neural networks model with time delays and general activation functions

IEEE Transactions on Circuits and Systems I: Regular Papers

Volumn 58, Issue 8, 2011, Pages 1877-1887

  Feng, Chunhua ^a   Plamondon, Rjean ^b   O'Reilly, Christian ^b  

^a GUANGXI NORMAL UNIVERSITY   (China)

^b ÉCOLE POLYTECHNIQUE DE MONTRÉAL   (Canada)

Author keywords

Linear matrix inequality; oscillations; recurrent neural networks; time delay

Indexed keywords

LINEAR MATRIX INEQUALITIES; TIME DELAY;

ACTIVATION FUNCTIONS; DELAY SYSTEMS; NETWORK MODELING; OSCILLATIONS;

RECURRENT NEURAL NETWORKS;

EID: 79960993629     PISSN: 15498328     EISSN: None     Source Type: Journal    
DOI: 10.1109/TCSI.2011.2112550     Document Type: Article

References (51)

1. J. Bélair, S. J. Campbell, and V. D. Driessche, "Frustration, stability and delay-induced oscillations in a neural network model," SIAM J. Appl. Math., vol. 56, no. 1, pp. 245-255, 1996. (Pubitemid 126598148)
2. J. Cao and J. Wang, "Global exponential stability and periodicity of recurrent neural networks with time delays," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 5, pp. 920-931, 2005. (Pubitemid 40773619)
3. J. Cao and J. Wang, "Global asymptotic stability of a general class of recurrent neural networks with time-varying delays," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 50, no. 1, pp. 34-44, 2003.
4. N. Chafee, "A bifurcation problem for a functional differential equation of finitely retarded type," J. Math. Anal. Appl., vol. 35, no. 2, pp. 312-348, 1971.
5. B. Chen and J. Wang, "Global exponential periodicity and global exponential stability of a class of recurrent neural networks with various activation functions and time-varying delays," Neural Netw., vol. 20, no. 10, pp. 1067-1080, 2007. (Pubitemid 350180352)
6. W.-H. Chen and W. X. Zheng, "Global exponential stability of impulsive neural networks with variable delay: An LMI approach," IEEE Trans. Circuits Syst. II, Exp. Brief, vol. 56, no. 6, pp. 1248-1259, 2009.
7. C. Feng, C. O'Reilly, and R. Plamondon, "Permanent oscillations in a 3-node recurrent neural network model," Neurocomputing, vol. 74, no. 1-3, pp. 274-283, 2010.
8. C. Feng, R. Plamondon, and C. O'Reilly, "On some necessary and sufficient conditions for a recurrent neural network model with time delays to generate oscillations," IEEE Trans. Neural Netw., vol. 21, no. 8, pp. 1197-1205, 2010.
9. A. Fichera and A. Pagano, "Application of neural dynamic optimization to combustion-instability control," Appl. Energy, vol. 83, no. 3, pp. 253-264, 2006. (Pubitemid 41697706)
10. M. Forti and A. Tesi, "New conditions for global stability of neural networks with application to linear and quadratic programming problems," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 42, no. 7, pp. 354-366, 1995.
11. B. Gao and W. Zhang, "Equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function," IEEE Trans. Neural Netw., vol. 19, no. 5, pp. 782-794, 2008. (Pubitemid 351728815)
12. Z. Guo and L. Huang, "LMI conditions for global robust stability of delayed neural networks with discontinuous neuron activations," Appl. Math. Comput., vol. 215, no. 3, pp. 889-900, 2009.
13. S. Guo and L. Huang, "Periodic oscillation for a class of neural networks with variable coefficients," Nonlinear Anal. Real World Appl., vol. 6, no. 3, pp. 545-561, 2005. (Pubitemid 40232070)
14. Y. He, G. Liu, and D. Rees, "New delay-dependent stability criteria for neural networks with time-varying delay," IEEE Trans. Neural Netw., vol. 18, no. 1, pp. 310-314, 2007. (Pubitemid 46062938)
15. J. Hale, Theory of Functional Differential Equations. New York,: Springer-Verlag, 1977.
16. S. Hu and J. Wang, "Global stability of a class of continuous-time recurrent neural networks," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 9, pp. 1334-1347, 2002.
17. Y. Isomura, Y. F. Tsukamoto, and M. Takada, "A network mechanism underlying hippocampal seizure-like synchronous oscillations," Neurosci. Res., vol. 61, no. 3, pp. 227-233, 2008.
18. H. Jin and M. Zacksenhouse, "Oscillatory neural networks for robotic yo-yo control," IEEE Trans. Neural Netw., vol. 14, no. 2, pp. 317-325, 2003.
19. V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations. Boston, MA: Kluwer, 1992, pp. 104-106.
20. L. Li and L. Huang, "Dynamical behaviors of a class of recurrent neural networks with discontinuous neuron activations," Appl. Math. Model., vol. 33, no. 12, pp. 4326-4336, 2009.
21. Y. Liu, Z. Wang, and X. Liu, "Global exponential stability of a generalized recurrent neural networks with discrete and distributed delays," Neural Netw., vol. 19, no. 5, pp. 667-675, 2006.
22. S. Long and D. Xu, "Delay dependent stability analysis for impulsive neural networks with time varying delays," Neurocomputing, vol. 71, no. 7-9, pp. 1705-1713, 2008. (Pubitemid 351373170)
23. C. Y. Lu, "A delay-range-dependent approach to design state estimator for discrete-time recurrent neural networks with interval time-varying delay," IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 11, pp. 1163-1167, 2008.
24. A. N. Michel and D. Liu, Qualitative Analysis and Synthesis of Recurrent Neural Networks. New York: Marcel Dekker, Inc., 2002.
25. T. Nishikawa, Y. C. Lai, and F. C. Hoppensteadt, "Capacity of oscillatory associative-memory networks with error-free retrieval," Phys. Rev. Lett., vol. 92, no. 10, pp. 108-101, 2004.
26. R. Plamondon, "A kinematic theory of rapid human movements: Part I: Movement representation and generation," Biol. Cybern., vol. 72, no. 4, pp. 295-307, 1995.
27. R. Plamondon, "A kinematic theory of rapid human movements: Part II: Movement time and control," Biol. Cybern., vol. 72, no. 4, pp. 309-320, 1995.
28. R. Plamondon, "A kinematic theory of rapid human movements: Part III: Kinetic outcomes," Biol. Cybern., vol. 78, no. 2, pp. 133-145, 1998. (Pubitemid 128636398)
29. R. Plamondon, C. Feng, and A. Woch, "A kinematic theory of rapid human movements: Part IV: A formal mathematical proof and new insights," Biol. Cybern., vol. 89, no. 2, pp. 126-138, 2003.
30. F. Qiu, B. Cui, and W. Wu, "Global exponential stability of high order recurrent neural network with time-varying delays," Appl. Math. Model., vol. 33, no. 1, pp. 198-210, 2009.
31. R. Rakkiyappan and P. Balasubramaniam, "Delay-dependent asymptotic stability for stochastic delayed recurrent neural networks with time varying delays," Appl. Math. Comput., vol. 198, pp. 526-533, 2008. (Pubitemid 351400727)
32. A. Ruiz, D. H. Owens, and S. Townley, "Existence, learning, and replication of periodic motion in recurrent neural networks," IEEE Trans. Neural Netw., vol. 9, no. 4, pp. 651-661, 1998. (Pubitemid 128750026)
33. H. Shao, "Delay-dependent approaches to globally exponential stability for recurrent neural networks," IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 6, pp. 591-595, 2008.
34. Y. Shen, "LMI-based stability criteria with auxiliary matrices for delayed recurrent neural networks," IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 8, pp. 881-815, 2008.
35. Q. Song, "Exponential stability of recurrent neural networks with both time-varying delays and general activation functions via LMI approach," Neurocomputing, vol. 71, no. 13-15, pp. 2823-2830, 2008.
36. M. Steriade and I. Timofeev, "Neuronal plasticity in thalamocortical networks during sleep and waking oscillations," Neuron, vol. 37, no. 4, pp. 563-576, 2003. (Pubitemid 36262979)
37. Y. Tan and M. Tan, "Global asymptotical stability of continuous-time delayed neural networks without global Lipschitz activation functions," Commun. Nonlinear Sci. Numer. Simul., vol. 14, no. 11, pp. 3715-3722, 2009.
38. S. Townley, A. Ilchmann, M. G.Weib, W. McClements, A. C. Ruiz, D. H. Owens, and D. Pratzel-Wolters, "Existence and learning of oscillations in recurrent neural networks," IEEE Trans. Neural Netw., vol. 11, no. 1, pp. 205-214, 2000. (Pubitemid 32132901)
39. L. Wang and Z. Xu, "Sufficient and necessary conditions for global exponential stability of discrete-time recurrent neural networks," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 6, pp. 1373-1380, 2006. (Pubitemid 44218358)
40. M. Wu, F. Liu, P. Shi, Y. He, and R. Yokoyama, "Improved freeweighting matrix approach for stability analysis of discrete-time recurrent neural networks with time-varying delay," IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 7, pp. 690-694, 2008.
41. J. Xu, Y. Y. Cao, Y. Sun, and J. Tang, "Absolute exponential stability of recurrent neural networks with generalized activation function," IEEE Trans. Neural Netw., vol. 19, no. 6, pp. 1075-1089, 2008.
42. D. Yang, X. Liao, Y. Chen, S. Guo, and H. Wang, "New delay-dependent global asymptotic stability criteria of delayed Hopfield neural networks," Nonlinear Anal. RealWorld Appl., vol. 9, no. 5, pp. 1894-1904, 2008.
43. C. K. Young and J. J. Eggermont, "Coupling of mesoscopic brain oscillations: Recent advances in analytical and theoretical perspectives," Prog. Neurobiol., vol. 89, no. 1, pp. 61-78, 2009.
44. J. Yu, K. Zhang, S. Fei, and T. Li, "Simplified exponential stability analysis for recurrent neural networks with discrete and distributed time-varying delays," Appl. Math. Comput., vol. 205, no. 1, pp. 465-474, 2008.
45. H. Zhang, Z. Wang, and D. Liu, "Global asymptotic stability of recurrent neural networks with multiple time-varying delays," IEEE Trans. Neural Netw., vol. 19, no. 5, pp. 855-873, 2008. (Pubitemid 351728821)
46. H. Zhang, Z.Wang, and D. Liu, "Robust exponential stability of recurrent neural networks with multiple time-varying delays," IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 8, pp. 730-734, 2007. (Pubitemid 47250016)
47. Y. Zhang and Q. G. Wang, "Stationary oscillation for high-order hopfield neural networks with time delays and impulses," J. Comput. Appl. Math., vol. 231, no. 1, pp. 473-477, 2009.
48. Y. Zhang, "Stationary oscillation for cellular neural networks with time delays and impulses," Math. Comput. Simul., vol. 79, no. 10, pp. 3174-3178, 2009.
49. Y. Zhang, W. Ma, and B. Cai, "From Zhang neural network to Newton iteration for matrix inversion," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 7, pp. 1405-1415, 2009.
50. C. D. Zheng, H. Zhang, and Z.Wang, "Delay-dependent globally exponential stability criteria for static neural networks: An LMI approach," IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 7, pp. 605-609, 2009.
51. W. Zhou and J. M. Zurada, "Discrete-time recurrent neural networks with complex-valued linear threshold neurons," IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 8, pp. 669-673, 2009.