Existence and learning of oscillations in recurrent neural networks

IEEE Transactions on Neural Networks

Volumn 11, Issue 1, 2000, Pages 205-214

  Townley, S ^a   Ilchmann, A ^a   Weiss M G ^b   Mcclements, W ^a   Ruiz, A C ^a   Owens, D H ^c   Pratzel Wolters D ^b  

^a UNIVERSITY OF EXETER   (United Kingdom)

^b Rheinland Pfälzische Technische Universität Kaiserslautern Landau   (Germany)

^c UNIVERSITY OF SHEFFIELD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

ADAPTIVE CONTROL SYSTEMS; CONVERGENCE OF NUMERICAL METHODS; LEARNING ALGORITHMS; LEARNING SYSTEMS;

MONOTONE DYNAMICAL SYSTEMS;

RECURRENT NEURAL NETWORKS;

EID: 0033640658     PISSN: 10459227     EISSN: None     Source Type: Journal    
DOI: 10.1109/72.822523     Document Type: Article

References (30)

1. H. Asada and J. J. E. Slotine, Robot Analysis and Control. New York: Wiley, 1986.
2. A. Atiya and P. Baldi, "Oscillations and synchronizations in neural networks: An exploration of the labeling hypothesis," Int. J. Neural Syst., vol. 1, pp. 103-124, 1989.
3. A. Babloyantz, C. Lourenco, and A. J. Sepulchre, "Control of chaos in delay differential equations in a network of oscillators and in model cortex," Phys. D, vol. 86, pp. 274-283, 1995.
4. R. Blach and D. H. Owens, "Replication of periodic signals using a neural-type structured network," Centre for Systems and Control Engineering, University of Exeter, Int. Rep. 92/03, 1992.
5. S. Boyd and S. S. Sastry, "Necessary and sufficient conditions for parameter convergence in adaptive control," Automatica, vol. 22, pp. 629-639, 1986.
6. K. Doya and S. Yoshizawa, "Adaptive neural oscillator using continuous time backpropagation learning," Neural Networks, vol. 2, pp. 375-385, 1989.
7. B. Doyon, M. Cessac, M. Quoy, and M. Samuelides, "Control of the transition to chaos in neural networks with random connectivity," Int. J. Bifurcation Chaos, vol. 3, pp. 279-291, 1993.
8. S. Hastings, J. J. Tyson, and D. Webster, "Existence of periodic solutions for negative feedback equations," J. Differential Equations, vol. 25, pp. 39-64, 1977.
9. M. Hirsch, "Systems of differential equations which are competitive or cooperative III," Nonlinearity, vol. 1, pp. 51-71, 1988.
10. R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge: Cambridge Univ. Press, 1985.
11. Z. Li and J. J. Hopfield, "Modeling the olfactory bulb and its neural oscillatory processings," Biol. Cybern., vol. 61, pp. 379-392, 1989.
12. J. Mallet-Paret and H. Smith, "The Poincaré-Bendixson theorm for monotone cyclic feedback systems," J. Differential Equations, vol. 2, pp. 376-421, 1990.
13. J. Mallet-Paret and G. R. Sell, "The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay," J. Differential Equations, vol. 125, pp. 441-489, 1996.
14. 1 = [A + B(t)]x, with skew symmetric matric B(t)," SIAM J. Contr. Optimization, vol 15, pp. 163-176, 1977.
15. K. S. Narendra and A. Annaswamy, Stable Adaptive Systems. Englewood Cliffs, NJ: Prentice-Hall, 1987.
16. K. S. Narendra and K. Parthasarathy, "Identification and control of dynamical systems using neural networks," IEEE Trans. Neural Networks, vol. 1, pp. 4-26, 1990.
17. B. A. Pearlmutter, "Gradient calculations for dynamic recurrent neural networks: A survey," IEEE Trans. Neural Networks, vol. 6, pp. 1212-1228, 1995.
18. R. Reinke, "Adaptive regeln zum lernen und reproduzieren periodischer signale mit dynamischen netzwerken," Ph.D. dissertation, Universität Kaiserslautern.
19. W. J. Rugh, Linear Systems Theory. Englewood Cliffs, NJ: Prentice-Hall, 1996.
20. 2 approximation properties of recurrent neural networks," in Proc. 4th European Contr. Conf. (Electron. Publication WE-E G1), Brussels, July 1997.
21. _, "Existence, learning and replication of periodic motions in recurrent neural networks," IEEE Trans. Neural Networks, vol. 9, pp. 651-661, 1998.
22. S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1979.
23. H. Smith, "Convergene and oscillatory activation dynamics for cascades of neural nets with nearest neighbor competitive or cooperative interactions," Neural Networks, vol. 4, pp. 41-46, 1991.
24. _, Monotone Dynamical Systems. Providence, RI: Amer. Math. Soc., 1995.
25. R. V. Sole and L. M. de la Pride, "Controlling chaos in discrete neural networks," Phys. Lett. A, vol. 199, pp. 65-69, 1995.
26. R. Waschler, A learning controller for lineariable nonlinear systems.
27. R. V. Sole and L. M. de la Prida, "Controlling chaos in discrete neural networks," Phys. Lett. A, vol. 199, pp. 65-69, 1995.
28. M. G. Weiss, "Learning oscillations using adaptive control," Proc. ICANN 97, vol. LNCS 1327, pp. 331-336, 1997.
29. _, "Learning periodic signals with recurrent neural networks," Ph.D. dissertation, Fachbereich Math., Universität Kaiserslautern, 1999.
30. H. Yang and T. S. Dillon, IEEE Trans. Neural Networks, vol. 5, pp. 719-729, 1994.