Extreme stability and almost periodicity in continuous and discrete neuronal models with finite delays

ANZIAM Journal

Volumn 44, Issue 2, 2002, Pages 261-282

  Mohamad, S ^a,b   Gopalsamy, K ^b  

^a UNIVERSITI BRUNEI DARUSSALAM   (Brunei Darussalam)

^b FLINDERS UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 3042798176     PISSN: 14461811     EISSN: None     Source Type: Journal    
DOI: 10.1017/S1446181100013936     Document Type: Article

References (33)

1. C. T. H. Baker and A. Tang, "Generalised Halanay inequalities for Volterra functional differential equations and discretised versions", Report 299, Manchester Centre for Computational Mathematics, The University of Manchester, England, 1996.
2. A. S. Besicovitch, Almost Periodic Functions (Dover, New York, 1954).
3. V. E. Bondarenko, "Self-organization processes in chaotic neural networks under external periodic force", Int. J. Bifur. Chaos 7 (1997) 1887-1895.
4. D. S. Broomhead and A. Iserles (eds.), The dynamics of numerics and the numerics of dynamics (Clarendon Press, Oxford, 1992).
5. E. R. Caianiello and de Luca, "Decision equation for binary systems; application to neuronal behaviour", Kybernetik 3 (1966) 33-40.
6. F. Chapeau-Blondeau and G. Chauvet, "Stable, oscillatory and chaotic regimes in the dynamics of small neural networks with delay", Neural Networks 5 (1992) 735-744.
7. C. Corduneanu, Almost Periodic Functions (Interscience, New York, 1968).
8. R. Eckhorn, R. Bauer, W. Jordan, W. Kruse, M. Munk and H. J. Rertboeck, "Coherent oscillations: A mechanism for feature linking in the visual cortex", Biol. Cybern. 60 (1988) 121-130.
9. A. K. Engel, A. K. Kreiter, P. König and W. Singer, "Synchronization of oscillatory neural responses between striate and extrastriate visual cortical areas of the cat", Proc. Natl. Acad. Sci. 88 (1991) 6048-6052.
10. A. M. Fink, Almost Periodic Differential Equations (Springer, New York, 1974).
11. W. J. Freeman, "The physiology of perception", Sci. Amer. February (1991).
12. W. J. Freeman, Y. Yau and B. Burke, "Central pattern generating and recognizing in olfactory bulb: A correlation learning rule". Neural Networks 1 (1988) 277-288.
13. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics (Kluwer Academic Publishers, The Netherlands, 1992).
14. K. Gopalsamy and X. Z. He, "Delay-independent stability in bidirectional associative memory networks", IEEE Trans. Neural Networks 5 (1994) 998-1002.
15. K. Gopalsamy and X. Z. He, "Stability in asymmetric Hopfield nets with transmission delays", Physica D 76 (1994) 344-358.
16. K. Gopalsamy and X. Z. He, "Dynamics of an almost periodic logistic integrodifferential equation", Methods Appl. Anal. 2 (1995) 38-66.
17. K. Gopalsamy, X. Z. He and L. Wen, "Global attractivity and oscillations in an almost periodic delay logistic equation". Nonlinear Times Digest 1 (1994) 9-24.
18. C. M. Gray, P. König, A. K. Engel and W. Singer, "Oscillatory responses in cat visual cortex exhibit intercolumnar synchronization which reflects global stimulus properties". Nature 388 (1989) 334-337.
19. A. Halanay, Differential Equations (Academic Press, New York, 1966).
20. A. Hjelmfelt and J. Ross, "Pattern recognition, chaos and multiplicity in neural networks and excitable systems", Proc. Nat. Academy Sci. 91 (1994) 63-67.
21. V. I. Istratescu, Fixed Point Theory (D. Riedel Publ. Co., Dordrecht, 1981).
22. P. König and J. B. Schillen, "Stimulus-dependent assembly formation of oscillatory responses: I. Synchronization", Neural Comput. 3 (1991) 155-166.
23. L. P. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method with Applications (Academic Press, New York, 1961).
24. R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations (World Scientific, River Edge, NJ, 1994).
25. J. E. Moreira and D. M. Auto, "Intermittency in a neural network with variable threshold", Europhys. Lett. 21 (1993) 639.
26. E. Ott, C. Grebogi and J. A. Yorke, "Controlling chaos", Phys. Rev. Lett. 64 (1990) 1196-1199.
27. A. Rescigno, R. B. Stein, R. L. Stein and R. E. Poppele, "A neuronal model for the discharge patterns produced by cyclic inputs", Bull. Math. Biophys. 32 (1970) 337-353.
28. C. A. Skarda and W. J. Freeman, "How brains make chaos in order to make sense of the world", Brain Behavioural Sci. 10 (1987) 161-195.
29. A. M. Stuart and A. R. Humphries, Dynamical systems and numerical analysis (Cambridge University Press, Cambridge, 1996).
30. X. Wang and E. K. Blum, "Discrete-time versus continuous-time models of neural networks", J. Comput. Syst. Sci. 45 (1992) 1-19.
31. Y. Yau, W. J. Freeman, B. Burke and Q. Yang, "Pattern recognition by a distributed neural network: An industrial application", Neural Networks 4 (1991) 103-121.
32. T. Yoshizawa, "Extreme stability and almost periodic solutions of functional-differential equations". Arch. Rat. Mech. Anal. 17 (1964) 148-170.
33. T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions (Springer, New York, 1975).