Delay induced periodicity in a neural netlet of excitation and inhibition

Physica D: Nonlinear Phenomena

Volumn 89, Issue 3-4, 1996, Pages 395-426

  Gopalsamy, K ^a   Leung, I ^a  

^a FLINDERS UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 22244473030     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/0167-2789(95)00203-0     Document Type: Article

References (56)

1. Amari S. and Arbib M.A. (1982), Competition and Cooperation in neural networks, in: Lecture Notes in Biomathematics, Vol. 45 (Springer, Berlin).
2. An der Heiden U. (1980), Analysis of neural networks, in: Lecture Notes in Biomathematics, Vol. 35 (Springer, Berlin).
3. Arrowsmith D.K. and Place C.M. (1990), An introduction to Dynamical systems (Cambridge Univ. Press, Cambridge).
4. Babcock K.L. and Westervelt R.M. (1986), Complex dynamics in simple neural circuits, in: Neural Networks for Computing, J.S. Denker, ed. (Amer. Inst. Phys., New York) pp. 288-293.
5. Babcock K.L. and Westervelt R.M. (1987), Dynamics of simple electronic neural networks with added inertia, Physica D 23, 464-469.
6. Babcock K.L. and Westervelt R.M. (1986), Dynamics of simple electronic neural networks, Physica D 28, 305-316.
7. Baird B. (1990), Bifurcation and category learning in network models of oscillating cortex, Physica D 42, 365-384.
8. Belair J. (1993), Stability in a model of a delayed neural network, J. Dynam. Diff. Eqns. 5, 607-623.
9. Belair J. (1992), Stability in delayed neural networks, in: Ordinary and Delay Differential Equations, J. Wiener and J. Hale, eds (Longman Scientific and Technical, New York) pp. 6-9.
10. Bellman R. and Cooke K.L. (1963), Differential-Difference Equations (Academic Press, New York).
11. Bunimovich L.A. and Sinai Ya.G. (1988), Spacetime chaos in coupled map lattices, Nonlinearity 1, 491-516.
12. Burton T.A. (1991), Neural networks with memory, J. Appl. Math. Stoch. Anal. 4, 313-322.
13. Burton T.A. (1993), Averaged neural networks, Neural Networks 6, 677-680.
14. Chapeau-Blondeau F. and Chauvet G. (1992), Stable, oscillatory and chaotic regimes in the dynamics of small neural networks with delay, Neural Networks 5, 735-743.
15. Cohen M.A. (1990), The stability of sustained oscillations in symmetric cooperative competitive networks, Neural Networks 3, 609-612.
16. Coolen A.C.C. and Gielen C.C.A.M. (1988), Delays in neural networks, Europhys. Lett. 7, 281-285.
17. Destexhe A. (1994), Stability of periodic oscillations in a network of neurons with time delay, Phys. Lett. A 187, 309-316.
18. Domany E., van Hemmen J.L. and Schulten K. (1991), Models of Neural Networks (Springer, Berlin).
19. Gopalsamy K. (1992), Stability and Oscillations in Delay Differential Equations of Population Dynamics (Kluwer, Dordrecht).
20. Gopalsamy, K. and He X.Z. (1994), Delay independent stability in bidirectional associative memory networks, IEEE Trans. Neural Networks 5, no. 6.
21. Gopalsamy, K. and He X.Z. (1994), Stability in asymmetric Hopfield nets with transmission delays, Physica D 76, 344-358.
22. Gray C.M., Konig P., Engel A.K. and Singer W. (1990), in: Synergetics of Cognition, H. Haken and M. Stadler, eds (Springer, New York).
23. Halanay A. (1966), Differential Equations, Mathematics in Science and Engineering, Vol. 23 (Academic Press, New York).
24. Hale J. and Koçak H. (1991), Dynamics and Bifurcations, Texts in Applied Mathematics, Vol. 3 (Springer, New York).
25. Hartline D.K. and Gassie Jr. D.V. (1979), Pattern generation in the lobster (Panulirus) stomatogastric ganglion I. Pyloric neuron kinetics and synaptic interactions, Biol. Cyber 33, 209-222.
26. Harmon L.D. (1964), Neuromines: Action of a reciprocally inhibitory pair, Science 146, 1323-1325.
27. Hassard B.D., Kazarinoff N.D. and Wan Y.H. (1981), Theory and Applications of Hopf Bifurcation (Cambridge Univ. Press, Cambridge).
28. Hayashi Y. (1994), Oscillatory neural network and learning continuously transformed patterns, Neural Networks 7, 219-231.
29. Herz A., Sulzer B., Kühn R., and van Hemmen J.L. (1988), The Hebb rule: storing static and dynamic objects in an associative network, Europhys. Lett. 7, 663-669.
30. Hopfield J.J. (1984), Neurons with graded response have collective computational properties like those of two state neurons, Proc. Natl. Acad. Sci. 81, 3088-3092.
31. Kaczmarek L.K. and Babloyantz (1977), Spatiotemporal patterns in epileptic seizures, Biol. Cybern. 26, 199-208.
32. Kleinfield D. and Sompolinsky H. (1988), Associate neural network model for the generation of temporal patterns, Biophys. J. 54, 1039-1051.
33. Kling U. and Szekeley G (1968), Simulation of rhythmic activities I. Function of networks with cyclic inhibitions, Kybernetic 5, 89-103.
34. Llinas R.R. (1988), The intrinsic electrophysiological properties of mammalian neurons: insight into central nervous system function, Science 242, 1654-1664.
35. Marcus C.M., Waugh F.R. and Westervelt R.M. (1991), Nonlinear dynamics and stability of analog neural networks, Physica D 51, 234-247.
36. Marcus C.M. and Westervelt R.M. (1989a), Dynamics of analog networks with time delay, Advances in Neural Information Processing, D.S. Touretsky, eds (Morgan Kauffmann, San Menlo) pp. 568-576.
37. Marcus, C.M. and Westervelt R.M. (1989b), Stability of analog networks with time delay, Phys. Rev. A 39, 347-359.
38. Milton J.G., Longtin A., Beuter A., Mackey M.A. and Glass L. (1989), Complex dynamics and bifurcations in neurology, J. Theor. Biol. 138, 129-147.
39. Muller B. and Reinhardt J. (1991), Neural Networks (Springer, Berlin).
40. Nagumo J. and Sato S. (1972), On the response characteristics of a mathematical neuron model, Kybernetik 10, 155-164.
41. Pavlidis T. (1973), Biological Oscillators (Academic Press, London).
42. Prufer M. (1985), Turbulence in multistep methods for initial value problems, SIAM J. Appl. Math. 45, 32-69.
43. Robertson M. and Pearson K.G. (1985), Neural circuits in the flight system of the locust, J. Neurophysiol. 53, 110-128.
44. Scott, A.C. (1977), Neurophysics (Wiley-Interscience, New York).
45. Sattinter D.H. (1973), Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, Vol. 309 (Springer, Berlin).
46. Stech H.W. (1979), The Hopf bifurcation: A stability result and applications, J. Math. Anal. Appl. 71, 525-546.
47. Ushiki S. (1982), Central difference scheme and chaos, Physica 4D, 407-424.
48. Vibert J.F., Khachayar P. and Noureddine A. (1994), Interneural delay modification synchronizes biologically plausible neural networks, Neural Networks 7, 589-607.
49. Wiggins S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, Vol. 2 (Springer, New York).
50. Wilson D.M. and Waldron I. (1968), Models for the generation of the motor output pattern in flying locusts, Proc. IEEE 56, 1058-1064.
51. 2, SIAM J. Appl. Math. 34, 167-175.
52. Wang X. (1991), Period doubling to chaos in a simple neural network: an analytic proof, Complex Systems 5, 425-441.
53. Wang X. and Bloom E.K. (1992), Discrete-time versus continuous time models of neural networks, J. Comp. Sys. Sci. 45, 1-9.
54. Yamaguti M. and Matano H. (1979), Euler's finite difference scheme and chaos, Proc. Japan Acad. Ser. A 55, 78-80.
55. Yamaguti M. and Ushiki S. (1981), Chaos in numerical analysis of ordinary differential equations, Physica D3, 618-626.
56. Yee H.C., Sweby P.K. and Griffiths D.F. (1991), Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations I: The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics, J. Comput. Phys. 97, 249-310.