Dynamics of a two-neuron system with discrete and distributed delays

Physica D: Nonlinear Phenomena

Volumn 191, Issue 3-4, 2004, Pages 323-342

  Ruan, Shigui ^a,b   Filfil, Rami S ^b  

^a Department of Biology   (United States)

^b DALHOUSIE UNIVERSITY   (Canada)

Author keywords

Discrete and distributed delays; Neural feedback; Neural interaction history; Two neuron system

Indexed keywords

BIFURCATION (MATHEMATICS); COMPUTER SIMULATION; DYNAMICS; FEEDBACK; OSCILLATIONS; STABILITY;

NEURAL FEEDBACK; TWO-NEURON SYSTEMS;

NEURAL NETWORKS;

EID: 1842635257     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2003.12.004     Document Type: Article

References (28)

1. Babcock K.L., Westervelt R.M., Dynamics of simple electronic neural networks. Physica D. 28:1987;305-316.
2. T.A. Burton, Volterra Integral and Differential Equations, Academic Press, New York, 1983.
3. Chen Y., Wu J., Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network. Physica D. 134:1999;185-199.
4. Cooke K.L., Grossman Z., Discrete delays, distributed delay and stability switches. J. Math. Anal. Appl. 86:1982;592-627.
5. J.D. Cowan, A statistical mechanics of neuron activity, in: M. Gerstenhaber (Ed.), Some Mathematical Questions in Biology, American Mathematical Society, Providence, RI, 1970, pp. 1-58.
6. B. Ermentrout, XPPAUT5.0 - The Differential Equations Tool, University of Pittsburgh, Pittsburgh, 2001.
7. Faria T., On a planar system modeling a neuron network with memory. J. Diff. Eqns. 168:2000;129-149.
8. S. Gao, L. Huang, Exponential stability and periodic solutions of neural networks with continuously distributed delays, Phys. Rev. E 67 (2003) 011902; erratum 069901.
9. Giannakopoulos F., Zapp A., Bifurcations in a planar system of differential delay equations modeling neural activity. Physica D. 159:2001;215-232.
10. Gopalsamy K., He X.-Z., Stability in asymmetric Hopfield nets with transmission delays. Physica D. 76:1994;344-358.
11. Gopalsamy K., Leung I., Delay-induced periodicity in a neural netlet of excitation and inhibition. Physica D. 89:1996;395-426.
12. J.K. Hale, S.M. Verduyn Lunel, Introduction to functional differential equations, in: Applied Mathematics Sciences, vol. 99, Springer-Verlag, New York, 1993.
13. Hopfield J.J., Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. U.S.A. 81:1984;3088-3092.
14. Koruga D., Molecular networks as a sub-neural factor of neural networks. BioSystems. 23:1990;297-303.
15. Liao X., Wong K.-W., Wu Z., Bifurcation analysis on a two-neuron system with distributed delays. Physica D. 149:2001;123-141.
16. Liao X., Wu Z., Yu J., Stability switches and bifurcation analysis of neural network with continuously delay. IEEE Trans. Syst. Man. Cybernet. 29:1999;692-696.
17. N. MacDonald, Time Lag in Biological Models, Lecture Notes in Biomathematics 27, Springer-Verlag, Heidelberg, 1978.
18. R.K. Miller, Nonlinear Volterra Integral Equations, Benjamin, New York, 1971.
19. Ogûztöreli M.N., Activity analysis of neural networks. Biol. Cybernet. 34:1979;159-169.
20. Ogûztöreli M.N., Steil G.M., Caelli T.M., Control mechanisms of a neural network. Biol. Cybernet. 54:1986;21-28.
21. Olien L., Belair J., Bifurcation, stability and monotonicity properties of a delayed neural network model. Physica D. 102:1997;349-363.
22. Ruan S., Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Quart. Appl. Math. 59:2001;159-173.
23. Ruan S., Wei J., Periodic solutions of planar systems with two delays. Proc. Roy. Soc. Edinburgh, Section A. 129:1999;1017-1032.
24. Sokolove P., Computer simulation of after-inhibition in crayfish slowly adapting stretch receptor. Biophys. J. 12:1972;1429-1451.
25. Tank D.W., Hopfield J.J., Neural computation by concentrating information in time. Proc. Natl. Acad. Sci. U.S.A. 84:1987;1896-1991.
26. Wang L., Zou X., Harmless delays in Cohen-Grossberg neural networks. Physica D. 170:2002;162-173.
27. Wei J., Ruan S., Stability and bifurcation in a neural network model with two delays. Physica D. 130:1999;255-272.
28. J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Walter de Gruyter, Berlin, 2001.