×

Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function. (English) Zbl 1012.92005

Summary: A simple neural network model with discrete time delays is investigated. The linear stability of this model is discussed by analyzing the associated characteristic transcendental equation. For the case with inhibitory influence from the past state, it is found that Hopf bifurcation occurs when this influence varies and passes through a sequence of critical values. The stability of bifurcating periodic solutions and the direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Chaotic behavior of a single delayed neuron equation with non-monotonously increasing transfer function has been observed in computer simulations. Some waveform diagrams, phase portraits, power spectra and plots of the largest Lyapunov exponent will also be given.

MSC:

92B20 Neural networks for/in biological studies, artificial life and related topics
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
37N25 Dynamical systems in biology
34K60 Qualitative investigation and simulation of models involving functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc Natl Acad Sci USA, 81, 10, 3088-3092 (1984) · Zbl 1371.92015
[2] Kurten, K. E.; Clark, J. W., Chaos in neural systems, Phys Lett A, 144, 7, 413-418 (1986)
[3] Babcock, K. L.; Westervelt, R. M., Dynamics of simple electronic neural networks, Physica D, 28, 3, 305-316 (1987)
[4] Freeman, W. J., Simulation of chaotic EEG patterns with a dynamic model of the olfactory system, Biol Cybern, 56, 2/3, 139-150 (1987)
[5] Riedel, U.; Kuhn, R.; Van Hemmen, J. L., Temporal sequences and chaos in neural nets, Phys Rev A, 38, 2, 1105-1108 (1988)
[6] Kepler, T. B.; Datt, S.; Meyer, R. B.; Abbott, L. F., Chaos in a neural network circuit, Physica D, 46, 3, 449-457 (1990) · Zbl 0709.94690
[7] Das II, P. K.; Schieve, W. C.; Zeng, Z. J., Chaos in an effective four-neuron neural network, Phys Lett A, 161, 1, 60-66 (1991)
[8] Matsumoto, T.; Chua, L. O.; Komuro, M., The double scroll, IEEE Trans Circuits Syst, 32, 798-818 (1985) · Zbl 0625.58013
[9] Gilli, M., Strange attractors in delayed cellular neural networks, IEEE Trans Circuits Syst, 40, 849-853 (1993) · Zbl 0844.58056
[10] Zou, F.; Nossek, J. A., Bifurcation and chaos in cellular neural networks, IEEE Trans Circuits Syst, 40, 166-172 (1993) · Zbl 0782.92003
[11] Gopalsmay, K.; Leung, Issic K. C., Convergence under dynamical thresholds with delays, IEEE Trans Neural Networks, 8, 341-348 (1994)
[12] Xioafeng Liao, Zhongfu Wu, Juebang Yu. Hopf bifurcation analysis of a neural system with a continuously distributed delay. International symposium on signal processing and intelligent system. Guangzhou, China, November 1999; Xioafeng Liao, Zhongfu Wu, Juebang Yu. Hopf bifurcation analysis of a neural system with a continuously distributed delay. International symposium on signal processing and intelligent system. Guangzhou, China, November 1999
[13] Xiaofeng Liao, Zhongfu Wu, Juebang Yu. Stability switches and bifurcation analysis of a neural network with continuous delay. IEEE Trans SMC-I. 1999;29:692-6; Xiaofeng Liao, Zhongfu Wu, Juebang Yu. Stability switches and bifurcation analysis of a neural network with continuous delay. IEEE Trans SMC-I. 1999;29:692-6
[14] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002
[15] Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer: Springer New York · Zbl 0787.34002
[16] Hansel, D.; Sompolinsky, H., Synchronization and computation in a chaotic neural network, Phys Rev Lett, 68, 718-721 (1992)
[17] Aihara, K.; Takabe, T.; Toyoda, M., Chaotic neural networks, Phys Lett A, 144, 333-340 (1990)
[18] Babloyantz, A.; Lourenco, C., Computation with chaos: a paradigm for cortical activity, Proc Natl Acad Sci USA, 91, 9027-9031 (1994)
[19] Freeman, W. J., Tutorial on neurobiology: from single neurons to brain chaos, Int J Bif Chaos, 2, 451-482 (1992) · Zbl 0900.92036
[20] Kaneko, K., Relevance of dynamical clustering to biological networks, Physica D, 75, 55-73 (1994) · Zbl 0859.92001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.