Bifurcation Analysis of a Network of Three Neurons with Time Delays

Abstract:

The paper studies the dynamical behaviors of a recurrent neural network model consisting of three neurons with time delays through theoretical analysis and numerical simulations. The local stability of the trivial equilibrium of the network is analyzed and the sufficient conditions of the existence of Hopf bifurcation are given by discussing the associated characteristic equation. The direction and stability of the bifurcated periodic oscillations arising from Hopf bifurcation, which depend on the nonlinear terms of the network, are determined by means of the normal form and the center manifold theorem. Afterwards, numerical examples are performed to validate the theoretical analysis. The case studies of numerical simulations reach nice agreement with the theoretical results.

• Periodical: Applied Mechanics and Materials (Volumes 105-107)
• Pages: 147-150
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.105-107.147
• Online since: September 2011
• Authors: Xiao Chen Mao
• Keywords: Periodic Oscillations, Recurrent Neural Networks, Stability, Time Delay

References

[1] C. M. Marcus and R. M. Westervelt: Phys. Rev. A, Vol. 39 (1989), p.347

[2] X. C. Mao and H. Y. Hu: Int. J. Bifurcat. Chaos, Vol. 18 (2008) No.10, p.3053

[3] X. Xu: J. Phys. A, Vol. 41 (2008) No.3, pp.035102-1

[4] A. Ruiz, D. H. Owens and S. Townley: IEEE. T. Neural Network, Vol. 9 (1998) No.4, p.651

[5] A. Hajihosseini, G. R. Rokni Lamooki, B. Beheshti and F. Maleki: Neurocomputing, Vol. 73 (2010) No.4-6, p.991

DOI: 10.1016/j.neucom.2009.08.018

[6] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan: Theory and Application of Hopf Bifurcation (Cambridge University Press, Britain 1981)