Bifurcation Analysis in a Two-neuron System with Different Time Delays

Authors: Changjin Xu

Abstract:

In this paper, we consider a two-neuron system with time-delayed connections between neurons. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulation results are given to support the theoretical predictions. Finally, main conclusions are given.

Keywords: Two-neuron system, delay, stability, Hopf bifurcation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1084980

References:

[1] K. L. Babcock and R. M. Westervelt, Dynamics of simple electric neural networks. Physica D 28 (1987) 305-316.
[2] K. L. Babcock and R. M. Westervelt, Dynamics of simple electric neural networks with added intertial. Physica D 23 (1986) 464-469.
[3] K. Gopalsamy and I. Leung, Delay-induced periodicity in a neural network of excitation and inhibition. Physica D 89 (1996) 395-426.
[4] K. Gopalsamy and I. Leung, Converge under dynamical thresholds with delays. IEEE Trans. Neural Netw. 8 (1997) 341-348.
[5] K. Gopalsamy, I. Leung and P. Liu, Global Hopf bifurcation in a neural netlet. Appl. Math. Comput. 94 (1998) 171-192.
[6] J. Hale, Theory of Functional Differential Equation. Springer-Verlag, 1977.
[7] B. Hassard, D. Kazarino and Y. Wan, Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press, 1981.
[8] X. F. Liao, S.W. Li and G. R. Chen, Bifurcation analysis on a two-neuron system with distributed dedays in the domain. Neural Netw. 17 (2004) 545-561.
[9] X. F. Liao, K. W. Wong and Z. F. Wu, Bifurcation analysis on a twoneuron system with distributed dedays. Physica D 149 (2001) 123-141.
[10] X. F. Liao, K. W. Wong, C. S. Leung and Z. F. Wu, Hopf bifurcation and chaos in a single delayed neuron equation with nonmonotonic activation function. Chaos, Solitons and Fractals 12 (2001) 1535-1547.
[11] X. F. Liao, Z. F. Wu and J. B. Yu, Hopf bifurcation analysis of a neural systems with a continuously distributed delay. Proceeding of the International Symposium on Signal Processing and Intelligent System. Guangzhou, China, (1999) 102-106.
[12] X. F. Liao, Z. F. Wu and J. B. Yu, Stability switches and bifurcation analysis of a neural network with continuous delay. IEEE Trans. syst. man cyber. A 29 (1999) 692-696.
[13] C. M. Marcus and R. M. Westervelt, Stability of analog networks with delay. Phys. Rev. A 39 (1989) 347-359.
[14] L. Olien and J. Belair, Bifurcation, stability and monotonicity properties of a delayed neural model. Physica D 102 (1997) 349-363.
[15] S. Ruan and J. Wei, On the zero of some transcendential functions with applications to stability of delay differential equations with two delays. Dyn. Contin. Discrete Impuls. Syst. Ser. A 10 (2003) 863-874.
[16] F. H. Tu, X. F. Liao and W. Zhang, Delay-dependent asymptotic stability of a two-neuron system with different time delays. Chaos, Solitons and Frctals 28 (2006) 437-447.