Hopf bifurcation in a general n-neuron ring network with n time delays

Document Type : Research Article

Authors

Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

Abstract

In this paper, we consider a general ring network consisting of n neurons and n time delays. By analyzing the associated characteristic equation, a classification according to n is presented. It is investigated that Hopf bifur-cation occurs when the sum of the n delays passes through a critical value.In fact, a family of periodic solutions bifurcate from the origin, while the zero solution loses its asymptotically stability. To illustrate our theoretical results, numerical simulation is given.

Keywords


1. Campbell, S.A. Stability and bifurcation of a simple neural network with multiple time delays, Fields Institute Communications 21 (1999), 65-79.
2. Campbell, S.A., Ruan, S. and Wei, J. Qualitative analysis of a neural network model with multiple time delays, Int. J. of Bifurcation and Chaos 9 (1999), 1585-1595.
3. Cao, J., Yu, W. and Qu, Y. A new complex network model and convergence dynamics for reputation computation in virtual organizations, Phys. Lett.A 356 (2006), 414-425.
4. Driver, R.D. Ordinary and delay differential equations, Springer, 1977.
5. Eccles, J. C., Ito, M. and Szenfagothai, J. The cerebellum as neuronal machine, Springer, New York, 1967.
6. Guckenheimer, J. and Holmes, P. Nonlinear oscillation, dynamical system and bifurcations of vector felds, Springer, 1993.
7. Hopfield, J. Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. 79 (1982), 2554-2558.
8. Hu, H. and Huang, L. Stability and Hopf bifurcation analysis on a ring of four neurons with delays, Appl.Math. Comput. 213 (2009), 587-599.
9. Javidmanesh, E., Afsharnezhad, Z. and Effati, S. Existence and stability analysis of bifurcating periodic solutions in a delayed five-neuron BAM neural network model, Nonlinaer Dyn. 72 (2013), 149-164.
10. Lee, S.M., Kwonb, O.M. and Park, J.H. A novel delay-dependent crite-rion for delayed neural networks of neutral type, Phys. Lett. A 374 (2010), 1843-1848.
11. Li, X. and Cao, J. Delay-dependent stability of neural networks of neutral type with time delay in the leakage term, Nonlinearity 23 (2010), 1709-1726.
12. Marcus, C.M. and Westervelt, R.M. Stability of analog neural network with delay, Phys. Rev. A 39 (1989), 347-359.
13. Perko, L. Differential equations and dynamical system, Springer 1991.
14. Ruan, S. and Wei, J. On the zeros of transcendental functions with appli-cations to stability of delay differential equations with two delays, Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematics and Analytical 10 (2003), 863-874.
CAPTCHA Image