Stability and Hopf bifurcation analysis on a two-neuron network with discrete and distributed delays

Abstract In this paper, a two-neuron network with both discrete and distributed delays is considered. Local stability of this system is investigated by analyzing its associated transcendental characteristic equation. By taking the discrete time delay as a bifurcation parameter, it is found that this system undergoes a sequence of Hopf bifurcations. Moreover, formulae for determining the direction of Hopf bifurcation and the stability and period of bifurcating periodic solutions are derived. Finally, numerical simulations are given to illustrate the theoretical analysis.

[1]  Lihong Huang,et al.  Hopf bifurcation analysis of two neurons with three delays , 2007 .

[2]  Sanling Yuan,et al.  Bifurcation analysis in a predator-prey system with time delay , 2006 .

[3]  Junjie Wei,et al.  Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays , 2005 .

[4]  Jinde Cao,et al.  Global Asymptotical Stability of Recurrent Neural Networks With Multiple Discrete Delays and Distributed Delays , 2006, IEEE Transactions on Neural Networks.

[5]  Lihong Huang,et al.  Hopf bifurcation analysis for a two-neuron network with four delays☆ , 2007 .

[6]  Jianhong Wu,et al.  Slowly Oscillating Periodic Solutions for a Delayed Frustrated Network of Two Neurons , 2001 .

[7]  S. Ruan,et al.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays , 2003 .

[8]  Zidong Wang,et al.  Robust stability analysis of generalized neural networks with discrete and distributed time delays , 2006 .

[9]  José Carlos Príncipe,et al.  An analysis of the gamma memory in dynamic neural networks , 1994, IEEE Trans. Neural Networks.

[10]  Jacques Bélair,et al.  Bifurcations, stability, and monotonicity properties of a delayed neural network model , 1997 .

[11]  Kwok-Wo Wong,et al.  Chaos and its synchronization in two-neuron systems with discrete delays , 2004 .

[12]  S. Ruan,et al.  Stability and bifurcation in a neural network model with two delays , 1999 .

[13]  Jianhong Wu,et al.  Introduction to Neural Dynamics and Signal Transmission Delay , 2001 .

[14]  José Carlos Príncipe,et al.  The gamma model--A new neural model for temporal processing , 1992, Neural Networks.

[15]  Lihong Huang,et al.  Linear Stability and Hopf bifurcation in a Two-Neuron Network with Three Delays , 2004, Int. J. Bifurc. Chaos.

[16]  X. Liao,et al.  Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays , 2003 .

[17]  J J Hopfield,et al.  Neural computation by concentrating information in time. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[18]  M. Gilli.  Strange attractors in delayed cellular neural networks , 1993 .

[19]  K. Gopalsamy,et al.  Delay induced periodicity in a neural netlet of excitation and inhibition , 1996 .

[20]  Jinde Cao,et al.  Stability and Hopf Bifurcation in a Simplified BAM Neural Network With Two Time Delays , 2007, IEEE Transactions on Neural Networks.

[21]  Shigui Ruan,et al.  Dynamics of a two-neuron system with discrete and distributed delays , 2004 .

[22]  Ju H. Park.  An analysis of global robust stability of uncertain cellular neural networks with discrete and distributed delays , 2007 .

[23]  S. Arik.  Global robust stability analysis of neural networks with discrete time delays , 2005 .