Bifurcation analysis and spatio-temporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling

In this paper, a delayed neural network with unidirectional coupling is considered which consists of two two-dimensional nonlinear differential equation systems with exponential decay where one system receives a delayed input from the other system. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the centre manifold theorem. We also investigate the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay-differential equations combined with representation theory of Lie groups. Then the global continuation of phase-locked periodic solutions is investigated. Numerical simulations are given to illustrate the results obtained.

[1]  Shangjiang Guo.  Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay , 2005 .

[2]  D. V. Reddy,et al.  Time delay effects on coupled limit cycle oscillators at Hopf bifurcation , 1998, chao-dyn/9810023.

[3]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[4]  Sen,et al.  Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators , 1998, Physical review letters.

[5]  T. Faria.  On a Planar System Modelling a Neuron Network with Memory , 2000 .

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

[7]  Seon Hee Park,et al.  MULTISTABILITY IN COUPLED OSCILLATOR SYSTEMS WITH TIME DELAY , 1997 .

[8]  Schuster,et al.  Collective frequencies and metastability in networks of limit-cycle oscillators with time delay. , 1991, Physical review letters.

[9]  Margarete Z. Baptistini,et al.  On the Existence and Global Bifurcation of Periodic Solutions to Planar Differential Delay Equations , 1996 .

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

[11]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[12]  P. Brown,et al.  EEG–EMG, MEG–EMG and EMG–EMG frequency analysis: physiological principles and clinical applications , 2002, Clinical Neurophysiology.

[13]  K. Ikeda,et al.  High-dimensional chaotic behavior in systems with time-delayed feedback , 1987 .

[14]  Shangjiang Guo,et al.  Equivariant Hopf bifurcation for neutral functional differential equations , 2008 .

[15]  Jianhong Wu.  SYMMETRIC FUNCTIONAL DIFFERENTIAL EQUATIONS AND NEURAL NETWORKS WITH MEMORY , 1998 .

[16]  S. A. Campbell,et al.  Stability and Synchronization of a Ring of Identical Cells with Delayed Coupling , 2004 .

[17]  Charles M. Gray,et al.  Synchronous oscillations in neuronal systems: Mechanisms and functions , 1994, Journal of Computational Neuroscience.

[18]  N. C. Majee,et al.  Temporal dynamics of a two-neuron continuous network model with time delay , 1997 .

[19]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[20]  Lihong Huang,et al.  Hopf bifurcating periodic orbits in a ring of neurons with delays , 2003 .

[21]  Peter A. Tass,et al.  Phase Resetting in Medicine and Biology: Stochastic Modelling and Data Analysis , 1999 .

[22]  Yongli Song,et al.  Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks , 2009, Biological Cybernetics.

[23]  S. Bungay,et al.  Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling , 2005 .

[24]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[25]  H. Schuster,et al.  Mutual Entrainment of Two Limit Cycle Oscillators with Time Delayed Coupling , 1989 .

[26]  R. Westervelt,et al.  Stability of analog neural networks with delay. , 1989, Physical review. A, General physics.

[27]  Lihong Huang,et al.  Nonlinear Waves in Networks of Neurons with Delayed Feedback: Pattern Formation and Continuation , 2003, SIAM J. Math. Anal..

[28]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations and Applications to Bogdanov-Takens Singularity , 1995 .

[29]  Pauline van den Driessche,et al.  Delayed Coupling Between Two Neural Network Loops , 2004, SIAM J. Appl. Math..

[30]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation , 1995 .

[31]  P. Tass.  Phase Resetting in Medicine and Biology , 1999 .

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

[33]  Yuan Yuan,et al.  Stability Switches and Hopf Bifurcations in a Pair of Delay-Coupled Oscillators , 2007, J. Nonlinear Sci..

[34]  Sue Ann Campbell,et al.  Stability, Bifurcation, and Multistability in a System of Two Coupled Neurons with Multiple Time Delays , 2000, SIAM J. Appl. Math..

[35]  S. Reppert,et al.  Coordination of circadian timing in mammals , 2002, Nature.

[36]  Maoan Han,et al.  Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays , 2004 .

[37]  S. Ruan.  Absolute stability, conditional stability and bifurcation in Kolmogrov-type predator-prey systems with discrete delays , 2001 .

[38]  L. Glass,et al.  Chaos in multi-looped negative feedback systems. , 1990, Journal of theoretical biology.

[39]  A. B. Bonds,et al.  Cooperative synchronized assemblies enhance orientation discrimination. , 2004, Proceedings of the National Academy of Sciences of the United States of America.