Dynamics of a delayed two-coupled oscillator with excitatory-to-excitatory connection

In this paper, we consider a neural network model consisting of two coupled oscillators with delayed feedback and excitatory-to-excitatory connection. We study how the strength of the connections between the oscillators affects the dynamics of the neural network. We give a full classification of all equilibria in the parameter space and obtain its linear stability by analyzing the characteristic equation of the linearized system. We also investigate the spatio-temporal patterns of bifurcated periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. Moreover, the stability and bifurcation direction of the bifurcated periodic solutions are obtained by employing center manifold reduction and normal form theory. Some numerical simulations are provided to illustrate the theoretical results.

[1]  John S. Denker,et al.  Neural Networks for Computing , 1998 .

[2]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[3]  J. Cowan,et al.  Excitatory and inhibitory interactions in localized populations of model neurons. , 1972, Biophysical journal.

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

[5]  V. I. Kryukov,et al.  The metastable and unstable states in the brain , 1988, Neural Networks.

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

[7]  D. Hansel,et al.  Phase Dynamics for Weakly Coupled Hodgkin-Huxley Neurons , 1993 .

[8]  G. Ermentrout,et al.  Multiple pulse interactions and averaging in systems of coupled neural oscillators , 1991 .

[9]  M. Kawato,et al.  Synergism and antagonism of neurons caused by an electrical synapse , 1979, Neuroscience Letters.

[10]  Lihong Huang,et al.  Regular dynamics in a delayed network of two neurons with all-or-none activation functions , 2005 .

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

[12]  Jianhong Wu,et al.  Existence and attraction of a phase-locked oscillation in a delayed network of two neurons , 2001, Differential and Integral Equations.

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

[14]  R. L. Dobrushin,et al.  Stochastic cellular systems : ergodicity, memory, morphogenesis , 1990 .

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

[16]  R. M. Westervelt,et al.  Complex dynamics in simple neural circuits , 1987 .

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

[18]  M. Golubitsky,et al.  Hopf Bifurcation in the presence of symmetry , 1985 .

[19]  Xiang-Ping Yan,et al.  Hopf bifurcation and stability for a delayed tri-neuron network model , 2006 .

[20]  Kwok-Wo Wong,et al.  Hopf bifurcation of a Two-Neuron Network with Different Discrete Time Delays , 2005, Int. J. Bifurc. Chaos.

[21]  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..

[22]  Junjie Wei,et al.  Global existence of periodic solutions in a tri-neuron network model with delays , 2004 .

[23]  G. Ermentrout,et al.  Amplitude response of coupled oscillators , 1990 .

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

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

[26]  Robert M. Westervelt,et al.  Stability and dynamics of simple electronic neural networks with added inertia , 1986 .

[27]  J. Hindmarsh,et al.  A model of the nerve impulse using two first-order differential equations , 1982, Nature.

[28]  Shangjiang Guo,et al.  Two-parameter bifurcations in a network of two neurons with multiple delays , 2008 .