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Research Papers

Stability and Bifurcation Analysis of a Network of Four Neurons With Time Delays

[+] Author and Article Information
Xiaochen Mao

MOE Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China; Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, 210098 Nanjing, China

Haiyan Hu

MOE Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China

J. Comput. Nonlinear Dynam 5(4), 041001 (Jun 29, 2010) (6 pages) doi:10.1115/1.4000317 History: Received September 19, 2008; Revised April 22, 2009; Published June 29, 2010; Online June 29, 2010

This paper reveals the dynamical behaviors of a bidirectional neural network consisting of four neurons with delayed nearest-neighbor and shortcut connections. The criterion of the global asymptotic stability of the trivial equilibrium of the network is derived by means of a suitable Lyapunov functional. The local stability of the trivial equilibrium is investigated by analyzing the distributions of roots of the associated characteristic equation. The sufficient conditions for the existence of nontrivial synchronous and asynchronous equilibria and periodic oscillations arising from codimension one bifurcations are obtained. Multistability near the codimension two bifurcation points is presented. Numerical simulations are given to validate the theoretical analysis.

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Figure 1

Architecture of a bidirectional neural network consisting of four neurons with delayed nearest-neighbor and shortcut connections

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Figure 2

The stable region (shading) of the trivial equilibrium of the network when c=2 and α=−0.2. Solid (dashed-dotted) curve corresponds to parameter values where Δ0(λ,τ)(Δ1(λ,τ),Δ2(λ,τ)) has a pair of pure imaginary roots. Solid (dashed-dotted) vertical line corresponds to parameter values where Δ0(λ,τ)(Δ1(λ,τ),Δ2(λ,τ)) has a zero root.

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Figure 3

The output of the network when c=0.5,  α=0.5,  and τ=1: (a) synchronous periodic oscillations when β=1.02; (b) a nontrivial synchronous equilibrium of the network when β=1.1 under initial conditions (0.2, 0.4, 0.3, and 0.1)

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Figure 4

The coexistence of asynchronous periodic oscillations of the network when c=1,  α=1,  β=−1,  and τ=1.6 under different initial conditions: (a) initial condition (0.2, 0.4, −0.3, and −0.1); (b) initial condition (0.2, −0.4, −0.3, and 0.1); (c) initial condition (0.2, 0.4, 0.2, and 0.4); and (d) initial condition (0.2, 0, 0, and 0)

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Figure 5

The coexistence of synchronous periodic oscillations and a pair of nontrivial asynchronous equilibria when c=2,  α=0.8,  β=0.56,  and τ=0.32 under initial conditions (0.2, 0.4, 0.3, and 0.1), (0.2, −0.4, 0.3, and −0.1), and (−0.2, 0.4, −0.3, and 0.1)

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Figure 6

The output of the network when c=2,  α=0.5,  β=1.1,  and τ=0.326 under different initial conditions: (a) two pairs of nontrivial asynchronous equilibria under initial conditions (0.2, 0.4, −0.3, and −0.1), (−0.2, −0.4, 0.3, and 0.1), (0.2, −0.4, −0.3, and 0.1), and (−0.2, 0.4, 0.3, and −0.1); (b) synchronous periodic oscillations under initial condition (0.2, 0.4, 0.3, and 0.1)

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