Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays

Song, Yongli; Han, Maoan; Wei, Junjie

doi:10.1016/j.physd.2004.10.010

Cited by 216 publications

(81 citation statements)

References 34 publications

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“…The delay can cancel or amplify multiple spikes thus leading to the neural information being selectively processed. The theoretical study of the dynamics of simple units organized into networks with delayed couplings revealed a rich variety of possible scenarios of transition to a global oscillatory behavior induced by the delay (see, e.g., Bungay and Campbell 2007;Campbell et al 2005;Guo 2005; Guo and Huang 2003;Guo 2007;Huang and Wu 2003;Song et al 2005;Wu et al 1999;Wu 1998;Yuan and Campbell 2004;Yuan 2007;Wei et al 2002;Wei and Velarde 2004 and references therein). The emerging oscillations can exhibit different spatio-temporal patterns sensitive to the delay.…”

Section: Introductionmentioning

confidence: 99%

Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

2009

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A model of time-delay recurrently coupled spatially segregated neural assemblies is here proposed. We show that it operates like some of the hierarchical architectures of the brain. Each assembly is a neural network with no delay in the local couplings between the units. The delay appears in the long range feedforward and feedback inter-assemblies communications. Bifurcation analysis of a simple four-units system in the autonomous case shows the richness of the dynamical behaviors in a biophysically plausible parameter region. We find oscillatory multistability, hysteresis, and stability switches of the rest state provoked by the time delay. Then we investigate the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric local Hopf bifurcation theory of delay differential equations and derive the equation describing the flow on the center manifold that enables us determining the direction of Hopf bifurcations and stability of the bifurcating periodic orbits. We also discuss computational properties of the system due to the delay when an external drive of the network mimicks external sensory input.

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Section: Introductionmentioning

confidence: 99%

Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

2009

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“…If there is a critical value of τ such that a certain root of (9) has zero real part, then at this critical value the stability of the zero equilibrium (0, 0) of system (8) will switch, and under certain conditions a family of small amplitude periodic solutions can bifurcate from the zero equilibrium (0, 0); that is, a Hopf bifurcation occurs at the zero equilibrium (0, 0). Now, we look for the conditions under which the characteristic Equation (9) has a pair of purely imaginary roots, see [24]. Clearly, iω(ω > 0) is a root of Equation (9) …”

Section: T MX T Nx T a X T A X T X T A X T X T X T Dx T Ex T B X T mentioning

confidence: 99%

Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

Wang¹,

Liu²

2012

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A delayed Lotka-Volterra two-species predator-prey system of population allelopathy with discrete delay is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, some numerical simulations are carried out for illustrating the theoretical results.

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“…k is defined by (13) and μ ∈ R, drop the bar for the simplification of notations, then system (3) can be written as an FDE in…”

Section: Direction and Stability Of The Hopf Bifurcationmentioning

confidence: 99%

“…For more information, one can see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In 1986 and 1987, Babcock and Westervelt [24,25] had analyzed the stability and dynamics of the following simple neural network model of two neurons with inertial coupling: 4 , dx 3 dt = −2ξx 3 …”

Section: Introductionmentioning

confidence: 99%

Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay

2012

Advances in Artificial Neural Systems

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A four-dimensional neural network model with delay is investigated. With the help of the theory of delay differential equation and Hopf bifurcation, the conditions of the equilibrium undergoing Hopf bifurcation are worked out by choosing the delay as parameter. Applying the normal form theory and the center manifold argument, we derive the explicit formulae for determining the properties of the bifurcating periodic solutions. Numerical simulations are performed to illustrate the analytical results.

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Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays

Cited by 216 publications

References 34 publications

Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks

Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay

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