Stability and dynamics of simple electronic neural networks with added inertia

Babcock, K. L.; Westervelt, R. M.

doi:10.1016/0167-2789(86)90152-1

Cited by 277 publications

(72 citation statements)

References 10 publications

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“…In this paper, we have considered a general neural network model with distributed delays, which is a generalization of previous models studied in [2][3][4][5][6][7]. By using the average time delays as a bifurcation parameter, we have shown that a Hopf bifurcation occurs when the parameter passes through a critical value.…”

Section: Discussionmentioning

confidence: 99%

“…From then on, much attention has been paid to the dynamics of neural networks and neural networks have also found numerous applications in various fields such as optimization and signal processing problem. Based on the Hopfield neural network model, many simplified models have been established (see [1][2][3][4][5][6], and references therein). These studies of simplified models are very useful and insightful, since the complexities found in simple models can often be carried over to large-scale networks in some way thereby yielding much better understanding of the latter from a careful study of the former [7].…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation and stability analysis of a neural network model with distributed delays

Cheng

Cao

2006

Nonlinear Dyn

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In this paper, the dynamics of a generalized two-neuron model with self-connections and distributed delays are investigated, together with the stability of the equilibrium. In particular, the conditions under which the Hopf bifurcation occurs at the equilibrium are obtained for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. Explicit algorithms for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [20]. Some numerical simulations are given to illustrate the effectiveness of the results found. The obtained results are new and they complement previously known results.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bifurcation and stability analysis of a neural network model with distributed delays

Cheng

Cao

2006

Nonlinear Dyn

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“…For a number of two neuron models and their linear stability analysis, we refer to the works of Marcus et al [18], Babcock and Westervelt [19,20], Gopalsamy and Leung [21]. A single neural oscillator modeled by (1) without delay (namely, = 0) relaxes toward the set of equilibria.…”

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confidence: 99%

Synchronization for two coupled oscillators with inhibitory connection

Xiao

Guo

2010

Math Methods in App Sciences

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Communicated by M. LachowiczIn this paper we present an oscillatory neural network composed of two coupled neural oscillators with inhibitory connections. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neurons. Regarding time delays s as the bifurcation parameter, we not only obtain the existence of Hopf bifurcations but also investigate the bifurcation direction and stability of bifurcated periodic solutions by employing normal form theory and center manifold reduction. Finally, numerical simulations are provided to illustrate the theoretical results.

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“…For a more detailed interpretation of the parameters, one can see [24,25]. In 1997, Lin and Li [26] made a detailed investigation on the bifurcation direction of periodic solution for system (1).…”

Section: Introductionmentioning

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 dynamics of simple electronic neural networks with added inertia

Cited by 277 publications

References 10 publications

Bifurcation and stability analysis of a neural network model with distributed delays

Bifurcation and stability analysis of a neural network model with distributed delays

Synchronization for two coupled oscillators with inhibitory connection

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

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