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【期刊论文】Stability Analysis of Cohen-Grossberg Neural Networks
郭上江, Shangjiang Guo and Lihong Huang
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL.17, NO.1, JANUARY 2006,-0001,():
-1年11月30日
Without assuming boundedness and differentiability of the activation functions and any symmetry of interconnections, we employ Lyapunov functions to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, and even global exponential stability of equilibria for the Cohen–Grossberg neural networks with and without delays. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria and can be applied to neural networks, including Hopfield neural networks, bidirectional association memory neural networks, and cellular neural networks.
Equilibrium,, global asymptotic stability (, GAS), ,, Lyapunov functions,, neural networks,, time delays.,
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【期刊论文】Non-linear waves in a ring of neurons
郭上江, SHANGJIANG GUO† AND LIHONG HUANG
IMA Journal of Applied Mathematics(2006)71, 496-518,-0001,():
-1年11月30日
In this paper, we study the effect of synaptic delay of signal transmission on the pattern formation and some properties of non-linear waves in a ring of identical neurons. First, linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Regarding the delay as a bifurcation parameter, we obtained the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Second, global continuation conditions for Hopf bifurcating periodic orbits are derived by using the equivariant degree theory developed by Geba et al. and independently by Ize & Vignoli. Third, we show that the coincidence of these periodic solutions is completely determined either by a scalar delay differential equation if the number of neurons is odd, or by a system of two coupled delay differential equations if the number of neurons is even. Fourth, we summarize some important results about the properties of Hopf bifurcating periodic orbits, including the direction of Hopf bifurcation, stability of the Hopf bifurcating periodic orbits, and so on. Fifth, in an excitatory ring network, solutions of most initial conditions tend to stable equilibria, the boundary separating the basin of attraction of these stable equilibria contains all of periodic orbits and homoclinic orbits. Finally, we discuss a trineuron network to illustrate the theoretical results obtained in this paper and conclude that these theoretical results are important to complement the experimental and numerical observations made in living neurons systems and artificial neural networks, in order to understand the mechanisms underlying the system dynamics better.
a ring of neurons, Hopf bifurcation, global ontinuation, Lie group.,
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【期刊论文】Global continuation of nonlinear waves in a ring of neurons
郭上江, Shangjiang Guo and Lihong Huang
Proceedings of the Royal Society of Edinburgh, 135A, 999-1015, 2005,-0001,():
-1年11月30日
In this paper, we consider a ring of neurons with self-feedback and delays. As a result of our approach based on global bifurcation theorems of delay differential equations coupled with representation theory of Lie groups, the coexistence of its asynchronous periodic solutions (i.e. mirror-reflecting waves, standing waves and discrete waves), bifurcated simultaneously from the trivial solution at some critical values of the delay, will be established for delay not only near to but also far away from the critical values. Therefore, we can obtain wave solutions of large amplitudes. In addition, we consider the coincidence of these periodic solutions.
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【期刊论文】Regular dynamics in a delayed network of two neurons with all-or-none activation functions
郭上江, Shangjiang Guo a, *, Lihong Huang a, Jianhong Wu b
Physica D 206(2005)32-48,-0001,():
-1年11月30日
We consider a delayed network of two neurons with both self-feedback and interaction described by an all-or-none threshold function. The model describes a combination of analog and digital signal processing in the network and takes the form of a system of delay differential equations with discontinuous nonlinearity.We show that the dynamics of the network can be understood in terms of the iteration of a one-dimensional map, and we obtain simple criteria for the convergence of solutions, the existence, multiplicity and attractivity of periodic solutions. © 2005 Elsevier B.V. All rights reserved.
Neural networks, Delayed feedback, One-dimensional map, Convergence, Periodic solutions
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【期刊论文】Periodic oscillation for a class of neural networks with variable coefficients☆
郭上江, Shangjiang Guo*, Lihong Huang
Nonlinear Analysis: RealWorld Applications 6(2005)545-561,-0001,():
-1年11月30日
In this paper, we study a class of neural networks with variable coefficients which includes delayed Hopfield neural networks, bidirectional associative memory networks and cellular neural networks as its special cases. By matrix theory and inequality analysis, we not only obtain some new sufficient conditions ensuring the existence, uniqueness, global attractivity and global exponential stability of the periodic solution but also estimate the exponentially convergent rate. Our results are less restrictive than previously known criteria and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Moreover, these conclusions are presented in terms of system parameters and can be easily verified for the globally Lipschitz and the spectral radius being less than 1. Therefore, our results have an important leading significance in the design and applications of periodic oscillatory neural circuits for neural networks with delays. © 2005 Elsevier Ltd. All rights reserved.
Neural networks, Periodic solution, Global attractor, A positively invariant set, Convergent rate
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