Abstract
This chapter first introduces the continuous-time Hopfield neural networks. The existence and uniqueness of equilibrium, as well as its stability and instability, of continuous-time Hopfield networks are analyzed, and less conservative yet more general results are established. Then, in light of the continuous-time architecture of Hopfield networks, the impulsive Hopfield neural networks with transmission delays are formulated and explained. Many evolutionary processes, particularly biological systems, that exhibit impulsive dynamical behaviors, can be described by the impulsive Hopfield neural networks. Fundamental issues such as the global exponential stability, the existence and uniqueness of the equilibrium of such impulsive Hopfield networks are established. A numerical example is given for illustration and interpretation of the theoretical results.
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P. Baldi and A. F. Atiya, “How delays affect neural dynamics and learning,” IEEE Trans. Neural Networks, vol. 5, pp. 612–621, 1994.
G. A. Carpenter and M. A. Cohen, “Computing with neural networks,” Science, vol. 235, pp. 1226–1227, 1987.
M. A. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,” IEEE Trans. Sys. Man and Cybern., vol. 13, pp. 815–826, 1983.
J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Nati. Acad. Sci., vol. 79, pp. 2554–2558, 1982.
J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proc. Nati. Acad. Sci., vol. 81, pp. 3088–3092, 1984.
Y. Fang and T. G. Kincaid, “Stability analysis of dynamical neural networks,” IEEE Trans. Neural Networks, vol. 7, pp. 996–1006, 1996.
K. Gopalsamy and X.-Z. He, “Delay-independent stability in bidirectional associative memory networks,” IEEE Trans. Neural networks, vol. 5, pp.998–1002, 1994.
K. Gopalsamy and X.-Z. He, “Stability in asymmetric Hopfield nets with transmission delays,” Physica D, vol. 76, pp. 344–358, 1994.
L. T. Grujic and A. N. Michel, “Exponential stability and trajectory bounds of neural networks under structural variations,” IEEE Trans. Circ. Syst., vol. 38, pp. 1182–1192, 1991.
Z.-H. Guan, Y.-Q. Liu, and X.-C. Wen, “Decentralized stabilization of singular and time-delay large-scale control systems with impulsive solutions,” IEEE Trans. Auto. Contr., vol. 40, pp. 1437–1441, 1995.
A. Guez, V. Protopopsecu, and J. Barhen, “On the stability, storage capacity and design of nonlinear continuous neural networks,” IEEE Trans. Syst. Man Cybern, vol. 18, pp. 80–87, 1988.
M. W. Hirch, “Convergent activation dynamics in continuous time networks,” Neural Networks, vol. 2, pp. 331–349, 1989.
B. Hou and J. Qian, “Stability analysis for neural dynamics with time-varying delays,” IEEE Trans. Neural Networks, vol. 9, pp. 221–223, 1998.
K. J. Hunt, D. Sbarbaro, R. Zbikowski, and P. J. Gawthrop, “Neural networks for control systems — a survey,” Automatica, vol. 28, pp. 1083–1112, 1992.
L. Jin and M. M. Gupta, “Globally asymptotical stability of discrete-time analog neural networks,” IEEE Trans. Neural Networks, vol. 7, pp. 1024–1031, 1996.
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, 1989. Theory of Impulse Differential Equations. Singapore: World Scientific Pub.
J. H. Li, A. N. Michel, and W. Porod, “Qualitative analysis and synthesis of a class of neural networks,” IEEE Trans. Circ. Syst., vol. 35, pp. 976–985, 1988.
D. D. Bainov and P. S. Simeonov, 1989. Stability Theory of Differential Equations with Impulse Effects: Theory and Applications. Ellis Horwood.
X. J. Liang and L. D. Wu, “Global exponential stability of Hopfield neural network and its applications,” Science in China (Series A), vol. 25, pp. 523–532, 1995.
X. X. Liao, “Stability of Hopfield neural networks,” Science in China, (Series A), vol. 23, pp. 1025–1035, 1992.
Y.-Q. Liu and Z.-H. Guan, 1996. Stability, Stabilization and Control of Measure Large-Scale Systems with Impulses. Guangzhou: The South China University of Technology Press.
K. Matsuoka, “Stability conditions for nonlinear continuous neural networks with asymmetric connection weights,” Neural Networks, vol. 5, pp. 495–499, 1992.
A. N. Michel and D. L. Gray, “Analysis and synthesis of neural networks with lower block triangular interconnecting structure,” IEEE Trans. Circ. Syst., vol. 37, pp. 1267–1283, 1990.
A. N. Michel, J. A. Farrel, and W. Porod, “Qualitative analysis of neural networks,” IEEE Trans. Circ. Syst., vol. 36, pp. 229–243, 1989.
S. G. Pandit and S. G. Deo, 1982. Differential Systems Involving Impulses. New York: Spring-Verlag.
J. Si and A. N. Michel, “Analysis and synthesis of a class of discrete-time neural networks with nonlinear interconnections,” IEEE Trans. Circ. Syst. I, vol. 41, pp. 52–58, 1994.
H. Yang and T. S. Dillon, “Exponential stability and oscillation of Hopfield graded response neural network,” IEEE Trans. Neural Networks, vol. 5, pp. 719–729, 1994.
Y. Zhang, S. M. Zhong, and Z. L. Li, “Periodic solution solutions and stability of Hopfield neural networks with variable delays,” Int. J. Systems Science, vol. 27, pp. 895–901, 1996.
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Guan, ZH., Hu, B., Shen, X.(. (2019). Delayed Hybrid Impulsive Neural Networks. In: Introduction to Hybrid Intelligent Networks. Springer, Cham. https://doi.org/10.1007/978-3-030-02161-0_2
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