Complete stability of delayed recurrent neural networks with Gaussian activation functions. (English) Zbl 1432.34093
Summary: This paper addresses the complete stability of delayed recurrent neural networks with Gaussian activation functions. By means of the geometrical properties of Gaussian function and algebraic properties of nonsingular \(M\)-matrix, some sufficient conditions are obtained to ensure that for an \(n\)-neuron neural network, there are exactly \(3^k\) equilibrium points with \(0\leq k\leq n\), among which \(2^k\) and \(3^k-2^k\) equilibrium points are locally exponentially stable and unstable, respectively. Moreover, it concludes that all the states converge to one of the equilibrium points; i.e., the neural networks are completely stable. The derived conditions herein can be easily tested. Finally, a numerical example is given to illustrate the theoretical results.
MSC:
| 34K20 | Stability theory of functional-differential equations |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 34K21 | Stationary solutions of functional-differential equations |
| 34K25 | Asymptotic theory of functional-differential equations |
References:
| [1] | Arik, S., Stability analysis of delayed neural networks, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47, 1089-1092 (2000) · Zbl 0992.93080 |
| [2] | Bao, G.; Zeng, Z. G., Analysis and design of associative memories based on recurrent neural network with discontinuous activation functions, Neurocomputing, 77, 101-107 (2012) |
| [3] | Chen, T. P.; Chen, H., Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks, IEEE Transactions on Neural Networks, 6, 904-910 (1995) |
| [4] | Chen, W. H.; Zheng, W. X., A new method for complete stability analysis of cellular neural networks with time delay, IEEE Transactions on Neural Networks, 21, 1126-1139 (2010) |
| [5] | Cheng, C.-Y.; Lin, K.-H.; Shih, C.-W., Multistability in recurrent neural networks, SIAM Journal on Applied Mathematics, 66, 1301-1320 (2006) · Zbl 1106.34048 |
| [6] | Cheng, C.-Y.; Lin, K.-H.; Shih, C.-W., Multistability and convergence in delayed neural networks, Physica D: Nonlinear Phenomena, 225, 61-74 (2007) · Zbl 1132.34058 |
| [7] | Cheng, C.-Y.; Lin, K.-H.; Shih, C.-W.; Tseng, J. P., Multistability for delayed neural networks via sequential contracting, IEEE Transactions on Neural Networks and Learning Systems, PP, 1-14 (2015) |
| [8] | Cheng, C.-Y.; Shih, C.-W., Complete stability in multistable delayed neural networks, Neural Computation, 21, 719-740 (2009) · Zbl 1178.68401 |
| [9] | Cohen, M.; Grossberg, S., Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Transactions on Systems, Man and Cybernetics, 815-826 (1983) · Zbl 0553.92009 |
| [10] | Di Marco, M.; Forti, M.; Grazzini, M.; Pancioni, L., Necessary and sufficient condition for multistability of neural networks evolving on a closed hypercube, Neural Networks, 54, 38-48 (2014) · Zbl 1322.93011 |
| [11] | Feng, Z.; Zheng, W., On extended dissipativity of discrete-time neural networks with time delay, IEEE Transactions on Neural Networks and Learning Systems, 26, 3293-3300 (2015) |
| [12] | Forti, M., Some extensions of a new method to analyze complete stability of neural networks, IEEE Transactions on Neural Networks, 13, 1230-1238 (2002) |
| [13] | Forti, M.; Tesi, A., A new method to analyze complete stability of PWL cellular neural networks, International Journal of Bifurcation and Chaos, 11, 655-676 (2001) · Zbl 1090.37567 |
| [14] | Hartman, E. J.; Keeler, J. D.; Kowalski, J. M., Layered neural networks with Gaussian hidden units as universal approximations, Neural Computation, 2, 210-215 (1990) |
| [15] | Huang, Y. J.; Zhang, H. G.; Wang, Z., Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions, Neurocomputing, 91, 21-28 (2012) |
| [16] | Igelnik, B.; Pao, Y.-H., Stochastic choice of basis functions in adaptive function approximation and the functional-link net, IEEE Transactions on Neural Networks, 6, 1320-1329 (1995) |
| [17] | Kamimura, R., Cooperative information maximization with Gaussian activation functions for self-organizing maps, IEEE Transactions on Neural Networks, 17, 909-918 (2006) |
| [18] | Kaslik, E.; Sivasundaram, S., Multistability in impulsive hybrid Hopfield neural networks with distributed delays, Nonlinear Analysis. Real World Applications, 12, 1640-1649 (2011) · Zbl 1221.34219 |
| [19] | Lee, C.-C.; Chung, P.-C.; Tsai, J.-R.; Chang, C.-I., Robust radial basis function neural networks, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 29, 674-685 (1999) |
| [20] | Lee, S.; Kil, R. M., A Gaussian potential function network with hierarchically self-organizing learning, Neural Networks, 4, 207-224 (1991) |
| [21] | Lee, S.-W.; Moraga, C., A Cosine-Modulated Gaussian activation function for Hyper-Hill neural networks, (3rd international conference on signal processing Vol. 2 (1996), IEEE), 1397-1400 |
| [22] | Lin, F.-C.; Ko, L.-W.; Chuang, C.-H.; Su, T.-P.; Lin, C.-T., Generalized EEG-based drowsiness prediction system by using a self-organizing neural fuzzy system, IEEE Transactions on Circuits and Systems. I. Regular Papers, 59, 2044-2055 (2012) · Zbl 1468.93107 |
| [23] | Liu, P.; Zeng, Z. G.; Wang, J., Multistability analysis of a general class of recurrent neural networks with non-monotonic activation functions and time-varying delays, Neural Networks, 79, 117-127 (2016) · Zbl 1417.34178 |
| [24] | Liu, P.; Zeng, Z. G.; Wang, J., Multistability of recurrent neural networks with nonmonotonic activation functions and mixed time delays, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 46, 512-523 (2016) |
| [25] | Nie, X. B.; Cao, J. D., Multistability of second-order competitive neural networks with nondecreasing saturated activation functions, IEEE Transactions on Neural Networks, 22, 1694-1708 (2011) |
| [26] | Nie, X. B.; Zheng, W. X., Multistability of neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays, Neural Networks, 65, 65-79 (2015) · Zbl 1394.68305 |
| [27] | Ozyildirim, B. M.; Avci, M., Generalized classifier neural network, Neural Networks, 39, 18-26 (2013) |
| [28] | Savitha, R.; Suresh, S.; Sundararajan, N., Metacognitive learning in a fully complex-valued radial basis function neural network, Neural Computation, 24, 1297-1328 (2012) |
| [29] | Savitha, R.; Suresh, S.; Sundararajan, N., Projection-based fast learning fully complex-valued relaxation neural network, IEEE Transactions on Neural Networks and Learning Systems, 24, 529-541 (2013) |
| [30] | Shayer, L. P.; Campbell, S. A., Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays, SIAM Journal on Applied Mathematics, 61, 673-700 (2000) · Zbl 0992.92013 |
| [31] | Vuković, N.; Miljković, Z., A growing and pruning sequential learning algorithm of hyper basis function neural network for function approximation, Neural Networks, 46, 210-226 (2013) · Zbl 1296.68148 |
| [32] | Wang, L. L.; Chen, T., Complete stability of cellular neural networks with unbounded time-varying delays, Neural Networks, 36, 11-17 (2012) · Zbl 1258.34164 |
| [33] | Wang, L. L.; Chen, T., Multistability of neural networks with Mexican-hat-type activation functions, IEEE Transactions on Neural Networks and Learning Systems, 23, 1816-1826 (2012) |
| [34] | Wang, L. L.; Chen, T., Multiple \(\mu \)-stability of neural networks with unbounded time-varying delays, Neural Networks, 53, 109-118 (2014) · Zbl 1307.93365 |
| [35] | Wang, L. L.; Chen, T., Multistability and complete convergence analysis on high-order neural networks with a class of nonsmooth activation functions, Neurocomputing, 152, 222-230 (2015) |
| [36] | Wang, L. L.; Lu, W. L.; Chen, T. P., Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions, Neural Networks, 23, 189-200 (2010) · Zbl 1409.34025 |
| [37] | Wang, J. L.; Wu, H. N., Synchronization and adaptive control of an array of linearly coupled reaction-diffusion neural networks with hybrid coupling, IEEE Transactions on Cybernetics, 44, 1350-1361 (2014) |
| [38] | Wen, S. P.; Huang, T. W.; Zeng, Z. G.; Chen, Y. R.; Li, P., Circuit design and exponential stabilization of memristive neural networks, Neural Networks, 63, 48-56 (2015) · Zbl 1323.93065 |
| [39] | Zeng, Z. G.; Huang, D.-S.; Wang, Z. F., Memory pattern analysis of cellular neural networks, Physics Letters A, 342, 114-128 (2005) · Zbl 1222.92013 |
| [40] | Zeng, Z. G.; Wang, J., Complete stability of cellular neural networks with time-varying delays, IEEE Transactions on Circuits and Systems. I. Regular Papers, 53, 944-955 (2006) · Zbl 1374.34292 |
| [41] | Zeng, Z. G.; Wang, J.; Liao, X. X., Stability analysis of delayed cellular neural networks described using cloning templates, IEEE Transactions on Circuits and Systems. I. Regular Papers, 51, 2313-2324 (2004) · Zbl 1374.34293 |
| [42] | Zeng, Z. G.; Zheng, W. X., Multistability of neural networks with time-varying delays and concave-convex characteristics, IEEE Transactions on Neural Networks and Learning Systems, 23, 293-305 (2012) |
| [43] | Zeng, Z. G.; Zheng, W. X., Multistability of two kinds of recurrent neural networks with activation functions symmetrical about the origin on the phase plane, IEEE Transactions on Neural Networks and Learning Systems, 24, 1749-1762 (2013) |
| [44] | Zhang, X. M.; Han, Q. L., New Lyapunov-Krasovskii functionals for global asymptotic stability of delayed neural networks, IEEE Transactions on Neural Networks, 20, 533-539 (2009) |
| [45] | Zhang, B.; Lam, J.; Xu, S., Stability analysis of distributed delay neural networks based on relaxed Lyapunov-Krasovskii functionals, IEEE Transactions on Neural Networks and Learning Systems, 26, 1480-1492 (2015) |
| [46] | Zhang, H. G.; Wang, Z. S.; Liu, D. R., A comprehensive review of stability analysis of continuous-time recurrent neural networks, IEEE Transactions on Neural Networks and Learning Systems, 25, 1229-1262 (2014) |
| [47] | Zhu, Q.; Cao, J., Robust exponential stability of Markovian jump impulsive stochastic Cohen-Grossberg neural networks with mixed time delays, IEEE Transactions on Neural Networks, 21, 1314-1325 (2010) |
| [48] | Zhu, Q.; Cao, J., Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 41, 341-353 (2011) |
| [49] | Zhu, Q.; Cao, J., Stability analysis of Markovian jump stochastic BAM neural networks with impulse control and mixed time delays, IEEE Transactions on Neural Networks and Learning Systems, 23, 467-479 (2012) |
| [50] | Zhu, Q.; Cao, J.; Rakkiyappan, R., Exponential input-to-state stability of stochastic Cohen-Grossberg neural networks with mixed delays, Nonlinear Dynamics, 79, 1085-1098 (2015) · Zbl 1345.92019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
