Nie, Xiaobing; Cao, Jinde; Fei, Shumin Multistability and instability of competitive neural networks with Mexican-hat-type activation functions. (English) Zbl 1474.34358 Abstr. Appl. Anal. 2014, Article ID 901519, 20 p. (2014). Summary: We investigate the existence and dynamical behaviors of multiple equilibria for competitive neural networks with a class of general Mexican-hat-type activation functions. The Mexican-hat-type activation functions are not monotonously increasing, and the structure of neural networks with Mexican-hat-type activation functions is totally different from those with sigmoidal activation functions or nondecreasing saturated activation functions, which have been employed extensively in previous multistability papers. By tracking the dynamics of each state component and applying fixed point theorem and analysis method, some sufficient conditions are presented to study the multistability and instability, including the total number of equilibria, their locations, and local stability and instability. The obtained results extend and improve the very recent works. Two illustrative examples with their simulations are given to verify the theoretical analysis. Cited in 5 Documents MSC: 34D05 Asymptotic properties of solutions to ordinary differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics × Cite Format Result Cite Review PDF Full Text: DOI OA License References: [1] Meyer-Baese, A.; Pilyugin, S. S.; Chen, Y., Global exponential stability of competitive neural networks with different time scales, IEEE Transactions on Neural Networks, 14, 3, 716-719 (2003) · doi:10.1109/TNN.2003.810594 [2] Lu, H.; He, Z., Global exponential stability of delayed competitive neural networks with different time scales, Neural Networks, 18, 3, 243-250 (2005) · Zbl 1078.68126 · doi:10.1016/j.neunet.2004.11.009 [3] Lu, H.; Amari, S., Global exponential stability of multitime scale competitive neural networks with nonsmooth functions, IEEE Transactions on Neural Networks, 17, 5, 1152-1164 (2006) · doi:10.1109/TNN.2006.875995 [4] Nie, X.; Cao, J., Exponential stability of competitive neural networks with time-varying and distributed delays, Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering, 222, 6, 583-594 (2008) · doi:10.1243/09596518JSCE575 [5] Cao, J.; Wan, Y., Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays, Neural Networks, 53, 165-172 (2014) · Zbl 1322.93087 [6] Cao, J.; Alofi, A.; Al-Mazrooei, A.; Elaiw, A., Synchronization of switched interval networks and applications to chaotic neural networks, Abstract and Applied Analysis, 2013 (2013) · Zbl 1421.93067 · doi:10.1155/2013/940573 [7] Yang, X.; Cao, J.; Yang, Z., Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM Journal on Control and Optimization, 51, 3486-3510 (2013) · Zbl 1281.93052 [8] Zeng, Z.; Wang, J., Multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks, Neural Computation, 18, 4, 848-870 (2006) · Zbl 1107.68086 · doi:10.1162/089976606775774624 [9] Zeng, Z.; Wang, J., Multiperiodicity of discrete-time delayed neural networks evoked by periodic external inputs, IEEE Transactions on Neural Networks, 17, 5, 1141-1151 (2006) · doi:10.1109/TNN.2006.877533 [10] Cao, J.; Feng, G.; Wang, Y., Multistability and multiperiodicity of delayed Cohen-Grossberg neural networks with a general class of activation functions, Physica D, 237, 13, 1734-1749 (2008) · Zbl 1161.34044 · doi:10.1016/j.physd.2008.01.012 [11] Lin, K.-H.; Shih, C.-W., Multiple almost periodic solutions in nonautonomous delayed neural networks, Neural Computation, 19, 12, 3392-3420 (2007) · Zbl 1146.68419 · doi:10.1162/neco.2007.19.12.3392 [12] Wang, L.; Lu, W.; Chen, T., Multistability and new attraction basins of almost-periodic solutions of delayed neural networks, IEEE Transactions on Neural Networks, 20, 10, 1581-1593 (2009) · doi:10.1109/TNN.2009.2027121 [13] Kaslik, E.; Sivasundaram, S., Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis, Neural Networks, 24, 4, 370-377 (2011) · Zbl 1225.93073 · doi:10.1016/j.neunet.2010.12.008 [14] Kaslik, E.; Sivasundaram, S., Multiple periodic solutions in impulsive hybrid neural networks with delays, Applied Mathematics and Computation, 217, 10, 4890-4899 (2011) · Zbl 1225.34071 · doi:10.1016/j.amc.2010.11.025 [15] Huang, Z.; Song, Q.; Feng, C., Multistability in networks with self-excitation and high-order synaptic connectivity, IEEE Transactions on Circuits and Systems I: Regular Papers, 57, 8, 2144-2155 (2010) · Zbl 1468.34099 · doi:10.1109/TCSI.2009.2037401 [16] Nie, X.; Cao, J., Multistability of second-order competitive neural networks with nondecreasing saturated activation functions, IEEE Transactions on Neural Networks, 22, 11, 1694-1708 (2011) · doi:10.1109/TNN.2011.2164934 [17] Nie, X.; Huang, Z., Multistability and multiperiodicity of high-order competitive neural networks with a general class of activation functions, Neurocomputing, 82, 1-13 (2012) · doi:10.1016/j.neucom.2011.09.032 [18] Cheng, C.-Y.; Lin, K.-H.; Shih, C.-W., Multistability in recurrent neural networks, SIAM Journal on Applied Mathematics, 66, 4, 1301-1320 (2006) · Zbl 1106.34048 · doi:10.1137/050632440 [19] Cheng, C.-Y.; Lin, K.-H.; Shih, C.-W., Multistability and convergence in delayed neural networks, Physica D, 225, 1, 61-74 (2007) · Zbl 1132.34058 · doi:10.1016/j.physd.2006.10.003 [20] Nie, X.; Cao, J., Multistability of competitive neural networks with time-varying and distributed delays, Nonlinear Analysis: Real World Applications, 10, 2, 928-942 (2009) · Zbl 1167.34383 · doi:10.1016/j.nonrwa.2007.11.014 [21] Cheng, C.-Y.; Shih, C.-W., Complete stability in multistable delayed neural networks, Neural Computation, 21, 3, 719-740 (2009) · Zbl 1178.68401 · doi:10.1162/neco.2008.03-07-492 [22] Huang, G.; Cao, J., Delay-dependent multistability in recurrent neural networks, Neural Networks, 23, 2, 201-209 (2010) · Zbl 1400.34115 · doi:10.1016/j.neunet.2009.10.004 [23] Wang, L.; Lu, W.; Chen, T., Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions, Neural Networks, 23, 2, 189-200 (2010) · Zbl 1409.34025 · doi:10.1016/j.neunet.2009.11.010 [24] Zeng, Z.; Zheng, W., Multistability of neural networks with time-varying delays and concaveconvex characteristics, IEEE Transactions on Neural Networks and Learning Systems, 23, 293-305 (2012) [25] Zeng, Z.; Huang, T.; Zheng, W. X., Multistability of recurrent neural networks with time-varying delays and the piecewise linear activation function, IEEE Transactions on Neural Networks, 21, 8, 1371-1377 (2010) · doi:10.1109/TNN.2010.2054106 [26] Huang, G.; Cao, J., Multistability in bidirectional associative memory neural networks, Physics Letters A, 372, 16, 2842-2854 (2008) · Zbl 1220.92002 · doi:10.1016/j.physleta.2007.12.053 [27] Huang, Z.; Wang, X.; Feng, C., Multiperiodicity of periodically oscillated discrete-time neural networks with transient excitatory self-connections and sigmoidal nonlinearities, IEEE Transactions on Neural Networks, 21, 10, 1643-1655 (2010) · doi:10.1109/TNN.2010.2067225 [28] Lu, W.; Wang, L.; Chen, T., On attracting basins of multiple equilibria of a class of cellular neural networks, IEEE Transactions on Neural Networks, 22, 3, 381-394 (2011) · doi:10.1109/TNN.2010.2102048 [29] Nie, X.; Cao, J.; Fei, S., Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions, Neurocomputing, 119, 281-291 (2013) [30] Wang, Y.; Cao, J., Multiperiodicity evoked by periodic external inputs in Cohen-Grossbergtype BAM networks with discrete and distributed delays, British Journal of Mathematics and Computer Science, 2, 94-113 (2012) [31] Cao, J.; Wang, Y., Bi-periodicity evoked by periodic external inputs in delayed Cohen-Grossberg-type bidirectional associative memory networks, Physica Scripta, 81, 5 (2010) · Zbl 1192.82060 · doi:10.1088/0031-8949/81/05/055803 [32] Huang, G.; Cao, J., Multistability of neural networks with discontinuous activation function, Communications in Nonlinear Science and Numerical Simulation, 13, 10, 2279-2289 (2008) · Zbl 1221.34131 · doi:10.1016/j.cnsns.2007.07.005 [33] Wang, 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) · doi:10.1109/TNNLS.2012.2210732 This reference list is based on information provided by the publisher or from digital mathematics libraries. 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