Multistability of neural networks with discontinuous activation function. (English) Zbl 1221.34131

Summary: The multistability is studied for two-dimensional neural networks with multilevel activation functions. It is shown that the system has \(n^{2}\) isolated equilibrium points which are locally exponentially stable, where the activation function has \(n\) segments. Furthermore, evoked by periodic external input, \(n^{2}\) periodic orbits which are locally exponentially attractive, can be found. These results are extended to \(k\)-neuron networks, which really enlarge the capacity of the associative memories. Examples and simulation results are used to illustrate the theory.


34D05 Asymptotic properties of solutions to ordinary differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
Full Text: DOI


[1] Cao, J.; Wang, J., Global asymptotic and robust stability of recurrent neural networks with time delays, IEEE Trans Circuits Syst Part I, 52, 2, 417-426 (2005) · Zbl 1374.93285
[2] Cao, J.; Wang, J., Absolute exponential stability of recurrent neural networks with time delays and Lipschitz-continuous activation functions, Neural Network, 17, 3, 379-390 (2004) · Zbl 1074.68049
[3] Cao, J.; Liang, J., Boundedness and stability for Cohen-Grossberg neural networks with time-varying delays, J Math Anal Appl, 296, 2, 665-685 (2004) · Zbl 1044.92001
[4] Cao, J.; Huang, D. S.; Qu, Y., Global robust stability of delayed recurrent neural networks, Chaos Solitons Fractals, 23, 1, 221-229 (2005) · Zbl 1075.68070
[5] Cao, J.; Chen, T., Globally exponentially robust stability and periodicity of delayed neural networks, Chaos Solitons Fractals, 22, 4, 957-963 (2004) · Zbl 1061.94552
[6] Yu, W.; Cao, J., Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with time delays, Nonlinear Anal, 351, 1 - 2, 64-78 (2006) · Zbl 1234.34047
[7] Guo, S.; Huang, L., Linear stability and Hopf bifurcation in a two-neuron network with three delays, Int J Bifur Chaos, 14, 8, 2790-2810 (2004) · Zbl 1062.34078
[8] Song, Y.; Han, M.; Wei, J., Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays, Physica D, 200, 185-204 (2005) · Zbl 1062.34079
[9] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272 (1999) · Zbl 1066.34511
[10] Forti, M.; Manetti, S.; Marini, M., Necessary and sufficient condition for absolute stability of neural networks, IEEE Trans Circuits Syst Part I, 41, 491-494 (1994) · Zbl 0925.92014
[11] Forti, M.; Tesi, A., New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Trans Circuits Syst Part I, 42, 354-366 (1995) · Zbl 0849.68105
[12] Tank, D. W.; Hopfield, J. J., Simple “neural” optimization networks: An AD converter signal decision circuit and a linear programming network, IEEE Trans Circuits Syst, 33, 533-541 (1986)
[13] Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc Natl Acad Sci, 81, 3088-3092 (1984) · Zbl 1371.92015
[14] Forti, M.; Nistri, P., Global convergence of neural networks with discontinuous neuron activation, IEEE Trans Circuits Syst Part I, 50, 1421-1435 (2003) · Zbl 1368.34024
[15] Forti, M.; Nistri, P.; Papini, D., Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE Trans Neural Network, 16, 1449-1463 (2005)
[16] Lu, W.; Chen, T., Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions, Neural Network, 18, 231-242 (2005) · Zbl 1078.68127
[17] Juang, J.; Lin, S., Cellular neural networks: mosaic pattern and spatial chaos, SIAM J Appl Math, 60, 891-915 (2000) · Zbl 0947.34038
[18] Shih, C., Pattern formation and spatial chaos for cellular neural networks with asymmetric templates, Int J Bifur Chaos Appl Sci Eng, 8, 1907-1936 (1998) · Zbl 1002.92512
[19] Shih, C., Influence of boundary conditions on pattern formation and spatial chaos in lattice systems, SIAM J Appl Math, 61, 335-368 (2000) · Zbl 0985.37091
[20] Zeng, Z.; Wang, J.; Liao, X., Stability analysis of delayed cellular neural networks described using cloning templates, IEEE Trans Circuits Syst Part I, 51, 2313-2324 (2004) · Zbl 1374.34293
[21] Zeng, Z.; Wang, J., Multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks, Neural Comput, 18, 848-870 (2006) · Zbl 1107.68086
[22] Cheng, C.; Lin, K.; Shih, C., Multistability in recurrent neural networks, SIAM J Appl Math, 66, 1301-1320 (2006) · Zbl 1106.34048
[23] Forti, M., A note on neural networks with multiple equilibrium points, IEEE Trans Circuits Syst Part I, 43, 487-491 (1996)
[24] Yi, Z.; Tan, K., Multistability of discrete-time recurrent neural networks with unsaturating piecewise linear activation functions, IEEE Trans Neural Network, 15, 329-336 (2004)
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.