Multistability and convergence in delayed neural networks. (English) Zbl 1132.34058

The authors investigate a general delayed neural network
\[ \frac{dx_i(t)}{dt}=-\mu_i x_i(t)+\sum_{j=1}^n\alpha_{ij}g_j(x_j(t))+ \sum_{j=1}^n\beta_{ij}g_j(x_j(t-\tau_{ij}))+I_i, \]
where \(i=1,\dots,n\); \(\mu_i>0\); \(\alpha_{ij}\) and \(\beta_{ij}\) are connection weights from neuron \(j\) to neuron \(i\); \(g_j(\cdot)\) are activation functions; \(0\leq\tau_{ij}\leq\tau\) are time lags; \(I_i\) stands for an independent bias current source. Two classes of activation function are considered. Class \(A\) contains bounded smooth sigmoidal functions, and class \(B\) contains nondecreasing functions with saturations.
The authors derive conditions for the existence of \(3^n\) equilibria. The parameter conditions motivated by a geometrical observation. The basins of attraction for \(2^n\) stable stationary solutions are established. The theory is extended to the existence of \(2^n\) limit cycles for the \(n\)-dimensional network with time-periodic inputs. The strongly order preserving property and quasiconvergence are also discussed. The study illustrates distinct dynamical behaviors between systems with activation functions of different classes. Numerical simulations are presented to illustrate the presented theory.
This paper is an extention of the recent paper by authors [SIAM J. Appl. Math. 66, No. 4, 1301–1320 (2006; Zbl 1106.34048)].


34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics


Zbl 1106.34048
Full Text: DOI


[1] Chua, L.O.; Yang, L., Cellular neural networks: theory, IEEE trans. circuits syst., 35, 1257-1272, (1988) · Zbl 0663.94022
[2] Foss, J.; Longtin, A.; Mensour, B.; Milton, J., Multistability and delayed recurrent loops, Phys. rev. lett., 76, 708-711, (1996)
[3] Hopfield, J., Neurons with graded response have collective computational properties like those of two sate neurons, Proc. natl. acad. sci. USA, 81, 3088-3092, (1984) · Zbl 1371.92015
[4] Morita, M., Associative memory with non-monotone dynamics, Neural netw., 6, 115-126, (1993)
[5] Hahnloser, R.L.T., On the piecewise analysis of networks of linear threshold neurons, Neural netw., 11, 691-697, (1998)
[6] Marcus, C.M.; Westervelt, R.M., Stability of analog neural networks with delay, Phys. rev. A, 39, 347-359, (1989)
[7] Chua, L.O.; Yang, L., Cellular neural networks: applications, IEEE trans. circuits syst., 35, 1273-1290, (1988)
[8] Roska, T.; Chua, L.O., Cellular neural networks with nonlinear and delay-type template, Int. J. circ. theor. appl., 20, 469-481, (1992) · Zbl 0775.92011
[9] Chua, L.O., CNN: A paradigm for complexity, (1998), World Scientific · Zbl 0916.68132
[10] Baldi, P.; Atiya, A.F., How delays affect neural dynamics and learning, IEEE trans. neural netw., 5, 612-621, (1994)
[11] Bélair, J.; Campbell, S.A.; van den Driessche, P., Frustration, stability, and delay-induced oscillations in a neural network model, SIAM J. appl. math., 56, 245-255, (1996) · Zbl 0840.92003
[12] Olien, L.; Bélair, J., Bifurcations, stability, and monotonicity properties of a delayed neural network model, Physica D, 102, 349-363, (1997) · Zbl 0887.34069
[13] van den Driessche, P.; Zou, X., Global attractivity in delayed Hopfield neural network models, SIAM J. appl. math., 58, 1878-1890, (1998) · Zbl 0917.34036
[14] Shayer, L.P.; Campbell, S.A., Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays, SIAM J. appl. math., 61, 673-700, (2000) · Zbl 0992.92013
[15] Campbell, S.A.; Edwards, R.; van den Driessche, P., Delayed coupling between two neural network loops, SIAM J. appl. math., 65, 316-335, (2004) · Zbl 1072.92003
[16] Cao, J., Global stability analysis in delayed cellular neural networks, Phys. rev. E, 59, 5940-5944, (1999)
[17] Cao, J., New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Phys. lett. A, 307, 136-147, (2003) · Zbl 1006.68107
[18] Cao, J.; Li, Q., On the exponential stability and periodic solutions of delayed cellular neural networks, J. math. anal. appl., 252, 50-64, (2000) · Zbl 0976.34067
[19] Civalleri, P.P.; Gilli, M., On stability of cellular neural networks with delay, IEEE trans. circuits syst., 40, 157-165, (1993) · Zbl 0792.68115
[20] Feng, C.; Plamondon, R., On the stability analysis of delayed neural networks systems, Neural netw., 14, 1181-1188, (2001)
[21] Liao, X.; Chen, G.; Sanchez, E.N., Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach, Neural netw., 15, 855-866, (2002)
[22] Joy, M., Results concerning the absolute stability of delayed neural networks, Neural netw., 13, 613-616, (2000)
[23] Liao, X.; Li, C., An LMI approach to asymptotical stability of multi-delayed neural networks, Physica D, 200, 139-155, (2005) · Zbl 1078.34052
[24] Mohamad, S.; Gopalsamy, K., Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. math. comput, 135, 17-38, (2003) · Zbl 1030.34072
[25] Zhang, Q.; Wei, X.; Xu, J., Global exponential convergence analysis of delayed neural networks with time-varying delays, Phys. lett. A, 318, 537-544, (2003) · Zbl 1098.82616
[26] Zeng, Z.; Huang, D.S.; Wang, Z., Memory pattern analysis of cellular neural networks, Phys. lett. A, 342, 114-128, (2005) · Zbl 1222.92013
[27] Cheng, C.Y.; Lin, K.H.; Shih, C.W., Multistability in recurrent neural networks, SIAM J. appl. math., 66, 4, 1301-1320, (2006) · Zbl 1106.34048
[28] Hirsch, M., Convergent activation dynamics in continuous-time networks, Neural netw., 2, 331-349, (1989)
[29] Wu, J.H., Introduction to neural dynamics and signal transmission delay, (2001), Walter de Gruyter Berlin · Zbl 0977.34069
[30] Liao, X.; Wang, J., Global dissipativity of continuous-time recurrent neural networks with time delay, Phys. rev. E, 68, 016118, (2003)
[31] C.Y. Cheng, Multistability and Convergence in Delayed Neural Networks, Ph.D. Dissertation, National Chiao Tung University, Hsinchu, Taiwan, 2006
[32] Smith, H.L., ()
[33] Smith, H.L.; Thieme, H.R., Strongly order preserving semiflows generated by functional differential equations, J. differential equations, 93, 332-363, (1991) · Zbl 0735.34065
[34] van den Driessche, P.; Wu, J.; Zou, X., Stabilization role of inhibitory self-connections in a delayed neural network, Physica D, 150, 84-90, (2001) · Zbl 1007.34072
[35] Chu, T.; Zhang, Z.; Wang, Z., A decomposition approach to analysis of competitive – cooperative neural networks with delay, Phys. lett. A, 312, 339-347, (2003) · Zbl 1050.82541
[36] Cosner, C., Comparison principles for systems that embed in cooperative systems with applications to diffusive lotka – volterra models, Dyn. contin. discrete impuls. syst., 3, 283-303, (1997) · Zbl 0885.35015
[37] Wu, J.; Zhao, X.Q., Permanence and convergence in multi-species competition systems with delay, Proc. amer. math. soc., 126, 1709-1714, (1998) · Zbl 0894.34062
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