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Convergence and periodicity of solutions for a discrete-time network model of two neurons. (English) Zbl 1045.92002
Summary: Considered is a class of difference systems with McCulloch-Pitts nonlinearity, which includes the discrete version of an artificial neural network of two neurons with piecewise constant arguments. Some interesting results are obtained for the convergence and periodicity of solutions of the systems. Most importantly, multiple periodic solutions exist. Our results have potential applications in neural networks.

92B20General theory of neural networks (mathematical biology)
39A10Additive difference equations
68T05Learning and adaptive systems
Full Text: DOI
[1] Y. Chen and J. Wu, Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Differential and Integral Equations (to appear). · Zbl 1023.34065
[2] Gopalsamy, K.; Leung, I.: Delay induced periodicity in a neural netlet of excitation and inhibition. Phys. D 89, 395-426 (1996) · Zbl 0883.68108
[3] Olien, L.; Belair, J.: Bifurcations, stability, and monotonicity properties of a delayed neural network model. Phys. D 102, 349-363 (1997) · Zbl 0887.34069
[4] Pakdaman, K.; Grota-Ragazzo, C.; Malta, C. P.; Arino, O.; Vibert, J. -F.: Effect of delay on the boundary of the basin of attraction in a system of two neurons. Neural networks 11, 509-519 (1998)
[5] Ruan, S.; Wei, J.: Periodic solutions of planar systems with two delays. Proc. roy. Soc. Edinburgh 129, 1017-1032 (1999) · Zbl 0946.34062
[6] Wei, J.; Ruan, S.: Stability and bifurcation in a neural network model with two delays. Phys. D 130, 255-272 (1999) · Zbl 1066.34511
[7] De Groat, R. D.; Hunt, L. R.; Linebarger, D. A.; Verma, M.: Discrete-time nonlinear system stability. IEEE trans. Circuits systems I fund. Theory appl. 39, 834-840 (1992)
[8] Huang, L.; Wu, J.: Dynamics of inhibitory artificial neural networks with threshold nonlinearity. Fields inst. Commun. 29, 235-243 (2001) · Zbl 0973.92002
[9] L. Huang and J. Wu, The role of threshold in preventing delay-induced oscillations of frustrated neural networks with McCulloch-Pitts nonlinearity, Game Theory and Algebra (to appear). · Zbl 1043.34086
[10] Pakdaman, K.; Malta, C. P.; Grota-Ragazzo, C.; Arino, O.; Vibert, J. -F.: Transient oscillations in continuous-time excitatory ring neural networks with delay. Phys. rev. E 55, 3234-3248 (1997)
[11] Pakdaman, K.; Grota-Ragazzo, C.; Malta, C. P.: Transient regime duration in continuous-time neural networks with delay. Phys. rev. E 58, 3623-3627 (1998)
[12] Ushio, T.: Limitation of delay feedback control in nonlinear discrete-time systems. IEEE. trans. Circuits syst. I 43, 815-816 (1996)
[13] Zhou, Z.; Yu, J. S.; Huang, L. H.: Asymptotic behavior of delay difference systems. Computers math. Applic. 42, No. 3--5, 283-290 (2001) · Zbl 0998.39004
[14] Busenberg, S.; Cooke, K. L.: V.lakshmikanthan models of vertically transmitted diseases with sequential-continuous dynamics. Nonlinear phenomena in mathematical science (1982) · Zbl 0512.92018
[15] Cooke, K. L.; Wiener, J.: Retarded differential equations with piecewise constant delays. J. math. Anal. appl. 99, 265-297 (1984) · Zbl 0557.34059
[16] Shah, S. M.; Wiener, J.: Advanced differential equations with piecewise constant argument deviations. Internat. J. Math. math. Sci. 6, 671-703 (1983) · Zbl 0534.34067
[17] Aftabizadeh, A. R.; Wiener, J.: Differential inequalities for delay differential equations with piecewise constant. Appl. math. Comput. 24, 183-194 (1987) · Zbl 0629.34074
[18] Aftabizadeh, A. R.; Wiener, J.; Xu, Jm.: Oscillatory and periodic solutions of delay differential equations with piecewise constant argument. Proc. amer. Math. soc. 99, 673-679 (1987) · Zbl 0631.34078
[19] Shen, Jh.; Stavroulakis, I. P.: Oscillatory and nonoscillatory delay equations with piecewise constant argument. J. math. Anal. appl. 248, 385-401 (2000) · Zbl 0966.34063
[20] Wang, Y.; Yan, J.: Necessary and sufficient condition for the global attractivity of the trivial solution of a delay equation with continuous and piecewise constant arguments. Appl. math. Lett. 10, No. 5, 91-96 (1997) · Zbl 0894.34069
[21] Wiener, J.; Shah, S. M.: Functional-differential equations with piecewise constant argument. Indian J. Math. 29, 131-158 (1987) · Zbl 0652.34071
[22] Cooke, K. L.; Wiener, J.: A survey of differential equations with piecewise constant arguments. Lecture notes in mathematics 1475 (1991) · Zbl 0737.34045