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Stability, bifurcation and global existence of a Hopf-bifurcating periodic solution for a class of three-neuron delayed network models. (English) Zbl 1133.34036
The paper studies a network consisting of three-neurons described by the following system of delay differential equations $$\align y_1'(t) &= -ky_1(t) +\beta \operatorname{tanh} (y_1(t-\tau))+ w_{12} \operatorname{tanh} [y_2(t-\tau)] + w_{13} \operatorname{tanh} [y_3(t-\tau)] + I_1, \\ y_2'(t) &= -ky_2(t) +w_{21} \operatorname{tanh} (y_1(t-\tau))+ \beta \operatorname{tanh} [y_2(t-\tau)] + w_{23} \operatorname{tanh} [y_3(t-\tau)] + I_2, \\ y_3'(t) &= -ky_3(t) +w_{31} \operatorname{tanh} (y_1(t-\tau))+ w_{32} \operatorname{tanh} [y_2(t-\tau)] + \beta \operatorname{tanh} [y_3(t-\tau)] + I_3, \endalign$$ where $k>0$, $w_{ij}$ and $\beta$ are weights of synaptic connections, $\tau>0$ is the fixed delay time, $I_i$ are constant inputs. The authors provide sufficient conditions for linear stability and occurrence of Hopf bifurcation. Also, sufficient conditions for the existence of multiple periodic solutions are obtained.

34K13Periodic solutions of functional differential equations
34K18Bifurcation theory of functional differential equations
92B20General theory of neural networks (mathematical biology)
34K20Stability theory of functional-differential equations
Full Text: DOI
[1] Cohen, M.; Grossberg, S.: Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE trans. Syst. man. Cybern. 13, No. 5, 815-826 (1983) · Zbl 0553.92009
[2] Hopfield, J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. natl. Acad. sci. USA biophysics 81, 3088-3092 (1984)
[3] Gopalsamy, K.; He, X.: Stability in asymmetric Hopfield nets with transmission delays. Physica D 76, 344-358 (1994) · Zbl 0815.92001
[4] Hou, C. H.; Quian, J. X.: Stability analysis for neural dynamics with time-varying delays. IEEE trans. Neural networks 9, No. 1, 221-223 (1998)
[5] Peng, J. G.; Qiao, H.; Xu, Z. B.: A new approach to stability of neural networks with time varying delays. Neural networks 15, No. 1, 95-103 (2002)
[6] Arik, S.; Tavsanoglu, V.: Equilibrium analysis of delayed cnns. IEEE trans. Circuits syst. I 45, No. 2, 168-171 (1998) · Zbl 0917.68223
[7] Cao, J. D.: A set of stability criteria for delayed cellular neural networks. IEEE trans. Circuits syst. I 48, No. 4, 494-498 (2001) · Zbl 0994.82066
[8] Cao, J. D.: New results concerning exponential stability and periodic solutions of delayed cellular neural networks. Phys. lett. A 307, No. 2--3, 136-147 (2003) · Zbl 1006.68107
[9] Chua, L. O.; Yang, L.: Cellular neural networks: theory. IEEE trans. Circuits syst. 35, No. 10, 1257-1272 (1988) · Zbl 0663.94022
[10] Mohamad, S.; Gopalsamy, K.: Exponential stability of continuous-time and discrete time cellular neural networks with delays. Appl. math. Comput. 135, No. 1, 17-38 (2003) · Zbl 1030.34072
[11] Marcus, C. M.; Westervelt, R. M.: Stability of analog neural networks with delay. Phys. rev. A 39, 347-359 (1989)
[12] Cohen, M.; Grossberg, S.: Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE trans. Syst. man. Cybern. 13, No. 5, 815-826 (1983) · Zbl 0553.92009
[13] Den Driessche, P. Van; Zou, X.: Global attractivity in delayed Hopfield neural network models. SIAM J. Appl. math. 58, 1878-1890 (1998) · Zbl 0917.34036
[14] Cao, J.; Tao, Q.: Estimation on domain of attraction and convergence rate of Hopfield continuous feedback neural networks. J. comput. Syst. sci. 62, 528-534 (2001) · Zbl 0987.68071
[15] Cao, J.: Global exponential stability of Hopfield neural networks. Internat. J. Syst. sci. 32, No. 2, 233-236 (2001) · Zbl 1011.93091
[16] Cao, J.; Wang, J.: Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays. Neural netw. 17, 379-390 (2004) · Zbl 1074.68049
[17] Cao, J.; Wang, J.; Liao, X.: Novel stability criteria of delayed cellular neural networks. Internat. J. Neural syst. 13, No. 5, 367-375 (2003)
[18] Zhou, D. M.; Cao, J. D.: Globally exponential stability conditions for cellular neural networks with time varying delays. Appl. math. Comput. 13, No. 2--3, 487-496 (2002) · Zbl 1034.34093
[19] Kosko, B.: Adaptive bidirectional associative memories. Appl. optics 26, No. 23, 4947-4960 (1987)
[20] Cao, J. D.; Liang, J. L.; Lam, J.: Exponential stability of high order bidirectional associative memory neural networks with time delays. Physica D 199, No. 3--4, 425-436 (2004) · Zbl 1071.93048
[21] Gilli, M.: Strange attractors in delayed cellular neural networks. IEEE trans. Circuits syst. I 40, 849-853 (1993) · Zbl 0844.58056
[22] Gopalsamy, K.; Leung, I.: Delay induced periodicity in a neural network of excitation and inhibition. Physica D 89, 395-426 (1996) · Zbl 0883.68108
[23] Olien, L.; Belair, J.: Bifurcations stability and monotonicity properties of a delayed neural network model. Physica D 102, 349-363 (1997) · Zbl 0887.34069
[24] Pakdaman, K.; Grotta-Ragazzo, C.; Malta, C. P.: Transient regime duration in continuous-time neural networks with delay. Phys. rev. E 58, No. 3, 3623-3627 (1998)
[25] Pakdaman, K.: Effect of delay on the boundary of the basin of attraction in a system in two neurons. Neural netw. 11, 509-519 (1998)
[26] Liao, X.; Wong, K. W.; Leung, C. S.; Wu, Z.: Hopf bifurcation and chaos in a single delayed neuron equation with nonmonotonic activation function. Chaos solitons fractals 21, 1535-1547 (2001) · Zbl 1012.92005
[27] Chen, Y.; Wu, J.: Slowly oscillating periodic solutions for a delayed frustrated network of two neurons. J. math. Anal. appl. 259, 188-205 (2001) · Zbl 0998.34058
[28] Faria, T.: On a planar system modelling a neuron network with memory. J. differential equations 168, 129-149 (2000) · Zbl 0961.92002
[29] Giamnakopoulos, F.; Zapp, A.: Bifurcation in a planar system of differential equations modelling neural activity. Physica D 159, 215-232 (2001) · Zbl 0984.92505
[30] Shayer, L. P.; Campbell, S. A.: Stability, bifurcations, and multistability of two coupled neurons with multiple time delays. SIAM J. Appl. math. 61, No. 2, 673-700 (2000) · Zbl 0992.92013
[31] Gopalsamy, K.; Leung, I. K. C.; Liu, P.: Global Hopf-bifurcation in a neural netlet. Appl. math. Comput. 94, 171-192 (1998) · Zbl 0946.34065
[32] Majee, N. C.; Roy, A. B.: Temporal dynamics of a two-neuron continuous network model with time delay. Appl. math. Modelling 21, 673-679 (1997) · Zbl 0893.68126
[33] Liao, X.; Wong, K. -W.; Wu, Z.: Asymptotic stability criteria for a two-neuron network with different time delays. IEEE trans. Neural networks 14, No. 1, 222-227 (2003)
[34] Wei, J.; Ruan, S.: Stability and bifurcation in a neural network model with two delays. Physica D 130, 255-272 (1999) · Zbl 1066.34511
[35] Baldi, P.; Atiya, A.: How delays affect neural dynamics and learning. IEEE trans. Neural netw. 5, 610-621 (1994)
[36] Wu, J.: Symmetric functional differential equations and neural networks with memory. Trans. amer. Math. soc. 350, 4799-4838 (1998) · Zbl 0905.34034
[37] Campbell, S. A.: Stability and bifurcation of a simple network with multiple time delays. Fields inst. Commun. 21, 65-79 (1999) · Zbl 0926.92003
[38] J. Wei, M. Velarde, Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos (2004) (in press) · Zbl 1080.34064
[39] Wei, J.; Li, M. Y.: Global existence of periodic solutions in a tri-neuron network model with delays. Physica D 198, 106-119 (2004) · Zbl 1062.34077
[40] Li, M. Y.; Muldowney, J.: On Bendixson’s criterion. J. differential equations 106, 27-39 (1994) · Zbl 0786.34033
[41] Ruan, S.; Wei, J.: On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion. IMA J. Math. appl. Med. biol. 18, 41-52 (2001) · Zbl 0982.92008
[42] Hale, J. K.; Lunel, S. M. Verduyn: An introduction to functional differential equations. Appl. math 99 (1993) · Zbl 0787.34002
[43] Kuang, Y.: Delay differential equation with applications in population dynamics. (1993) · Zbl 0777.34002
[44] Butler, G.; Waltson, P.: Persistence in dynamical system. J. differential equations 63, 256-263 (1986)
[45] Potapov, A.; Ali, M. K.: Robust chaos in neural networks. Phys. lett. A 227, 310-322 (2000) · Zbl 1167.82359
[46] Belair, J.; Campbell, S. A.; Den Driessche, P. Van: Frustration, stability and delay-induced oscillations in a neural network model. SIAM J. Appl. math. 56, 245-255 (1996) · Zbl 0840.92003