Global existence of periodic solutions in a tri-neuron network model with delays. (English) Zbl 1062.34077

Summary: We consider a delayed differential system that models a network of three neurons with memory. Using a global Hopf bifurcation theorem for FDEs due to J. Wu and a Bendixson’s criterion for high-dimensional ODEs due to Li and Muldowney, we obtain a group of sufficient conditions for the system to have multiple periodic solutions when the sum of delays is sufficiently large.


34K13 Periodic solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
Full Text: DOI


[1] Chen, Y.; Wu, J., Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network, Physica D, 134, 185-199 (1999) · Zbl 0942.34062
[2] Chen, Y.; Wu, J., Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Differential Integral Equations, 14, 1181-1236 (2001) · Zbl 1023.34065
[3] Chen, Y.; Wu, J., Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, J. Math. Anal. Appl., 259, 188-208 (2001) · Zbl 0998.34058
[4] Faria, T., On a planar system modelling a neuron network with memory, J. Differential Equations, 168, 129-149 (2000) · Zbl 0961.92002
[5] Giamnakopoulos, F.; Zapp, A., Bifurcation in a planar system of differential delay equations modelling neural activity, Physica D, 159, 215-232 (2001) · Zbl 0984.92505
[6] Olien, L.; Bélair, J., Bifurcation stability and monotonicity properties of a delayed neural network model, Physica D, 102, 349-363 (1997) · Zbl 0887.34069
[7] Gopalsamy, K.; Leung, I., Delay induced periodicity in a neural network of excitation and inhibition, Physica D, 89, 395-426 (1996) · Zbl 0883.68108
[8] Majee, N. C.; Roy, A. B., Temporal dynamics of a two-neuron continuous network model with time delay, Appl. Math. Modeling, 21, 673-679 (1997) · Zbl 0893.68126
[9] Liao, X.; Wong, K.; Wu, Z., Bifurcation analysis on a two-neuron system with distribution delays, Physica D, 149, 123-141 (2001) · Zbl 1348.92035
[10] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272 (1999) · Zbl 1066.34511
[11] Baldi, P.; Atiya, A., How delays affect neural dynamics and learning, IEEE. Tran. NN, 5, 610-621 (1994)
[12] Wu, J., Symmetric functional differential equations and neural networks with memory, Trans. Am. Math. Soc., 350, 4799-4838 (1998) · Zbl 0905.34034
[13] Campbell, S. A., Stability and bifurcation of a simple neural network with multiple time delays, Fields Inst. Commun., 21, 65-79 (1999) · Zbl 0926.92003
[14] Campbell, S. A.; Ruan, S.; Wei, J., Qualitative analysis of a neural network model with multiple time delays, Int. J. Bifu. Chaos, 9, 1585-1595 (1999) · Zbl 1192.37115
[15] J. Wei, M. Velarde, Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos (2004), in press.; 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
[16] Ruan, S.; Wei, J., On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretions, IMA. J. Math. Appl. Medi. Biol., 18, 41-52 (2001) · Zbl 0982.92008
[17] Li, M. Y.; Muldowney, J., On Bendixson’s criterion, J. Differential Equations, 106, 27-39 (1994) · Zbl 0786.34033
[18] Beretta, E.; Solimano, F.; Tacheuchi, Y., Negative criteria for the existence of periodic solutions in a class of delay- differential equations, Nonlinear Anal., 50, 941-966 (2002) · Zbl 1087.34542
[19] Fiedler, M., Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czech. Math. J., 99, 392-402 (1974) · Zbl 0345.15013
[20] Muldowney, J. S., Compound matrices and ordinary differential equations, Rocky Mountain J. Math., 20, 857-871 (1990) · Zbl 0725.34049
[21] Erbe, L. H.; Geba, K.; Krawcewicz, W.; Wu, J., \(S^1\)-degree and global Hopf bifurcation theory of functional differential equations, J. Differential Equations, 98, 277-298 (1992) · Zbl 0765.34023
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