×

Hopf bifurcation and stability for a delayed tri-neuron network model. (English) Zbl 1175.37086

Summary: A neural network model with three neurons and a single time delay is considered. Its linear stability is investigated and Hopf bifurcations are demonstrated by analyzing the corresponding characteristic equation. In particular, the explicit formulae determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form theory and the center manifold theorem. In order to illustrate our theoretical analysis, some numerical simulations are also included in the end.

MSC:

37N25 Dynamical systems in biology
34K13 Periodic solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
92C20 Neural biology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Babcock, K. L.; Westervelt, R. M., Dynamics of simple electronic neural networks, Physica D, 28, 305-316 (1987)
[2] Campbell, S. A.; Ruan, S.; Wei, J., Qualitative analysis of a neural network model with multiple time delays, Internat. J. Bifurcation Chaos, 9, 1585-1595 (1999) · Zbl 1192.37115
[3] Cao, J., On stability analysis in delayed celler neural networks, Phys. Rev. E, 59, 5940-5944 (1999)
[4] Cao, J.; Wang, J., Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation and time delays, Neural Networks, 17, 379-390 (2004) · Zbl 1074.68049
[5] Chen, Y.; Wu, J., Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, J. Math. Anal. Appl., 259, 188-288 (2001) · Zbl 0998.34058
[6] Faria, T., On a planar system modelling a neuron network with memory, J. Differential Equations, 168, 129-149 (2000) · Zbl 0961.92002
[7] Gopalsamy, K.; Leung, I., Delay induced periodicity in a neural netlet of excitation and inhibition, Physica D, 89, 395-426 (1996) · Zbl 0883.68108
[8] Guo, S.; Huang, L., Hopf bifurcating periodic orbits in a ring of neurons with delays, Physica D, 183, 19-44 (2003) · Zbl 1041.68079
[9] Hale, J. K., Theory of Functional Differential Equations (1977), Springer: Springer New York · Zbl 0425.34048
[10] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002
[11] Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Nat. Acad. Sci. USA, 81, 3088-3092 (1984) · Zbl 1371.92015
[12] Li, X.; Wei, J., On the zeros of a fourth degree exponential polynomial with application to a neural network model with delays, Chaos, Solitons Fractals, 26, 519-526 (2005) · Zbl 1098.37070
[13] Liu, Z.; Yuan, R., Stability and bifurcation in a harmonic oscillator with delays, Choas, Solitons Fractals, 23, 551-562 (2005) · Zbl 1078.34050
[14] Marcus, C. M.; Westervelt, R. M., Stability of analog neural network with delay, Phys. Rev. A, 39, 347-359 (1989)
[15] 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
[16] Olien, L.; Bélair, J., Bifurcations, stability, and monotonicity properties of a delayed neural network model, Physica D, 102, 349-363 (1997) · Zbl 0887.34069
[17] Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics Continuous, Discrete Impulsive Systems, Ser. A: Math. Anal., 10, 863-874 (2003) · Zbl 1068.34072
[18] Song, Y.; Han, M.; Wei, J., Stability and Hopf bifurcation on a simplified BAM neural network with delays, Physica D, 200, 185-204 (2005) · Zbl 1062.34079
[19] Wei, J.; Li, M. Y., Global existence of periodic solutions in a tri-neuro network model with delays, Physica D, 198, 106-119 (2004) · Zbl 1062.34077
[20] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272 (1999) · Zbl 1066.34511
[21] Wei, J.; Velarde, M., Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos, 143, 940-953 (2004) · Zbl 1080.34064
[22] Wu, J., Introduction to Neural Dynamics and Signal Transmission Delay (2001), Walther de Gruyter: Walther de Gruyter Berlin · Zbl 0977.34069
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.