Hopf bifurcating periodic orbits in a ring of neurons with delays. (English) Zbl 1041.68079

Summary: In this paper, we consider a ring of neurons with self-feedback and delays. The linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Based on the normal form approach and the center manifold theory, we derive the formula for determining the properties of Hopf bifurcating slowly oscillating periodic orbits for a ring of neurons with delays, including the direction of Hopf bifurcation, stability of the Hopf bifurcating slowly oscillating periodic orbits, and so on. Moreover, by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, we not only investigate the effect of synaptic delay of signal transmission on the pattern formation, but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns.


68T05 Learning and adaptive systems in artificial intelligence
92B20 Neural networks for/in biological studies, artificial life and related topics
Full Text: DOI


[1] Alexander, J.C., Bifurcation of zeros of parameterized functions, J. funct. anal., 29, 37-53, (1978) · Zbl 0385.47038
[2] Alexander, J.C.; Auchmuty, G., Global bifurcations of phase-locked oscillators, Arch. ration. mech. anal., 93, 253-270, (1986) · Zbl 0596.92010
[3] an der Heiden, U., Delays in physiological systems, J. math. biol., 8, 345-364, (1978) · Zbl 0429.92009
[4] P. Anderson, O. Gross, T. Lomo, et al., Participation of inhibitory interneurons in the control of hippocampal cortical output, in: M. Brazier (Ed.), The Interneuron, University of California Press, Los Angeles, CA, 1969.
[5] Bélair, J.; Campbell, S.A.; van den Driessche, P., Frustration, stability and delay-induced oscillations in a neural network model, SIAM J. appl. math., 46, 245-255, (1996) · Zbl 0840.92003
[6] S.A. Campbell, Stability and bifurcation of a simple neural network with multiple time delays, in: S. Ruan, G.S.K. Wolkowicz, J. Wu (Eds.), Differential Equations with Application to Biology, vol. 21, Fields Institute Communications, AMS, Providence, RI, 1998, pp. 65-79. · Zbl 0926.92003
[7] Chen, Y.; Huang, Y.; Wu, J., Desynchronization of large scale delayed neural networks, Proc. am. math. soc., 128, 8, 2365-2371, (2000) · Zbl 0945.34056
[8] Cohen, M.A.; Grossberg, S., Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE trans. syst. man cybernet., 13, 815-826, (1983) · Zbl 0553.92009
[9] J.M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, vol. 20, Springer, New York, 1977. · Zbl 0363.92014
[10] O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.-O. Walther, Delay Equations, Functional-, Complex-, and Nonlinear Analysis, Springer, New York, 1995. · Zbl 0826.34002
[11] Erbe, L.H.; Krawcewicz, W.; Geba, K.; Wu, J., S1-degree and global Hopf bifurcation theory of functional differential equations, J. diff. eq., 98, 227-298, (1992) · Zbl 0765.34023
[12] J.C. Eccles, M. Ito, J. Szenfagothai, The Cerebellum as Neuronal Machine, Springer, New York, 1967.
[13] Faria, T.; Magalháes, L.T., Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. diff. eq., 122, 181-200, (1995) · Zbl 0836.34068
[14] B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Lecture Notes in Mathematics, vol. 1309, Springer, New York, 1988. · Zbl 0644.34038
[15] M. Golubitsky, I. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, Springer, New York, 1988. · Zbl 0691.58003
[16] Gopalsamy, K.; Leung, I., Delay induced periodicity in a neural network of excitation and inhibition, Physica D, 89, 395-426, (1996) · Zbl 0883.68108
[17] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. · Zbl 0787.34002
[18] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. · Zbl 0474.34002
[19] Hirsch, M.W., Convergent activation dynamics in continuous-time networks, Neural networks, 2, 331-349, (1989)
[20] Hopfield, J.J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. natl. acad. sci. USA, 81, 3088-3092, (1984) · Zbl 1371.92015
[21] Huang, L.; Wu, J., Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation, SIAM J. math. anal., 34, 4, 836-860, (2003) · Zbl 1038.34076
[22] Levinger, B.W., A folk theorem in functional differential equations, J. diff. eq., 4, 612-619, (1968) · Zbl 0174.13902
[23] Marcus, C.M.; Westervelt, R.M., Stability of analog neural networks with delay, Phys. rev. A, 39, 347-359, (1989)
[24] Olien, L.; Bélair, J., Bifurcations, stability and monotonicity properties of a delayed neural network model, Physica D, 102, 349-363, (1997) · Zbl 0887.34069
[25] Othmer, H.G.; Scriven, L.E., Instability and dynamics pattern in cellular networks, J. theor. biol., 32, 507-537, (1971)
[26] Pakdaman, K.; Grotta-Ragazzo, C.; Malta, C.P., Transient regime duration in continuous-time neural networks with delay, Phys. rev. E, 58, 3623-3627, (1998)
[27] Pakdaman, K.; Grotta-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)
[28] Sattinger, D.H., Bifurcation and symmetry breaking in applied mathematics, Bull. am. math. soc., 3, 779-819, (1980) · Zbl 0448.35011
[29] Szenfagothai, J., The module-concept in cerebral cortex architecture, Brain res., 95, 475-496, (1967)
[30] Tank, D.W.; Hopfield, J.J., Neural computation by concentrating information in time, Proc. natl. acad. sci. USA, 84, 1896-1900, (1987)
[31] van Gils, S.A.; Valkering, T., Hopf bifurcation and symmetry: standing and traveling waves in a circular-chain, Jpn. J. appl. math., 3, 207-222, (1986) · Zbl 0642.58038
[32] Wu, J., Symmetric functional differential equations and neural networks with memory, Trans. am. math. soc., 350, 12, 4799-4838, (1998) · Zbl 0905.34034
[33] Wu, J., Synchronization and stable phase-locking in a network of neurons with memory, Math. comp. model., 30, 1-2, 117-138, (1999) · Zbl 1043.92500
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.