zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

MSC:
68T05Learning and adaptive systems
92B20General theory of neural networks (mathematical biology)
WorldCat.org
Full Text: DOI
References:
[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] Der Heiden, U. An: 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.; Den Driessche, P. Van: 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, No. 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)
[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)
[21] Huang, L.; Wu, J.: Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation. SIAM J. Math. anal. 34, No. 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) · Zbl 0869.68086
[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, No. 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, No. 1-2, 117-138 (1999) · Zbl 1043.92500