Desynchronization of large scale delayed neural networks

Authors:
Yuming Chen, Ying Sue Huang and Jianhong Wu

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2365-2371

MSC (2000):
Primary 34K15, 92B20

Published electronically:
February 25, 2000

MathSciNet review:
1709744

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We consider a ring of identical neurons with delayed nearest neighborhood inhibitory interaction. Under general conditions, such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay differential equation with negative feedback. Despite the fact that the slowly oscillatory periodic solution of the scalar equation is stable, we show that the associated synchronous solution is unstable if the size of the network is large.

**[1]**Jacques Bélair, Sue Ann Campbell, and P. van den Driessche,*Frustration, stability, and delay-induced oscillations in a neural network model*, SIAM J. Appl. Math.**56**(1996), no. 1, 245–255. MR**1372899**, 10.1137/S0036139994274526**[2]**Y. Chen, J. Wu and T. Krisztin, Connecting orbits from synchronous periodic solutions to phase-locked periodic solutions in a delay differential system, to appear in Journal of Differential Equations, 1999.**[3]**Shui-Nee Chow and Hans-Otto Walther,*Characteristic multipliers and stability of symmetric periodic solutions of 𝑥(𝑡)=𝑔(𝑥(𝑡-1))*, Trans. Amer. Math. Soc.**307**(1988), no. 1, 127–142. MR**936808**, 10.1090/S0002-9947-1988-0936808-2**[4]**Odo Diekmann, Stephan A. van Gils, Sjoerd M. Verduyn Lunel, and Hans-Otto Walther,*Delay equations*, Applied Mathematical Sciences, vol. 110, Springer-Verlag, New York, 1995. Functional, complex, and nonlinear analysis. MR**1345150****[5]**Jack K. Hale and Sjoerd M. Verduyn Lunel,*Introduction to functional-differential equations*, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993. MR**1243878****[6]**J. J. Hopfield, Neurons with graded response have collective computational properties like two-stage neurons, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 3088-3092.**[7]**Anatoli Ivanov, Bernhard Lani-Wayda, and Hans-Otto Walther,*Unstable hyperbolic periodic solutions of differential delay equations*, Recent trends in differential equations, World Sci. Ser. Appl. Anal., vol. 1, World Sci. Publ., River Edge, NJ, 1992, pp. 301–316. MR**1180119****[8]**T. Krisztin, H. O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, Fields Institute Monographs, Vol. 11, American Mathematical Society, Providence, 1999.**[9]**C. M. Marcus and R. M. Westervelt,*Stability of analog neural networks with delay*, Phys. Rev. A (3)**39**(1989), no. 1, 347–359. MR**978323**, 10.1103/PhysRevA.39.347**[10]**Hal L. Smith,*Monotone dynamical systems*, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR**1319817****[11]**Jianhong Wu,*Symmetric functional-differential equations and neural networks with memory*, Trans. Amer. Math. Soc.**350**(1998), no. 12, 4799–4838. MR**1451617**, 10.1090/S0002-9947-98-02083-2**[12]**Xianwen Xie,*Uniqueness and stability of slowly oscillating periodic solutions of delay equations with bounded nonlinearity*, J. Dynam. Differential Equations**3**(1991), no. 4, 515–540. MR**1129559**, 10.1007/BF01049098**[13]**Xianwen Xie,*Uniqueness and stability of slowly oscillating periodic solutions of delay equations with unbounded nonlinearity*, J. Differential Equations**103**(1993), no. 2, 350–374. MR**1221910**, 10.1006/jdeq.1993.1054**[14]**Xianwen Xie,*The multiplier equation and its application to 𝑆-solutions of a differential delay equation*, J. Differential Equations**95**(1992), no. 2, 259–280. MR**1165423**, 10.1016/0022-0396(92)90032-I**[15]**J. Wu, T. Faria and Y. S. Huang, Absolute synchronization and stable phase-locking in a network of neurons with memory, Mathematical and Computer Modeling 30 (1999), 117-138.

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Additional Information

**Yuming Chen**

Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3

Email:
yumingc@mathstat.yorku.ca

**Ying Sue Huang**

Affiliation:
Department of Mathematics, Pace University, Pleasantville, New York 10570

Email:
huang@risc.dac.pace.edu

**Jianhong Wu**

Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3

Email:
wujh@mathstat.yorku.ca

DOI:
https://doi.org/10.1090/S0002-9939-00-05635-5

Keywords:
Synchronization,
slow oscillation,
delay,
neural network

Received by editor(s):
September 19, 1998

Published electronically:
February 25, 2000

Additional Notes:
This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 2000
American Mathematical Society