×

Almost periodic solutions for Wilson-Cowan type model with time-varying delays. (English) Zbl 1264.34086

Summary: Wilson-Cowan model of neuronal population with time-varying delays is considered in this paper. Some sufficient conditions for the existence and delay-based exponential stability of a unique almost periodic solution are established. The approaches are based on constructing Lyapunov functionals and the well-known Banach contraction mapping principle. The results are new, easily checkable, and complement existing periodic ones.

MSC:

34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
92B20 Neural networks for/in biological studies, artificial life and related topics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] H. R. Wilson and J. D. Cowan, “Excitatory and inhibitory interactions in localized populations of model neurons,” Biophysical Journal, vol. 12, no. 1, pp. 1-24, 1972.
[2] H. R. Wilson and J. D. Cowan, “A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue,” Kybernetik, vol. 13, no. 2, pp. 55-80, 1973. · Zbl 0281.92003
[3] C. van Vreeswijk and H. Sompolinsky, “Chaos in neuronal networks with balanced excitatory and inhibitory activity,” Science, vol. 274, no. 5293, pp. 1724-1726, 1996.
[4] S. E. Folias and G. B. Ermentrout, “New patterns of activity in a pair of interacting excitatory-inhibitory neural fields,” Physical Review Letters, vol. 107, no. 22, Article ID 228103, 2011.
[5] K. Mantere, J. Parkkinen, T. Jaaskelainen, and M. M. Gupta, “Wilson-Cowan neural-network model in image processing,” Journal of Mathematical Imaging and Vision, vol. 2, no. 2-3, pp. 251-259, 1992. · Zbl 0797.68172
[6] L. H. A. Monteiro, M. A. Bussab, and J. G. C. Berlinck, “Analytical results on a Wilson-Cowan neuronal network modified model,” Journal of Theoretical Biology, vol. 219, no. 1, pp. 83-91, 2002.
[7] B. Pollina, D. Benardete, and V. W. Noonburg, “A periodically forced Wilson-Cowan system,” SIAM Journal on Applied Mathematics, vol. 63, no. 5, pp. 1585-1603, 2003. · Zbl 1035.92007
[8] R. Decker and V. W. Noonburg, “A periodically forced Wilson-Cowan system with multiple attractors,” SIAM Journal on Mathematical Analysis, vol. 44, no. 2, pp. 887-905, 2012. · Zbl 1318.92005
[9] P. L. William, “Equilibrium and stability of wilson and cowan’s time coarse graining model,” in Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems (MTNS ’12), Melbourne, Australia, July 2012.
[10] Z. Huang, S. Mohamad, X. Wang, and C. Feng, “Convergence analysis of general neural networks under almost periodic stimuli,” International Journal of Circuit Theory and Applications, vol. 37, no. 6, pp. 723-750, 2009.
[11] Y. Liu and Z. You, “Multi-stability and almost periodic solutions of a class of recurrent neural networks,” Chaos, Solitons and Fractals, vol. 33, no. 2, pp. 554-563, 2007. · Zbl 1136.34311
[12] C. Y. He, Almost Periodic Differential Equation, Higher Education, Beijing, China, 1992.
[13] H. Zhenkun, F. Chunhua, and S. Mohamad, “Multistability analysis for a general class of delayed Cohen-Grossberg neural networks,” Information Sciences, vol. 187, pp. 233-244, 2012. · Zbl 1256.93086
[14] H. Zhenkun and Y. N. Raffoul, “Biperiodicity in neutral-type delayed difference neural networks,” Advances in Difference Equations, vol. 2012, article 5, 2012. · Zbl 1278.39021
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.