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Stability properties of neural networks with non-instantaneous impulses

1 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
2 Department of Applied Mathematics and Modeling, University of Plovdiv ”Paisii Hilendarski”, 4000 Plovdiv, Bulgaria
3 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
4 Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA

Special Issues: Differential Equations in Mathematical Biology

In this paper, we consider neural networks in the case when the neurons are subject to a certain impulsive state displacement at fixed moments and the duration of this displacement is not negligible small (these are known as non-instantaneous impulses). We examine some stability properties of the equilibrium of the model. Several sufficient conditions for uniform Lipschitz stability of the equilibrium of neural networks with time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. These sufficient conditions are explicitly expressed in terms of the parameters of the system and hence they are easily verifiable. The case of non-Lipschitz activation functions is also studied. The theory is illustrated on particular nonlinear neural networks.
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Keywords nonlinear neural networks; non-instantaneous impulses; Lipschitz stability

Citation: Ravi Agarwal, Snezhana Hristova, Donal O’Regan, Radoslava Terzieva. Stability properties of neural networks with non-instantaneous impulses. Mathematical Biosciences and Engineering, 2019, 16(3): 1210-1227. doi: 10.3934/mbe.2019058


  • 1. R. Agarwal, S. Hristova and D. O'Regan, Non-Instantaneous Impulses in Differential Equations, Springer, 2017.
  • 2. H. Akca, R. Alassar, V. Covachev, Z. Covacheva and E. A. Al-Zahrani, Continuous-time additive Hopfield-type neural networks with impulses, J. Math. Anal. Appl., 290 (2004), 436–451.
  • 3. H. Akca, R. Alassar, Y. M. Shebadeh and V. Covachev, Neural networks: Modelling with impulsive differential equations, Proc. Dynamical Syst. Appl., (2004), 32–47.
  • 4. N. T. Carnevale and M. L. Hines, The NEURON Book, Cambridge, UK, Cambridge University Press, 2009.
  • 5. F. M. Dannan and S. Elaydi, Lipschitz stability of nonlinear systems of differential equations, J. Math. Anal. Appl., 113, (1986), 562–577.
  • 6. K. Gopalsamy, Stability of artificial neural networks with impulses, Appl. Math. Comput., 154 (2004), 783–813.
  • 7. J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. USA, 79 (1982), 2554–2558.
  • 8. S. Hristova, Qualitative investigations and approximate methods for impulsive equations, Nova Sci. Publ. Inc., New York, 2009.
  • 9. S. Hristova and R. Terzieva, Lipschitz stability of differential equations with non-instantaneous impulses, Adv. Differ. Equ., 2016, 322.
  • 10. Y. Huang, H. Zhang and Z. Wang, Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions, Neurocomputing, 91 (2012), 21–28.
  • 11. R. D. King, S. M. Garrett and G. M. Coghill, On the use of qualitative reasoning to simulate and identify metabolic pathways, Bioinformatics, 21 (2005), 2017–2026.
  • 12. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  • 13. X. Li and J.Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64 (2016), 63–69.
  • 14. X. Li and S. Song, Stabilization of Delay Systems: Delay-Dependent Impulsive Control, IEEE Transactions on Automatic Control, 62 (2017), 406–411.
  • 15. C. Li and G. Feng, Delay-interval-dependent stability of recurrent neural networks with timevarying delay, Neurocomputing, 72 (2009), 1179–1183.
  • 16. X. Li, D. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 99 (2019), 361–368.
  • 17. W. McCulloch and W. H. Pitts, A logical calculus of the ideas immanent in nervous activity, Bill. Math. Bioph., 5 (1943), 115–133.
  • 18. A. Rahimi and B. Recht, Weighted sums of random kitchen sinks: Replacing minimization with randomization in learning, Adv. Neural Information Processing Syst, 21 (2008), 1313–1320.
  • 19. R. I. Watson Sr., The great psychologists, J.B. Lippincott Co., New York, 1978.


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