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Neural Computing and Applications

, Volume 29, Issue 10, pp 815–822 | Cite as

A Hopfield neural network with multi-compartmental activation

  • Marat U. Akhmet
  • Meltem Karacaören
Original Article
  • 115 Downloads

Abstract

The Hopfield network is a form of recurrent artificial neural network. To satisfy demands of artificial neural networks and brain activity, the networks are needed to be modified in different ways. Accordingly, it is the first time that, in our paper, a Hopfield neural network with piecewise constant argument of generalized type and constant delay is considered. To insert both types of the arguments, a multi-compartmental activation function is utilized. For the analysis of the problem, we have applied the results for newly developed differential equations with piecewise constant argument of generalized type beside methods for differential equations and functional differential equations. In the paper, we obtained sufficient conditions for the existence of an equilibrium as well as its global exponential stability. The main instruments of investigation are Lyapunov functionals and linear matrix inequality method. Two examples with simulations are given to illustrate our solutions as well as global exponential stability.

Keywords

Hopfield neural networks Equilibrium Exponential stability Piecewise constant argument of generalized type Constant delay Linear matrix inequality 

Notes

Acknowledgments

The authors wish to express their sincere gratitude to the referees for the helpful criticism and valuable suggestions, which helped to improve the paper significantly.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-stage neurons. Proc Natl Acad Sci USA 81:3088–3092CrossRefzbMATHGoogle Scholar
  2. 2.
    Cao J (2000) Estimation of the domain of attraction and the convergence rate of a Hopfield associative memory and an application. J Comput Syst Sci 60:179–186MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cao J (2001) Global exponential stability of Hopfield neural networks. Int J Syst Sci 32:233–236MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cao J (2004) An estimation of the domain of attraction and convergence rate for Hopfield continuous feedback neural networks. Phys Lett A 325:370–374MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chua LO, Roska T (1990) Cellular neural networks with nonlinear and delay type template elements. In: Proceedings of 1990 IEEE int workshop on cellular neural networks and their applications, pp 12–25Google Scholar
  6. 6.
    Chua LO, Roska T (1992) Cellular neural networks with nonlinear and delay type template elements and non-uniform grids. Int J Circ Theory Appl 20:449–451CrossRefzbMATHGoogle Scholar
  7. 7.
    Chua LO, Yang L (1988) Cellular neural networks: theory. IEEE Trans Circ Syst 35:1257–1272MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chua LO, Yang L (1988) Cellular neural networks: applications. IEEE Trans Circ Syst 35:1273–1290MathSciNetCrossRefGoogle Scholar
  9. 9.
    Forti M, Tesi A (1995) New conditions for global stability of neural networks with application to linear and quadratic programming problems. IEEE Trans Circ Syst I 42:354–366MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Tank D, Hopfield JJ (1986) Simple neural optimization networks: an A/D converter signal decision circuit and a linear programming circuit. IEEE Trans Circ Syst 33:533–541CrossRefGoogle Scholar
  11. 11.
    Cao J, Li X (2005) Stability in delayed Cohen–Grossberg neural networks: LMI optimization approach. Phys D 212(1–2):54–65MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gopalsamy K, He X (1994) Delayed-independent stability in bidirectional associative memory networks. IEEE Trans Neural Netw 5:998–1002CrossRefGoogle Scholar
  13. 13.
    Li X, Chen Z (2009) Stability properties for Hopfield neural networks with delays and impulsive perturbations. Nonlinear Anal Real World Appl 10:3253–3265MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liao X, Chen GR, Sanchenz EN (2002) Delay dependent exponential stability analysis of delayed neural networks: an LMI approach. Neural Netw 15:855–866CrossRefGoogle Scholar
  15. 15.
    Mohamad S, Gopalsamy K, Akça (2008) Exponential stability of artificial neural networks with distributed delays and large impulses. Nonlinear Anal Real World Appl 9:872–888MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Xu S, Lam J (2006) A new approach to exponential stability of neural networks with time-varying delays. Neural Netw 19:76–83CrossRefzbMATHGoogle Scholar
  17. 17.
    Zhang J, Jin X (2000) Global stability analysis in delayed Hopfield neural network models. Neural Netw 13:745–753CrossRefGoogle Scholar
  18. 18.
    Zhang Q, Wei X, Xu J (2003) Global asymptotic stability of Hopfield neural networks with transmission delays. Phys Lett A 318:399–405MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Driver RD (1979) Can the future influence the present? Phys Rev D 19:1098–1107MathSciNetCrossRefGoogle Scholar
  20. 20.
    Baldi P, Atiya AF (1994) How delays affect neural dynamics and learning. IEEE Trans Neural Netw 5(4):612–621CrossRefGoogle Scholar
  21. 21.
    Civalleri PP, Gilli M, Pandolfi L (1993) On the stability of cellular neural networks with delay. IEEE Trans Circ Syst I 40:157–165CrossRefzbMATHGoogle Scholar
  22. 22.
    Marcus CM, Westervelt RM (1989) Stability of analog neural networks with delay. Phys Rev A 39:347MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhou D, Cao J (2002) Globally exponential stability conditions for cellular neural networks with time-varying delays. Appl Math Comput 13:487–496MathSciNetzbMATHGoogle Scholar
  24. 24.
    Arik S (2003) Global asymptotic stability of a larger class of neural networks with constant time delay. Phys Lett A 311:504–511CrossRefzbMATHGoogle Scholar
  25. 25.
    Arik S (2004) An analysis of exponential stability of delayed neural networks with time varying delays. Neural Netw 17:1027–1031CrossRefzbMATHGoogle Scholar
  26. 26.
    Guo S, Huang L (2005) Periodic oscillation for a class of neural networks with variable coefficients. Nonlinear Anal Real World Appl 6:545–561MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Liu B (2007) Almost periodic solutions for Hopfield neural networks with continuously distributed delays. Math Comput Simul 73:327–335MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Liu Y, You Z, Cao L (2006) On the almost periodic solution of generalized Hopfield neural networks with time-varying delays. Neurocomputing 69:1760–1767CrossRefGoogle Scholar
  29. 29.
    Wang Z, Shu H et al (2006) Robust stability analysis of generalized neural networks with discrete and distributed time delays. Chaos Solitons Fractals 30:886–896MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Liu X, Jiang N (2009) Robust stability analysis of generalized neural networks with multiple discrete delays and multiple distributed delays. Neurocomputing 72:1789–1796CrossRefGoogle Scholar
  31. 31.
    Cao J, Liang J, Lam J (2004) Exponential stability of high-order bidirectional associative memory neural networks with time delays. Phys D 199:425–436MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Cao J, Song Q (2006) Stability in Cohen–Grossberg type BAM neural networks with time-varying delays. Nonlinearity 19(7):1601–1617MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Cao J, Wang J (2005) Global asymptotic and robust stability of recurrent neural networks with time delays. IEEE Trans Circ Syst I 52(2):417–426MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Cao J, Wang J (2005) Global exponential stability and periodicity of recurrent neural networks with time delay. IEEE Trans Circ Syst I 52:920–931MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Song Q, Cao J (2006) Stability analysis of Cohen–Grossberg neural network with both time-varying and continuously distributed delays. J Comput Appl Math 197:188–203MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Gopalsamy K, He X (1994) Stability in asymmetric Hopfield nets with delays transmission delays. Phys D 76:344–358MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Lou XY, Cui BT (2006) Global asymptotic stability of delay BAM neural networks with impulses. Chaos Solitons Fractals 29:1023–1031MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ye H, Michel AN, Wang KN (1995) Qualitative analysis of Cohen Grossberg neural Networks with multiple delays. Phys Rev E 51:2611–2618MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zhang Z, Liu K (2011) Existence and global exponential stability of a periodic solution to interval general bidirectional associative memory (BAM) neural networks with multiple delays on time scales. Neural Netw 24:427–439CrossRefzbMATHGoogle Scholar
  40. 40.
    Zhang Z, Liu W, Zhou D (2012) Global asymptotic stability to a generalized Cohen Grossberg BAM neural networks of neutral type delays. Neural Netw 25:94–105CrossRefzbMATHGoogle Scholar
  41. 41.
    Zhang Z, Cao J, Zhou D (2014) Novel LMI-based condition on global asymptotic stability for a class of CohenGrossberg BAM networks with extended activation functions. IEEE Trans Neural Netw Learn Syst 25(6):1161–1172CrossRefGoogle Scholar
  42. 42.
    Wiener J (1993) Generalized solutions of functional differential equations. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  43. 43.
    Akhmet MU (2006) On the integral manifolds of the differential equations with piecewise constant argument of generalized type. In: Agarval RP, Perera K (eds) Proceedings of the conference on differential and difference equations at the Florida Institute of Technology. Hindawi Publishing Corporation, Cario, pp 11–20Google Scholar
  44. 44.
    Akhmet MU (2011) Nonlinear hybrid continuous/discrete-time. Models Atlantis Press, AmsterdamCrossRefzbMATHGoogle Scholar
  45. 45.
    Yang X (2006) Existence and exponential stability of almost periodic solutions for cellular neural networks with piecewise constant argument. Acta Math Appl Sinica 29:789–800MathSciNetGoogle Scholar
  46. 46.
    Zhu H, Huang L (2004) Dynamics of a class of nonlinear discrete-time neural networks. Comput Math Appl 48:85–94MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Akhmet MU, Aruğaslan D, Yılmaz E (2010) Stability analysis of recurrent neural networks with piecewise constant argument of generalized type. Neural Netw 23:305–311CrossRefzbMATHGoogle Scholar
  48. 48.
    Akhmet MU, Aruğaslan D, Yılmaz E (2010) Stability in cellular neural networks with a piecewise constant argument. J Comput Appl Math 233:2365–2373MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Akhmet MU, Yılmaz E (2014) Neural networks with discontinuous/impact activations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  50. 50.
    Akhmet MU (2014) Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Commun Pure Appl Anal 13(2):929–947MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Yu T et al (2016) Stability analysis of neural networks with periodic coefficients and piecewise constant arguments. J Franklin Inst 353(2):409–425MathSciNetCrossRefGoogle Scholar
  52. 52.
    Pinto M (2009) Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments. Math Comput Model 49:1750–1758MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Alwan MS, Xinzhi L, Wei-Chau X (2013) Comparison principle and stability of differential equations with piecewise constant arguments. J Franklin Inst 350:211–230MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Mohamad S, Gopalsamy K (2003) Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Appl Math Comput 135:17–38MathSciNetzbMATHGoogle Scholar
  55. 55.
    Liao X, Chen G, Sanchez EN (2002) Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach. Neural Netw 15:855–866CrossRefGoogle Scholar
  56. 56.
    Wang H, Song Q, Duana C (2010) LMI criteria on exponential stability of BAM neural networks with both time-varying delays and general activation functions. Math Comput Simul 81:837–850MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Huang X, Cao J, Huang D (2005) LMI-based approach for delay-dependent exponential stability analysis of BAM neural networks. Chaos Solitons Fractals 24:885–898MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Nie X, Cao J (2009) Stability analysis for the generalized Cohen–Grossberg neural networks with inverse Lipschitz neuron activations. Comput Math Appl 57:1522–1536MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Sanchez EN, Perez JP (1999) Input-to-state stability (ISS) analysis for dynamic neural networks. IEEE Trans Circ Syst I 46:1395–1398MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  1. 1.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey

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