Skip to main content
SpringerLink
Log in

Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Chua LO (1998) CNN: a paradigm for complexity. World Scientific, Singapore

    Book  MATH  Google Scholar 

  2. Chua LO, Yang L (1988) Cellular neural networks: theory. IEEE Trans Circuits Syst 35:1257–1272

    Article  MathSciNet  MATH  Google Scholar 

  3. Chua LO, Yang L (1988) Cellular neural networks: applications. IEEE Trans Circuits Syst 35:1273–1290

    Article  MathSciNet  Google Scholar 

  4. Chua LO, Roska T (1990) Cellular neural networks with nonlinear and delay-type template elements. In: Proceedings of IEEE international workshop on cellular neural networks and their applications, pp 12–25

  5. Chua LO, Roska T (1992) Cellular neural networks with nonlinear and delay type template elements and non-uniform grids. Int J Circuit Theory Appl 20:449–451

    Article  MATH  Google Scholar 

  6. Hsu C-H, Lin S-S, Shen W (1999) Traveling waves in cellular neural networks. Int J Bifurcat Chaos 9:1307–1319

    Article  MathSciNet  MATH  Google Scholar 

  7. Weng P, Wu J (2003) Deformation of traveling waves in delayed cellular neural networks. Int J Bifurcat Chaos 13:797–813

    Article  MathSciNet  MATH  Google Scholar 

  8. Zou F, Schwartz S, Nossek J (1990) Cellular neural network design using a learning algorithm. In: Proceedings of IEEE international workshop on cellular neural networks and their applications, pp 73–81

  9. Li L, Fang Z, Yang Y (2012) A shunting inhibitory cellular neural network with continuously distributed delays of neutral type. Nonlinear Anal Real World Appl 13:1186–1196

    Article  MathSciNet  MATH  Google Scholar 

  10. Rosko T, Boros T, Thiran P, Chua LO (1990) Detecting simple motion using cellular neural networks. In: Proceedings of IEEE international workshop on cellular neural networks and their applications, pp 127–138

  11. Bouzerdoum A, Pinter RB (1993) Shunting inhibitory cellular neural networks: derivation and stability analysis. IEEE Trans Circuits Syst I Fund Theory Appl 40:215–221

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen L, Zhao H (2008) Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients. Chaos Solitons Fractals 35:351–357

    Article  MathSciNet  MATH  Google Scholar 

  13. Ding HS, Liang J, Xiao TJ (2008) Existence of almost periodic solutions for SICNNs with time-varying delays. Phys Lett A 372:5411–5416

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen A, Cao J (2002) Almost periodic solution of shunting inhibitory CNNs with delays. Phys Lett A 298:161–170

    Article  MathSciNet  MATH  Google Scholar 

  15. Ou C (2009) Almost periodic solutions for shunting inhibitory cellular neural networks. Nonlinear Anal Real World Appl 10:2652–2658

    Article  MathSciNet  MATH  Google Scholar 

  16. Cherif F (2012) Existence and global exponential stability of pseudo almost periodic solution for SICNNs with mixed delays. J Appl Math Comput 39:235–251

    Article  MathSciNet  MATH  Google Scholar 

  17. Hu M, Wang L (2011) Existence and exponential stability of almost periodic solution for Cohen–Grossberg SICNNs with impulses. World Acad Sci Eng Technol 52:941–950

    Google Scholar 

  18. Xia Y, Cao J, Huang Z (2007) Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses. Chaos Solitons Fractals 34:1599–1607

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang X, Cao J (2003) Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay. Phys Lett A 314:222–231

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhou Q, Xiao B, Yu Y (2006) Existence and stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays. Electron J Differ Equ 2006:1–10

    Article  MathSciNet  MATH  Google Scholar 

  21. Akhmet M (2006) On the integral manifolds of the differential equations with piecewise constant argument of generalized type. In: Agarwal RP, Perera K (eds) Proceedings of the conference on differential and difference equations at the Florida Institute of Technology, August 1–5, 2005. Hindawi Publishing Corporation, Melbourne, pp 11–20

    Google Scholar 

  22. Akhmet M (2011) Nonlinear hybrid continuous/discrete time models. Atlantis Press, Amsterdam

    Book  MATH  Google Scholar 

  23. Aftabizadeh AR, Wiener J (1988) Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument. Appl Anal 26:327–333

    Article  MathSciNet  MATH  Google Scholar 

  24. Busenberg S, Cooke KL (1982) Models of vertically transmitted diseases with sequential-continuous dynamics. In: Lakshmikantham V (ed) Nonlinear phenomena in mathematical sciences. Academic Press, New York, pp 179–187

    Chapter  Google Scholar 

  25. Cooke KL, Wiener J (1987) Neutral differential equations with piecewise constant argument. Boll Un Mat Ital 7:321–346

    MathSciNet  MATH  Google Scholar 

  26. Cooke KL, Wiener J (1984) Retarded differential equations with piecewise constant delays. J Math Anal Appl 99:265–297

    Article  MathSciNet  MATH  Google Scholar 

  27. Dai L (2008) Nonlinear dynamics of piecewise constant systems and implementation of piecewise constant arguments. World Scientific, Singapore

    Book  MATH  Google Scholar 

  28. Dai L, Singh MC (1994) On oscillatory motion of spring-mass systems subjected to piecewise constant forces. J Sound Vib 173:217–232

    Article  MATH  Google Scholar 

  29. Seifert G (2000) Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence. J Differ Equ 164:451–458

    Article  MathSciNet  MATH  Google Scholar 

  30. Wiener J (1993) Generalized solutions of functional differential equations. World Scientific, Singapore

    Book  MATH  Google Scholar 

  31. Zhu H, Huang L (2004) Dynamics of a class of nonlinear discrete-time neural networks. Comput Math Appl 48:85–94

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang X (2006) Existence and exponential stability of almost periodic solutions for cellular neural networks with piecewise constant argument. Acta Math Appl Sin 29:789–800

    MathSciNet  Google Scholar 

  33. Jun YX, Jian WZ (2008) Asymptotic behavior of a neural network model with three piecewise constant arguments. Hunan Daxue Xuebao 35:59–62 (Chinese)

    MathSciNet  Google Scholar 

  34. Murray JD (2002) Mathematical biology: I. An introduction. Springer, New York

    MATH  Google Scholar 

  35. Hoppensteadt FC, Peskin CS (1992) Mathematics in medicine and the life sciences. Springer, New York

    Book  MATH  Google Scholar 

  36. Buck J (1988) Synchronous rhythmic flashing of fireflies. II. Q Rev Biol 63:265–290

    Article  Google Scholar 

  37. Driver RD (1979) Can the future influence the present? Phys Rev D 19:1098–1107

    Article  MathSciNet  Google Scholar 

  38. Akhmet MU, Yılmaz E (2010) Impulsive Hopfield-type neural network system with piecewise constant argument. Nonlinear Anal Real World Appl 11:2584–2593

    Article  MathSciNet  MATH  Google Scholar 

  39. Akhmet MU, Aruğaslan D, Yılmaz E (2010) Stability in cellular neural networks with piecewise constant argument. J Comput Appl Math 233:2365–2373

    Article  MathSciNet  MATH  Google Scholar 

  40. Akhmet MU, Aruğaslan D, Yılmaz E (2010) Stability analysis of recurrent neural networks with piecewise constant argument of generalized type. Neural Netw 23:805–811

    Article  MATH  Google Scholar 

  41. Shepherd GM (2004) The synaptic organization of the brain. Oxford University Press, New York

    Book  Google Scholar 

  42. Vida I, Bartos M, Jonas P (2006) Shunting inhibition improves robustness of gamma oscillations in hippocampal interneuron networks by homogenizing firing rates. Neuron 49:107–117

    Article  Google Scholar 

  43. Mitchell SJ, Silver RA (2003) Shunting inhibition modulates neuronal gain during synaptic excitation. Neuron 38:433–445

    Article  Google Scholar 

  44. Borg-Graham LJ, Monier C, Frégnac Y (1998) Visual input evokes transient and strong shunting inhibition in visual cortical neurons. Nature 393:369–373

    Article  Google Scholar 

  45. Bouzerdoum A, Nabet B, Pinter RB (1991) Analysis and analog implementation of directionally sensitive shunting inhibitory cellular neural networks. In: Visual information processing: from neurons to chips, vol SPIE-1473, pp 29–38

  46. Bouzerdoum A, Pinter RB (1992) Nonlinear lateral inhibition applied to motion detection in the fly visual system. In: Pinter RB, Nabet B (eds) Nonlinear vision. CRC Press, Boca Raton, pp 423–450

    Google Scholar 

  47. Bouzerdoum A, Pinter RB (1990) A shunting inhibitory motion detector that can account for the functional characteristics of fly motion-sensitive interneurons. Proc Int Jt Conf Neural Netw 1:149–153

    Google Scholar 

  48. Pinter RB, Olberg RM, Warrant E (1989) Luminance adaptation of preferred object size in identified dragonfly movement detectors. In: Proceedings of IEEE international conference SMC, pp 682–686

  49. Bouzerdoum A (1993) The elementary movement detection mechanism in insect vision. Philos Trans R Soc London B 339:375–384

    Article  Google Scholar 

  50. Pinter RB (1983) Product term nonlinear lateral inhibition enhances visual selectivity for small objects and edges. J Theor Biol 110:525–531

    Article  Google Scholar 

  51. Jernigan ME, Belshaw RJ, McLean GF (1991) Nonlinear lateral inhibition and image processing. In: Nabet B, Pinter RB (eds) Sensory neural networks: lateral inhibition. CRC Press, Boca Raton, pp 27–45

    Google Scholar 

  52. Fukushima K, Miyake S, Ito T (1983) Neocognitron: a neural network model for a mechanism of visual pattern recognition. IEEE Trans Syst Man Cybern 13:826–834

    Article  Google Scholar 

  53. Arulampalam G, Bouzerdoum A (2001) Application of shunting inhibitory artificial neural networks to medical diagnosis. Proceedings of seventh Australian and New Zealand intelligent information systems conference. Perth, Western Australia, pp 89–94

    Google Scholar 

  54. Liao X, Chen G, Sanchez EN (2002) Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach. Neural Netw 15:855–866

    Article  Google Scholar 

  55. Yi Z, Heng PA, Fu AWC (1999) Estimate of exponential convergence rate and exponential stability for neural networks. IEEE Trans Neural Netw 10:1487–1493

    Article  Google Scholar 

  56. Wen S, Huang T, Zeng Z, Chen Y, Li P (2015) Circuit design and exponential stabilization of memristive neural networks. Neural Netw 63:48–56

    Article  MATH  Google Scholar 

  57. He Y, Wu M, She J-H (2006) Delay-dependent exponential stability of delayed neural networks with time-varying delay. IEEE Trans Circuits Syst II Exp Briefs 53:553–557

    Article  Google Scholar 

  58. Dan S, Yang SX, Feng W (2013) Lag synchronization of coupled delayed chaotic neural networks by periodically intermittent control. Abstr Appl Anal 2013:501461

    MathSciNet  MATH  Google Scholar 

  59. Jiang H, Zhang L, Teng Z (2005) Existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays. IEEE Trans Neural Netw 16:1340–1351

    Article  Google Scholar 

  60. Wang L (2010) Existence and global attractivity of almost periodic solutions for delayed high-ordered neural networks. Neurocomputing 73:802–808

    Article  Google Scholar 

  61. Wen SP, Zeng ZG, Huang TW, Li CJ (2015) Passivity and passification of stochastic impulsive memristor-based piecewise linear system with mixed delays. Int J Robust Nonlinear Control 25:610–624

    Article  MathSciNet  MATH  Google Scholar 

  62. Wen S, Zeng Z, Huang T, Zhang Y (2014) Exponential adaptive lag synchronization of memristive neural networks via fuzzy method and applications in pseudorandom number generators. IEEE Trans Fuzzy Syst 22:1704–1713

    Article  Google Scholar 

  63. Wen S, Zeng Z, Huang T, Meng Q, Yao W (2015) Lag synchronization of switched neural networks via neural activation function and applications in image encryption. IEEE Trans Neural Netw Learn Syst 26:1493–1502

    Article  MathSciNet  Google Scholar 

  64. Pasemann F, Hild M, Zahedi K (2003), SO(2)-networks as neural oscillators. In: Mira J, Álvarez JR (eds) Computational methods in neural modeling, lecture notes in computer science, vol 2686, pp 144–151

  65. Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. The MIT Press, Cambridge

    Google Scholar 

  66. Kimura H, Akiyama S, Sakurama K (1999) Realization of dynamic walking and running of the quadruped using neural oscillator. Auton Robots 7:247–258

    Article  Google Scholar 

  67. Wang X (1992) Discrete-time dynamics of coupled quasi-periodic and chaotic neural network oscillators. Proceedings of international joint conference on neural networks. Baltimore, Maryland, pp 517–522

    Google Scholar 

  68. Burton TA (1985) Stability and periodic solutions of ordinary and functional differential equations. Academic Press, Orlando

    MATH  Google Scholar 

  69. Hale J (1971) Functional differential equations. Springer, New York

    Book  MATH  Google Scholar 

  70. Kuang Y (1993) Delay differential equations: with applications in population dynamics. Academic Press, Boston

    MATH  Google Scholar 

  71. Seifert G (2003) Second-order neutral delay-differential equations with piecewise constant time dependence. J Math Anal Appl 281:1–9

    Article  MathSciNet  MATH  Google Scholar 

  72. Seifert G (2003) Almost periodic solutions of certain neutral functional differential equations. Commun Appl Anal 7:437–442

    MathSciNet  MATH  Google Scholar 

  73. Wang G (2007) Periodic solutions of a neutral differential equation with piecewise constant arguments. J Math Anal Appl 326:736–747

    Article  MathSciNet  MATH  Google Scholar 

  74. Wang L, Yuan R, Zhang CY (2011) A spectrum relation of almost periodic solution of second order scalar functional differential equations with piecewise constant argument. Acta Math Sin Engl Ser 27:2275–2284

    Article  MathSciNet  MATH  Google Scholar 

  75. Akhmet MU (2014) Quasilinear retarded differential with functional dependence on piecewise constant argument. Commun Pure Appl Anal 13:929–947

    Article  MathSciNet  MATH  Google Scholar 

  76. Corduneanu C (2009) Almost periodic oscillations and waves. Springer, New York

    Book  MATH  Google Scholar 

  77. Samoilenko AM, Perestyuk NA (1995) Impulsive differential equations. World Scientific, Singapore

    Book  MATH  Google Scholar 

  78. Halanay A, Wexler D (1971) Qualitative theory of impulsive systems. Mir, Moscow (Russian)

    MATH  Google Scholar 

  79. Wexler D (1966) Solutions périodiques et presque-périodiques des systémes d’équations différetielles linéaires en distributions. J Differ Equ 2:12–32

    Article  MathSciNet  MATH  Google Scholar 

  80. Akhmetov MU, Perestyuk NA, Samoilenko AM (1983) Almost-periodic solutions of differential equations with impulse action. Akad Nauk Ukrain SSR Inst Mat. Preprint 26:49 (Russian)

  81. Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 81:3088–3092

    Article  Google Scholar 

  82. Cohen MA, Grossberg S (1993) Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans Syst Man Cybern SMC 13:815–826

    MathSciNet  MATH  Google Scholar 

Download references

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. The second author is supported by the 2219 Scholarship Programme of TÜBİTAK, the Scientific and Technological Research Council of Turkey.

Author information

Authors and Affiliations

  1. Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    Marat Akhmet

  2. Neuroscience Institute, Georgia State University, Atlanta, GA, 30303, USA

    Mehmet Onur Fen

  3. Laboratoire de Mathématiques, Image et Applications, Pôle Sciences et Technologies, Université de La Rochelle, Avenue Michel Crépeau, 17042, La Rochelle, France

    Mokhtar Kirane

Authors
  1. Marat Akhmet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehmet Onur Fen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mokhtar Kirane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marat Akhmet.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Akhmet, M., Fen, M.O. & Kirane, M. Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument. Neural Comput & Applic 27, 2483–2495 (2016). https://doi.org/10.1007/s00521-015-2019-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-015-2019-4

Keywords

Search

Navigation