Research article

Bifurcation of a feed forward neural network with delay and application in image contrast enhancement

  • Received: 29 December 2018 Accepted: 05 June 2019 Published: 10 October 2019
  • This paper is concerned with how the singularity and delay in a feed forward neural network affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem, the mode interactions in two parameters bifurcations with a single zero and a pair of imaginary roots are considered. The codimension two normal form with Hopf-pitchfork bifurcations are given. Then, the bifurcation diagrams and phase portraits are obtained by computing the normal form. Furthermore, we find some interesting dynamical behaviors of the original system, such as the coexistence of two unstable nontrivial equilibria and a pair of stable periodic orbits, which are verified both theoretically and numerically. Through numerical simulation, we also find that this model has a special signal enhancement property in Hopf bifurcation state. Using this feed-forward neural network, we show that the gray scale picture contrast is strongly enhanced even if this one is initially very small.

    Citation: Wenlong Wang, Chunrui Zhang. Bifurcation of a feed forward neural network with delay and application in image contrast enhancement[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 387-403. doi: 10.3934/mbe.2020021

    Related Papers:

  • This paper is concerned with how the singularity and delay in a feed forward neural network affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem, the mode interactions in two parameters bifurcations with a single zero and a pair of imaginary roots are considered. The codimension two normal form with Hopf-pitchfork bifurcations are given. Then, the bifurcation diagrams and phase portraits are obtained by computing the normal form. Furthermore, we find some interesting dynamical behaviors of the original system, such as the coexistence of two unstable nontrivial equilibria and a pair of stable periodic orbits, which are verified both theoretically and numerically. Through numerical simulation, we also find that this model has a special signal enhancement property in Hopf bifurcation state. Using this feed-forward neural network, we show that the gray scale picture contrast is strongly enhanced even if this one is initially very small.


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    [1] M. Cohen and S. Grossberg, Absolute stability of global pattern formation and parallelmemory storage by competitive neural networks, IEEE T. Syst. Man. Cy-s., 13 (1983), 815-825.
    [2] J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, P. Natl. Acad. Sci. USA, 81 (1984), 3088-3092.
    [3] J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Physica D, 130 (1999), 255-272.
    [4] L. Li and Y. Yuan, Dynamics in three cells with multiple time delays, Nonlinear Anal-Real., 9 (2008), 725-746.
    [5] C. Zhang, B. Zheng and L.Wang, Multiple Hopf bifurcations of symmetric BAM neural network model with delay, Appl. Math. Lett., 22 (2009), 616-622.
    [6] Y. Song and J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predatorprey system, Math. Anal. Appl., 301 (2005), 1-21.
    [7] T. Dong, X. Liao, T.Huang,et al., Hopf-pitchfork bifurcation in an inertial two- neuron system with time delay, Neurocomputing, 97 (2012), 223-232.
    [8] C. Zhang, X. Zhang and Y. Zhang, Dynamic properties of feed-forward neural networks and application in contrast enhancement for image, CHAOS Soliton. Fract., 114 (2018), 281-290.
    [9] A. Algaba, E. Freire, E. Gamero, et al., A tame degenerate Hopf-pitchfork bifurcation in a modified van der pol-duffing oscillator, Nonlinear Dyn., 22 (2000), 249-269.
    [10] A. Algaba, E. Freire, E.Gamero, et al., A three-parameter study of a degenerate case of the Hopfpitchfork bifurcation, Nonlinearity, 12 (1999), 1177-1206.
    [11] A. Algaba, E. Freire and E.Gamero, Hypernormal form for the Hopf-zero bifurcation, Int. J. Bifurcat. Chaos, 8 (2011), 18575-1887.
    [12] J. Carr, Applications of centre manifold Theory, Springer-Verlag, New York, 1981.
    [13] L. Li and Y. Yuan, Dynamics in three cells with multiple time delays, Nonlinear Anal-Real., 9 (2008), 725-746.
    [14] J. S. W. Lamb, M. A. Teixeira and K. N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R3, J. Differ. Equations, 219 (2005), 78-115.
    [15] H. Wang and J. Wang, Hopf-pitchfork bifurcation in a two-neuron system with discrete and distributed delays, Math. Method Appl. Sci., 38 (2015), 4967-4981.
    [16] J. Wei and M. Li, Global existence of periodic solutions in a tri-neuron network model with delays, Physica D, 198 (2004), 106-119.
    [17] X. Wu and L. Wang, Zero-Hopf bifurcation analysis in delayed differential equations with two delays, J. Franklin I., 354 (2017), 1484-1513.
    [18] B. Zheng, H. Yin and C. Zhang, Equivariant Hopf-Pitchfork Bifurcation of Symmetric Coupled Neural Network with Delay, I. J. Bifurcation Chaos, 26 (2016), 11650205.
    [19] N. J. McCullen, T. Mullin and M. Golubitsky, Sensitive Signal Detection Using a Feed-Forward Oscillator Network, Phys. Rev. lett., 98 (2007), 254101.
    [20] Y. Qin, J. Wang, C. Men, et al., Vibrational resonance in feed-forward network, Chaos, 21 (2011), 023133.
    [21] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1998.
    [22] T. Faria and L. Magalhaes, Normal forms for retarded functional differential equation with parameters and applications to Hopf bifurcation, J. Differ. Equations, 122 (1995), 181-200.
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