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March  2019, 18(2): 809-824. doi: 10.3934/cpaa.2019039

Long term behavior of a random Hopfield neural lattice model

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

* Corresponding author

Received  March 2018 Revised  May 2018 Published  October 2018

Fund Project: This work was partially supported by Simons Foundation, USA (Collaboration Grants for Mathematicians No. 429717) and NSF of China (Grant No. 11571125)

A Hopfield neural lattice model is developed as the infinite dimensional extension of the classical finite dimensional Hopfield model. In addition, random external inputs are considered to incorporate environmental noise. The resulting random lattice dynamical system is first formulated as a random ordinary differential equation on the space of square summable bi-infinite sequences. Then the existence and uniqueness of solutions, as well as long term dynamics of solutions are investigated.

Citation: Xiaoying Han, Peter E. Kloeden, Basiru Usman. Long term behavior of a random Hopfield neural lattice model. Communications on Pure & Applied Analysis, 2019, 18 (2) : 809-824. doi: 10.3934/cpaa.2019039
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P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastic and Dynamics, 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

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S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751.  doi: 10.1109/81.473583.  Google Scholar

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I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, Heidelberg, 2002. doi: 10.1007/b83277.  Google Scholar

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

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F. Fandoli and B. Schmalfuss, Random attractor for the 3d stochastic navier-stokes equation with multiplicative noise, Stochastics Stochastics, 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

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F. Favata and R. Walker, A study of the application of kohonen-type neural networks to the traveling salesman problem, Springger-Verlag: Biological Cybernetics, 64 (1991), 463-468.   Google Scholar

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E. GrusenG. Kayakutlu and T. U. Daim, Using artificial neural network models in stock market index prediction, Elsevier: Expert System with Application, 38 (2011), 10389-10397.   Google Scholar

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T. Kimoto, K. Asakawa, M. Yoda and M. Takeoka, Stock market prediction system with modular neural networks, IEE Xplore, 1-6. Google Scholar

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N. Mani and B. Srinivasan, Application of artificial neural network model for optical character recognition, IEEE Xplore, 100 (1997), 2517-2520.   Google Scholar

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G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[19]

W. Shen, Lifted lattice, hyperbolic structure, and topological disorder in coupled map lattices, SIAM J. Appl. Math., 56 (1996), 1379-1399.  doi: 10.1137/S0036139995282670.  Google Scholar

[20]

E. V. Vlerk and B. Wang, Attractors for lattice fitzhugh-nagumo systems, Phys. D, 221 (2005), 317-336.  doi: 10.1016/j.physd.2005.10.006.  Google Scholar

[21]

B. Wang, Dynamics of system of infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[22]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070.  Google Scholar

[23]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2007), 605-624.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastic and Dynamics, 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[3]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[4]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractor for asymptotically compact nonautonomous dynamical systems, Nonlinear Analysis TMA, 6 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[5]

S. N. Chow, Lattice Dynamical Systems, in: Dynamical System, in Lecture notes in Math. Vol. 1822, Springer-Verlag, Berlin, pp. 1-102, 2003. doi: 10.1007/978-3-540-45204-1_1.  Google Scholar

[6]

S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751.  doi: 10.1109/81.473583.  Google Scholar

[7]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, Heidelberg, 2002. doi: 10.1007/b83277.  Google Scholar

[8]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[9]

R. D. Dony and S. Haykin, Neural networks approaches to image compression, IEEE, 83 (1995), 288-303.   Google Scholar

[10]

H. Engler, Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity, Mathematische Zeitschrift, 202 (1989), 251-259.  doi: 10.1007/BF01215257.  Google Scholar

[11]

F. Fandoli and B. Schmalfuss, Random attractor for the 3d stochastic navier-stokes equation with multiplicative noise, Stochastics Stochastics, 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[12]

F. Favata and R. Walker, A study of the application of kohonen-type neural networks to the traveling salesman problem, Springger-Verlag: Biological Cybernetics, 64 (1991), 463-468.   Google Scholar

[13]

E. GrusenG. Kayakutlu and T. U. Daim, Using artificial neural network models in stock market index prediction, Elsevier: Expert System with Application, 38 (2011), 10389-10397.   Google Scholar

[14]

X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6265-0.  Google Scholar

[15]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci. U.S.A, 81 (1984), 3088-3092.   Google Scholar

[16]

T. Kimoto, K. Asakawa, M. Yoda and M. Takeoka, Stock market prediction system with modular neural networks, IEE Xplore, 1-6. Google Scholar

[17]

N. Mani and B. Srinivasan, Application of artificial neural network model for optical character recognition, IEEE Xplore, 100 (1997), 2517-2520.   Google Scholar

[18]

G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[19]

W. Shen, Lifted lattice, hyperbolic structure, and topological disorder in coupled map lattices, SIAM J. Appl. Math., 56 (1996), 1379-1399.  doi: 10.1137/S0036139995282670.  Google Scholar

[20]

E. V. Vlerk and B. Wang, Attractors for lattice fitzhugh-nagumo systems, Phys. D, 221 (2005), 317-336.  doi: 10.1016/j.physd.2005.10.006.  Google Scholar

[21]

B. Wang, Dynamics of system of infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[22]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070.  Google Scholar

[23]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2007), 605-624.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

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