American Institute of Mathematical Sciences

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February  2020, 25(2): 799-813. doi: 10.3934/dcdsb.2019268

Attractors of Hopfield-type lattice models with increasing neuronal input

Xiaoli Wang 1,, , Peter E. Kloeden 2, and  Xiaoying Han 2,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
2. 

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36832, USA

* Corresponding author: Xiaoli Wang

Dedicated to Juan J. Nieto on the occasion of his 60th birthday

Received  September 2018 Revised  February 2019 Published  November 2019

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

Two Hopfield-type neural lattice models are considered, one with local $ n $-neighborhood nonlinear interconnections among neurons and the other with global nonlinear interconnections among neurons. It is shown that both systems possess global attractors on a weighted space of bi-infinite sequences. Moreover, the attractors are shown to depend upper semi-continuously on the interconnection parameters as $ n \to \infty $.

Keywords: Hopfield neural model, lattice dynamical systems, global neuronal interactions, global attractors, upper semi-continuity of global attractors.
Mathematics Subject Classification: Primary: 34A35, 34C35, 34D45, 37L60; Secondary: 47J05, 58F10, 92B20.
Citation: Xiaoli Wang, Peter E. Kloeden, Xiaoying Han. Attractors of Hopfield-type lattice models with increasing neuronal input. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 799-813. doi: 10.3934/dcdsb.2019268
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T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.  Google Scholar

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X. Han, P. E. Kloeden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, submitted. Google Scholar

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X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

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J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. U.S.A., 79 (1982), 2554-2558.  doi: 10.1073/pnas.79.8.2554.  Google Scholar

[23]

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J. J. Hopfield and D. W. Tank, "Neural" computation of decisions in optimization problems, Biol. Cybernet., 52 (1985), 141-152.  doi: 10.1007/BF00339943.  Google Scholar

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J. C. Juang, Stability analysis of Hopfield-type neural networks, IEEE Trans. Neural Networks, 10 (2002), 1366-1374.  doi: 10.1109/72.809081.  Google Scholar

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[29]

A. N. Michel, J. A. Farrell and W. Porod, Qualitative analysis of neural networks, IEEE Trans. Circuits and Systems, 36 (1989), 229-243.  doi: 10.1109/31.20200.  Google Scholar

[30]

A. N. Michel and J. A. Farrell, Associative memories via artificial neural networks, IEEE Control Syst., 10 (1990), 6-17.  doi: 10.1109/37.55118.  Google Scholar

[31]

W. M. Schouten and H. J. Hupkes, Nonlinear stability of pulse solutions for the discrete Fitzhugh-Nagumo equation with infinite range interactions, Discrete Contin. Dyn. Syst., 39 (2019), 5017-5083.  doi: 10.3934/dcds.2019205.  Google Scholar

[32]

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

[33]

E. S. V. Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D, 212 (2005), 317-336.  doi: 10.1016/j.physd.2005.10.006.  Google Scholar

[34]

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

[35]

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

[36]

X. Wang, P. E. Kloeden and M. Yang, Asymptotic behaviour of a neural field lattice model with delays, submitted. Google Scholar

[37]

D. Yu, Z. Mao, Q. Zhou and T. P. Leung, Qualitative analysis of Hopfield neural networks, Control Theory Appl., 12 (1995), 382-388.   Google Scholar

[38]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

[39]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Phys. D, 178 (2003), 51-61.  doi: 10.1016/S0167-2789(02)00807-2.  Google Scholar

[40]

X. Zhou, F. Yin and S. Zhou, Uniform exponential attractors for second order non-autonomous lattice dynamical systems, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 587-606.  doi: 10.1007/s10255-017-0684-z.  Google Scholar

show all references

References:
[1]

H. Aka, R. Alassar, V. Covachev, Z. Covacheva and E. Al-Zahrani, Continuous-time additive Hopfield-type neural networks with impulses, J. Math. Anal. Appl., 290 (2004), 436-451.  doi: 10.1016/j.jmaa.2003.10.005.  Google Scholar

[2]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.  Google Scholar

[3]

P. W. Bates, X. Chen and A. J. J. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.  Google Scholar

[4]

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

[5]

J. Bell, Some threshold results for models of myelinated nerves, Math. Biosci., 54 (1981), 181-190.  doi: 10.1016/0025-5564(81)90085-7.  Google Scholar

[6]

J. Bruck and J. W. Goodman, A generalized convergence theorem for neural networks, IEEE Trans. Inform. Theory, 34 (1988), 1089-1092.  doi: 10.1109/18.21239.  Google Scholar

[7]

T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.  Google Scholar

[8]

S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-756.  doi: 10.1109/81.473583.  Google Scholar

[9]

S. N. Chow, J. Mallet-Paret and E. S. V. Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput. Dynam., 4 (1996), 109-178.   Google Scholar

[10]

S. N. Chow, Lattice dynamical systems, in Dynamical Systems, Lecture Notes in Math., 1822, Springer, Berlin, 2003, 1–102. doi: 10.1007/978-3-540-45204-1_1.  Google Scholar

[11]

L. O. Chua and L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits and Systems, 35 (1988), 1273-1290.  doi: 10.1109/31.7601.  Google Scholar

[12]

L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits and Systems, 35 (1988), 1257-1272.  doi: 10.1109/31.7600.  Google Scholar

[13]

A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing, John Wiley & Sons, Inc., New York, NY, 1993. doi: 10.5860/choice.31-2156.  Google Scholar

[14]

K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Mathematics, 596, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0091636.  Google Scholar

[15]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction-diffusion systems, Phys. D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I.  Google Scholar

[16]

M. Gobbino and M. Sardella, On the connectedness of attractors for dynamical systems, J. Differential Equations, 133 (1997), 1-14.  doi: 10.1006/jdeq.1996.3166.  Google Scholar

[17]

Z. Guan, G. Chen and Y. Qin, On equilibria, stability, and instability of Hopfield neural networks, IEEE Trans. Neural Networks, 11 (2000), 534-540.  doi: 10.1109/72.839023.  Google Scholar

[18]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.  Google Scholar

[19]

X. Han and P. E. Kloeden, Asymptotic behaviour of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.  Google Scholar

[20]

X. Han, P. E. Kloeden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, submitted. Google Scholar

[21]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[22]

J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. U.S.A., 79 (1982), 2554-2558.  doi: 10.1073/pnas.79.8.2554.  Google Scholar

[23]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Nat. Acad. Sci. U.S.A., 81 (1984), 3088-3092.  doi: 10.1073/pnas.81.10.3088.  Google Scholar

[24]

J. J. Hopfield and D. W. Tank, "Neural" computation of decisions in optimization problems, Biol. Cybernet., 52 (1985), 141-152.  doi: 10.1007/BF00339943.  Google Scholar

[25]

J. C. Juang, Stability analysis of Hopfield-type neural networks, IEEE Trans. Neural Networks, 10 (2002), 1366-1374.  doi: 10.1109/72.809081.  Google Scholar

[26]

R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.  Google Scholar

[27] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar
[28]

R. J. McEliece, E. C. Posner, E. R. Rodemich and S. S. Venkatesh, The capacity of the Hopfield associative memory, IEEE Trans. Inform. Theory, 33 (1987), 461-482.  doi: 10.1109/TIT.1987.1057328.  Google Scholar

[29]

A. N. Michel, J. A. Farrell and W. Porod, Qualitative analysis of neural networks, IEEE Trans. Circuits and Systems, 36 (1989), 229-243.  doi: 10.1109/31.20200.  Google Scholar

[30]

A. N. Michel and J. A. Farrell, Associative memories via artificial neural networks, IEEE Control Syst., 10 (1990), 6-17.  doi: 10.1109/37.55118.  Google Scholar

[31]

W. M. Schouten and H. J. Hupkes, Nonlinear stability of pulse solutions for the discrete Fitzhugh-Nagumo equation with infinite range interactions, Discrete Contin. Dyn. Syst., 39 (2019), 5017-5083.  doi: 10.3934/dcds.2019205.  Google Scholar

[32]

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

[33]

E. S. V. Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D, 212 (2005), 317-336.  doi: 10.1016/j.physd.2005.10.006.  Google Scholar

[34]

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

[35]

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

[36]

X. Wang, P. E. Kloeden and M. Yang, Asymptotic behaviour of a neural field lattice model with delays, submitted. Google Scholar

[37]

D. Yu, Z. Mao, Q. Zhou and T. P. Leung, Qualitative analysis of Hopfield neural networks, Control Theory Appl., 12 (1995), 382-388.   Google Scholar

[38]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

[39]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Phys. D, 178 (2003), 51-61.  doi: 10.1016/S0167-2789(02)00807-2.  Google Scholar

[40]

X. Zhou, F. Yin and S. Zhou, Uniform exponential attractors for second order non-autonomous lattice dynamical systems, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 587-606.  doi: 10.1007/s10255-017-0684-z.  Google Scholar

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