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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  February 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (2) : 799-813. doi: 10.3934/dcdsb.2019268
References:
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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.

[2]

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[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.

[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.

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

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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.

[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.

[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.

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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. 

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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.

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

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

[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.

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

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

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

[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.

[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.

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[37]

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

[38]

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

[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.

[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.

show all references

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

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

[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.

[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.

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[37]

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

[38]

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

[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.

[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.

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