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

Attractors of Hopfield-type lattice models with increasing neuronal input

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 $.

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
References:
[1]

H. AkaR. AlassarV. CovachevZ. 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. BatesX. 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. BatesK. 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. CaraballoX. HanB. 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. ChowJ. 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. GuanG. 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. HanW. 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. McElieceE. C. PosnerE. 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. MichelJ. 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. YuZ. MaoQ. 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. ZhouF. 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. AkaR. AlassarV. CovachevZ. 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. BatesX. 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. BatesK. 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. CaraballoX. HanB. 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. ChowJ. 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. GuanG. 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. HanW. 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. McElieceE. C. PosnerE. 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. MichelJ. 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. YuZ. MaoQ. 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. ZhouF. 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

[1]

Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521

[2]

David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96

[3]

Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120

[4]

Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. Totally dissipative dynamical processes and their uniform global attractors. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1989-2004. doi: 10.3934/cpaa.2014.13.1989

[5]

Antônio Luiz Pereira, Severino Horácio da Silva. Continuity of global attractors for a class of non local evolution equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1073-1100. doi: 10.3934/dcds.2010.26.1073

[6]

Monica Conti, Vittorino Pata. On the regularity of global attractors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1209-1217. doi: 10.3934/dcds.2009.25.1209

[7]

Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803

[8]

Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445

[9]

Noriaki Yamazaki. Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935-944. doi: 10.3934/proc.2003.2003.935

[10]

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

[11]

Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079

[12]

Sergey Dashkovskiy, Oleksiy Kapustyan, Iryna Romaniuk. Global attractors of impulsive parabolic inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1875-1886. doi: 10.3934/dcdsb.2017111

[13]

Rodrigo Samprogna, Cláudia B. Gentile Moussa, Tomás Caraballo, Karina Schiabel. Trajectory and global attractors for generalized processes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3995-4020. doi: 10.3934/dcdsb.2019047

[14]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891

[15]

Alessia Berti, Valeria Berti, Ivana Bochicchio. Global and exponential attractors for a Ginzburg-Landau model of superfluidity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 247-271. doi: 10.3934/dcdss.2011.4.247

[16]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[17]

Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative lattice dynamical systems with delays. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 643-663. doi: 10.3934/dcds.2008.21.643

[18]

Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51

[19]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31

[20]

Jörg Härterich, Matthias Wolfrum. Describing a class of global attractors via symbol sequences. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 531-554. doi: 10.3934/dcds.2005.12.531

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (68)
  • HTML views (42)
  • Cited by (0)

Other articles
by authors

[Back to Top]