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Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations
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 |
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 $.
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] |
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. 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. |
[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. 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.
|
[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
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. 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. |
[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. 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.
|
[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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