Attractors of Hopfield-type lattice models with increasing neuronal input

Xiaoli Wang; Peter E. Kloeden; Xiaoying Han

doi:10.3934/dcdsb.2019268

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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

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

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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Attractors of Hopfield-type lattice models with increasing neuronal input

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