American Institute of Mathematical Sciences

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:

show all references

References:
 [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