# American Institute of Mathematical Sciences

June  2020, 28(2): 1037-1048. doi: 10.3934/era.2020056

## Asymptotic behaviour of a neural field lattice model with delays

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong, University of Science and Technology, Wuhan 430074, China

* Corresponding author: Meihua Yang

Received  March 2020 Revised  April 2020 Published  June 2020

Fund Project: The authors are supported by the NSFC grant 11971184

The asymptotic behaviour of an autonomous neural field lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice system is reformulated as an infinite dimensional ordinary delay differential equation on weighted sequence state space $\ell_\rho^2$ under some appropriate assumptions. Then the global existence and uniqueness of its solution and its formulation as a semi-dynamical system on a suitable function space are established. Finally, the asymptotic behaviour of solution of the system is investigated, in particular, the existence of a global attractor is obtained.

Citation: Xiaoli Wang, Peter Kloeden, Meihua Yang. Asymptotic behaviour of a neural field lattice model with delays. Electronic Research Archive, 2020, 28 (2) : 1037-1048. doi: 10.3934/era.2020056
##### References:

show all references

##### References:
 [1] Xiaoli Wang, Meihua Yang, Peter E. Kloeden. Sigmoidal approximations of a delay neural lattice model with Heaviside functions. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2385-2402. doi: 10.3934/cpaa.2020104 [2] 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 [3] 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, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51 [4] 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 [5] Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure & Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038 [6] Shi-Liang Wu, Cheng-Hsiung Hsu. Entire solutions with merging fronts to a bistable periodic lattice dynamical system. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2329-2346. doi: 10.3934/dcds.2016.36.2329 [7] Fang-Di Dong, Wan-Tong Li, Li Zhang. Entire solutions in a two-dimensional nonlocal lattice dynamical system. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2517-2545. doi: 10.3934/cpaa.2018120 [8] Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899 [9] Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101 [10] Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197 [11] Chin-Chin Wu. Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2813-2827. doi: 10.3934/dcds.2017121 [12] 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 [13] Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445 [14] David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499 [15] Canan Çelik. Dynamical behavior of a ratio dependent predator-prey system with distributed delay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 719-738. doi: 10.3934/dcdsb.2011.16.719 [16] 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 [17] P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692 [18] Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 [19] Leong-Kwan Li, Sally Shao, K. F. Cedric Yiu. Nonlinear dynamical system modeling via recurrent neural networks and a weighted state space search algorithm. Journal of Industrial & Management Optimization, 2011, 7 (2) : 385-400. doi: 10.3934/jimo.2011.7.385 [20] Chiun-Chuan Chen, Ting-Yang Hsiao, Li-Chang Hung. Discrete N-barrier maximum principle for a lattice dynamical system arising in competition models. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 153-187. doi: 10.3934/dcds.2020007

Impact Factor: 0.263