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

Asymptotic behaviour of a neural field lattice model with delays

Xiaoli Wang 1,2, , Peter Kloeden 1, and  Meihua Yang 1,2,,

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.

Keywords: Neural lattice model, sigmoidal function, delay dynamical system, autonomous dynamical system, global attractors.
Mathematics Subject Classification: Primary: 34A33, 34D05; Secondary: 35B41, 92B20.
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:
[1]

S.-I. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybernet., 27 (1977), 77-87.  doi: 10.1007/BF00337259.  Google Scholar

[2]

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

[3]

T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.  Google Scholar

[4]

S. Coombes, P. B. Graben, R. Potthast and J. Wright, Neural Fields. Theory and Applications, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-54593-1.  Google Scholar

[5]

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

[6]

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

[7]

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

[8] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar
[9]

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

[10]

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

show all references

References:
[1]

S.-I. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybernet., 27 (1977), 77-87.  doi: 10.1007/BF00337259.  Google Scholar

[2]

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

[3]

T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.  Google Scholar

[4]

S. Coombes, P. B. Graben, R. Potthast and J. Wright, Neural Fields. Theory and Applications, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-54593-1.  Google Scholar

[5]

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

[6]

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

[7]

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

[8] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar
[9]

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

[10]

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

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