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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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418.![]() ![]() |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418.![]() ![]() |
[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. |
[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. |
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