Asymptotic behaviour of a neural field lattice model with delays

Xiaoli Wang; Peter Kloeden; Meihua Yang; Xiaoli Wang; Peter Kloeden; Meihua Yang

doi:10.3934/era.2020056

Electronic Research Archive

2020, Volume 28, Issue 2: 1037-1048. doi: 10.3934/era.2020056

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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

Received: 01 March 2020 Revised: 01 April 2020
Primary: 34A33, 34D05; Secondary: 35B41, 92B20

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.
- Neural lattice model,
- sigmoidal function,
- delay dynamical system,
- autonomous dynamical system,
- global attractors
Citation: Xiaoli Wang, Peter Kloeden, Meihua Yang. Asymptotic behaviour of a neural field lattice model with delays[J]. Electronic Research Archive, 2020, 28(2): 1037-1048. doi: 10.3934/era.2020056

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Abstract

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.

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