April  2020, 19(4): 2385-2402. doi: 10.3934/cpaa.2020104

Sigmoidal approximations of a delay neural lattice model with Heaviside functions

1. 

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

2. 

Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany

*Corresponding author

Dedicated to Professor Tomás Caraballo on occasion of his 60th Birthday

Received  April 2019 Revised  October 2019 Published  January 2020

Fund Project: This work was partially supported by the NSF of China Grant No. 11971184.

The approximation of Heaviside coefficient functions in delay neural lattice models with delays by sigmoidal functions is investigated. The solutions of the delay sigmoidal models are shown to converge to a solution of the delay differential inclusion as the sigmoidal parameter goes to zero. In addition, the existence of global attractors is established and compared for the various systems.

Citation: 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
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]

J. P. Aubin and A. Cellina, Differential Inclusions, Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[3]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Inter. J. Bifur. & Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[4]

T. CaraballoF. 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

[5]

S. Coombes and M. R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM J. Appl. Dyn. Syst., 3 (2004), 574-600.  doi: 10.1137/040605953.  Google Scholar

[6]

G. Faye, Traveling fronts for lattice neural field equations, Physica D, 378/379 (2018) 20–32. doi: 10.1016/j.physd.2018.04.004.  Google Scholar

[7]

Xiaoying Han and P. E. Kloeden, Asymptotic behaviour of a neural field lattice model with a Heaviside operator, Physica D: Nonlinear Phenomena, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.  Google Scholar

[8]

Xiaoying Han and P. E. Kloeden, Sigmoidal approximations of Heaviside functions in neural lattice models, submitted. Google Scholar

[9]

Xiaoying HanWenxian Shen and Shengfan Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[10]

A. IlievN. Kyurkchiev and S. Markov, On the approximation of the step function by some sigmoid functions, Mathematics and Computers in Simulation, 133 (2017), 223-234.  doi: 10.1016/j.matcom.2015.11.005.  Google Scholar

[11]

P. E. Kloeden and V. S. Kozyakin, The inflation of attractors and discretization: the autonomous case, Nonlinear Anal. TMA, 40 (2000), 333–343. doi: 10.1016/S0362-546X(00)85020-8.  Google Scholar

[12]

P. E. Kloeden and V. S. Kozyakin, The inflation and perturbation of nonautonomous difference equations and their pullback attractors, Proceedings of the Sixth International Conference on Difference Equations, 139–152, CRC, Boca Raton, FL, 2004.  Google Scholar

[13]

G. P. Szegö and G. Treccani, Semigruppi di Trasformazioni Multivoche, Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 1969.  Google Scholar

[14]

Xiaoli Wang, P. E. Kloeden and Meihua Yang, Asymptotic behaviour of a neural field lattice model with delays, submitted. Google Scholar

[15]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica 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]

J. P. Aubin and A. Cellina, Differential Inclusions, Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[3]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Inter. J. Bifur. & Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[4]

T. CaraballoF. 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

[5]

S. Coombes and M. R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM J. Appl. Dyn. Syst., 3 (2004), 574-600.  doi: 10.1137/040605953.  Google Scholar

[6]

G. Faye, Traveling fronts for lattice neural field equations, Physica D, 378/379 (2018) 20–32. doi: 10.1016/j.physd.2018.04.004.  Google Scholar

[7]

Xiaoying Han and P. E. Kloeden, Asymptotic behaviour of a neural field lattice model with a Heaviside operator, Physica D: Nonlinear Phenomena, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.  Google Scholar

[8]

Xiaoying Han and P. E. Kloeden, Sigmoidal approximations of Heaviside functions in neural lattice models, submitted. Google Scholar

[9]

Xiaoying HanWenxian Shen and Shengfan Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[10]

A. IlievN. Kyurkchiev and S. Markov, On the approximation of the step function by some sigmoid functions, Mathematics and Computers in Simulation, 133 (2017), 223-234.  doi: 10.1016/j.matcom.2015.11.005.  Google Scholar

[11]

P. E. Kloeden and V. S. Kozyakin, The inflation of attractors and discretization: the autonomous case, Nonlinear Anal. TMA, 40 (2000), 333–343. doi: 10.1016/S0362-546X(00)85020-8.  Google Scholar

[12]

P. E. Kloeden and V. S. Kozyakin, The inflation and perturbation of nonautonomous difference equations and their pullback attractors, Proceedings of the Sixth International Conference on Difference Equations, 139–152, CRC, Boca Raton, FL, 2004.  Google Scholar

[13]

G. P. Szegö and G. Treccani, Semigruppi di Trasformazioni Multivoche, Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 1969.  Google Scholar

[14]

Xiaoli Wang, P. E. Kloeden and Meihua Yang, Asymptotic behaviour of a neural field lattice model with delays, submitted. Google Scholar

[15]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61.  doi: 10.1016/S0167-2789(02)00807-2.  Google Scholar

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