American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Initial value problem for fractional Volterra integro-differential equations with Caputo derivative
  • DCDS-B Home
  • This Issue
  • Next Article
    Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation
doi: 10.3934/dcdsb.2021001

Feedback synchronization of FHN cellular neural networks

Leslaw Skrzypek and  Yuncheng You ,

Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA

* Corresponding author: Yuncheng You

Received  July 2020 Revised  October 2020 Published  December 2020

In this work we study the synchronization of ring-structured cellular neural networks modeled by the lattice FitzHugh-Nagumo equations with boundary feedback. Through the uniform estimates of solutions and the analysis of dissipative dynamics, the synchronization of this type neural networks is proved under the condition that the boundary gap signal exceeds the adjustable threshold.

Keywords: Cellular neural network, discrete FitzHugh-Nagumo equations, feedback synchronization, dissipative dynamics.
Mathematics Subject Classification: 34A33, 34D06, 37L60, 37N25, 92B20.
Citation: Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021001
References:
[1]

B. Ambrosio and M. A. Aziz-Alaoui, Synchronization and control of a network of coupled reaction-diffusion systems of generalized FitzHugh-Nagumo type, ESAIM: Proceedings, 39 (2013), 15-24.  doi: 10.1051/proc/201339003.  Google Scholar

[2]

A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002.  Google Scholar

[3]

A. Cattani, FitzHugh-Nagumo equations with generalized diffusive coupling, Math. Biosci. Eng., 11 (2014), 203-215.  doi: 10.3934/mbe.2014.11.203.  Google Scholar

[4]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems - Part I, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751.  doi: 10.1109/81.473583.  Google Scholar

[5]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems - Part II, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 752-756.   Google Scholar

[6]

L. O. Chua, M. Hasler, G. S. Moschytz and J. Neirynck, Autonomous cellular neural networks: A unified paradigm for pattern formation and active wave propagation, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 559-577.  doi: 10.1109/81.473564.  Google Scholar

[7] L. O. Chua and T. Roska, Cellular Neural Networks and Visual Computing, Cambridge University Press, Cambridge, UK, 2002.  doi: 10.1017/CBO9780511754494.  Google Scholar
[8]

L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits and Systems, 35 (1988), 1257-1272.  doi: 10.1109/31.7600.  Google Scholar

[9]

L. O. Chua and L. Yang, Cellular neural networks: Application, IEEE Trans. Circuits and Systems, 35 (1988), 1273-1290.  doi: 10.1109/31.7601.  Google Scholar

[10]

S. M. Dickson, Stochastic Neural Network Dynamics: Synchronization and Control, Ph. D. Dissertation, Loughborough University, UK, 2014. Google Scholar

[11]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[12]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences, 81 (1984), 3088-3092.  doi: 10.1073/pnas.81.10.3088.  Google Scholar

[13]

M. M. Ibrahim and I. H. Jung, Complex synchronization of a ring-structured network of FitzHugh-Nagumo neurons with single and dual state gap junctions under ionic gates and external electrical disturbance, IEEE Access, 7 (2019), 57894-57906.  doi: 10.1109/ACCESS.2019.2913872.  Google Scholar

[14]

S. Indolia, A. K. Goswami, S. P. Mishra and P. Asopa, Conceptual understanding of convolutional neural networks - a deep learning approach, Procedia Computer Science, 132 (2018), 679-688.  doi: 10.1016/j.procs.2018.05.069.  Google Scholar

[15]

C. Phan, L. Skrzypek and Y. You, Dynamics and synchronization of complex neural networks with boundary coupling, preprint, arXiv: 2004.09988, 2020. Google Scholar

[16]

C. Phan and Y. You, Synchronization of boundary coupled Hindmarsh-Rose neuron network, Nonlinear Anal. Real World Appl., 55 (2020), 103139, 13pp. doi: 10.1016/j.nonrwa.2020.103139.  Google Scholar

[17]

C. Quiñinao and J. D. Touboul, Clamping and synchronization in the strongly coupled FitzHugh-Nagumo model, SIAM J. Appl. Dyn. Syst., 19 (2020), 788-827.  doi: 10.1137/19M1283884.  Google Scholar

[18]

H. Serrano-Guerrero et al., Chaotic synchronization in star coupled networks of three-dimensional cellular neural networks and its applications in communications, International J. Nonlinear Science and Numerical Simulation, 11 (2010), 571-580.   Google Scholar

[19]

L. Skrzypek and Y. You, Dynamics and synchronization of boundary coupled FitzHugh-Nagumo neural networks, Appl. Math. Comput., 388 (2021), 125545, 13 pp. doi: 10.1016/j.amc.2020.125545.  Google Scholar

[20]

A. Slavova, Applications of some mathematical methods in the analysis of cellular neural networks, J. Comput. Appl. Math., 114 (2000), 387-404.  doi: 10.1016/S0377-0427(99)00277-0.  Google Scholar

[21]

A. Slavova, Cellular Neural Networks: Dynamics and Modeling, Kluwer Academic Publishers, Dordrecht, 2003. doi: 10.1007/978-94-017-0261-4.  Google Scholar

[22]

X. F. Wang, Complex networks, topology, dynamics and synchronization, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 885-916.  doi: 10.1142/S0218127402004802.  Google Scholar

[23]

D. Q. Wei, X. S. Luo and Y. L. Zou, Firing activity of complex space-clamped FitzHugh-Nagumo neural networks, European Physical Journal B, 63 (2008), 279-282.  doi: 10.1140/epjb/e2008-00227-5.  Google Scholar

[24]

Z. Yong et al., The synchronization of FitzHugh-Nagumo neuron network coupled by gap junction, Chinese Physics B, 17 (2008), 2297-2303.   Google Scholar

show all references

References:
[1]

B. Ambrosio and M. A. Aziz-Alaoui, Synchronization and control of a network of coupled reaction-diffusion systems of generalized FitzHugh-Nagumo type, ESAIM: Proceedings, 39 (2013), 15-24.  doi: 10.1051/proc/201339003.  Google Scholar

[2]

A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002.  Google Scholar

[3]

A. Cattani, FitzHugh-Nagumo equations with generalized diffusive coupling, Math. Biosci. Eng., 11 (2014), 203-215.  doi: 10.3934/mbe.2014.11.203.  Google Scholar

[4]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems - Part I, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751.  doi: 10.1109/81.473583.  Google Scholar

[5]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems - Part II, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 752-756.   Google Scholar

[6]

L. O. Chua, M. Hasler, G. S. Moschytz and J. Neirynck, Autonomous cellular neural networks: A unified paradigm for pattern formation and active wave propagation, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 559-577.  doi: 10.1109/81.473564.  Google Scholar

[7] L. O. Chua and T. Roska, Cellular Neural Networks and Visual Computing, Cambridge University Press, Cambridge, UK, 2002.  doi: 10.1017/CBO9780511754494.  Google Scholar
[8]

L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits and Systems, 35 (1988), 1257-1272.  doi: 10.1109/31.7600.  Google Scholar

[9]

L. O. Chua and L. Yang, Cellular neural networks: Application, IEEE Trans. Circuits and Systems, 35 (1988), 1273-1290.  doi: 10.1109/31.7601.  Google Scholar

[10]

S. M. Dickson, Stochastic Neural Network Dynamics: Synchronization and Control, Ph. D. Dissertation, Loughborough University, UK, 2014. Google Scholar

[11]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[12]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences, 81 (1984), 3088-3092.  doi: 10.1073/pnas.81.10.3088.  Google Scholar

[13]

M. M. Ibrahim and I. H. Jung, Complex synchronization of a ring-structured network of FitzHugh-Nagumo neurons with single and dual state gap junctions under ionic gates and external electrical disturbance, IEEE Access, 7 (2019), 57894-57906.  doi: 10.1109/ACCESS.2019.2913872.  Google Scholar

[14]

S. Indolia, A. K. Goswami, S. P. Mishra and P. Asopa, Conceptual understanding of convolutional neural networks - a deep learning approach, Procedia Computer Science, 132 (2018), 679-688.  doi: 10.1016/j.procs.2018.05.069.  Google Scholar

[15]

C. Phan, L. Skrzypek and Y. You, Dynamics and synchronization of complex neural networks with boundary coupling, preprint, arXiv: 2004.09988, 2020. Google Scholar

[16]

C. Phan and Y. You, Synchronization of boundary coupled Hindmarsh-Rose neuron network, Nonlinear Anal. Real World Appl., 55 (2020), 103139, 13pp. doi: 10.1016/j.nonrwa.2020.103139.  Google Scholar

[17]

C. Quiñinao and J. D. Touboul, Clamping and synchronization in the strongly coupled FitzHugh-Nagumo model, SIAM J. Appl. Dyn. Syst., 19 (2020), 788-827.  doi: 10.1137/19M1283884.  Google Scholar

[18]

H. Serrano-Guerrero et al., Chaotic synchronization in star coupled networks of three-dimensional cellular neural networks and its applications in communications, International J. Nonlinear Science and Numerical Simulation, 11 (2010), 571-580.   Google Scholar

[19]

L. Skrzypek and Y. You, Dynamics and synchronization of boundary coupled FitzHugh-Nagumo neural networks, Appl. Math. Comput., 388 (2021), 125545, 13 pp. doi: 10.1016/j.amc.2020.125545.  Google Scholar

[20]

A. Slavova, Applications of some mathematical methods in the analysis of cellular neural networks, J. Comput. Appl. Math., 114 (2000), 387-404.  doi: 10.1016/S0377-0427(99)00277-0.  Google Scholar

[21]

A. Slavova, Cellular Neural Networks: Dynamics and Modeling, Kluwer Academic Publishers, Dordrecht, 2003. doi: 10.1007/978-94-017-0261-4.  Google Scholar

[22]

X. F. Wang, Complex networks, topology, dynamics and synchronization, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 885-916.  doi: 10.1142/S0218127402004802.  Google Scholar

[23]

D. Q. Wei, X. S. Luo and Y. L. Zou, Firing activity of complex space-clamped FitzHugh-Nagumo neural networks, European Physical Journal B, 63 (2008), 279-282.  doi: 10.1140/epjb/e2008-00227-5.  Google Scholar

[24]

Z. Yong et al., The synchronization of FitzHugh-Nagumo neuron network coupled by gap junction, Chinese Physics B, 17 (2008), 2297-2303.   Google Scholar

[1]

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457
[2]

Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic & Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052
[3]

Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216
[4]

Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203
[5]

Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118
[6]

Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205
[7]

Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134
[8]

Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251
[9]

Fuzhi Li, Dongmei Xu. Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3517-3542. doi: 10.3934/dcdsb.2020244
[10]

Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1203-1223. doi: 10.3934/dcdsb.2016.21.1203
[11]

Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations & Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028
[12]

John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851
[13]

Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172
[14]

Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072
[15]

Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150
[16]

Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027
[17]

Ying Sue Huang, Chai Wah Wu. Stability of cellular neural network with small delays. Conference Publications, 2005, 2005 (Special) : 420-426. doi: 10.3934/proc.2005.2005.420
[18]

Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2229-2243. doi: 10.3934/dcdss.2020397
[19]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[20]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (38)
  • HTML views (192)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences