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December  2021, 26(12): 6047-6056. 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 2021 Early access  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, 2021, 26 (12) : 6047-6056. 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

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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

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