December  2021, 26(12): 6047-6056. doi: 10.3934/dcdsb.2021001

Feedback synchronization of FHN cellular neural networks

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.

Citation: Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete and 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.

[2]

A. ArenasA. Diaz-GuileraJ. KurthsY. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002.

[3]

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

[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.

[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. 

[6]

L. O. ChuaM. HaslerG. 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.

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

[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.

[10]

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

[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.

[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.

[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.

[14]

S. IndoliaA. K. GoswamiS. 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.

[15]

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

[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.

[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.

[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. 

[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.

[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.

[21]

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

[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.

[23]

D. Q. WeiX. 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.

[24]

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

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.

[2]

A. ArenasA. Diaz-GuileraJ. KurthsY. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002.

[3]

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

[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.

[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. 

[6]

L. O. ChuaM. HaslerG. 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.

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

[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.

[10]

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

[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.

[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.

[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.

[14]

S. IndoliaA. K. GoswamiS. 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.

[15]

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

[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.

[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.

[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. 

[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.

[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.

[21]

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

[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.

[23]

D. Q. WeiX. 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.

[24]

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

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