Synchronization of FitzHugh-Nagumo reaction-diffusion systems via one-dimensional linear control law

• Autorzy: Ouannas Adel , Mesdoui Fatiha , Momani Shaher , Batiha Iqbal , Grassi Giuseppe

Identyfikatory

DOI

10.24425/acs.2021.137421

• Języki publikacji: EN

Abstrakty

EN

The Fitzhugh-Nagumo model (FN model), which is successfully employed in modeling the function of the so-called membrane potential, exhibits various formations in neuronal networks and rich complex dynamics. This work deals with the problem of control and synchronization of the FN reaction-diffusion model. The proposed control law in this study is designed to be uni-dimensional and linear law for the purpose of reducing the cost of implementation. In order to analytically prove this assertion, Lyapunov’s second method is utilized and illustrated numerically in one- and/or two-spatial dimensions.

Słowa kluczowe

EN

FitzHugh-Nagumo; synchronization; uni-dimensional control; linear control; reaction-diffusion system; neuronal networks; Lyapunov’s second method

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2021
• Tom: Vol. 31, No. 2
• Strony: 333--345
• Opis fizyczny: Bibliogr. 33 poz., rys., wzory

Twórcy

autor

Ouannas Adel

• Laboratory of Dynamical Systems and Control, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria.

autor

Mesdoui Fatiha

• Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan

autor

Momani Shaher

• Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan
• Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

autor

Batiha Iqbal

• Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 2600, Jordan
• Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

autor

Grassi Giuseppe

• Dipartimento Ingegneria Innovazione, Universita del Salento, 73100 Lecce, Italy

Bibliografia

• [1] S.K. Agrawal and S. Das: A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters. Nonlinear Dynamics, 73(1), (2013), 907-919, DOI: 10.1007/s11071-013-0842-7.
• [2] B. Ambrosio and M.A. Aziz-Alaoui: Synchronization and control of coupled reaction-diffusion systems of the FitzHugh-Nagumo type. Computers & Mathematics with Applications, 64(5), (2012), 934-943, DOI: 10.1016/j.camwa.2012.01.056.
• [3] B. Ambrosio, M.A. Aziz-Alaoui, and V.L.E. Phan: Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems, 23(9), (2018), 3787-3797, DOI: 10.3934/dcdsb.2018077.
• [4] B. Ambrosio, M. A. Aziz-Alaoui, and V.L.E. Phan: Large time behaviour and synchronization of complex networks of reaction-diffusion systems of FitzHugh-Nagumo type. IMA Journal of Applied Mathematics, 84(2), (2019), 416-443, DOI: 10.1093/imamat/hxy064.
• [5] M. Aqil, K.-S. Hong, and M.-Y. Jeong: Synchronization of coupled chaotic FitzHugh-Nagumo systems. Communications in Nonlinear Science and Numerical Simulation, 17(4), (2012), 1615-1627, DOI: 10.1016/j.cnsns.2011.09.028.
• [6] S. Bendoukha, S. Abdelmalek, and M. Kirane: The global existence and asymptotic stability of solutions for a reaction-diffusion system. Nonlinear Analysis: Real World Applications. 53, (2020), 103052, DOI: 10.1016/j.nonrwa.2019.103052.
• [7] X.R. Chen and C.X. Liu: Chaos synchronization of fractional order unified chaotic system via nonlinear control. International Journal of Modern Physics B, 25(03), (2011), 407-415, DOI: 10.1142/S0217979211058018.
• [8] D. Eroglu, J.S.W. Lamb, and Y. Pereira: Synchronisation of chaos and its applications. Contemporary Physics, 58(3), (2017), 207-243, DOI: 10.1080/00107514.2017.1345844.
• [9] R. Fitzhugh: Thresholds and Plateaus in the Hodgkin-Huxley Nerve Equations. The Journal of General Physiology, 43(5), (1960), 867-896, DOI: 10.1085/jgp.43.5.867.
• [10] P. Garcia, A. Acosta, and H. Leiva: Synchronization conditions for masterslave reaction diffusion systems. EPL, 88(6), (2009), 60006.
• [11] A.L. Hodgkin and A.F. Huxley: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol, 117, (1952), 500-544, DOI: 10.1113/jphysiol.1952.sp004764.
• [12] T. Kapitaniak: Continuous control and synchronization in chaotic systems. Chaos, Solitons & Fractals, 6 (1995), 237-244, DOI: 10.1016/0960-0779(95)80030-K.
• [13] A.C.J. Luo: Dynamical System Synchronization. Springer-Verlag, New York. 2013.
• [14] D. Mansouri, S. Bendoukha, S. Abdelmalek, and A. Youkana: On the complete synchronization of a time-fractional reaction-diffusion system with the Newton-Leipnik nonlinearity. Applicable Analysis, 100(3), (2021), 675-694, DOI: 10.1080/00036811.2019.1616694.
• [15] F. Mesdoui, A. Ouannas, N. Shawagfeh, G. Grassi, and V.-T. Pham: Synchronization Methods for the Degn-Harrison Reaction-Diffusion Systems. IEEE Access., 8 (2020), 91829-91836, DOI: 10.1109/ACCESS.2020.2993784.
• [16] F. Mesdoui, N. Shawagfeh, and A. Ouannas: Global synchronization of fractional-order and integer-order N component reaction diffusion systems: Application to biochemical models. Mathematical Methods in the Applied Sciences, 44(1), (2021), 1003-1012, DOI: 10.1002/mma.6807.
• [17] J. Nagumo, S. Arimoto, and S. Yoshizawa: An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 50(10), (1962), 2061-2070, DOI: 10.1109/JRPROC.1962.288235.
• [18] L.H. Nguyen and K.-S. Hong: Synchronization of coupled chaotic FitzHugh-Nagumo neurons via Lyapunov functions. Mathematics and Computers in Simulation, 82(4), (2011), 590-603, DOI: 10.1016/j.matcom.2011.10.005.
• [19] Z.M. Odibat: Adaptive feedback control and synchronization of nonidentical chaotic fractional order systems. Nonlinear Dynamics, 60(4), (2010), 479-487, DOI: 10.1007/s11071-009-9609-6.
• [20] Z.M. Odibat, N. Corson, M.A. Aziz-Alaoui, and C. Bertelle: Synchronization of chaotic fractional-order systems via linear control. International Journal of Bifurcation and Chaos, 20(1), (2010), 81-97, DOI: 10.1142/S0218127410025429.
• [21] A. Ouannas, M. Abdelli, Z. Odibat, X. Wang, V.-T. Pham, G. Grassi, and A. Alsaedi: Synchronization Control in Reaction-Diffusion Systems: Application to Lengyel-Epstein System. Complexity, (2019), Article ID 2832781, DOI: 10.1155/2019/2832781.
• [22] A. Ouannas, Z. Odibat, N. Shawagfeh, A. Alsaedi, and B. Ahmad: Universal chaos synchronization control laws for general quadratic discrete systems. Applied Mathematical Modelling, 45 (2017), 636-641, DOI: 10.1016/j.apm.2017.01.012.
• [23] A. Ouannas, Z. Odibat, and N. Shawagfeh: A new Q-S synchronization results for discrete chaotic systems. Differential Equations and Dynamical Systems, 27(4), (2019), 413-422, DOI: 10.1007/s12591-016-0278-x.
• [24] N. Parekh, V.R. Kumar, and B.D. Kulkarni: Control of spatiotemporal chaos: A study with an autocatalytic reaction-diffusion system. Pramana - J. Phys., 48(1), (1997), 303-323, DOI: 10.1007/BF02845637.
• [25] L.M. Pecora and T.L. Carroll: Synchronization in chaotic systems. Physical Review Letter, bf 64(8), (1990), 821-824, DOI: 10.1103/Phys-RevLett.64.821.
• [26] M. Srivastava, S.P. Ansari, S.K. Agrawal, S. Das, and A.Y.T. Leung: Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method. Nonlinear Dynamics, 76 (2014), 905-914, DOI: 10.1007/s11071-013-1177-0.
• [27] J. Wang, T. Zhang, and B. Deng: Synchronization of FitzHugh-Nagumo neurons in external electrical stimulation via nonlinear control. Chaos, Solitons & Fractals, 31(1), (2007), 30-38, DOI: 10.1016/j.chaos.2005.09.006.
• [28] J. Wang, Z. Zhang, and H. Li: Synchronization of FitzHugh-Nagumo systems in EES via H1 variable universe adaptive fuzzy control. Chaos, Solitons & Fractals, 36(5), (2008), 1332-1339, DOI: 10.1016/j.chaos.2006.08.012.
• [29] L. Wang and H. Zhao: Synchronized stability in a reaction-diffusion neural network model. Physics Letters A, 378(48), (2014), 3586-3599, DOI: 10.1016/j.physleta.2014.10.019.
• [30] J. Wei and M. Winter: Standingwaves in the FitzHugh-Nagumo system and a problem in combinatorial geometry. Mathematische Zeitschrift, 254(2), (2006), 359-383, DOI: 10.1007/s00209-006-0952-8.
• [31] X. Wei, J. Wang, and B. Deng: Introducing internal model to robust output synchronization of FitzHugh-Nagumo neurons in external electrical stimulation. Communications in Nonlinear Science and Numerical Simulation, 14(7), (2009), 3108-3119, DOI: 10.1016/j.cnsns.2008.10.016.
• [32] F. Wu, Y. Wang, J. Ma, W. Jin, and A. Hobiny: Multi-channels coupling-induced pattern transition in a tri-layer neuronal network. Physica A: Statistical Mechanics and its Applications, 493 (2018), 54-68, DOI: 10.1016/j.physa.2017.10.041.
• [33] K.-N. Wu, T. Tian, and L. Wang: Synchronization for a class of coupled linear partial differential systems via boundary control. Journal of the Franklin Institute, 353(16), (2016), 4062-4073, DOI: 10.1016/j.jfranklin.2016.07.019.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).