Synchronization of coupled chaotic FitzHugh-Nagumo neurons via Lyapunov functions

Mathematics and Computers in Simulation

Volumn 82, Issue 4, 2011, Pages 590-603

  Nguyen, Le Hoa ^a   Hong, Keum Shik ^a  

^a Pusan National University   (South Korea)

Author keywords

Coupling coefficient; FitzHugh Nagumo model; Lyapunov function; Nonlinear control; Synchronization

Indexed keywords

CHAOTIC DYNAMICS; CONTROL LAWS; COUPLING COEFFICIENT; ELECTRICAL STIMULATIONS; FEEDBACK CONTROL LAW; FITZHUGH-NAGUMO MODEL; LYAPUNOV; LYAPUNOV FUNCTION METHOD; NON LINEAR CONTROL; SELF SYNCHRONIZATION; TRANSVERSE LYAPUNOV EXPONENTS;

CONTROL THEORY; LYAPUNOV FUNCTIONS; SYNCHRONIZATION;

LYAPUNOV METHODS;

EID: 84855277497     PISSN: 03784754     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matcom.2011.10.005     Document Type: Article

References (50)

1. J. Aguirre, E. Mosekilde, and M.A.F. Sanjuan Analysis of the noise-induced bursting-spiking transition in a pancreatic beta-cell model Phys. Rev. E 69 2004 041910
2. K. Aihara, G. Matsumoto, and M. Ichiwaka An alternating periodic-chaotic sequence observed in neural oscillations Phys. Lett. A 111 1985 251 255
3. M. Aqil, K.-S. Hong, and M.-Y. Jeong Synchronization of coupled chaotic FitzHugh-Nagumo systems Commun. Nonlinear Sci. Numer. Simul. 17 4 2012 1615 1627 10.1016/j.cnsns.2011.09.028
4. P. Ashwin, and J.R. Terry Blowout bifurcation in a system of coupled chaotic lasers Phys. Rev. E 58 1998 7186 7189 (Pubitemid 128596511)
5. P. Badola, S.S. Tambe, and B.D. Kulkarni Driving systems with chaotic signals Phys. Rev. A 46 1992 6735 6737
6. V.N. Belykh, I.V. Belykh, M. Colding-Jørgensen, and E. Mosekilde Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models Eur. Phys. J. E 3 2000 205 219
7. S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, and C.S. Zhou The synchronization of chaotic systems Phys. Rep. 336 2002 1 101
8. S. Bowong, M. Kakmeni, and R. Koina Chaos synchronization and duration time of a class of uncertain chaotic systems Math. Comput. Simul. 71 3 2006 212 228
9. H.A. Braun, H. Wissing, K. Schäfer, and M.C. Hirsch Oscillation and noise determine signal transduction in shark multimodel sensory cells Nature 367 1994 270 273 (Pubitemid 24051142)
10. J.M. Casado Synchronization of two Hodgkin-Huxley neurons due to internal noise Phys. Lett. A 310 2003 400 406
11. T.R. Chay Chaos in a three-variable model of an excitable cell Physica D16 1985 233 242
12. M.H. Chou, and Y.T. Lin Exotic dynamic behavior of the forced FitzHugh-Nagumo equations Comput. Math. Appl. 32 1996 109 124
13. S. Coombes, and A.H. Osbaldestin Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator Phys. Rev. E 62 2000 4057 4066
14. O. Cornejo-Pérez, and R. Femat Unidirectional synchronization of Hodgkin-Huxley neurons Chaos Soliton Fract. 25 2005 43 53 (Pubitemid 40220060)
15. B. Deng, J. Wang, and X. Fei Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control Chaos Soliton Fract. 29 2006 182 189
16. E. Eckhorn Neural mechanisms of scene segmentation: recording from the visual cortex suggests basic circuits or linking field models IEEE Trans. Neural Netw. 10 1999 464 479
17. Y.-S. Fan, and T.R. Chay Generation of periodic and chaotic bursting in an excitable cell model Biol. Cybern. 71 1984 417 431
18. R. Fitzhugh Thresholds and plateaus in the Hodgkin-Huxley nerve equations J. Gen. Physiol. 43 1960 867 896
19. M.R. Guevara, L. Glass, and A. Shrier Phase locking, period-doubling bifurcation, and irregular dynamics in periodically stimulated cardiac cells Science 214 1981 1350 1353 (Pubitemid 12135120)
20. J.F. Heagy, T.L. Carroll, and L.M. Pecora Desynchronization by periodic orbits Phys. Rev. E 52 1995 1253 1256
21. J.L. Hindmarsh, and R.M. Rose A model of neuronal bursting using three coupled first order differential equations Proc. R. Soc. Lond. B 221 1984 87 102
22. 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
23. H.K. Khalil Nonlinear Systems third ed. 2002 Prentice Hall New York
24. T. Kostova, R. Ravindran, and M. Schonbek FitzHugh-Nagumo revisited: types of bifurcations, periodical forcing and stability regions by a Lyapunov functional Int. J. Bifurcat. Chaos 14 3 2004 913 925 (Pubitemid 40242179)
25. G.M. Mahmoud, and E.E. Mahmoud Synchronization and control of hyperchaotic complex Lorenz system Math. Comput. Simul. 80 12 2010 2286 2296
26. G. Matsumoto, K. Aihara, M. Ichikawa, and A. Tasaki Periodic and nonperiodic response of membrane potentials in squid giant axons during sinusoidal current stimulation J. Theoret. Neurobiol. 3 1984 1 14 (Pubitemid 15186713)
27. C. Morris, and H. Lecar Voltage oscillations in the barnacle giant muscle fiber Biophys. J. 35 1981 193 213 (Pubitemid 11079105)
28. E. Mosekilde, A.L. Fradkov, and I.I. Blekhman Chaos synchronization and control-Preface Math. Comput. Simul. 58 4-6 2002 283 284
29. E. Mosekilde, B. Lading, S. Yanchuk, and Y. Maistrenko Bifurcation structure of a model of bursting pancreatic cells BioSystems 63 2001 3 13 (Pubitemid 32928484)
30. J. Nagumo, S. Arimoto, and S. Yoshizawa An active pulse transmission line simulating nerve axon Proc. IRE 50 1962 2061 2070
31. A.B. Neiman, and D.F. Russell Synchronization of noise-induced bursts in noncoupled sensory neurons Phys. Rev. Lett. 88 2002 138103
32. E. Ott, and J.C. Sommerer Blowout bifurcations: the occurrence of riddled basins and on-off intermittency Phys. Lett. A 188 1994 39 47
33. L.M. Pecora, and T.L. Carroll Master stability functions for synchronized coupled systems Phys. Rev. Lett. 80 1998 2109 2112 (Pubitemid 128621504)
34. L.M. Pecora, and T.L. Carroll Synchronization in chaotic systems Phys. Rev. Lett. 64 1990 821 825
35. R.E. Plant Bifurcation and resonance in a model for bursting nerve cells J. Math. Biol. 11 1981 15 32 (Pubitemid 11182676)
36. D.E. Postnov, L.S. Ryazanova, E. Mosekilde, and O.V. Sosnovtseva Neural synchronization via potassium signaling Int. J. Neural Syst. 16 2 2006 99 109 (Pubitemid 44099467)
37. D.E. Postnov, L.S. Ryazanova, R.A. Zhirin, E. Mosekilde, and O.V. Sosnovtseva Noise controlled synchronization in potassium coupled neural models Int. J. Neural Syst. 17 2 2007 105 113 (Pubitemid 46915350)
38. M. Rehan, K.-S. Hong, and M. Aqil Synchronization of multiple chaotic FitzHugh-Nagumo neurons with gap junctions under external electrical stimulation Neurocomputing 74 16 2011 3296 3304
39. M. Rehan, and K.-S. Hong LMI-based robust adaptive synchronization of FitzHugh-Nagumo neurons with unknown parameters under certain external electrical stimulation Phys. Lett. A 375 15 2011 1666 1670
40. M.G. Rosenblum, A.S. Pikovsky, and J. Kurths Phase synchronization of chaotic oscillators Phys. Rev. Lett. 76 1996 1804 1807 (Pubitemid 126638871)
41. N.F. Rulkov, and M.M. Sushchik Robustness of synchronized chaotic oscillations Int. J. Bifurcat. Chaos 7 1997 625 643 (Pubitemid 127338603)
42. N.F. Rulkov, M.M. Sushchik, and L.S. Tsimring Generalized synchronization of chaos in directionally coupled chaotic systems Phys. Rev. E 51 1995 980 994
43. H. Salarieh, and A. Alasty Adaptive chaos synchronization in Chua's systems with noisy parameters Math. Comput. Simul. 79 3 2008 233 241
44. C.J. Thompson, D.C. Bardos, Y.S. Yang, and K.H. Yoyner Nonlinear cable models for cell exposed to electric field I. General theory and space-clamped solutions Chaos Soliton Fract. 10 11 1999 1825 1842
45. L.L. Tian, D.H. Li, and X.F. Sun Nonlinear-estimator-based robust synchronization of Hodgkin-Huxley neurons Neurocomputing 72 2008 186 196
46. X.J. Wan, and J.T. Sun Adaptive-impulsive synchronization of chaotic systems Math. Comput. Simul. 81 8 2011 1609 1617
47. J. Wang, Z. Zhang, and H. Li Synchronization of FitzHugh-Nagumo systems in EES via H-infinity variable universe adaptive fuzzy control Chaos Soliton Fract. 36 2008 1332 1339 (Pubitemid 351172782)
48. A. Wolf, J.B. Swift, H.L. Swinney, and J.A. Vastano Determining Lyapunov exponents from a time series Physica D 16 1985 285 317
49. C.W. Wu, and L.O. Chua A unified framework for synchronization and control of dynamical system Int. J. Bifurcat. Chaos 10 1994 979 998
50. S. Yanchuk, Y. Maistrenko, B. Lading, and E. Mosekilde Effects of a parameter mismatch on the synchronization of two coupled chaotic oscillators Int. J. Bifurcat. Chaos 10 2000 2629 2648 (Pubitemid 33218740)