Desynchronization and inhibition of Kuramoto oscillators by scalar mean-field feedback

Mathematics of Control, Signals, and Systems

Volumn 24, Issue 1-2, 2012, Pages 169-217

  Franci, Alessio ^a   Chaillet, Antoine ^a   Panteley, Elena ^a   Lamnabhi Lagarrigue, Françoise ^b  

^a UNIV PARIS SUD   (France)

^b CNRS   (France)

Author keywords

Desynchronization; Kuramoto oscillators; Mean field feedback; Oscillations inhibition

Indexed keywords

COLLECTIVE OSCILLATIONS; DEEP BRAIN STIMULATION; DESYNCHRONIZATION; FEEDBACK TOPOLOGY; FIXED POINTS; FORMAL DEFINITION; GLOBALLY ASYMPTOTICALLY STABLE; MEAN-FIELD; NEURONAL POPULATIONS; PARKINSON'S DISEASE; PROPORTIONAL FEEDBACK; SCALAR CONTROL; SIMPLIFIED MODELS; SYNCHRONIZED OSCILLATOR;

SURGERY;

NATURAL FREQUENCIES;

EID: 84859105913     PISSN: 09324194     EISSN: 1435568X     Source Type: Journal    
DOI: 10.1007/s00498-011-0072-9     Document Type: Article

References (60)

1. D Aeyels JA Rogge 2004 Existence of partial entrainment and stability of phase locking behavior of coupled oscillators Prog Theor Phys 112 6 921 942 10.1143/PTP.112.921 1071.34033 (Pubitemid 40366573)
2. D Angeli L Praly 2011 Stability robustness in the presence of exponentially unstable isolated equilibria IEEE Trans Automat Control 56 7 1582 1592 2848270 10.1109/TAC.2010.2091170
3. AL Benabid P Pollak C Gervason D Hoffmann DM Gao M Hommel JE Perret J de Rougemont 1991 Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus Lancet 337 403 406 10.1016/0140-6736(91) 91175-T
4. II Blekhman AL Fradkov H Nijmeijer A Yu Pogromsky 1997 On self synchronization and controlled synchronization Syst Control Lett 31 299 305 10.1016/S0167-6911(97)00047-9 0901.93028
5. Brown E, Holmes P, Moehlis J (2003) Globally coupled oscillator networks. In: Sreenivasan K, Kaplan E, Marsden J (eds) Perspectives and Problems in Nonlinear Science: A Celebratory Volume in Honor of Larry Sirovich. Springer, New York
6. Chen G (2003) Chaotification via feedback: the discrete case. In: Lecture Notes in Control and Information Sciences, vol 292. Springer, Berlin
7. G Chen L Yang 2003 Chaotifying a continuous-time system near a stable limit cycle Chaos Solitons Fractals 15 2 245 253 1926745 10.1016/S0960-0779(02) 00096-6 1047.34057
8. N Chopra MW Spong 2009 On exponential synchronization of Kuramoto oscillators IEEE Trans Automat Control 54 2 353 357 2491964 10.1109/TAC.2008. 2007884
9. F Dörfler F Bullo 2011 On the critical coupling for Kuramoto oscillators SIAM J Appl Dyn Syst 10 1070 1099 2837521 10.1137/10081530X
10. GB Ermentrout 1990 Oscillator death in populations of "all to all" coupled nonlinear oscillators Phys D 41 2 219 231 1049128 10.1016/0167-2789(90)90124-8 0693.34040
11. GB Ermentrout 1996 Type I membranes, phase resetting curves, and synchrony Neural Comput 8 5 979 1001 10.1162/neco.1996.8.5.979
12. GB Ermentrout N Kopell 1990 Oscillator death in systems of coupled neural oscillators SIAM J Appl Math 50 1 125 146 1036235 10.1137/0150009 0686.34033 (Pubitemid 20661621)
13. Fradkov AL (2007) Cybernetical physics. From control of chaos to quantum control. Springer, Berlin
14. Franci A, Chaillet A, Pasillas-Lépine W (2010) Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback. Autom Spec Issue Biol Syst 47(6):1193-1202, (extended version available at: http://hal.archives-ouvertes.fr/hal-00526066/ )
15. Franci A, Panteley E, Chaillet A, Lamnabhi-Lagarrigue F (2011) Desynchronization of coupled phase oscillators, with application to the Kuramoto system under mean-field feedback. In: Proc. 50th. IEEE Conf Decision Contr, Orlando, FL, USA, December 2011 (in press)
16. Gao Y, Chau K (2002) Chaotification of permanent-magnet synchronous motor drives using time-delay feedback. In: IEEE Annual Conf. of Industrial Electronics Soc., pp 762-766
17. Golubitsky M, Guillemin V (1973) Stable mappings and their singularities graduate texts in mathematics, vol 14. Springer-Verlag, New York
18. Guckenheimer J, Holmes P (2002) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. In: Applied Mathematical Sciences, 7th edn, vol 42. Springer, New-York
19. Hale JK (1969) Ordinary differential equations. Interscience John Wiley, New York
20. C Hammond R Ammari B Bioulac L Garcia 2008 Latest view on the mechanism of action of deep brain stimulation Mov Disord 23 15 2111 2121 10.1002/mds.22120
21. Hammond C, Bergman H, Brown P (2007) Pathological synchronization in Parkinson's disease: networks, models and treatments. Trends Neurosci 30(7):357-364 (July INMED/TINS special issue-physiogenic and pathogenic oscillations: the beauty and the beast) (Pubitemid 46994777)
22. C Hauptmann O Popovych PA Tass 2005 Delayed feedback control of synchronization in locally coupled neuronal networks Neurocomputing 65 759 767 10.1016/j.neucom.2004.10.072 (Pubitemid 40613615)
23. C Hauptmann O Popovych PA Tass 2005 Effectively desynchronizing deep brain stimulation based on a coordinated delayed feedback stimulation via several sites: a computational study Biol Cybern 93 463 470 2203127 10.1007/s00422-005-0020-1 1132.92318 (Pubitemid 41744300)
24. C Hauptmann O Popovych PA Tass 2005 Multisite coordinated delayed feedback for an effective desynchronization of neuronal networks Stoch Dyn 5 2 307 319 2147291 10.1142/S0219493705001420 1070.92007 (Pubitemid 44103071)
25. Hirsch M, Pugh C, Shub M (1977) Invariant manifolds. Lecture notes in mathematics. Springer-Verlag, Berlin
26. Hirsch MW, Smale S (1974) Differential equations, dynamical systems, and linear algebra. In: Pure and applied mathematics. Harcourt Brace Jovanovich, Accademic Press, Massachusetts
27. Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge
28. Isidori A (1999) Nonlinear control systems II. Springer Verlag, London
29. Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. MIT Press, Cambridge
30. Jadbabaie A, Motee N, Barahona M (2004) On the stability of the Kuramoto model of coupled nonlinear oscillators. In: Proc. American Control Conf., pp 4296-4301
31. Khalil H (2001) Nonlinear systems 3. Prentice Hall, New York
32. Kuramoto Y (1984) Chemical oscillations, waves, and turbulence. Springer, Berlin
33. Lee J (2006) Introduction to smooth manifolds. Graduate texts in mathematics. Springer-Verlag, Berlin
34. J Lopez-Azcarate M Tainta MC Rodriguez-Oroz M Valencia R Gonzalez J Guridi J Iriarte JA Obeso J Artieda M Alegre 2010 Coupling between beta and high-frequency activity in the human subthalamic nucleus may be a pathophysiological mechanism in Parkinson's disease J Neurosci 30 19 6667 6677 10.1523/JNEUROSCI.5459-09.2010
35. M Luo Y Wu J Peng 2009 Washout filter aided mean field feedback desynchronization in an ensemble of globally coupled neural oscillators Biol Cybern 101 241 246 2545706 10.1007/s00422-009-0334-5
36. R Orsi L Praly I Mareels 2001 Sufficient conditions for the existence of an unbounded solution Automatica 37 10 1609 1617 10.1016/S0005-1098(01)00114-5 0996.93075 (Pubitemid 32737017)
37. E Panteley A Loría A Teel 2001 Relaxed persistency of excitation for uniform asymptotic stability IEEE Trans Automat Control 46 12 1874 1886 1878208 10.1109/9.975471 1032.93070 (Pubitemid 34126233)
38. D Pazó MA Zaks J Kurths 2003 Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators Chaos 13 1 309 318 1964975 10.1063/1.1518430 1080.37541 (Pubitemid 36419928)
39. Pikovsky A, Rosenblum M, Kurths J (2001) Synchronization: a universal concept in nonlinear sciences. Cambridge Nonlinear Science Series, Cambridge
40. D Plenz ST Kital 1999 A basal ganglia pacemaker formed by the subthalamic nucleus and external globus pallidus Nature 400 6745 677 682 10.1038/23281 (Pubitemid 29382573)
41. OV Popovych C Hauptmann PA Tass 2005 Effective desynchronization by nonlinear delayed feedback Phys Rev Lett 94 164102 10.1103/PhysRevLett.94.164102 (Pubitemid 40620309)
42. OV Popovych C Hauptmann PA Tass 2006 Control of neuronal synchrony by nonlinear delayed feedback Biol Cybern 95 69 85 2237184 10.1007/s00422-006-0066- 8 1169.93338
43. OV Popovych C Hauptmann PA Tass 2006 Desynchronization and decoupling of interacting oscillators by nonlinear delayed feedback Int J Bifurc Chaos 16 7 1977 1987 2263290 10.1142/S0218127406015830 1168.34322 (Pubitemid 44370619)
44. OV Popovych PA Tass 2010 Synchronization control of interacting oscillatory ensembles by mixed nonlinear delayed feedback Phys Rev E 82 2 026204 2736438 10.1103/PhysRevE.82.026204
45. OV Popovych V Krachkovskyi PA Tass 2007 Twofold impact of delayed feedback on coupled oscillators Int J Bifurc Chaos 17 7 2517 2530 2349747 10.1142/S0218127407018592 1163.34045 (Pubitemid 47376635)
46. K Pyragas OV Popovich PA Tass 2008 Controlling synchrony in oscillatory networks with a separate stimulation-registration setup Eur Phys Lett. 80 4 40002:1 40002:6
47. M Rosa S Marceglia D Servello G Foffani L Rossi M Sassi S Mrakic-Sposta R Zangaglia C Pacchetti M Porta A Priori 2010 Time dependent subthalamic local field potential changes after DBS surgery in Parkinson's disease Exp Neurol 222 184 190 10.1016/j.expneurol.2009.12.013
48. M Rosenblum A Pikovsky 2004 Delayed feedback control of collective synchrony: an approach to suppression of pathological brain rhythms Phys Rev E 70 4 041904 2130392 10.1103/PhysRevE.70.041904 (Pubitemid 40015654)
49. MG Rosenblum AS Pikovsky 2004 Controlling synchronization in an ensemble of globally coupled oscillators Phys Rev Lett 92 114102 10.1103/PhysRevLett.92. 114102
50. MG Rosenblum N Tukhlina A Pikovsky L Cimponeriu 2006 Delayed feedback suppression of collective rhythmic activity in a neuronal ensemble Int J Bifurc Chaos 16 7 1989 1999 2263291 10.1142/S0218127406015842 1154.92307 (Pubitemid 44370620)
51. Sarlette A (2009) Geometry and Symmetries in Coordination Control. PhD thesis, University of Liège, Belgium
52. SV Sarma M Cheng Z Williams R Hu E Eskandar EN Brown 2010 Comparing healthy and Parkinsonian neuronal activity in sub-thalamic nucleus using point process models IEEE Trans Biomed Eng 57 6 1297 1305 10.1109/TBME.2009.2039213
53. R Sepulchre DA Paley NE Leonard 2007 Stabilization of planar collective motion: all-to-all communication IEEE Trans Automat Control 52 5 811 824 2322479 10.1109/TAC.2007.898077 (Pubitemid 46796067)
54. J Sijbrand 1985 Properties of center manifolds Trans Am Math Soc 289 2 431 469 783998 10.1090/S0002-9947-1985-0783998-8 0577.34039
55. SH Strogatz 2000 From Kuramoto to Crawford: exploring the onset of synchronization in population of coupled oscillators Phys D 143 1 20 1783382 10.1016/S0167-2789(00)00094-4 0983.34022
56. N Tukhlina M Rosenblum 2008 Feedback suppression of neural synchrony in two interacting populations by vanishing stimulation J Biol Phys 34 301 314 10.1007/s10867-008-9081-4
57. N Tukhlina M Rosenblum A Pikovsky J Kurths 2007 Feedback suppression of neural synchrony by vanishing stimulation Phys Rev E 75 1 011918 2324650 10.1103/PhysRevE.75.011918 (Pubitemid 46130696)
58. J Volkmann M Joliot A Mogilner AA Ioannides F Lado E Fazzini U Ribary R Llinás 1996 Central motor loop oscillations in Parkinsonian resting tremor revealed by magnetoencephalography Neurology 46 1359 1370 (Pubitemid 26159299)
59. Winfree AT (1980) The geometry of biological times. Springer, New-York
60. Zhang H, Liu D, Wang Z (2009) Controlling chaos: suppression, synchronization and chaotification. In: Communications and Control Engineering. Springer-Verlag, New York