Biological Cybernetics

, Volume 95, Issue 1, pp 69–85 | Cite as

Control of Neuronal Synchrony by Nonlinear Delayed Feedback

  • Oleksandr V. Popovych
  • Christian Hauptmann
  • Peter A. Tass
Original Paper

Abstract

We present nonlinear delayed feedback stimulation as a technique for effective desynchronization. This method is intriguingly robust with respect to system and stimulation parameter variations. We demonstrate its broad applicability by applying it to different generic oscillator networks as well as to a population of bursting neurons. Nonlinear delayed feedback specifically counteracts abnormal interactions and, thus, restores the natural frequencies of the individual oscillatory units. Nevertheless, nonlinear delayed feedback enables to strongly detune the macroscopic frequency of the collective oscillation. We propose nonlinear delayed feedback stimulation for the therapy of neurological diseases characterized by abnormal synchrony.

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References

  1. Ahissar E (1998) Temporal-code to rate-code conversion by neuronal phase-locked loops. Neural Comput 10:597–650CrossRefPubMedGoogle Scholar
  2. Ahissar E, Vaadia E (1990) Oscillatory activity of single units in a somatosensory cortex of an awake monkey and their possible role in texture analysis. Proc Natl Acad Sci USA 87:8935–8939PubMedCrossRefGoogle Scholar
  3. Alberts WW, Wright EJ, Feinstein B (1969) Cortical potentials and parkinsonian tremor. Nature 221:670–672PubMedCrossRefGoogle Scholar
  4. Atay FM (2003) Distributed delays facilitate amplitude death of coupled oscillators. Phys Rev Lett 91:094101CrossRefPubMedGoogle Scholar
  5. Atay FM, Jost J, Wende A (2004) Delays, connection topology, and synchronization of coupled chaotic maps. Phys Rev Lett 92:144101CrossRefPubMedGoogle Scholar
  6. Bellman R, Cooke KL (1963) Differential-difference equations. Academic, New YorkGoogle Scholar
  7. Benabid AL, Pollak P, Louveau A, Henry S, de Rougemont J (1987) Combined (thalamotomy and stimulation) stereotactic surgery of the vim thalamic nucleus for bilateral parkinson’s disease. Appl Neurophysiol 50:344–346PubMedCrossRefGoogle Scholar
  8. Benabid AL, Pollak P, Gervason C, Hoffmann D, Gao DM, Hommel M, Perret JE, de Rougemount J (1991) Longterm suppression of tremor by chronic stimulation of ventral intermediate thalamic nucleus. The Lancet 337:403–406CrossRefGoogle Scholar
  9. Benabid AL, Benazzous A, Pollak P (2002) Mechanisms of deep brain stimulation. Mov Disord 17:73–74CrossRefGoogle Scholar
  10. Daido H (1997) Order function theory of macroscopic phase-locking in globally and weakly coupled limit-cycle oscillators. Int J Bifurcat Chaos 7(4):807–829CrossRefGoogle Scholar
  11. Dolan K, Witt A, Spano ML, Neiman A, Moss F (1999) Surrogates for finding unstable periodic orbits in noisy data sets. Phys Rev E 59:5235–5241CrossRefGoogle Scholar
  12. Eckhorn R, Bauer R, Jordan W, Brosch M, Kruse W, Munk M, Reitboeck HJ (1988) Coherent oscillations: a mechanism of feature linking in the visual cortex? Multiple electrode and correlation analyses in the cat. Biol Cybern 60(2):121–130CrossRefPubMedGoogle Scholar
  13. Ernst U, Pawelzik K, Geisel T (1995) Synchronization induced by temporal delays in pulse-coupled oscillators. Phys Rev Lett 74:1570–1573CrossRefPubMedGoogle Scholar
  14. Ernst U, Pawelzik K, Geisel T (1998) Delay-induced multistable synchronization of biological oscillators. Phys Rev E 57:2150–2162CrossRefGoogle Scholar
  15. Filali M, Hutchison WD, Palter VN, Lozano AM, Dostrovsky JO (2004) Stimulation-induced inhibition of neuronal firing in human subthalamic nucleus. Exp Brain Res 156(3):274–281CrossRefPubMedGoogle Scholar
  16. Garcia L, Audin J, D’Alessandro G, Bioulac B, Hammond C (2003) Dual effect of high-frequency stimulation on subthalamic neuron activity. J Neurosci 23:8743–8751PubMedGoogle Scholar
  17. Gray CM, Singer W (1989) Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc Natl Acad Sci USA 86(5):1698–1702PubMedCrossRefGoogle Scholar
  18. Grill WM, McIntyre CC (2001) Extracellular excitation of central neurons: implications for the mechanisms of deep brain stimulation. Thalamus Relat Syst 1:269–277CrossRefGoogle Scholar
  19. Hansel D, Mato G, Meunier C (1993) Phase dynamics of weakly coupled Hodgkin–Huxley neurons. Europhys Lett 23:367–372CrossRefGoogle Scholar
  20. Hauptmann C, Mackey MC (2003) Stimulus dependent onset latency of the inhibitory recurrent activity. Biol Cybern 88:459–467PubMedGoogle Scholar
  21. Hauptmann C, Popovych O, Tass PA (2005a) Delayed feedback control of synchronization in locally coupled neuronal networks. Neurocomputing 65–66:759–767Google Scholar
  22. Hauptmann C, Popovych O, Tass PA (2005b) Effectively desynchronizing deep brain stimulation based on a coordinated delayed feedback stimulation via several sites: a computational study. Biol Cybern 93:463–470CrossRefGoogle Scholar
  23. Hauptmann C, Popovych O, Tass PA (2005c) Multisite coordinated delayed feedback for an effective desynchronization of neuronal networks. Stochast Dyn 5(2):307–319CrossRefGoogle Scholar
  24. Hellwig B (2000) A quantitative analysis of the local connectivity between pyramidal neurons in layers 2/3 of the rat visual cortex. Biol Cybern 82:111–121CrossRefPubMedGoogle Scholar
  25. Hoppensteadt FC (1997) An introduction to the mathematics of neurons: modeling in the frequency domain. Cambridge University Press, CambridgeGoogle Scholar
  26. Kim S, Park SH, Ryu CS (1997) Multistability in coupled oscillator systems with time delay. Phys Rev Lett 79:2911–2914CrossRefGoogle Scholar
  27. Kumar R, Lozano AM, Sime E, Lang AE (2003) Long-term follow-up of thalamic deep brain stimulation for essential and parkinsonian tremor. Neurology 61(11):1601–1604PubMedGoogle Scholar
  28. Kuramoto Y (1984) Chemical oscillations, waves, and turbulence. Springer, Berlin Heidelberg New YorkGoogle Scholar
  29. Kuznetsov YuA (1998) Elements of applied bifurcation theory. Springer, Berlin Heidelberg New YorkGoogle Scholar
  30. Lenz FA, Kwan HC, Martin RL, Tasker RR, Dostrovsky JO, Lenz YE (1994) Single unit analysis of the human ventral thalamic nuclear group. Tremor-related activity in functionally identified cells Brain 117:531–543Google Scholar
  31. Luhmann HJ, Mudrick-Donnon LA, Mittmann T, Heinemann U (1995) Ischaemia-induced long-term hyperexcitability in rat neocortex. Eur J Neurosci 7:180–191CrossRefPubMedGoogle Scholar
  32. Maistrenko Yu, Popovych O, Burylko O, Tass PA (2004) Mechanism of desynchronization in the finite-dimensional Kuramoto model. Phys Rev Lett 93:084102CrossRefPubMedGoogle Scholar
  33. Matthews PC, Strogatz SH (1990) Phase diagram for the collective behavior of limit-cycle oscillators. Phys Rev Lett 65:1701–1704CrossRefPubMedGoogle Scholar
  34. McIntyre CC, Grill WM (1999) Excitation of central nervous system neurons by nonuniform electric fields. Biophys J 76:878–888PubMedGoogle Scholar
  35. McIntyre CC, Savasta M, Kerkerian-Le, Goff L, Vitek JL (2004) Uncovering the mechanism(s) of action of deep brain stimulation: activation, inhibition, or both. Clin Neurophysiol 115(6):1239–1248CrossRefPubMedGoogle Scholar
  36. Meissner W, Leblois A, Hansel D, Bioulac B, Gross CE, Benazzouz A, Boraud T (2005) Subthalamic high frequency stimulation resets subthalamic firing and reduces abnormal oscillations. Brain 128:2372–2382CrossRefPubMedGoogle Scholar
  37. Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193–213PubMedCrossRefGoogle Scholar
  38. Nini A, Feingold A, Slovin H, Bergmann H (1995) Neurons in the globus pallidus do not show correlated activity in the normal monkey, but phase-locked oscillations appear in the MPTP model of parkinsonism. J Neurophysiol 74:1800–1805PubMedGoogle Scholar
  39. Nowak LG, Bullier J (1998a) Axons, but not cell bodies, are activated by electrical stimulation in cortical gray matter I evidence from chronaxie measurements. Exp Brain Res 118:477–488CrossRefGoogle Scholar
  40. Nowak LG, Bullier J (1998b) Axons, but not cell bodies, are activated by electrical stimulation in cortical gray matter II evidence from selective inactivation of cell bodies and axon initial segments. Exp Brain Res 118:489–500CrossRefGoogle Scholar
  41. Nunez PL (1981) Electric fields of the brain. Oxford University Press, New YorkGoogle Scholar
  42. Pare D, Curro’Dossi R, Steriade M (1990) Neuronal basis of the parkinsonian resting tremor: a hypothesis and its implications for treatment. Neuroscience 35:217–226CrossRefPubMedGoogle Scholar
  43. Pikovsky AS, Rosenblum MG, Kurths J (1996) Synchronization in a population of globally coupled chaotic oscillators. Europhys Lett 34(3):165–170CrossRefGoogle Scholar
  44. Pikovsky A, Rosenblum M, Kurths J (2001) Synchronization, a universal concept in nonlinear sciences. Cambridge University Press, CambridgeGoogle Scholar
  45. van der Pol B (1920) A theory of the amplitude of free and forced triode vibration. Radio Rev 1:704–754Google Scholar
  46. van der Pol B (1927) Forced oscillations in a circuit with non-linear resistance. Phil Mag Ser 7 3:65–80Google Scholar
  47. van der Pol B, van der Mark J (1928) The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Phil Mag Suppl 6:763–775Google Scholar
  48. Popovych OV, Hauptmann C, Tass PA (2005a) Effective desynchronization by nonlinear delayed feedback. Phys Rev Lett 94:164102CrossRefGoogle Scholar
  49. Popovych OV, Maistrenko YuL, Tass PA (2005b) Phase chaos in coupled oscillators. Phys Rev E 71:065201(R)CrossRefMathSciNetGoogle Scholar
  50. Pyragas K (1992) Continuous control of chaos by self-controlling feedback. Phys Lett A 170:421–428CrossRefGoogle Scholar
  51. Ranck JB (1975) Which elements are excited in electrical stimulation of mammalian central nervous system: a review. Brain Res 98:417–468CrossRefPubMedGoogle Scholar
  52. Reddy DVR, Sen A, Johnston GL (1998) Time delay induced death in coupled limit cycle oscillators. Phys Rev Lett 80:5109–5112CrossRefGoogle Scholar
  53. Reddy DVR, Sen A, Johnston GL (1999) Time delay effects on coupled limit cycle oscillators at Hopf bifurcation. Physica D 129(1–2):15–34CrossRefGoogle Scholar
  54. Reddy DVR, Sen A, Johnston GL (2000) Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators. Phys Rev Lett 85:3381–3384CrossRefPubMedGoogle Scholar
  55. Rinzel J, Ermentrout GB (1989) Analysis of neural excitability and oscillations. In: Koch CH, Segev I (eds) Methods in neuronal modelling from synapses to networks. MIT Press, Cambridge, pp 135–169Google Scholar
  56. Rodriguez-Oroz MC, Obeso JA, Lang AE, Houeto J-L, Pollak P, Rehncrona S, Kulisevsky J, Albanese A, Volkmann J, Hariz MI, , Speelman JD, Guridi J, Zamarbide I, Gironell A, Molet J, Pascual-Sedano B, Pidoux B, Bonnet AM, Agid Y, Xie J, Benabid A-L, Lozano AM, Saint-Cyr J, Romito L, Contarino MF, Scerrati M, Van Blercom N (2005) Bilateral deep brain stimulation in parkinson’s disease: a multicentre study with 4 years follow-up. Brain 128(10):2240–2249CrossRefPubMedGoogle Scholar
  57. Rosenblum MG, Pikovsky AS (2004a) Controlling synchronizatio n in an ensemble of globally coupled oscillators. Phys Rev Lett 92:114102CrossRefGoogle Scholar
  58. Rosenblum MG, Pikovsky AS (2004b) Delayed feedback control of collective synchrony: an approach to suppression of pathological brain rhythms. Phys Rev E 70:041904CrossRefGoogle Scholar
  59. Rosenblum MG, Pikovsky AS, Kurths J (1996) Phase synchronization of chaotic oscillators. Phys Rev Lett 76(11):1804–1807CrossRefPubMedGoogle Scholar
  60. Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57:397–398CrossRefGoogle Scholar
  61. Schuster HG, Wagner P (1989) Mutual entrainment of two limit cycle oscillators with time delayed coupling. Prog Theor Phys 81:939–945CrossRefGoogle Scholar
  62. Singer W, Gray CM (1995) Visual feature integration and the temporal correlation hypothesis. Annu Rev Neurosci 18:555–586CrossRefPubMedGoogle Scholar
  63. Steriade M, Jones EG, Llinas RR (1990) Thalamic oscillations and signaling. Wiley, New YorkGoogle Scholar
  64. Tass PA (1999) Phase resetting in medicine and biology: stochastic modelling and data analysis. Springer, Berlin Heidelberg New YorkGoogle Scholar
  65. Tass PA (2001a) Desynchronizing double-pulse phase resetting and application to deep brain stimulation. Biol Cybern 85:343–354CrossRefGoogle Scholar
  66. Tass PA (2001b) Effective desynchronization by means of double-pulse phase resetting. Europhys Lett 53:15–21CrossRefGoogle Scholar
  67. Tass PA (2001c) Effective desynchronization with a resetting pulse train followed by a single pulse. Europhys Lett 55:171–177CrossRefGoogle Scholar
  68. Tass PA (2002a) Desynchronization of brain rhythms with soft phase-resetting techniques. Biol Cybern 87:102–115CrossRefGoogle Scholar
  69. Tass PA (2002b) Effective desynchronization with bipolar double-pulse stimulation. Phys Rev E 66:036226CrossRefGoogle Scholar
  70. Tass PA (2003a) A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations. Biol Cybern 89:81–88CrossRefGoogle Scholar
  71. Tass PA (2003b) Stochastic phase resetting of two coupled phase oscillators stimulated at different times. Phys Rev E 67:051902CrossRefGoogle Scholar
  72. Terman D, Rubin JE, Yew AC, Wilson CJ (2002) Activity patterns in a model for the subthalamopallidal network of the basal ganglia. J Neurosci 22:2963–2976PubMedGoogle Scholar
  73. Traub RD, Miles R (1991) Neural networks of the hippocampus. Cambridge University Press, CambridgeGoogle Scholar
  74. VanWiggeren GD, Roy R (1998) Communication with chaotic lasers. Science 279(5354):1198–1200CrossRefPubMedGoogle Scholar
  75. Volkmann J (2004) Deep brain stimulation for the treatment of Parkinson’s disease. J Clin Neurophysiol 21:6–17CrossRefPubMedGoogle Scholar
  76. Wichmann T, Bergman H, Starr PA, Subramanian T, Watts RL, (1999) Comparison of MPTP-induced changes in spontaneous neuronal discharge in the internal pallidal segment and in the substantia nigra pars reticulata in primates. Exp Brain Res 125:397–409CrossRefPubMedGoogle Scholar
  77. Yeung MKS, Strogatz SH (1999) Time delay in the Kuramoto model of coupled oscillators. Phys Rev Lett 82:648–651CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Oleksandr V. Popovych
    • 1
    • 2
  • Christian Hauptmann
    • 1
    • 2
  • Peter A. Tass
    • 1
    • 2
    • 3
    • 4
  1. 1.Institute of MedicineResearch Center JülichJülichGermany
  2. 2.Virtual Institute of NeuromodulationResearch Center JülichJülichGermany
  3. 3.Department of Stereotaxic and Functional NeurosurgeryUniversity HospitalCologneGermany
  4. 4.Brain Imaging Center WestJülichGermany

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