Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Additive Noise Tunes the Self-Organization in Complex Systems

  • Axel HuttEmail author
  • Jérémie Lefebvre
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_696-1

Glossary

Additive noise

Random fluctuations that add to the phase space flow of model systems.

Center manifold theorem

Mathematical theorem describing the slaving principle in complex systems.

Slaving principle

Units in a complex system that interact nonlinearly with other units evolve on different time scales. Close to instability points, fast units obey the dynamics of slow units and are enslaved by them. Such units may be spatial modes in spatially extended systems or neural ensembles in neural populations.

Introduction

The dynamics of natural systems is complex, e.g., due to various processes and their interactions on different temporal and spatial scales. Several of such processes appear to be of random nature, i.e., they cannot be predicted by known laws. In this context, it is not necessary to know whether these processes are random in reality or whether we just do not know their deterministic law and they appear to be random. The insight that unknown laws of processes may be...

This is a preview of subscription content, log in to check access.

Bibliography

  1. Arnold L (1998) Random dynamical systems. Springer, BerlinCrossRefzbMATHGoogle Scholar
  2. Bloemker D (2003) Amplitude equations for locally cubic non-autonomous nonlinearities. SIAM J Appl Dyn Syst 2(2):464–486MathSciNetCrossRefGoogle Scholar
  3. Bloemker D, Hairer M, Pavliotis GA (2005) Modulation equations: stochastic bifurcation in large domains. Commun Math Phys 258:479–512ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. Boxler P (1989) A stochastic version of the center manifold theorem. Probab Theory Relat Fields 83:509–545MathSciNetCrossRefzbMATHGoogle Scholar
  5. Carr J (1981) Applications of center manifold theory. Springer, BerlinCrossRefzbMATHGoogle Scholar
  6. Doiron B, Lindner B, Longtin A, Maler L, Bastian J (2004) Oscillatory activity in electrosensory neurons increases with the spatial correlation of the stochastic input stimulus. Phys Rev Lett 93:048101ADSCrossRefGoogle Scholar
  7. Drolet F, Vinals J (1998) Adiabatic reduction near a bifurcation in stochastically modulated systems. Phys Rev E 57(5):5036–5043ADSCrossRefGoogle Scholar
  8. Drolet F, Vinals J (2001) Adiabatic elimination and reduced probability distribution functions in spatially extended systems with a fluctuating control parameter. Phys Rev E 64:026120ADSCrossRefGoogle Scholar
  9. Fertonani A, Pirulli C, Miniussi C (2011) Random noise stimulation improves neuroplasticity in perceptual learning. J Neurosci 31(43):15416–15423CrossRefGoogle Scholar
  10. Fuchs A, Kelso J, Haken H (1992) Phase transitions in the human brain: spatial mode dynamics. Int J Bifurcation Chaos 2(4):917CrossRefzbMATHGoogle Scholar
  11. Garcia-Ojalvo J, Sancho J (1999) Noise in spatially extended systems. Springer, New YorkCrossRefzbMATHGoogle Scholar
  12. Haken H (1983) Advanced synergetics. Springer, BerlinzbMATHGoogle Scholar
  13. Haken H (1985) Light II – laser light dynamics. North Holland, AmsterdamGoogle Scholar
  14. Haken H (1996) Slaving principle revisited. Phys D 97:95–103MathSciNetCrossRefzbMATHGoogle Scholar
  15. Haken H (2004) Synergetics. Springer, BerlinCrossRefzbMATHGoogle Scholar
  16. Herrmann C, Rach S, neuling T, Strüber D (2013) Transcranial alternating current stimulation: a review of the underlying mechanisms and modulation of cognitive processes. Front Hum Neurosci 7:279CrossRefGoogle Scholar
  17. Herrmann CS, Murray MM, Ionta S, Hutt A, Lefebvre J (2016) Shaping intrinsic neural oscillations with periodic stimulation. J Neurosci 36(19):5328–5339CrossRefGoogle Scholar
  18. Horsthemke W, Lefever R (1984) Noise-induced transitions. Springer, BerlinzbMATHGoogle Scholar
  19. Hutt A (2008) Additive noise may change the stability of nonlinear systems. Europhys Lett 84(34):003Google Scholar
  20. Hutt A, Buhry L (2014) Study of GABAergic extra-synaptic tonic inhibition in single neurons and neural populations by traversing neural scales: application to propofol-induced anaesthesia. J Comput Neurosci 37(3):417–437MathSciNetCrossRefGoogle Scholar
  21. Hutt A, Lefebvre J (2016) Stochastic center manifold analysis in scalar nonlinear systems involving distributed delays and additive noise. Markov Process Relat Fields 22:555–572MathSciNetzbMATHGoogle Scholar
  22. Hutt A, Longtin A, Schimansky-Geier L (2007) Additive global noise delays Turing bifurcations. Phys Rev Lett 98:230601ADSCrossRefGoogle Scholar
  23. Hutt A, Longtin A, Schimansky-Geier L (2008) Additive noise-induced Turing transitions in spatial systems with application to neural fields and the Swift–Hohenberg equation. Phys D 237:755–773MathSciNetCrossRefzbMATHGoogle Scholar
  24. Hutt A, Lefebvre J, Longtin A (2012) Delay stabilizes stochastic systems near an non-oscillatory instability. Europhys Lett 98:20004ADSCrossRefGoogle Scholar
  25. Hutt A, Mierau A, Lefebvre J (2016) Dynamic control of synchronous activity in networks of spiking neurons. PLoS One 11(9):e0161488.  https://doi.org/10.1371/journal.pone.0161488 CrossRefGoogle Scholar
  26. Jirsa V, Friedrich R, Haken H (1995) Reconstruction of the spatio-temporal dynamics of a human magnetoencephalogram. Phys D 89:100–122CrossRefzbMATHGoogle Scholar
  27. Lachaux JP, Rodriguez E, Martinerie J, Varela F (1999) Measuring phase synchrony in brain signals. Hum Brain Mapp 8:194–208CrossRefGoogle Scholar
  28. Lefebvre J, Hutt A (2013) Additive noise quenches delay-induced oscillations. Europhys Lett 102:60003ADSCrossRefGoogle Scholar
  29. Lefebvre J, Hutt A, LeBlanc V, Longtin A (2012) Reduced dynamics for delayed systems with harmonic or stochastic forcing. Chaos 22:043121ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. Lefebvre J, Hutt A, Knebel J, Whittingstall K, Murray M (2015) Stimulus statistics shape oscillations in nonlinear recurrent neural networks. J Neurosci 35(7):2895–2903CrossRefGoogle Scholar
  31. Lefebvre J, Hutt A, Frohlich F (2017) Stochastic resonance mediates the statedependent effect of periodic stimulation on cortical alpha oscillations. eLife 6:e32054Google Scholar
  32. Noilhan J, Planton S (1989) A simple parameterization of land surface processes for meteorological models. Mon Weather Rev 117:536–549ADSCrossRefGoogle Scholar
  33. Schanz M, Pelster A (2003) Synergetic system analysis for the delay-induced Hopf bifurcation in the Wright equation. SIAM J Appl Dyn Syst 2:277–296MathSciNetCrossRefzbMATHGoogle Scholar
  34. Schoener G, Haken H (1986) The slaving principle for Stratonovich stochastic differential equations. Z Phys B 63:493–504ADSCrossRefGoogle Scholar
  35. Terney D, Chaieb L, Moliadze V, Antal A, Paulus W (2008) Increasing human brain excitability by transcranial high-frequency random noise stimulation. J Neurosci 28(52):14147–14155CrossRefGoogle Scholar
  36. Xu C, Roberts A (1996) On the low-dimensional modelling of Stratonovich stochastic differential equations. Phys A 225:62–80MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media LLC 2018

Authors and Affiliations

  1. 1.Deutscher WetterdienstOffenbach am MainGermany
  2. 2.Krembil Research InstituteTorontoCanada