Journal of Computational Neuroscience

, Volume 21, Issue 3, pp 243–257 | Cite as

The K-complex and slow oscillation in terms of a mean-field cortical model

  • M. T. WilsonEmail author
  • D. A. Steyn-Ross
  • J. W. Sleigh
  • M. L. Steyn-Ross
  • L. C. Wilcocks
  • I. P. Gillies


We use a mean-field macrocolumn model of the cerebral cortex to offer an interpretation of the K-complex of the electroencephalogram to complement those of more detailed neuron-by-neuron models. We interpret the K-complex as a momentary excursion of the cortex from a stable low-firing state to an unstable high-firing state, and hypothesize that the related slow oscillation can be considered as the periodic oscillation between two meta-stable solutions of the mean-field model. By incorporating a Hebbian-style learning rule that links the growth in synapse strength to fluctuations in soma potential, we demonstrate a self-organization behaviour that draws the modelled cortex close to the edge of stability of the low-firing state. Furthermore, a very slow oscillation can occur in the excitability of the cortex that has similarities with the infra-slow oscillation of sleep.


K-complex Sleep Cortex Oscillation Electroencephalogram 


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  1. Amzica F, Steriade M (2002) The functional significance of Kcomplexes. Sleep Med. Rev. 6: 139–149.Google Scholar
  2. Bazhenov M, Timofeev I, Steriade M, Sejnowski TJ (2002) Model of thalamocortical slow-wave sleep oscillations and transitions to activated states. J. Neurosci. 22: 8691–8704.Google Scholar
  3. Bienenstock E, Lehmann D (1998) Regulated criticality in the brain? Adv. Comp. Syst. 1: 361–384.Google Scholar
  4. Bienenstock EL, Cooper LN, Munro PW (1982) Theory for the development of neuron selectivity: orientation specificity and binocular interation in visual cortex. J. Neurosci. 2: 32–48.Google Scholar
  5. Bojak I, Liley DTJ (2005) Modelling the effects of anaesthesia on the electroencephalogram. Phys. Rev. E 71: 41902.Google Scholar
  6. Colrain IM (2005) The K-complex: A 7-decade history. Sleep 28(2): 255–273.Google Scholar
  7. Compte A, Sanchez-Vives MV, McCormick DA, Wang XJ (2003) Cellular and network mechanisms of slow oscillatory activity (<1 Hz) and wave propagations in a cortical network model. J. Neurophysiol. 89: 2707–2725.Google Scholar
  8. Contreras D, Destexhe A, Sejnowski TJ, Steriade M (1997) Spatiotemporal patterns of spindle oscillations in cortex and thalamus. J. Neurosci. 17: 1179–1196.Google Scholar
  9. Destexhe A, Contreras D, Steriade M (1999) Spatiotemporal analysis of local field potentials and unit discharges in cat cerebral cortex during natural wake and sleep states. J. Neurosci. 19(11): 4595–4608Google Scholar
  10. Freeman WJ (1992) Predictions on neocortical dynamics derived from studies in paleocortex. In: E Basar, TH Bullock, eds., Induced Rhythms of the Brain, Birkhaeuser, Boston, pp. 183–199.Google Scholar
  11. Gardiner CW (2003) Handbook of Sochastic Methods for Physics, Chemistry and the Natural Sciences, Springer series in synergetics, Springer-Verlag, Berlin, Heidelberg, New York.Google Scholar
  12. Golomb D, Amitai Y (1997) Propagating neuronal discharges in neocortical slices: Computational and experimental study. J. Neurophysiol. 78(3): 1199–1211.Google Scholar
  13. Hebb DO (1949) The Organization of Behaviour, Wiley, New York.Google Scholar
  14. Hill S, Tononi G (2005) Modeling sleep and wakefulness in the thalamocortical system. J. Neurophysiol. 93: 1671–1698.Google Scholar
  15. Kramer MA, Kirsch HE, Szeri AJ (2005) Pathalogical pattern formation and epileptic seizures. J. Royal Soc. Interface 2: 113–127.Google Scholar
  16. Liley DTJ, Cadusch PJ, Wright JJ (1999) A continuum theory of electro-cortical activity. Neurocomputing 26–27: 795–800.Google Scholar
  17. Marshall L, Mölle M, Born J (2003) Spindle and slow wave rhythms at slow wave sleep transitions are linked to strong shifts in the cortical direct current potential. Neuroscience 121: 1047–1053.Google Scholar
  18. Massimini M, Huber R, Ferrarelli F, Hill S, Tononi G (2004) The sleep slow oscillation as a traveling wave. J. Neurosci. 24: 6862–6870.Google Scholar
  19. Numminen J, Makela JP, Hari R (1996) Distributions and sources of magnetoencephalographic K-complexes. Electroencephalograpr. Clin. Neurophysil. 99: 544–555.Google Scholar
  20. Nunez PL (1974) The brain wave function: A model for the EEG. Math. Biosci. 21: 279–297.Google Scholar
  21. Pace-Schott EF, Hobson JA (2002) The neurobiology of sleep: genetics, cellular physiology and subcortical networks. Nature Rev.: Neurosci. 3: 591–605.Google Scholar
  22. Patenaude C, Massicotte G, Lacaille J-C (2005) Cell-type specific GABA synaptic transmission and activity-dependent plasticity in rat hippocampal stratum radiatum interneurons. Eur. J. Neurosci. 22: 179–188.Google Scholar
  23. Rennie CJ, Wright JJ, Robinson PA (2000) Mechanisms for cortical electrical activity and emergence of gamma rhythm. J. Theor. Biol. 205: 17–35.Google Scholar
  24. Robinson PA, Rennie CJ, Wright JJ (1997) Propagation and stability of waves of electrical activity in the cerebral cortex. Phys. Rev. E 56: 826–840.Google Scholar
  25. Robinson PA, Wright JJ, Rennie CJ (1998) Synchronous oscillations in the cerebral cortex. Phys. Rev. E 57: 4578–4588.Google Scholar
  26. Sanchez-Vives MV, McCormick DA (2000) Cellular and network mechanisms of rhythmic recurrent activity in neocortex. Nat. Neurosci. 3: 1027–1034.Google Scholar
  27. Steriade M, Núnez A, Amzica F (1993) A novel slow (<1 Hz) oscillation of neocortical neurons in vivo: Depolarizing and hyperpolarizing components. J. Neurosci. 13: 3252–3265.Google Scholar
  28. Steriade M, Timofeev I, Grenier F (2001) Natural waking and sleep states: A view from inside neocortical neurons. J. Neurophysiol. 85: 1969–1985.Google Scholar
  29. Steyn-Ross DA, Steyn-Ross ML, Sleigh JW, Wilson MT, Gillies IP, Wright JJ (2005) The sleep cycle modelled as a cortical phase transition. J. Biophys. 31: 543–565.Google Scholar
  30. Steyn-Ross ML, Steyn-Ross DA, Sleigh JW (2004) Modelling general anaesthesia as a first-order phase transition in the cortex. Prog. Biophys. Mol. Biol. 85: 369–385.Google Scholar
  31. Steyn-Ross ML, Steyn-Ross DA, Sleigh JW, Wilson MT, Wilcocks LC (2005) A mechanism for learning and memory erasure in a white-noisedriven sleeping cortex. Phys. Rev. E 72: 061910Google Scholar
  32. Vanhatalo S, Palva JM, Holmes MD, Miller JW, Voipio J, Kaila K (2003) Infraslow oscillations modulate excitability and interictal epileptic activity in the human cortex during sleep. Proc. Natl. Acad. Sci. 101: 5053–5057.Google Scholar
  33. Wilson MT, Steyn-Ross ML, Steyn-Ross DA, Sleigh JW (2005) Predictions and simulations of cortical dynamics during natural sleep using a continuum approach. Phys. Rev. E 72: 051910Google Scholar
  34. Wright JJ, Liley DTJ (1996) Dynamics of the brain at global and microscopic scales: Neural networks and the EEG. Behav. Brain Sci. 19: 285–316.Google Scholar

Copyright information

© Springer Science Business Media, LLC 2006

Authors and Affiliations

  • M. T. Wilson
    • 1
    Email author
  • D. A. Steyn-Ross
    • 1
  • J. W. Sleigh
    • 2
  • M. L. Steyn-Ross
    • 1
  • L. C. Wilcocks
    • 1
  • I. P. Gillies
    • 1
  1. 1.Department of Physics and Electronic EngineeringUniversity of WaikatoHamiltonNew Zealand
  2. 2.Department of AnaestheticsWaikato HospitalHamiltonNew Zealand

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