Cognitive Neurodynamics

, Volume 6, Issue 4, pp 367–375 | Cite as

The phase response of the cortical slow oscillation

  • Arne Weigenand
  • Thomas Martinetz
  • Jens Christian Claussen
Research Article

Abstract

Cortical slow oscillations occur in the mammalian brain during deep sleep and have been shown to contribute to memory consolidation, an effect that can be enhanced by electrical stimulation. As the precise underlying working mechanisms are not known it is desired to develop and analyze computational models of slow oscillations and to study the response to electrical stimuli. In this paper we employ the conductance based model of Compte et al. (J Neurophysiol 89:2707–2725, 2003) to study the effect of electrical stimulation. The population response to electrical stimulation depends on the timing of the stimulus with respect to the state of the slow oscillation. First, we reproduce the experimental results of electrical stimulation in ferret brain slices by Shu et al. (Nature 423:288–293, 2003) from the conductance based model. We then numerically obtain the phase response curve for the conductance based network model to quantify the network’s response to weak stimuli. Our results agree with experiments in vivo and in vitro that show that sensitivity to stimulation is weaker in the up than in the down state. However, we also find that within the up state stimulation leads to a shortening of the up state, or phase advance, whereas during the up–down transition a prolongation of up states is possible, resulting in a phase delay. Finally, we compute the phase response curve for the simple mean-field model by Ngo et al. (EPL Europhys Lett 89:68002, 2010) and find that the qualitative shape of the PRC is preserved, despite its different mechanism for the generation of slow oscillations.

Keywords

Sleep Cortex Phase response Slow oscillation Synchronization 

References

  1. Achuthan S, Butera RJ, Canavier CC (2010) Synaptic and intrinsic determinants of the phase resetting curve for weak coupling. J Comput Neurosci 30(2):373–390PubMedCrossRefGoogle Scholar
  2. Akam T, Oren I, Mantoan L, Ferenczi E, Kullmann DM (2012) Oscillatory dynamics in the hippocampus support dentate gyrus-CA3 coupling. Nat Neurosci 15(5):763–768PubMedCrossRefGoogle Scholar
  3. Amzica F, Steriade M (1998) Electrophysiological correlates of sleep delta waves. Electroencephalogr Clin Neurophysiol 107(2):69–83PubMedCrossRefGoogle Scholar
  4. Biktasheva IV, Barkley D, Biktashev VN, Foulkes AJ (2010) Computation of the drift velocity of spiral waves using response functions. Phys Rev E 81(6):066202CrossRefGoogle Scholar
  5. Brunel N (2000) Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J Comput Neurosci 8(3):183–208PubMedCrossRefGoogle Scholar
  6. Clemens Z, Mölle M, Erőss L, Barsi P, Halász P, Born J (2007) Temporal coupling of parahippocampal ripples, sleep spindles and slow oscillations in humans. Brain 130:2868–2878PubMedCrossRefGoogle Scholar
  7. Compte A, Sanchez-Vives MV, McCormick DA, Wang X (2003) Cellular and network mechanisms of slow oscillatory activity (<1 hz) and wave propagations in a cortical network model. J Neurophysiol 89(5):2707–2725PubMedCrossRefGoogle Scholar
  8. Contreras D, Steriade M (1995) Cellular basis of EEG slow rhythms: a study of dynamic corticothalamic relationships. J Neurosci 15(1):604PubMedGoogle Scholar
  9. Deco G, Mart D, Ledberg A, Reig R, Sanchez Vives MV (2009) Effective reduced Diffusion-Models: a data driven approach to the analysis of neuronal dynamics. PLoS Comput Biol 5(12):e1000587PubMedCrossRefGoogle Scholar
  10. Diekelmann S, Born J (2010) The memory function of sleep. Nat Rev Neurosci 11(2):114–126PubMedGoogle Scholar
  11. Ermentrout B (2002) Simulating, analyzing, and animating dynamical systems. SIAM, PhiladelphiaCrossRefGoogle Scholar
  12. Ermentrout B, Terman D (2010) Mathematical foundations of neuroscience. Springer, BerlinCrossRefGoogle Scholar
  13. Fröhlich F, McCormick DA (2010) Endogenous electric fields may guide neocortical network activity. Neuron 67(1):129–143PubMedCrossRefGoogle Scholar
  14. Granada A, Hennig R, Ronacher B, Kramer A, Herzel H (2009) Phase response curves: elucidating the dynamics of coupled oscillators. In: Michael L. Johnson and Ludwig Brand (eds) Methods in enzymology, vol 454. Elsevier, pp 1–27Google Scholar
  15. Grannan ER, Kleinfeld D, Sompolinsky H (1993) Stimulus-dependent synchronization of neuronal assemblies. Neural Comput 5(4):550–569CrossRefGoogle Scholar
  16. Gray CM, Konig P, Engel AK, Singer W (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338(6213):334–337PubMedCrossRefGoogle Scholar
  17. Hájos N, Palhalmi J, Mann EO, Németh B, Paulsen O, Freund TF (2004) Spike timing of distinct types of GABAergic interneuron during hippocampal gamma oscillations in vitro. J Neurosci 24(41):9127–9137PubMedCrossRefGoogle Scholar
  18. Izhikevich EM (2000) Phase equations for relaxation oscillators. SIAM J Appl Math 60(5):1789–1804CrossRefGoogle Scholar
  19. Jirsa V (2008) Dispersion and time delay effects in synchronized spike-burst networks. Cogn Neurodyn 2(1):29–38PubMedCrossRefGoogle Scholar
  20. Ko TW, Ermentrout GB (2009) Phase-response curves of coupled oscillators. Phys Rev E 79:016–211CrossRefGoogle Scholar
  21. Kori H, Kawamura Y, Nakao H, Arai K, Kuramoto Y (2009) Collective-phase description of coupled oscillators with general network structure. Phys Rev E 80:036207CrossRefGoogle Scholar
  22. Kuramoto Y (2003) Chemical oscillations, waves, and turbulence. Chemistry series, Dover Publications (originally published: Springer Berlin, 1984)Google Scholar
  23. Levnajić Z, Pikovsky A (2010) Phase resetting of collective rhythm in ensembles of oscillators. Phys Rev E 82:056202CrossRefGoogle Scholar
  24. MacLean JN, Watson BO, Aaron GB, Yuste R (2005) Internal dynamics determine the cortical response to thalamic stimulation. Neuron 48(5):811–823PubMedCrossRefGoogle Scholar
  25. Marshall L, Mölle M, Hallschmid M, Born J (2004) Transcranial direct current stimulation during sleep improves declarative memory. J Neurosci 24(44):9985–9992PubMedCrossRefGoogle Scholar
  26. Marshall L, Helgadottir H, Mölle M, Born J (2006) Boosting slow oscillations during sleep potentiates memory. Nature 444(7119):610–613PubMedCrossRefGoogle Scholar
  27. Massimini M (2002) EEG slow (1 hz) waves are associated with nonstationarity of Thalamo–Cortical sensory processing in the sleeping human. J Neurophysiol 89:1205–1213CrossRefGoogle Scholar
  28. Massimini M, Huber R, Ferrarelli F, Hill S, Tononi G (2004) The sleep slow oscillation as a traveling wave. J Neurosci 24(31):6862PubMedCrossRefGoogle Scholar
  29. Massimini M, Ferrarelli F, Esser SK, Riedner BA, Huber R, Murphy M, Peterson MJ, Tononi G (2007) Triggering sleep slow waves by transcranial magnetic stimulation. Proc Natl Acad Sci USA 104(20):8496–8501PubMedCrossRefGoogle Scholar
  30. Mattia M, Sanchez-Vives M (2012) Exploring the spectrum of dynamical regimes and timescales in spontaneous cortical activity. Cogn Neurodyn 6(3):239–250Google Scholar
  31. Mayer J, Schuster HG, Claussen JC, Mölle M (2007) Corticothalamic projections control synchronization in locally coupled bistable thalamic oscillators. Phys Rev Lett 99(6):068102PubMedCrossRefGoogle Scholar
  32. Mejias JF, Kappen HJ, Torres JJ (2010) Irregular dynamics in up and down cortical states. PLoS One 5(11):e13651PubMedCrossRefGoogle Scholar
  33. Mölle M, Marshall L, Gais S, Born J (2002) Grouping of spindle activity during slow oscillations in human non-rapid eye movement sleep. J Neurosci 22(24):10941–10947PubMedGoogle Scholar
  34. Ngo HV, Köhler J, Mayer J, Claussen JC, Schuster HG (2010) Triggering up states in all-to-all coupled neurons. EPL Europhys Lett 89(6):68002CrossRefGoogle Scholar
  35. Perez Velazquez JL, Galán RF, Dominguez LG, Leshchenko Y, Lo S, Belkas J, Erra RG (2007) Phase response curves in the characterization of epileptiform activity. Phys Rev E 76:061912CrossRefGoogle Scholar
  36. Petersen CC, Hahn TT, Mehta M, Grinvald A, Sakmann B (2003) Interaction of sensory responses with spontaneous depolarization in layer ii/iii barrel cortex. Proc Natl Acad Sci USA 100(23):13638Google Scholar
  37. Peyrache A, Khamassi M, Benchenane K, Wiener SI, Battaglia FP (2009) Replay of rule-learning related neural patterns in the prefrontal cortex during sleep. Nat Neurosci 12(7):919–926PubMedCrossRefGoogle Scholar
  38. Sanchez-Vives MV, McCormick DA (2000) Cellular and network mechanisms of rhythmic recurrent activity in neocortex. Nat Neurosci 3(10):1027–1034PubMedCrossRefGoogle Scholar
  39. Sanchez-Vives MV, Descalzo VF, Reig R, Figueroa NA, Compte A, Gallego R (2008) Rhythmic spontaneous activity in the piriform cortex. Cereb Cortex 18(5):1179PubMedCrossRefGoogle Scholar
  40. Sanchez-Vives MV, Mattia M, Compte A, Perez-Zabalza M, Winograd M, Descalzo VF, Reig R (2010) Inhibitory modulation of cortical up states. J Neurophysiol 104(3):1314–1324PubMedCrossRefGoogle Scholar
  41. Seamari Y, Narvez JA, Vico FJ, Lobo D, Sanchez-Vives MV (2007) Robust off- and online separation of intracellularly recorded up and down cortical states. PLoS One 2(9):e888PubMedCrossRefGoogle Scholar
  42. Shu Y, Hasenstaub A, McCormick DA (2003) Turning on and off recurrent balanced cortical activity. Nature 423(6937):288–293PubMedCrossRefGoogle Scholar
  43. Somers D, Kopell N (1995) Waves and synchrony in networks of oscillators of relaxation and non-relaxation type. Physica D 89(1–2):169–183CrossRefGoogle Scholar
  44. Stickgold R (2005) Sleep-dependent memory consolidation. Nature 437(7063):1272–1278PubMedCrossRefGoogle Scholar
  45. Tass PA (1999) Phase resetting in medicine and biology: stochastic modelling and data analysis. Springer, BerlinGoogle Scholar
  46. Tsubo Y, Takada M, Reyes AD, Fukai T (2007) Layer and frequency dependencies of phase response properties of pyramidal neurons in rat motor cortex. Eur J Neurosci 25(11):3429–3441PubMedCrossRefGoogle Scholar
  47. Várkonyi PL, Holmes P (2008) On synchronization and traveling waves in chains of relaxation oscillators with an application to lamprey cpg. SIAM J Appl Dyn Syst 7:766–794CrossRefGoogle Scholar
  48. Winfree AT (2001) The geometry of biological time, 2nd edn. Springer, BerlinGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Arne Weigenand
    • 1
    • 2
  • Thomas Martinetz
    • 1
  • Jens Christian Claussen
    • 1
  1. 1.Institute for Neuro- and BioinformaticsUniversity of LuebeckLübeckGermany
  2. 2.Graduate School for Computing in Medicine and Life SciencesUniversity of LuebeckLübeckGermany

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