Advertisement

Journal of Computational Neuroscience

, Volume 22, Issue 2, pp 173–189 | Cite as

Generation of theta oscillations by weakly coupled neural oscillators in the presence of noise

  • Michael H. K. Bendels
  • Christian Leibold
Article

Abstract

Neuronal oscillations are a robust phenomenon occurring in a variety of brain regions despite considerable amounts of noise. In this article classical phase-response theory is generalized to the case of noisy weak-coupling regimes by deriving an iterated map for the asynchrony of spikes in an oscillation cycle. Two criteria are introduced to check the validity of our approximations: One criterion tests the assumption that all neurons fire exactly once per cycle, the other criterion tests for linearity. The framework is applied to stellate cells of the medial entorhinal cortex layer II. We find that rhythmogenesis is more robust in the case of excitatory noise as compared to inhibitory noise. It is shown that a network of stellate cells can also act as a generator of theta if the neurons are connected via a fast-oscillating network of inhibitory interneurons.

Keywords

Phase-response theory Synaptic noise Theta rhythm Stellate cell 

Notes

Acknowledgments

The authors thank Jan Benda and Roland Schaette for insightful comments on the manuscript and are grateful to Andreas Herz, Richard Kempter for ongoing support and Dietmar Schmitz for support and discussions on stellate cell physiology. This work has been supported by the Berliner NaFöG grant (MB), and the Deutsche Forschungsgemeinschaft (DFG) via SFB 618 Theoretical Biology (MB & CL) as well as via the Emmy-Noether grant (Ke 788/1-3,4) to Richard Kempter (CL) and the Bundesministerium für Bildung und Forschung (Bernstein Center for Computational Neuroscience Berlin, 01GQ0410).

References

  1. Acebrón JA, Bonilla LL (1998) Asymptotic description of transients and synchronized states of globally coupled oscillators. Physica D 114: 296–314.Google Scholar
  2. Acebrón JA, Spigler R (1998) Adaptive frequency model for phase-frequency synchronization in large populations of globally coupled nonlinear oscillators. Phys. Rev. Lett. 81: 2229–2232.Google Scholar
  3. Acker CD, Kopell N, White JA (2003) Synchronization of strongly coupled excitatory neurons: Relating network behavior to biophysics. J. Comput. Neurosci. 15: 71–90.Google Scholar
  4. Alonso A, Klink R (1993) Differential electroresponsiveness of stellate and pyramidal-like cells of medial entorhinal cortex layer II. J. Neurophysiol. 70: 128–143.Google Scholar
  5. Alonso A, Llinas RR (1989) Subthreshold Na+-dependent theta-like rhythmicity in stellate cells of entorhinal cortex layer II. Nature 342: 175–177.Google Scholar
  6. Axmacher N, Mormann F, Fernandez G, Elger CE, Fell J. (2006) Memory formation by neuronal synchronization. Brain Res. Brain Res. Rev. 52: 170–182.Google Scholar
  7. Berretta N, Jones RS (1996) A comparison of spontaneous EPSCs in layer II and layer IV-V neurons of the rat entorhinal cortex in vitro. J. Neurophysiol. 76: 1089–1100.Google Scholar
  8. Bonilla LL, Neu JC, Spigler R (1992) Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators. J. Stat. Phys. 67: 313–330.Google Scholar
  9. Bonilla LL, Pérez-Vicente CJ, Ritort F, Soler J (2000) Exact solutions and dynamics of globally coupled phase oscillators. arXiv:nlin.AO/0004016Google Scholar
  10. Buzsáki G (2002) Theta oscillations in the hippocampus. Neuron 33: 325–340.Google Scholar
  11. Buzsáki G, Draguhn A (2004) Neuronal oscillations in cortical networks. Science 304: 1926–1929.Google Scholar
  12. Chrobak JJ, Buzsáki G (1998) Gamma oscillations in the entorhinal cortex of the freely behaving rat. J. Neurosci. 18: 388–398.Google Scholar
  13. Cunningham MO, Davies CH, Buhl EH, Kopell N, Whittington MA (2003) Gamma oscillations induced by kainate receptor activation in the entorhinal cortex in vitro. J. Neurosci. 23: 9761–9769.Google Scholar
  14. Dhillon A, Jones RS (2000) Laminar differences in recurrent excitatory transmission in the rat entorhinal cortex in vitro. Neuroscience 99: 413–422.Google Scholar
  15. Dickson CT, Biella G, de Curtis M (2000) Evidence for spatial modules mediated by temporal synchronization of carbachol-induced gamma rhythm in medial entorhinal cortex. J. Neurosci. 20: 7846–7854.Google Scholar
  16. Ermentrout B (1996) Type I membranes, phase resetting curves, and synchrony. Neural. Comput. 8: 979–1001.Google Scholar
  17. Fisahn A, Pike FG, Buhl EH, Paulsen O (1998) Cholinergic induction of network oscillations at 40 Hz in the hippocampus in vitro. Nature 394: 186–189.Google Scholar
  18. Gerstner W, van Hemmen JL, Cowan JD (1996) What matters in neuronal locking? Neural Comput. 8: 1653–1676.Google Scholar
  19. Goel P, Ermentrout B (2002) Synchrony, stability, and firing patterns in pulse-coupled oscillators. Physica D: Nonlinear Phenomena 163: 191–216.Google Scholar
  20. Gutkin B, Ermentrout GB, Rudolph M (2003) Spike generating dynamics and the conditions for spike-time precision in cortical neurons. J. Comput. Neurosci. 15: 91–103.Google Scholar
  21. Gutkin B, Ermentrout GB, Reyes AD (2005) Phase-response curves give the responses of neurons to transient inputs. J. Neurophysiol. 94: 1623–1635.Google Scholar
  22. Hajos N, Katona I, Naiem SS, MacKie K, Ledent C, Mody I, Freund TF (2000) Cannabinoids inhibit hippocampal GABAergic transmission and network oscillations. Eur. J. Neurosci. 12: 3239–3249.Google Scholar
  23. Hasselmo ME, Fransen E, Dickson C, Alonso AA (2000) Computational modeling of entorhinal cortex. Ann. N. Y. Acad. Sci. 911: 418–446.Google Scholar
  24. Hormuzdi SG, Pais I, LeBeau FE, Towers SK, Rozov A, Buhl EH, Whittington MA, Monyer H (2001) Impaired electrical signaling disrupts gamma frequency oscillations in connexin 36-deficient mice. Neuron 31: 487–495.Google Scholar
  25. Jensen O, Lisman JE (1996) Hippocampal CA3 region predicts memory sequences: Accounting for the phase precession of place cells. Learn Mem. 3: 279–287.Google Scholar
  26. Jones RS (1993) Entorhinal-hippocampal connections: A speculative view of their function. Trends Neurosci. 16: 58–64.Google Scholar
  27. Klink R, Alonso A (1997) Muscarinic modulation of the oscillatory and repetitive firing properties of entorhinal cortex layer II neurons. J. Neurophysiol. 77: 1813–1828.Google Scholar
  28. Kohler C (1986) Intrinsic connections of the retrohippocampal region in the rat brain. II. The medial entorhinal area. J. Comp. Neurol. 246: 149–169.Google Scholar
  29. Kopell N, Ermentrout GB (2002) Mechanisms of phase-locking in pairs of coupled neural oscillators. In: B Fiedler (ed), Handbook on Dynamical Systems: Toward applications. Elsevier, Amsterdam, pp. 3–54.Google Scholar
  30. Kramis R, Vanderwolf CH, Bland BH (1975) Two types of hippocampal rhythmical slow activity in both the rabbit and the rat: Relations to behavior and effects of atropine, diethyl ether, urethane, and pentobarbital. Exp. Neurol. 49: 58–85.Google Scholar
  31. Kuramoto Y (1984) Chemical Oscillations, Waves and Turbulence. Springer, Berlin.Google Scholar
  32. Melamed O, Gerstner W, Maass W, Tsodyks M, Markram H (2004) Coding and learning of behavioral sequences. Trends Neurosci. 27: 11–14.Google Scholar
  33. Netoff TI, Banks MI, Dorval AD, Acker CD, Haas JS, Kopell N, White JA (2005) Synchronization in hybrid neuronal networks of the hippocampal formation. J. Neurophysiol. 93: 1197–1208.Google Scholar
  34. Neuenschwander S, Varela FJ (1993) Visually triggered neuronal oscillations in the pigeon: An autocorrelation study of tectal activity. Eur. J. Neurosci. 5: 870–881.Google Scholar
  35. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge, MA.Google Scholar
  36. Prechtl JC (1994) Visual motion induces synchronous oscillations in turtle visual cortex. Proc. Natl. Acad. Sci. USA 91: 12467–12471.Google Scholar
  37. Rinzel J, Ermentrout B (1998) Analysis of neural excitability and oscillations. In: C Koch, I Segev (eds). Methods in Neuronal Modeling: From Ions to Networks. MIT Press, Cambridge, MA, pp. 251–291.Google Scholar
  38. Ritz R, Sejnowski TJ (1997) Synchronous oscillatory activity in sensory systems: New vistas on mechanisms. Curr. Opin. Neurobiol. 7: 536–546.Google Scholar
  39. Sakaguchi H (1988) Cooperative phenomena in coupled oscillator systems. Prog. Theor. Phys. 79: 39–46.Google Scholar
  40. Singer W (1999) Neuronal synchrony: A versatile code for the definition of relations? Neuron 24: 49–65.Google Scholar
  41. Stewart M, Quirk GJ, Barry M, Fox SE (1992) Firing relations of medial entorhinal neurons to the hippocampal theta rhythm in urethane anesthetized and walking rats. Exp. Brain Res. 90: 21–28.Google Scholar
  42. Strogatz SH, Mirollo RF (1991) Stability of incoherence in a population of coupled oscillators. J. Stat. Phys. 63: 613–635.Google Scholar
  43. Traub RD, Whittington MA, Colling SB, Buzsáki G, Jefferys JG (1996) Analysis of gamma rhythms in the rat hippocampus in vitro and in vivo. J. Physiol. 493: 471–484.Google Scholar
  44. Uchida S, Maehara T, Hirai N, Okubo Y, Shimizu H (2001) Cortical oscillations in human medial temporal lobe during wakefulness and all-night sleep. Brain Res. 891: 7– 19.Google Scholar
  45. Wehr M, Laurent G (1996) Odour encoding by temporal sequences of firing in oscillating neural assemblies. Nature 384: 162–166.Google Scholar
  46. White JA, Klink R, Alonso A, Kay AR (1998) Noise from voltage-gated ion channels may influence neuronal dynamics in the entorhinal cortex. J. Neurophysiol. 80: 262–269.Google Scholar
  47. White JA, Rubinstein JT, Kay AR (2000) Channel noise in neurons. Trends Neurosci. 23: 131–137.Google Scholar
  48. Whittington MA, Traub RD, Jefferys JG (1995) Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation. Nature 373: 612–615.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Institute for Theoretical BiologyHumboldt-Universtität zu BerlinBerlinGermany
  2. 2.NeuroScience Research Center, CharitéUniversity Medicine BerlinBerlinGermany
  3. 3.Bernstein Center for Computational Neuroscience BerlinBerlinGermany

Personalised recommendations