Advertisement

Bulletin of Mathematical Biology

, Volume 59, Issue 3, pp 517–532 | Cite as

Hierarchically coupled ultradian oscillators generating robust circadian rhythms

  • Rafael A. Barrio
  • Limei Zhang
  • Philip K. Maini
Article

Abstract

Ensembles of mutually coupled ultradian cellular oscillators have been proposed by a number of authors to explain the generation of circadian rhythms in mammals. Most mathematical models using many coupled oscillators predict that the output period should vary as the square root of the number of participating units, thus being inconsistent with the well-established experimental result that ablation of substantial parts of the suprachiasmatic nuclei (SCN), the main circadian pacemaker in mammals, does not eliminate the overt circadian functions, which show no changes in the phases or periods of the rhythms. From these observations, we have developed a theoretical model that exhibits the robustness of the circadian clock to changes in the number of cells in the SCN, and that is readily adaptable to include the successful features of other known models of circadian regulation, such as the phase response curves and light resetting of the phase.

Keywords

Circadian Rhythm Circadian Clock Couple Oscillator Suprachiasmatic Nucleus Circadian Oscillation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aguilar-Roblero, R., J. L. Chávez and M. Díaz-Muñoz. 1996. Circadian modulation of intracellular Ca++ release channel (Ryanodine receptor) in the suprachiasmatic nuclei.Fifth Meeting Soc. for Research on Biological Rhythms, Jacksonville, Florida, p. 63 (abstract).Google Scholar
  2. Bos, N. P. A. and M. Mirmiran. 1990. Circadian rhythms in spontaneous neuronal discharges of the cultured suprachiasmatic nucleus.Brain Res. 511, 158–162.CrossRefGoogle Scholar
  3. Carpenter, G. A. and S. M. Grossberg. 1983. Mammalian circadian rhythms. InOscillations in Mathematical Biology, J. P. E. Hodgson (Ed), pp. 102–196. Berlin: Springer-Verlag.Google Scholar
  4. Davis, F. C. and R. A. Gorski. 1984. Unilateral lesions of the hamster suprachiasmatic nuclei: evidence for redundant control of circadian rhythms.J. Comp. Physiol. A 154, 221–232.CrossRefGoogle Scholar
  5. Drucker-Colín, R., R. Aguilar-Roblero, F. Garcia-Hernández, F. Fernández-Cancino and F. Bermúdez-Rattoni. 1984. Fetal suprachiasmatic nucleus transplants: diurnal rhythm recovery of lesioned rats.Brain Res. 311, 353–357.CrossRefGoogle Scholar
  6. Earnest, D. J., S. M. Digiorgio and C. D. Sladek. 1991. Effects of tetrodotoxin on the circadian pacemaker mechanism in suprachiasmatic explants in vitro.Brain Res. Bull. 26, 677–682.CrossRefGoogle Scholar
  7. Edery, I., J. E. Rutila and M. Roshbach. 1994. Phase shifting of the circadian clock by induction of the Drosophial period protein.Science 237, 237–240.Google Scholar
  8. Enright, J. T. 1980. Temporary precision in circadian systems: a reliable neuronal clock from unreliable components?Science 209, 1542–1545.Google Scholar
  9. Gekakis, N., L. Saez, A.-M. Delahaye-Brown, M. P. Myers, A. Sehgal, M. W. Young, C. J. Weitz. 1995. Isolation oftimeless by PER protein interaction: defective interaction betweentimeless protein and long-period mutant PER.Science 270, 811–815.Google Scholar
  10. Giaume, C. and K. D. McCarthy. 1996. Control of gap-junctional communication in astrocytic networks.Trends Neurosci. 19, 319–325.CrossRefGoogle Scholar
  11. Gillette, M. U. 1991. SCN electrophysiology in vitro: rhythmic activity and endogenous clock properties. InSuprachiasmatic Nucleus the Mind's Clock, D. C. Klein, R. Y. Moore and S. M. Reppert (Eds), ch. 6, pp. 125–143. New York: Oxford University Press.Google Scholar
  12. Gradshteyn, I. S. and I. M. Ryzhik. 1980.Table of Integrals, Series, and Products, ch. 8.17, pp. 921–925. New York: Academic Press.zbMATHGoogle Scholar
  13. Hardin, P. E., J. C. Hall and M. Rosbach. 1990. Feedback of the Drosophila period gene product on circadian cycling of its messenger RNA levels.Nature 343, 536–540.CrossRefGoogle Scholar
  14. Inouye, S. I. T. and H. Kawamura. 1979. Persistence of circadian rhythmicity in the mammalian hypothalamic “island” containing the suprachiasmatic nucleus.Proc. Natl. Acad. Sci. USA 76, 5962–5966.CrossRefGoogle Scholar
  15. Konopka, R. J. and S. Benzer. 1971. Clock mutants ofDrosophila melanogaster.Proc. Natl. Acad. Sci. 68, 2112.CrossRefGoogle Scholar
  16. Lee, C., V. Vaishali, T. Itsukaichi, K. Bae and I. Edery. 1996. Resetting theDrosophila clock by photic regulation of PER and a PER-TIM complex.Science 271, 1740–1741.Google Scholar
  17. Lehman, M. N., R. Silver, W. R. Gladstone, R. M. Kahn, M. Gibson and E. L. Bittman. 1987. Circadian rhythmicity restored by neural trasplant. Immunocytochemical characterization of the graft and its interaction with host brain.J. Neurosci. 7, 1626–1638.Google Scholar
  18. Matthews, P. C., R. E. Mirollo and S. H. Strogatz. 1991. Dynamics of a large system of coupled nonlinear oscillators.Physica D 52, 293–331.zbMATHMathSciNetCrossRefGoogle Scholar
  19. Meijer, J. H. and W. J. Rietveld. 1989. Neurophysiology of the suprachiasmatic circadian pacemaker in rodents.Physiol. Rev. 69, 671–707.Google Scholar
  20. Moore, R. Y. 1992. The suprachiasmatic nucleus and the circadian timing system. InCircadian Rhythms, Discussions in Neuroscience, P. J. Magistratti (Ed), Vol. 8, Nos. 2–3, ch. 5, pp. 26–33 Amsterdam: Elsevier Science Publishers B.V.Google Scholar
  21. Mosko, S. S. and R. Y. Moore. 1979. Neonatal suprachiasmatic nucleus lesions.Brain Res. 164, 17–38.CrossRefGoogle Scholar
  22. Myers, M. P., K. Wagner-Smith, A. Rothenfluh-Hilfiker and M. W. Young. 1996. Lightinduced degradation of theDrosophila circadian clock.Science 271, 1736–1740.Google Scholar
  23. Newman, G. C., F. E. Hospod, C. S. Patlak and R. Y. Moore. 1992. Analysis of in vitro glucose utilization in a circadian pacemaker model.J. Neurosci. 12, 2015–2021.Google Scholar
  24. Othmer, H. G. and M. Watanabe. 1992. Novel mechanism for generating circadian periods from fast oscillators. InAbst. of the 2nd Annual Conf. of the Japan Soc. for Industrial and Applied Math., pp. 1–2.Google Scholar
  25. Pavlidis, T. 1969. Populations of interacting oscillators and circadian rhythms.J. Theor. Biol. 22, 418–436.CrossRefGoogle Scholar
  26. Pavlidis, T. 1975. Spatial organization of chemical oscillators via an averaging operator.J. Chem. Phys. 63, 5269–5273.CrossRefGoogle Scholar
  27. Pavlidis, T. 1992. Mathematical models. InHandbook of Behavioral Neurobiology, J. F. Aschoff (Ed), Vol. 4, ch. 4, pp. 41–54. New York: Plenum Press.Google Scholar
  28. Pittendrigh, C. S. 1974. Circadian organization in cells and the circadian organization of the multicellualr system. InNeurosciences Third Study Program, S. O. Schmitt and F. G. Worden (Eds). Cambridge, MA: MIT Press.Google Scholar
  29. Prosser, R. E., J. D. Miller and H. C. Heller. 1990. Sertonin agonist phase-shifts in circadian clock in the suprachiasmatic nucleiin vitro.Brain Res. 534, 336–339.CrossRefGoogle Scholar
  30. Ralph, M. R., R. G. Foster, F. D. Davis and M. Menaker. 1990. Transplanted suprachiasmatic nucleus determines circadian period.Science Wash. DC 247, 975–978.Google Scholar
  31. Schwartz, W. J., R. A. Gross and M. T. Morton. 1987. The suprachiasmatic nuclei contain a tetrodotoxin-resistant circadian pacemaker.Proc. Natl. Acad. Sci. USA 84, 1694–1698.CrossRefGoogle Scholar
  32. Sehgal, A., J. L. Price, B. Mam and M. W. Young. 1994. Loss of circadian behavioral rhythms andper RNA oscillation in theDrosophila mutanttimeless.Science 263, 1603.Google Scholar
  33. Sehgal, A., A. Rothenfluh-Hilfiker, M. Hunter-Ensor, Y. Chen, M. P. Myers and M. W. Young. 1995. Rhythmic expression oftimeless: a basis for promoting circadian cycles inperiod gene autoregulation.Science 270, 808–810.Google Scholar
  34. Shibata, S., Y. Oomura, S. Y. Liou and S. Ueki. 1984. Electrophysiological studies of the development of suprachiasmatic neuronal activities in hypothalamic slice preparation.Brain Res. 13, 29–35.CrossRefGoogle Scholar
  35. Thomson, A. M., D. C. West and I. G. Vlachonikolis. 1984. Regular firing of suprachiasmatic neurons maintainedin vitro.Neurosci Lett. 52, 329–334.CrossRefGoogle Scholar
  36. Van den Pol, A. N. and T. P. Powley. 1979. A fine grained anatomical analysis of the role of the rat suprachiasmatic nucleus in circadian rhythms of feeding and drinking.Brain Res. 160, 307–326.CrossRefGoogle Scholar
  37. Van den Pol, A. N. 1980. The hypothalamic suprachiasmatic nucleus of rat: intrinsic anatomy.J. Comp. Neurol. 191, 661–702.CrossRefGoogle Scholar
  38. Van den Pol, A. N., S. M. Pinkbeiner and A. H. Cornell-Bell. 1992. Calcium excitability and oscillations in suprachiasmatic nucleus neurons and glia in vitro.J. Neurosci. 12, 2649–2664.Google Scholar
  39. Verkhratsky, A. and H. Kettenmann. 1966. Calcium signalling in glial cells.Trends Neurosci. 19, 346–352.CrossRefGoogle Scholar
  40. Vosshall, L. B., J. L. Price, A. Sehgal, L. Saez and M. W. Young. 1994. Block in nuclear localization ofperiod protein by a second clock mutation,timeless.Science 263, 1606.Google Scholar
  41. Watanabe, K., N. Koibuchi, H. Ohtake and S. Yamaoka. 1993. Circadian rhythms of vasopressin release in primary cultures of rat suprachiasmatic nucleus.Brain Res. 624, 115–120.CrossRefGoogle Scholar
  42. Welsh, D. K., D. E. Logothetis, M. Meister and S. Reppert. 1995. Individual neurons dissociated from rat suprachiasmatic nucleus express independently phased circadian firing rhythms.Neuron 14, 697–706.CrossRefGoogle Scholar
  43. Winfree, A. T. 1967. Biological rhythms and the behavior of population of coupled oscillators.J. Theor. Biol. 16 15–42.CrossRefGoogle Scholar
  44. Winfree, A. T. 1975. Unclocklike behavior of biological clocks.Nature 253, 315–319.CrossRefGoogle Scholar
  45. Zhang, L. and R. Aguilar-Roblero. 1995. Asymmetrical electrical activity between the suprachiasmatic nuclei in vitro.NeuroReport 6, 537–540.CrossRefGoogle Scholar
  46. Zhang, L., R. Aguilar-Roblero, R. A. Barrio and P. K. Maini. 1995. Rhythmic firing patterns in suprachiasmatic nucleus (SCN): the rôle of circuit interactions.Int. J. Bio-med. Comp. 38, 23–31.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1997

Authors and Affiliations

  • Rafael A. Barrio
    • 1
  • Limei Zhang
    • 2
  • Philip K. Maini
    • 3
  1. 1.Instftuto de FísicaUNAMMéxicoMexico
  2. 2.Departmeneto de Fisiología, Facultad de MedicinaUNAMMéxicoMexico
  3. 3.Centre for Mathematical Biology, Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations