Noise-induced Periodicity: Some Stochastic Models for Complex Biological Systems

  • Paolo Dai Pra
  • Giambattista Giacomin
  • Daniele Regoli
Part of the Springer INdAM Series book series (SINDAMS, volume 6)

Abstract

After a review of some examples of life science stochastic models, we propose a stylized model with characteristics inspired by the examples above, reproducing noise-induced pulsations as a collective macroscopic phenomenon.

References

  1. 1.
    Acebrón, J.A., López Bonilla, L., Pérez Vicente, C.J., Ritort, F., Spigler, R.: The kuramoto model: A simple paradigm for synchronization phenomena. Reviews of modern physics 77(1), 137 (2005)CrossRefGoogle Scholar
  2. 2.
    Bertini, L., Giacomin, G., Pakdaman, K.: Dynamical aspects of mean field plane rotators and the kuramoto model. J. Statist. Phys. 138, 270–290 (2010)CrossRefGoogle Scholar
  3. 3.
    Brunel, N., Hakim, V.: Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural computation 11(7), 1621–1671 (1999)CrossRefGoogle Scholar
  4. 4.
    Carletti, T., Villari, G.: A note on existence and uniqueness of limit cycles for Liénard systems. Journal of mathematical analysis and applications 307(2), 763–773 (2005)CrossRefGoogle Scholar
  5. 5.
    Dai Pra, P., Fischer, M., Regoli, D.: A Curie-Weiss model with dissipation. Journal of Statistical Physics 152, 37–53 (2013)CrossRefGoogle Scholar
  6. 6.
    Dai Pra, P., Giacomin, G., Regoli, D.: Periodic behavior in a Curie-Weiss model with noisy rates. In preparation (2013)Google Scholar
  7. 7.
    Elowitz, M.B., Leibler, S.: Asynthetic oscillatory network of transcriptional regulators. Nature 403(6767), 335–338 (2000)CrossRefGoogle Scholar
  8. 8.
    Garcia-Ojalvo, J., Elowitz, M.B., Strogatz, S.H.: Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. Proceedings of the National Academy of Sciences of the United States of America 101(30), 10955–10960 (2004)CrossRefGoogle Scholar
  9. 9.
    Giacomin, G., Pakdaman, K., Pellegrin, X., Poquet, C.: Transitions in Active Rotator Systems: Invariant Hyperbolic Manifold Approach. SIAM J. Math. Anal. 44(6), 4165–4194 (2012)CrossRefGoogle Scholar
  10. 10.
    Kuramoto, Y.: Chemical oscillations, waves, and turbulence. Courier Dover Publications, New York (2003)Google Scholar
  11. 11.
    McMillen, D., Kopell, N., Hasty, J., Collins, J.J.: Synchronizing genetic relaxation oscillators by intercell signaling. Proceedings of the National Academy of Sciences 99(2), 679–684 (2002)CrossRefGoogle Scholar
  12. 12.
    Pakdaman, K., Perthame, B., Salort, D.: Relaxation and self-sustained oscillations in the time elapsed neuron network model. SIAM J. Appl. Math. 73(3), 1260–1279 (2013)CrossRefGoogle Scholar
  13. 13.
    Pakdaman, K., Perthame, B., Salort, D.: Dynamics of a structured neuron population. Nonlinearity 23(1), 55–75 (2010)CrossRefGoogle Scholar
  14. 14.
    Romanczuk, P., Bär, M., Ebeling, W., Lindner, B., Schimansky-Geier, L.: Active brownian particles. The European Physical Journal Special Topics 202(1), 1–162 (2012)CrossRefGoogle Scholar
  15. 15.
    Sabatini, M., Villari, G.: Limit cycle uniqueness for a class of planar dynamical systems. Applied mathematics letters 19(11), 1180–1184 (2006)CrossRefGoogle Scholar
  16. 16.
    Sakaguchi, H., Shinomoto, S., Kuramoto, Y.: Phase transitions and their bifurcation analysis in a large population of active rotators with mean-field coupling. Progress of Theoretical Physics 79(3), 600–607 (1988)CrossRefGoogle Scholar
  17. 17.
    Scheutzow, M.: Periodic behavior of the stochastic brusselator in the mean-field limit. Probability theory and related fields 72(3), 425–462 (1986)CrossRefGoogle Scholar
  18. 18.
    Schweitzer, F.: Brownian agents and active particles. Collective dynamics in the natural and social sciences, With a foreword by J. Doyne Farmer. Springer Series in Synergetics. Springer-Verlag, Berlin Heidelberg New York (2003)Google Scholar
  19. 19.
    Shinomoto, S., Kuramoto, Y.: Phase transitions in active rotator systems. Progress of Theoretical Physics 75(5), 1105–1110 (1986)CrossRefGoogle Scholar
  20. 20.
    Touboul, J., Hermann, G., Faugeras, O.: Noise-induced behaviors in neural mean field dynamics. SIAM J. Applied Dynamical Systems 11(1) 49–81 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Paolo Dai Pra
    • 1
  • Giambattista Giacomin
    • 2
  • Daniele Regoli
    • 1
  1. 1.Dipartimento di MatematicaUniversità degli Studi di PadovaPadovaItaly
  2. 2.U.F.R. de MathématiquesUniversité Paris DiderotPARIS Cedex 13France

Personalised recommendations