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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
Bertini, L., Giacomin, G., Pakdaman, K.: Dynamical aspects of mean field plane rotators and the kuramoto model. J. Statist. Phys. 138, 270–290 (2010)
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)
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)
Dai Pra, P., Fischer, M., Regoli, D.: A Curie-Weiss model with dissipation. Journal of Statistical Physics 152, 37–53 (2013)
Dai Pra, P., Giacomin, G., Regoli, D.: Periodic behavior in a Curie-Weiss model with noisy rates. In preparation (2013)
Elowitz, M.B., Leibler, S.: Asynthetic oscillatory network of transcriptional regulators. Nature 403(6767), 335–338 (2000)
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)
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)
Kuramoto, Y.: Chemical oscillations, waves, and turbulence. Courier Dover Publications, New York (2003)
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)
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)
Pakdaman, K., Perthame, B., Salort, D.: Dynamics of a structured neuron population. Nonlinearity 23(1), 55–75 (2010)
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)
Sabatini, M., Villari, G.: Limit cycle uniqueness for a class of planar dynamical systems. Applied mathematics letters 19(11), 1180–1184 (2006)
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)
Scheutzow, M.: Periodic behavior of the stochastic brusselator in the mean-field limit. Probability theory and related fields 72(3), 425–462 (1986)
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)
Shinomoto, S., Kuramoto, Y.: Phase transitions in active rotator systems. Progress of Theoretical Physics 75(5), 1105–1110 (1986)
Touboul, J., Hermann, G., Faugeras, O.: Noise-induced behaviors in neural mean field dynamics. SIAM J. Applied Dynamical Systems 11(1) 49–81 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Pra, P.D., Giacomin, G., Regoli, D. (2014). Noise-induced Periodicity: Some Stochastic Models for Complex Biological Systems. In: Celletti, A., Locatelli, U., Ruggeri, T., Strickland, E. (eds) Mathematical Models and Methods for Planet Earth. Springer INdAM Series, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-02657-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-02657-2_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02656-5
Online ISBN: 978-3-319-02657-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)