Advertisement

Journal of Statistical Physics

, Volume 171, Issue 5, pp 878–896 | Cite as

Stochastic Resonance and First Arrival Time for Excitable Systems

  • Solomon Fekade Duki
  • Mesfin Asfaw Taye
Article
  • 96 Downloads

Abstract

We study the noise induced thermally activated barrier crossing of Brownian particles that hop in a piecewise linear potential. Using the exact analytic solutions and via numerical simulations not only we explore the dependence for the first passage time of a single particle but also we calculate the first arrival time for one particle out of N particles. The first arrival time decreases as the number of particles increases as expected. We then explore the thermally activated barrier crossing rate of the system in the presence of time varying signal. The dependence of signal to noise ratio SNR as well as the power amplification (\(\eta \)) on model parameters is explored. \(\eta \) and SNR depict a pronounced peak at particular noise strength. In the presence of N particles, \(\eta \) is considerably amplified as N steps up showing the weak periodic signal plays a vital role in controlling the noise induced dynamics of the system. Moreover, for the sake of generality, the viscous friction \(\gamma \) is considered to decrease exponentially when the temperature T of the medium increases (\(\gamma =Be^{-A T}\)) as proposed originally by Reynolds (Philos Trans R Soc Lond 177:157, 1886).

Keywords

Brownian motion Stochastic resonance Statistical physics 

Notes

Acknowledgements

We would like to thank Mulugeta Bekele for the interesting discussions we had. MA would like to thank Mulu Zebene for the constant encouragement. SFD’s research was supported in part by the Intramural Research Program of the National Institute of Health, National Library of Medicine.

References

  1. 1.
    Redner, S.: A Guide to First-Passage Processes. Cambridge University Press, Boston (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Kramer, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284 (1940)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Hanggi, P., Talkner, P., Borkovec, M.: Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251 (1990)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Park, P.J., Sung, W.: Dynamics of a polymer surmounting a potential barrier: the Kramers problem for polymers. J. Chem. Phys. 111, 5259 (1999)ADSCrossRefGoogle Scholar
  5. 5.
    Lee, S., Sung, W.: Coil-to-stretch transition, kink formation, and efficient barrier crossing of a flexible chain. Phys. Rev. E 63, 021115 (2001)ADSCrossRefGoogle Scholar
  6. 6.
    Hanggi, P., Marchesoni, F., Sodano, P.: Nucleation of thermal sine-Gordon Solitons: effect of many-body interactions. Phys. Rev. Lett. 60, 2563 (1988)ADSCrossRefGoogle Scholar
  7. 7.
    Marchesoni, F., Cattuto, C., Costantini, G.: Elastic strings in solids: thermal nucleation. Phys. Rev. B 57, 7930 (1998)ADSCrossRefGoogle Scholar
  8. 8.
    Hanggi, P., Marchesoni, F.: Artificial Brownian motors: controlling transport on the nanoscale. Rev. Mod. Phys. 81, 387 (2009)ADSCrossRefGoogle Scholar
  9. 9.
    Sebastian, K.L., Paul, Alok K.R.: Kramers problem for a polymer in a double well. Phys. Rev. E 62, 927 (2000)ADSCrossRefGoogle Scholar
  10. 10.
    Bekele, M., Ananthakrishna, G., Kumar, N.: Mean first passage time approach to the problem of optimal barrier subdivision for Kramer’s escape rate. Physica A 270, 149 (1999)ADSCrossRefGoogle Scholar
  11. 11.
    Asfaw, M., Lacalle, E.A., Shiferaw, Y.: The timing statistics of spontaneous calcium release in cardiac myocytes. PLoS ONE 8, e62967 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Chen, W., Asfaw, M., Shiferaw, Y.: The statistics of Calcium-mediated focal excitations on a one-dimensional cable. Biophys. J. 102, 461 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Lindnera, B., Garcia-Ojalvob, J., Neimand, A., Schimansky-Geier, L.: Effects of noise in excitable systems. Phys. Rep. 392, 321 (2004)ADSCrossRefGoogle Scholar
  14. 14.
    Benzi, R., Parisi, G., Sutera, A., Vulpiani, A.: Stochastic resonance in climatic change. Tellus 34, 10 (1982)ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Gammaitoni, L., Hanggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223 (1998)ADSCrossRefGoogle Scholar
  16. 16.
    Jung, P., Behn, U., Pantazelou, E., Moss, F.: Collective response in globally coupled bistable systems. Phys. Rev. A 46, R1709 (1992)ADSCrossRefGoogle Scholar
  17. 17.
    Lindner, J.F., Meadows, B.K., Ditto, W.L., Inchiosa, M.E., Bulsara, A.R.: Array enhanced stochastic resonance and spatiotemporal synchronization. Phys. Rev. Lett. 75, 3 (1995)ADSCrossRefGoogle Scholar
  18. 18.
    Lindner, J.F., Meadows, B.K., Ditto, W.L., Inchiosa, M.E., Bulsara, A.R.: Scaling laws for spatiotemporal synchronization and array enhanced stochastic resonance. Phys. Rev. E 53, 2081 (1996)ADSCrossRefGoogle Scholar
  19. 19.
    Marchesoni, F., Gammaitoni, L., Bulsara, A.R.: Spatiotemporal stochastic resonance in a \(\upphi \)4 model of Kink-Antikink nucleation. Phys. Rev. Lett. 76, 2609 (1996)ADSCrossRefGoogle Scholar
  20. 20.
    Dikshtein, I.E., Kuznetsov, D.V., Schimansky-Geier, L.: Stochastic resonance for motion of flexible macromolecules in solution. Phys. Rev. E. 65, 061101 (1996)ADSCrossRefGoogle Scholar
  21. 21.
    Goychuk, I., Hanggi, P.: Non-markovian stochastic resonance. Phys. Rev. Lett. 91, 070601 (2003)ADSCrossRefGoogle Scholar
  22. 22.
    Yasuda, H., et al.: Novel class of neural stochastic resonance and error-free information transfer. Phys. Rev. Lett. 100, 118103 (2008)ADSCrossRefGoogle Scholar
  23. 23.
    Vilar, J.M.G., Rubi, J.M.: Spatiotemporal stochastic resonance in the Swift-Hohenberg equation. Phys. Rev. Lett. 78, 2886 (1997)ADSCrossRefGoogle Scholar
  24. 24.
    Lindner, J.F., Bennett, M., Wiesenfeld, K.: Potential energy landscape and finite-state models of array-enhanced stochastic resonance. Phys. Rev. E 73, 031107 (2003)ADSCrossRefGoogle Scholar
  25. 25.
    Asfaw, M., Sung, W.: Stochastic resonance of a flexible chain crossing over a barrier. EPL 90, 3008 (2010)CrossRefGoogle Scholar
  26. 26.
    Asfaw, M.: Thermally activated barrier crossing and stochastic resonance of a flexible polymer chain in a piecewise linear bistable potential. Phys. Rev. E 82, 021111 (2010)ADSCrossRefGoogle Scholar
  27. 27.
    Reynolds, O.: On the theory of lubrication and is application to beauchamp, experiments. Philos. Trans. R. Soc. Lond. 177, 157 (1886)CrossRefGoogle Scholar
  28. 28.
    Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin (1984)zbMATHGoogle Scholar

Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2018

Authors and Affiliations

  1. 1.National Center for Biotechnology InformationNational Library of Medicine and National Institute of HealthBethesdaUSA
  2. 2.Department of PhysicsCalifornia State University Dominguez HillsCarsonUSA

Personalised recommendations