Abstract
This study of the effect of noise on bifurcations in a simple biological oscillator with a periodically modulated threshold uses the first-passage-time problem of the Ornstein–Uhlenbeck process with a periodic boundary to define the operator governing the transition of a threshold phase density. Stochastic phase-locking is analyzed numerically by evaluating the evolution of the probability density function of the threshold phase. A firing phase map in a noisy environment is extended to a stochastic kernel so that stochastic bifurcations can be investigated by spectral analysis of the kernel.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
F. Moss and P. V. E. McClintock, Noise in Nonlinear Dynamical Systems, Vols. 1–3 (Cambridge University Press, Cambridge, 1987).
L. Arnold, Random dynamical systems, in Dynamical Systems, R. Johnson, ed. (Springer, 1994), pp. 1–43.
J. Grasman and J. B. T. M. Roerdink, Stochastic and chaotic relaxation oscillations, J. Stat. Phys. 54:949–970 (1989).
T. Tateno, S. Doi, S. Sato, and L. M. Ricciardi, Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach, J. Stat. Phys. 78:917–935 (1995).
S. Doi, J. Inoue, and S. Kumagai, Spectral analysis of stochastic phase lockings and stochastic bifurcations in the sinusoidally forced van der Pol oscillator with additive noise, J. Stat. Phys. 90:1107–1127 (1998).
S. Kauffman, Measuring a mitotic oscillator: the arc discontinuity, Bull. Math. Biol. 36:171–182 (1974).
J. P. Keener, On cardiac arrhythmias: AV conduction block, J. Math. Biol. 12:215–225 (1981).
B. W. Knight, Dynamics of encoding in a population of neurons, J. Gen. Bhysiol. 59:734–766 (1972).
L. Glass and M. C. Mackey, A simple model for phase locking of biological oscillators, J. Math. Biology 7:339–352 (1979).
J. P. Keener, F. C. Hoppensteadt, and J. Rinzel, Integrate-and-fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math. 41:503–517 (1981).
A. Buoncore, A. G Nobile, and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. Appl. Prob. 19:784–800 (1987).
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd ed. (Springer-Verlag, 1994), pp. 112–122.
S. Chang and J. Wright, Transitions and distribution functions for chaotic systems, Phys. Rev. A 23:1419–1433 (1981).
F. R. Gantmacher, The Theory of Matrices, Vol. II (Chelsea Publishing Company, 1959), pp. 82–112.
W. Feller, An Introduction to Probability Theory and Its Application, Vol. 1 (John Wiley & Sons, 1968), pp. 372–423.
S. Karlin, A First Course in Stochastic Processes (Academic Press Inc., 1966), pp. 27–103.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tateno, T. Characterization of Stochastic Bifurcations in a Simple Biological Oscillator. Journal of Statistical Physics 92, 675–705 (1998). https://doi.org/10.1023/A:1023048923644
Issue Date:
DOI: https://doi.org/10.1023/A:1023048923644