Abstract
In this paper we develop a general modeling framework within which many models for systems which produce events at irregular times through a combination of probabilistic and deterministic dynamics can be comprehended. We state and prove new sufficient conditions for the global asymptotic behaviour of the density evolution in these systems, and apply our results to many previously published models for the cell division cycle. In addition, we develop a new interpretation for the statistics of action potential production in excitable cells.
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Foguel, S. R.: The Ergodic Theory of Markov Processes. New York: Van Nostrand Reinhold 1969
Gerstein, G. L., Mandelbrot, B.: Random walk model for the spike activity of a single neuron. Biophys. J. 4, 41–68 (1964)
Glass, L., Mackey, M. C.: A simple model for phase locking of biological oscillators. J. Math. Biol. 7, 339–352 (1979)
Glass, L., Mackey, M. C.: From Clocks to Chaos: The Rhythms of Life. Princeton: Princeton University Press 1988
Glass, L., Graves, C., Petrillo, G. Mackey, M. C.: Unstable dynamics of a periodically driven oscillator in the presence of noise. J. Theor. Biol. 86, 455–475 (1980)
Hannsgen, K. B., Tyson, J. J., Watson, L. T.: Steady state size distribution in probabilistic models of the cell division cycle. SIAM J. Appl. Math. 45, 523–540 (1985)
Komornik, J., Lasota, A.: Asymptotic decomposition of Markov operators. Bull. Pol. Acad. Sci., Math. 35, 321–327 (1987)
Lasota, A., Mackey, M. C.: Globally asymptotic properties of proliferating cell populations. J. Math. Biol. 19, 43–62 (1984)
Lasota, A., Mackey, M. C.: Probabilistic Properties of Deterministic Systems. Cambridge: Cambridge University Press 1985
Lasota, A., Mackey, M. C.: Stochastic perturbation of dynamical systems: The weak convergence of measures. J. Math. Anal. Appl. 138, 232–248 (1989)
Lasota, A., Tyrcha, J.: On the strong convergence to equilibrium for randomly perturbed dynamical systems. Ann. Pol. Math. 53, 79–89 (1991)
Mackey, M. C.: Ion Transport Through Biological Membranes. Berlin Heidelberg New York: Springer 1975
Mannard, A., Rajchgot, P., Polosa, C.: Effect of post-impulse depression on background firing of sympathetic preganglionic neurons. Brain Res. 126, 243–261 (1977)
Miklavcic, M.: Asymptotic periodicity of the iterates of positivity preserving operators. Trans. Am. Math. Soc. 307, 469–480 (1988)
Shields, R.: Transition probability and the origin of variation in the cell cycle. Nature 267, 704–707 (1977)
Sine, R.: Weakly constricted operators and Jamison convergence theorem. Proc. Am. Math. Soc. 106, 751–755 (1989)
Smith, J. A., Martin, L.: Do cells cycle? Proc. Natl. Acad. Sci., USA 70, 1263–1267 (1973)
Tuckwell, H. C.: Stochastic Processes in the Neurosciences. Philadelphia: Society for Industrial and Applied Mathematics 1989
Tyrcha, J.: Asymptotic stability in a generalized probabilistic deterministic model of the cell cycle. J. Math. Biol. 26, 465–475 (1988)
Tyson, J. J., Hannsgen, K. B.: Global asymptotic stability of the size distribution in probabilistic model of the cell cycle. J. Math. Biol. 22, 61–68 (1985)
Tyson, J. J., Hannsgen, K. B.: Cell growth and division: A deterministic probabilistic model of the cell cycle. J. Math. Biol. 23, 231–246 (1986)
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Lasota, A., Mackey, M.C. & Tyrcha, J. The statistical dynamics of recurrent biological events. J. Math. Biol. 30, 775–800 (1992). https://doi.org/10.1007/BF00176455
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DOI: https://doi.org/10.1007/BF00176455