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Random Perturbations

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Abstract

Up to this point, our “mathematics in the laboratory” approach to the investigation of biological systems has been very simplistic. We have assumed that very complex biological systems, ranging from ecosystems to the human brain, can be adequately described by measuring changes in just a few variables. Moreover, we have ignored the fact that all living dynamical systems are continuously subjected to large numbers of influences. For example, it has been estimated that a typical neuron in the human brain receives ≈ 104 synaptic connections from other neurons [597].

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Notes

  1. 1.

    The term independent means that a change in one variable does not lead to a change in other variables. We will give a more formal definition in Section 12.2.

  2. 2.

    Juan Luis Cabrera (PhD 1997), Venezuelan condensed matter biophysicist.

  3. 3.

    The ability of a bacterium to change its phenotype to achieve evolutionary advantage is reminiscent of a similar ability that has been observed in Homo sapiens, which has been known to change its hair color.

  4. 4.

    S = { H, T} is standard mathematical notation for the set S consisting of the elements H and T.

  5. 5.

    Technically, the probability is defined on the sigma field or algebra of the sample space (for an introduction, see [488]).

  6. 6.

    Achilles (fl. ca. 1250 b.c.e.), Greek hero of the Trojan War. According to legend, his only physical vulnerability was his heel, and he is said to have been killed in the Trojan War by Paris, who shot him in the heel with an arrow.

  7. 7.

    The Gaussian distribution is just one of a multitude of mathematical concepts (including the gauss, the cgs unit of magnetic field strength) named after Johann Carl Friedrich Gauss (1777–1855), German mathematician.

  8. 8.

    James Clerk Maxwell (1831–1879), Scottish mathematical physicist; Ludwig Eduard Boltzmann (1844–1906), Austrian physicist and philosopher.

  9. 9.

    Andrzej Aleksander Lasota (1932–2006), Polish mathematician.

  10. 10.

    Paul Pierre Lévy (1886–1971), French mathematician who introduced many concepts in probability theory used in this book, including alpha-stable distributions and Lévy flights.

  11. 11.

    Keep in mind that this statement is true when the mean values have been removed before the distributions are added.

  12. 12.

    Augustin–Louis Cauchy (1789–1857), French mathematician.

  13. 13.

    For Matlab programmers, a very useful software package has been developed by M. Veillette. It is freely downloadable from the Internet site http://math.bu.edu/people/mveillet/html/alphastablepub.html. Similar tools for Python have been developed within the package PyLevy.

  14. 14.

    The reader should keep in mind that correlation does not imply causality.

  15. 15.

    The functions xcorr() and acorr() take care of the fast Fourier transform part of the calculation of the autocorrelation and the padding with zeros. The option normed = True in xcorr() ensures that the autocorrelation function is normalized by the variance to give K xx (Δ).

  16. 16.

    It should be noted that mathematicians and investigators working in communication theory often use the two-sided power spectrum [35]. In this case, we have, for example,

    $$\displaystyle{\hat{w}(f) =\int _{ -\infty }^{\infty }c_{xx}(\varDelta )\cos (2\pi f\varDelta )\,d\varDelta = 2\int _{ 0}^{\infty }c_{xx}(\varDelta )\cos (2\pi f\varDelta )\,d\varDelta.}$$

    However, the use of the one-sided power spectrum is favored in the stochastic signal literature.

  17. 17.

    Leonard Salomon Ornstein (1880–1941), Dutch physicist; George Eugene Uhlenbeck (1900–1988), Dutch–American theoretical physicist.

  18. 18.

    It seems more reasonable to define colored noise as noise that is not delta-correlated. However, in current usage of the term, colored noise does not include exponentially correlated noise.

  19. 19.

    The major exception occurs when the two functions are not correlated, and hence C xy (Δ) = 0 for all Δ.

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Authors and Affiliations

  1. W.M. Keck Science Department, The Claremont Colleges, Claremont, CA, USA

    John Milton

  2. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Toru Ohira

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Milton, J., Ohira, T. (2014). Random Perturbations. In: Mathematics as a Laboratory Tool. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9096-8_12

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