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Liebovitch, 1998).

We see that the Hurst exponent, Fano and Allen factors, Levy exponent and power spectral scaling exponent are kindred statistical descriptors. They are most usefully applicable to systems with distributions that fail to be Gaussian or asymmetrically Poisson, the latter from random data sequences with only positive x values, thus backed up toward zero by a minimum inter-event interval or amplitude. These time series are sequentially dependent, not conventionally stationary, without finite central moments and with self-correlations that don't demonstrate Gaussian exponential decay with sample length or time. The following are some examples of the use of these measures in studies of biological dynamics. .

Examples of Biological Data with Divergent Distributions and Power Law Scaling

A paradigm challenging group of experiments involved models and measures of the distribution of characteristic open and closed times of membrane ion conductance channels. The usual approach to this problem assumed the existence of a small set of distinguishable channel types that were reflected in discrete conductance events with a small set of characteristic open and closed times. The distributions of each of could be fitted with its own, Markov process derived, exponential. With technical advances and improved temporal resolution, more characteristic times and their associated α = 2 exponentials were reported with as many as three not being unusual. Liebovitch (and Sullivan, 1987; 1989) used analogue to digital transformation of current recordings from the unselective corneal epithelial channels and voltage dependent potassium channels in cultured mouse hippocampal cells at temporal resolutions ranging from 170 to 5000 Hz and found similarly shaped, α < 2, nonconvergent distributions across temporal scales. This led these investigators to suggest that, related to the >16 recorded magnitudes of characteristic times, from picoseconds to months, in autonomous protein motion (Careri et al, 1975; Gurd and Rothgeb, 1979), that there was an “α stable” hierarchy

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