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of lifetimes of states, observable at almost any temporal resolution that methods
would allow.

Early and representative studies comparing the fit of the data with
hierarchical scaling functions versus a sum of a small number of Markovian
exponentials included studies of a calcium activated potassium channel in human
fibroblasts (Stockbridge and French, 1989) which yielded evidence to support both
models, as did studies of membrane conductances in corneal epithelial cells by
another group (Korn and Horn, 1988). In a systematic comparison of scaling and
Markov exponential modes of the gating kinetics of GABA activated chloride
channels, acetylcholine activated end plate potentials, calcium activated potassium
channels and fast chloride channels (McManus et al, 1988), it was found that the
latter fit the data best in most experiments. Similar results were reported in studies
of the glutamate and delayed rectifier potassium channel with respect to
distributions of open and closed times (Sansom et al, 1989). Space does not permit
a systematic account of the continuing debate and conflicting studies about these
representations and the implicit biophysics of discrete, finite versus continuous,
hierarchical channel event heterogeneity. It is interesting that recent experiments
making use of Hurst rescaled range analyses of time series of whole cell membrane
voltage fluctuations (without the assumptions and current renormalizing procedures
associated with patch clamping) have yielded additional evidence for multiply
correlated, Hurst > 0.5, α < 2 power law behavior of what some might regard more
generally as a protein relaxation time mediated hierarchical array of ion
conductance behaviors (Liebovitch and Todorov, 1996).

Following the discovery of (very) subsaturating (“far from equilibrium”) rat
brain levels of the common cofactor for tyrosine and tryptophan hydroxylases,
tetrahydrobiopterin (Bullard et al, 1978), studies of amino acid substrate saturation
functions and time courses determined at these low, physiological co-reactant
concentrations manifested patterns of hierarchical multiplicity and discontinuities
suggestive of bifurcations and time-dependent fluctuations with fractional
(hierarchical) time scaling exponents that were sensitive to psychotropic drugs

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