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Of course, most real biological dynamics are not uniformly mixing and so are
non-ergodic, but we shall see that the ways they fail to be ergodic (and thus remain
in the conceptual context of ergodic measures) are descriptively useful (Mandell
and Selz, 1997a). The emergence of many statistical approaches to characterizing
these motions have been accompanied by the expected controversies about which
is best or correct (see below) and have been applied to the problem of diagnosis
and clinical discrimination in a variety of neuroscience settings. In ideal abstract
chaotic dynamical systems called Axiom A (Russians called them "C systems"),
where most mathematical theorems are proven (Smale, 1967), all these measures,
if properly computed, are equivalent. In real life, as in the related case of ergodicity,
they are not, and since no single one is complete, the more (incomplete) measures
we use in our studies along with interest in the way that they differ, supplies more
useful information about the system. Though researching and elucidating the most
reliable and valid ways of computing these measures are a valuable goal, the
current debates focused on the superiority of a single particular measure,
constructed in a particular way in relationship to issues of insoluble absolutes like
"randomness" versus "deterministic chaos may not be particularly valuable for
uncovering new characteristics and potential mechanisms underlying a specific set
of real neurobiological observables.

Emphasizing diversity and relevance to the clinical biological sciences, we
note that quantifying patterns in ergodic (non-ergodic) measures have aided: the
discrimination between normal and abnormal opticokinetic nystagmus in neurology
patients (Aeson et al, 1997); localizing a two year old subcortical stroke in an EEG
of a patient with no other signs or neurological findings (Molnar et al, 1997); the
diagnosis of early (not late) multiple sclerosis, as a nonspecific long tract disorder,
in patients with mild optical neuritis using cardiac rate dynamics (Ganz and
Faustman, 1996); seizure prediction from minutes to hours before the event in
which subthreshold, pre-phase transition spatial diffusion and oscillations in
characteristic changes in these measures can be found (Martinerie et al, 1998;
Elger and Lehnertz, 1998; Pign et al, 1997; lasemidis et al, 1990); using these
measures on the EEG to differentially predict hereditary predisposition to alcoholism

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