which, in fact, permit definition of an adiabatically piece-
wise memoryless ergodic information source associated with
the paths xn . We call that source X.
We further suppose, in the manner of Wallace (2002a, b),
that the set of external signals is also highly structured and
forms another information source Y which interacts not only
with the system of interest globally, but specifically with its
universality class properties as characterized by X. Y is nec-
essarily associated with a set of paths yn .
We pair the two sets of paths into a joint path, zn ≡ (xn , yy)
and invoke an inverse coupling parameter, K , between the
information sources and their paths. This leads, by the ar-
guments above, to phase transition punctuation of I [K], the
mutual information between X and Y, under either the Joint
Asymptotic Equipartition Theorem or under limitation by
a distortion measure, through the Rate Distortion Theorem
(Cover and Thomas, 1991). Again, see Wallace (2002a, b)
for more details of the argument. The essential point is that
I [K] is a splitting criterion under these theorems, and thus
partakes of the homology with free energy density which we
have invoked above.
Activation of universality class tuning, our version of at-
tentional focusing, then becomes itself a punctuated event in
response to increasing linkage between the organism and an
external structured signal.
We note in passing, but without further calculation, that
another path to the fluctuating dynamic threshold might be
through a second order iteration similar to that just above,
but focused on the parameters defining the universality class
distributions of section 3.
Extending the global neuronal workspace model
The Rate Distortion and Joint Asymptotic Equipartition
Theorems are generalizations of the Shannon-McMillan Theo-
rem which examine the interaction of two information sources,
with and without the constraint of a fixed average distortion.
We conduct one more iteration, and require a generalization
of the SMT in terms of the splitting criterion for triplets as
opposed to single or double stranded patterns. The tool for
this is at the core of what is termed network information the-
ory [Cover and Thomas, 1991, Theorem 14.2.3]. Suppose we
have three (piecewise adiabatically memoryless) ergodic in-
formation sources, Y1 , Y2 and Y3 . We assume Y3 constitutes a
critical embedding context for Y1 and Y2 so that, given three
sequences of length n, the probability of a particular triplet
of sequences is determined by conditional probabilities with
respect to Y3 :
P (Y1 = y1 , Y2 = y2 , Y3 = y3) =
πn=ιP(yii\узі)Р(У2і\узі)Р(Узі).
(22)
That is, Y1 and Y2 are, in some measure, driven by their
interaction with Y3
Then, in analogy with previous analyses, triplets of se-
quences can be divided by a splitting criterion into two sets,
having high and low probabilities respectively. For large n the
number of triplet sequences in the high probability set will be
determined by the relation [Cover and Thomas, 1992, p. 387]
N(n) α exp[nI(Yι; Y2∖Y3)],
(23)
where splitting criterion is given by
I(Y1;Y2\Y3) ≡
H(Y3)+H(Y1\Y3)+H(Y2\Y3) -H(Y1,Y2,Y3)
We can then examine mixed cognitive/adaptive phase tran-
sitions analogous to learning plateaus (Wallace, 2002b) in the
splitting criterion I(Y1, Y2 \Y3), which characterizes the syn-
ergistic interaction between Y3 , taken as an embedding con-
text, and the cognitive processes characterized by Y1 and Y2 .
These transitions delineate the various stages of the chronic
infection, which are embodied in the slowly varying ‘piece-
wise adiabatically memoryless ergodic’ phase between tran-
sitions. Again, our results are exactly analogous to the El-
dredge/Gould model of evolutionary punctuated equilibrium.
We can, if necessary, extend this model to any number of
interacting information sources, Y1, Y2, ..., Ys conditional on
an external context Z in terms of a splitting criterion defined
by
I (Yι;...; Ys ∖Z) = H (Z)+ ∑ H (Yj ∖Z) - H (Y1,...,Ys,Z),
j=1
(24)
where the conditional Shannon uncertainties H(Yj ∖Z) are
determined by the appropriate direct and conditional proba-
bilities.
Note that this simple-seeming extension opens a Pandora’s
box in the study of ‘mind-body interaction’ and the impacts
of culture and history on individual cognition. We now have
a tool for examining the interpenetration of a broad range of
cognitive physiological, psychological, and social submodules
- not just neural substructures - with each other and with
embedding contextual cultural language so characteristic of
human hypersociality, all within the further context of struc-
tured psychosocial stress. Wallace (2003) analyzes the impli-
cations for understanding comorbid mind/body dysfunction,
and provides a laundry list of physiological, psychological, and
social cognitive modules associated with health and disease.
Bennett and Hacker (2003) define the ‘mereological fallacy’
in neuroscience as the assignment, to parts of an animal,
of those characteristics which are properties of the whole.
11