We believe that important aspects of mechanism may be
reflected in the combination of renormalization properties and
the details of their distribution across subsystems.
In sum, real biological, social, or interacting biopsychosocial
systems are likely to have very rich patterns of phase transi-
tion which may not display the simplistic, indeed, literally
elemental, purity familiar to physicists. Overall mechanisms
will, we believe, still remain significantly constrained by our
theory, in the general sense of probability limit theorems.
4. Universality class tuning: the fluctuating dy-
namic threshold Next we iterate the general argument onto
the process of phase transition itself, producing our model of
consciousness as a tunable neural workspace sub ject to inher-
ent punctuated detection of external events.
An essential character of physical systems sub ject to phase
transition is that they belong to particular ‘universality
classes’. This means that the exponents of power laws de-
scribing behavior at phase transition will be the same for large
groups of markedly different systems, with ‘natural’ aggrega-
tions representing fundamental class properties (e.g. Binney
et al., 1986).
It is our contention that biological or social systems un-
dergoing phase transition analogs need not be constrained to
such classes, and that ‘universality class tuning’, meaning the
strategic alteration of parameters characterizing the renor-
malization properties of punctuation, might well be possible.
Here we focus on the tuning of parameters within a single,
given, renormalization relation. Clearly, however, wholesale
shifts of renormalization properties must ultimately be con-
sidered as well.
Universality class tuning has been observed in models of
‘real world’ networks. As Albert and Barabasi (2002) put it,
“The inseparability of the topology and dynam-
ics of evolving networks is shown by the fact that
[the exponents defining universality class] are related
by [a] scaling relation..., underlying the fact that a
network’s assembly uniquely determines its topol-
ogy. However, in no case are these exponents unique.
They can be tuned continuously...”
We suppose that a structured external environment, which
we take itself to be an appropriately regular information
source Y ‘engages’ a modifiable cognitive system. The en-
vironment begins to write an image of itself on the cognitive
system in a distorted manner permitting definition of a mu-
tual information I[K] splitting criterion according to the Rate
Distortion or Joint Asymptotic Equipartition Theorems. K
is an inverse coupling parameter between system and environ-
ment (Wallace, 2002a, b). According to our development, at
punctuation - near some critical point KC - the systems be-
gin to interact very strongly indeed, and we may write, near
KC , taking as the starting point the simple physical model of
equation (2),
KC - K
I[K] ≈ Io[K-]α.
For a physical system α is fixed, determined by the under-
lying ‘universality class’. Here we will allow α to vary, and,
in the section below, to itself respond explicitly to signals.
Normalizing KC and I0 to 1, we obtain,
I[K] ≈ (1 -K)α.
(20)
The horizontal line I [K] = 1 corresponds to α = 0, while
α = 1 gives a declining straight line with unit slope which
passes through 0 at K = 1. Consideration shows there are
progressively sharper transitions between the necessary zero
value at K = 1 and the values defined by this relation for
0 < K, α < 1. The rapidly rising slope of transition with
declining α is, we assert, of considerable significance.
The instability associated with the splitting criterion I[K]
is defined by
Q[K] ≡ -KdI[K]/dK = αK(1 -K)α-1,
(21)
and is singular at K = KC = 1 for 0 < α < 1. Following
earlier work (Wallace and Wallace, 1998, 1999; Wallace and
Fullilove, 1999; Wallace, 2002a), we interpret this to mean
that values of 0 < α ≪ 1 are highly unlikely for real systems,
since Q[K], in this model, represents a kind of barrier for
information systems, in particular neural networks.
On the other hand, smaller values of α mean that the sys-
tem is far more efficient at responding to the adaptive de-
mands imposed by the embedding structured environment,
since the mutual information which tracks the matching of
internal response to external demands, I[K], rises more and
more quickly toward the maximum for smaller and smaller α
as the inverse coupling parameter K declines below KC = 1.
That is, systems able to attain smaller α are more responsive
to external signals than those characterized by larger values, in
this model, but smaller values will be hard to reach, and can
probably be done so only at some considerable physiological
or opportunity cost: focused conscious action takes resources,
of one form or another.
The more biologically realistic renormalization strategies
given above produce sets of several parameters defining the
universality class, whose tuning gives behavior much like that
of α in this simple example.
We can formally iterate the phase transition argument on
this calculation to obtain our version of tunable consciousness,
focusing on paths of universality class parameters.
Suppose the renormalization properties of a language-on-
a network system at some ‘time’ k are characterized by a
set of parameters Ak ≡ α1k, ..., αmk . Fixed parameter val-
ues define a particular universality class for the renormal-
ization. We suppose that, over a sequence of ‘times’, the
universality class properties can be characterized by a path
xn = A0, A1, ..., An-1 having significant serial correlations
10