(16)
and apply Landau’s spontaneous symmetry breaking argu-
ment to generate punctuated changes in the linkage between
the genetic information source XDi and the embedding epi-
genetic information source Z. Recall that Landau’s insight
was that certain phase transitions were usually in the context
of a significant symmetry change in the physical states of a
system.
Again, the biological renormalization schemes of the Ap-
pendix to Wallace and Wallace (2008) may now be imposed
on FR [K] itself, leading to a spectrum of highly punctuated
transitions in the overall system of interacting biological sub-
structures.
Since 1/K is proportional to the embedding metabolic free
energy, we assert that
[1] the greatest possible set of symmetries will be realized
for high developmental metabolic free energies, and
[2] phase transitions, related to total available developmen-
tal metabolic free energy, will be accompanied by fundamental
changes in the final topology of the system of interest - phe-
notype changes - recognizing that evolutionary selection acts
on phenotypes, not genotypes.
The relation 1/K = κMτ suggests the possibility of evolu-
tionary tradeoffs between development time and the rate of
available metabolic free energy.
8 More topology
It seems possible to extend this treatment using standard
topological arguments.
Taking T = 1/K in equations (6) and (14) as a product of
eigenvalues, we can define it as the determinant of a Hessian
matrix representing a Morse Function, f , on some underlying,
background, manifold, M, characterized in terms of (as yet
unspecified) variables X = (x1 , ..., xn), so that
1/K = det(Hi,j),
Hi,j ≡ ∂2f /∂xi∂xj .
(17)
See the Appendix for a brief outline of Morse Theory.
Thus κ, M , and the development time τ are seen as eigen-
values of H on the manifold M in an abstract space defined
by some set of variables X .
By construction H has everywhere only nonzero, and in-
deed, positive, eigenvalues, whose product thereby defines T
as a generalized volume. Thus, and accordingly, all critical
points of f have index zero, that is, no eigenvalues of H are
ever negative at any point, and hence at any critical point Xc
where df (Xc) = 0.
This defines a particularly simple topological structure for
M: If the interval [a, b] contains a critical value of f with
a single critical point Xc, then the topology of the set Mb
defined above differs from that of Ma in a manner determined
by the index i of the critical point. Mb is then homeomorphic
to the manifold obtained from attaching to Ma an i-handle,
the direct product of an i-disk and an (m - i)-disk.
One obtains, in this case, since i = 0, the two halves of
a sphere with critical points at the top and bottom (Mat-
sumoto, 2002; Pettini, 2007). This is, as in Ciliberti et al.
(2007), a simply connected object. What one does then is to
invoke the Seifert-Van Kampen Theorem (SVKT, Lee, 2000)
and patch together the various simply connected subcompo-
nents to construct the larger, complicated, topological object
representing the full range of possibilities.
The physical natures of κ, M , and τ thus impose constraints
on the possible complexity of this system, in the sense of the
SVKT.
9 Inherited epigenetic memory
The cognitive paradigm for gene expression invoked here re-
quires an internal picture of the world against which incoming
signals are compared - algorithmically combined according to
the rules of Section 3 - and then fed into a nonlinear decision
oscillator that chooses one (or a few) action(s) from a much
large repertoire of possibilities. Memory is inherent, and much
recent work, as described in the introduction, suggests that
epigenetic memory is indeed heritable.
The abduction of spinglass and other models from neural
network studies to the analysis of development and its evolu-
tion carries with it the possibility of more than one system of
memory. What Baars called ‘contexts’ channeling high level
animal cognition may often be the influence of cultural in-
heritance, in a large sense. Our formalism suggests a class
of statistical models that indeed greatly generalize those used
for measuring the ‘effects of cultural inheritance on human
behavior in populations’.
Epigenetic machinery, as a dual information source to a
cognitive process, serves as a heritable system, intermediate
between (relatively) hard-wired classical genetics, and a (usu-
ally) highly Larmarckian embedding cultural context. In par-
ticular, the three heritable systems interact, in our model,
through a crosstalk in which the epigenetic machinery acts as
a kind of intelligent catalyst for gene expression.
10 Multiple processes
The argument to this point has, in large measure, been di-
rectly abducted from recent formal studies of high level cog-
nition - consciousness - based on a Dretske-style informa-
tion theoretic treatment of Bernard Baars’ global workspace
model (Wallace, 2005; Atmanspacher, 2006). A defining and
grossly simplifying characteristic of that phenomenon is its
10