energy density as defined above, i.e., from the similarity with
the relation H = limn→∞ log[N (n)]/n.
Ash’s comment above then has an important corollary: If,
for a biological system, H1 > H2 , source 1 will require more
metabolic free energy than source 2.
constant κ, so that Mτ is total developmental free energy.
Then the probability of a particular Hα will be determined
by the standard relation (e.g., Landau and Lifshitz, 2007),
5 Symmetry arguments
A formal equivalence class algebra, in the sense of the Ap-
pendix, can now be constructed by choosing different origin
and end points S0 , S∞ and defining equivalence of two states
by the existence of a high probability meaningful path con-
necting them with the same origin and end. Disjoint partition
by equivalence class, analogous to orbit equivalence classes for
dynamical systems, defines the vertices of the proposed net-
work of cognitive dual languages, much enlarged beyond the
spinglass example. We thus envision a network of metanet-
works, in the sense of Ciliberti et al. Each vertex then repre-
sents a different equivalence class of information sources dual
to a cognitive process. This is an abstract set of metanetwork
‘languages’ dual to the cognitive processes of gene expression
and development.
This structure generates a groupoid, in the sense of We-
instein (1996). States aj , ak in a set A are related by the
groupoid morphism if and only if there exists a high prob-
ability grammatical path connecting them to the same base
and end points, and tuning across the various possible ways
in which that can happen - the different cognitive languages
- parametizes the set of equivalence relations and creates the
(very large) groupoid.
There is a hierarchy here. First, there is structure within the
system having the same base and end points, as in Ciliberti et
al. Second, there is a complicated groupoid structure defined
by sets of dual information sources surrounding the variation
of base and end points. We do not need to know what that
structure is in any detail, but can show that its existence has
profound implications.
First we examine the simple case, the set of dual informa-
tion sources associated with a fixed pair of beginning and end
states.
5.1 The first level
The spinglass model of Ciliberti et al. produced a simply con-
nected, but otherwise undifferentiated, metanetwork of gene
expression dynamics that could be traversed continuously by
single-gene transitions in the highly parallel w-space. Taking
the serial grammar/syntax model above, we find that not all
high probability meaningful paths from S0 to S∞ are actually
the same. They are structured by the uncertainty of the as-
sociated dual information source, and that has a homological
relation with free energy density.
Let us index possible dual information sources connecting
base and end points by some set A = ∪α. Argument by
abduction from statistical physics is direct: Given metabolic
energy density available at a rate M , and an allowed devel-
opment time τ, let K = 1∕κMτ for some appropriate scaling
(5)
P [Hβ] =
exp[-Hβ K ]
Pα exp[-HαK],
where the sum may, in fact, be a complicated abstract inte-
gral. The basic requirement is that the sum/integral always
converges. K is the inverse product of a scaling factor, a
metabolic energy density rate term, and a characteristic de-
velopment time τ . The developmental energy might be raised
to some power, e.g., K = 1∕(κ(Mτ)b), suggesting the possi-
bility of allometric scaling.
Thus, in this formulation, there must be structure within a
(cross sectional) connected component in the w-space of Cilib-
erti et al., determined in no small measure by available energy.
Some dual information sources will be ‘richer’/smarter than
others, but, conversely, must use more metabolic energy for
their completion.
The next generalization is crucial:
While we might simply impose an equivalence class struc-
ture based on equal levels of energy/source uncertainty, pro-
ducing a groupoid in the sense of the Appendix (and possibly
allowing a Morse Theory approach in the sense of Matsumoto,
2002 or Pettini, 2007), we can do more by now allowing both
source and end points to vary, as well as by imposing energy-
level equivalence. This produces a far more highly structured
groupoid that we now investigate.
5.2 The second level
Equivalence classes define groupoids, by standard mechanisms
(e.g., Weinstein, 1996; Brown, 1987; Golubitsky and Stewart,
2006). The basic equivalence classes - here involving both in-
formation source uncertainty level and the variation of S0 and
S∞, will define transitive groupoids, and higher order systems
can be constructed by the union of transitive groupoids, hav-
ing larger alphabets that allow more complicated statements
in the sense of Ash above.
Again, given an appropriately scaled, dimensionless, fixed,
inverse available metabolic energy density rate and devel-
opment time, so that K = 1∕κMτ, we propose that the
metabolic-energy-constrained probability of an information
source representing equivalence class Di, HDi , will again be
given by the classic relation
P [HDi] =
exp[-HDi K ]
Pj exp[-HDj K],