Figure 2: Given an initial developmental state S0 and a criti-
cal period C casting a path-dependent developmental shadow,
there are two directed homotopy equivalence classes of de-
formable paths leading, respectively, to final phenotype states
S1 and S2 that are expressions of the highly dynamic infor-
mation source solutions to equation (19). These equivalence
classes define a topological groupoid on the developmental
system.
15 Directed homotopy
First, we restrict the analysis to a two dimensional phenotype
space in the sense of Section 2 above, and begin development
at some S0 as in figure 2.
If one requires temporal path dependence - no reverse de-
velopment - then figure 2 shows two possible final states, S1
and S2 , separated by a critical point C that casts a path-
dependent developmental shadow in time. There are, conse-
quently, two separate ‘ways’ of reaching a final state in this
model. The Si thus represent (relatively) static phenotypic
expressions of the solutions to equation (19) that are, of them-
selves, highly dynamic information sources.
Elements of each ‘way’ can be transformed into each other
by continuous deformation without crossing the impenetrable
shadow cast by the critical period C.
These ways are the equivalence classes defining the system’s
topological structure, a groupoid analogous to the fundamen-
tal homotopy group in spaces that admit of loops (Lee, 2000)
rather than time-driven, one-way paths. That is, the closed
loops needed for classical homotopy theory are impossible for
this kind of system because of the ‘flow of time’ defining the
output of an information source - one goes from S0 to some
final state. The theory is thus one of directed homotopy, di-
homotopy, and the central question revolves around the con-
tinuous deformation of paths in development space into one
another, without crossing the shadow cast by the critical pe-
riod C. Goubault and Rausssen (2002) provide another in-
troduction to the formalism.
Thus the external signals U of equation (21), as a catalytic
mechanism, can define quite different developmental dihomo-
topies.
Such considerations suggest that a multitasking develop-
mental process that becomes trapped in a particular pat-
tern cannot, in general, expect to emerge from it in the ab-
sence of external forcing mechanisms or the stochastic res-
onance/mutational action of ‘noise’. Emerging from such a
trap involves large-scale topological changes, and this is the
functional equivalent of a first order phase transition in a
physical systems and requires energy.
The fundamental topological insight is that environmental
context - the U in equation (21) - can be imposed on the ‘nat-
ural’ groupoids underlying massively parallel gene expression.
This sort of behavior is, as noted in Wallace (2008b), central
to ecosystem resilience theory.
Apparently the set of developmental manifolds, and its sub-
sets of directed homotopy equivalence classes, formally clas-
sifies quasi-equilibrium states, and thus characterizes the dif-
ferent possible developmental resilience modes. Some of these
may be highly pathological.
Shifts between markedly different topological modes appear
to be necessary effects of phase transitions, involving analogs
to phase changes in physical systems.
It seems clear that both ‘normal development’ and possi-
ble pathological states can be represented as topological re-
silience/phase modes in this model, suggesting a real equiva-
lence between difficulties in carrying out gene expression and
its stabilization. This mirrors recent results on the relation
between programming difficulty and system stability in highly
parallel computing devices (Wallace, 2008a).
16 Epigenetic programming of arti-
ficial systems for biotechnology
Wallace (2009) examines how highly parallel ‘Self-X’ comput-
ing machines - self-programming, protecting, repairing, etc. -
are inevitably coevolutionary in the sense of Section 11 above,
since elements of a dynamic structural hierarchy always in-
teract, an effect that will asymptotically dominate system
behavior at great scale. The ‘farming’ paradigm provides a
model for programming such devices, that, while broadly sim-
ilar to the liquid state machines of Maas et al. (2002), differs
in that convergence is to an information source, a system-
atic dynamic behavior pattern, rather than to a computed
fixed ‘answer’. As the farming metaphor suggests, stabiliz-
ing complex coevolutionary mechanisms appears as difficult
as programming them. Sufficiently large networks of even the
most dimly cognitive modules will become emergently coevo-
lutionary, suggesting the necessity of ‘second order’ evolution-
ary programming that generalizes the conventional Nix/Vose
models.
Although we cannot pursue the argument in detail here,
very clearly such an approach to programming highly par-
allel coevolutionary machines - equivalent to deliberate epi-
genetic farming - should be applicable to a broad class of
artificial biological systems/machines for which some particu-
lar ongoing behavior is to be required, rather than some final
17