GENE EXPRESSION AND ITS DISCONTENTS Developmental disorders as dysfunctions of epigenetic cognition

where ∆t is a constant and σ a sigmodial function whose
value lies in the interval (-1, 1). In the spinglass limit σ is
the sign function, taking only the values ±1.

The networks of interest are those whose expression state
begins from a prespecified initial state S(0) at time t = 0 and
converge to a second prespecified stable equilibrium state S∞.
Such a network is termed viable, for obvious reasons. Such
viable networks, of course, comprise a tiny fraction of possible
ones, i.e., those that do not begin with S0 and end at S∞.

The model used by Ciliberti et al. is abstracted from sp-
inglass treatments of neural networks, as is made clear in
the seminal papers by the Reinitz group (e.g., Jaeger et al,
2004; Mjolsness et al., 1991; Reinitz and Sharp, 1995; Sharp
and Reinitz, 1998; Toulouse et al., 1986). Thus and conse-
quently, Ciliberti et al. are invoking an implicit cognitive
paradigm for gene expression (e.g., Cohen, 2006; Cohen and
Harel, 2007; Wallace and Wallace, 2008), and cognitive pro-
cess, as the philosopher Fred Dretske (1994) eloquently ar-
gues, is constrained by the necessary conditions imposed by
the asymptotic limit theorems of information theory.

The next sections use information theory methods to make
the transition from crossectional w-space into that of serially
correlated sequences of phenotypes, expanding the results of
Wallace and Wallace, (2008).

3 Cognition as  an information

source

Atlan and Cohen (1998) argue, in the context of immune cog-
nition, that the essence of cognitive function involves compar-
ison of a perceived signal with an internal, learned picture of
the world, and then, upon comparison, choosing a response
from a much larger repertoire of possible responses.

Such choice inherently involves information and informa-
tion transmission since it always generates a reduction in un-
certainty (e.g., Ash 1990, p. 21).

More formally, a pattern of incoming input - like the S(t)
of equation (1) - is mixed in a systematic algorithmic manner
with a pattern of internal ongoing activity - like the (wij ) ac-
cording to equation (1) - to create a path of combined signals
x = (a₀, a₁, ..., a_n, ...) - analogous to the sequence of S(t + ∆t)
of equation (1), with, say, n = t∕∆t. Each a_k thus represents
some functional composition of internal and external signals.

This path is fed into a highly nonlinear decision oscillator,
h, which generates an output h(x) that is an element of one
of two disjoint sets B0 and B1 of possible system responses.
Let

h(x) ∈ B0,

the pattern is not recognized, and if

h(x) ∈ B1,

the pattern is recognized, and some action bj , k+ 1 ≤ j ≤ m
takes place.

The principal ob jects of formal interest are paths x trigger-
ing pattern recognition-and-response. That is, given a fixed
initial state a₀ , examine all possible subsequent paths x be-
ginning with a₀ and leading to the event h(x) ∈ B₁. Thus
h(a0, ..., aj) ∈ B0 for all 0 < j < m, but h(a0, ..., a_m) ∈ B1 .

For each positive integer n, let N(n) be the number of
high probability grammatical and syntactical paths of length
n which begin with some particular a0 and lead to the condi-
tion h(x) ∈ B₁ . Call such paths ‘meaningful’, assuming, not
unreasonably, that N(n) will be considerably less than the
number of all possible paths of length n leading from a0 to
the condition h(x) ∈ B1.

While the combining algorithm, the form of the nonlin-
ear oscillator, and the details of grammar and syntax are all
unspecified in this model, the critical assumption which per-
mits inference of the necessary conditions constrained by the
asymptotic limit theorems of information theory is that the
finite limit

H ≡ lim lθg[N(n)l
n→∞   n

(2)

both exists and is independent of the path x.

Define such a pattern recognition-and-response cognitive
process as ergodic. Not all cognitive processes are likely to be
ergodic in this sense, implying that H, if it indeed exists at
all, is path dependent, although extension to nearly ergodic
processes seems possible (Wallace and Fullilove, 2008).

Invoking the spirit of the Shannon-McMillan Theorem, as
choice involves an inherent reduction in uncertainty, it is
then possible to define an adiabatically, piecewise station-
ary, ergodic (APSE) information source X associated with
stochastic variates Xj having joint and conditional probabili-
ties P (a0, ..., an) and P (an|a0, ..., an-1 ) such that appropriate
conditional and joint Shannon uncertainties satisfy the classic
relations

B₀ ≡ b₀, ..., b_k,

H[X] = lim ^log[N(ⁿ>^l =
n→∞ n

B1 ≡ bk+1,..., bm.

Assume a graded response, supposing that if                               _n^l_→^im_∞ ^H(^Xn|^X0^{, ..., X}n-1 ) ⁼

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