Biomedical Engineering Reference
In-Depth Information
ited before. From then on, it must cycle on the same loop forever. This means
that, strictly speaking, all attractors on any Boolean network are periodic limit
cycles. When
N
is very large, however, 2
N
is astronomically huge and these cy-
cles can become extremely long.
A surprising aspect of RBNs is the existence of two qualitatively different
behaviors for different parameter regimes. A parameter
q
1
can be defined that
corresponds to the probability that changing the value of one randomly selected
input to a randomly selected gate will result in a change in the output of that gate
(27). A qualitative change in the network behavior is observed as
q
1
is varied,
which can be accomplished by changing
K
or changing the weights of the dif-
ferent truth tables. For small values of
q
1
, typical networks have only a few at-
tractors; almost all of the gates wind up stuck on one value or the other and the
duration of the attractor cycles are short. For larger values of
q
1
, a number of
gates of order
N
remain active and the cycles are extremely long. The attractors
in the two regimes also have markedly different stability properties. In the case
of small
q
1
, small externally imposed perturbations, like changing the output
value of a single gate for one time step, have little effect. The system quickly
returns to the original attractor. For large
q
1
, on the other hand, small perturba-
tions often place the system in the basin of a different attractor. The regime in
which one observes short, stable cycles is called "ordered," and the region with
exponentially long, attractors that are sensitive to small perturbations is called
"chaotic." The latter term is meant to emphasize the erratic nature of the attrac-
tors over many times steps, but is not a rigorous description of the attractors
over the tremendously long times associated with their cycle durations.
RBNs at the critical value of
q
1
exhibit a unique balance of attractor stability
and flexibility (15). The discovery of these special and totally unanticipated
properties of critical RBNs is an indication of the power of the nonlinear dynam-
ics conceptual framework. Even though these specific RBN models are not
faithful representations of real biological processes, they reveal nonlinear dy-
namical structures that are likely to arise also in models that incorporate more
realistic details, and therefore suggest new ways of understanding of the inte-
grated behavior of the genome.
6.
DISCUSSION AND CONCLUSIONS
This chapter is intended only to establish some of the vocabulary of nonlin-
ear dynamics and give some indication of the rich behaviors that fall within its
domain. Many important phenomena have been neglected entirely to this point.
Three stand out as requiring some comment, however brief: the effects of sto-
chastic processes; the role of boundary conditions; and the phenomena of fre-
quency locking and synchronization.
