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in state space and forgets past distinctions. Finally, in a chaotic system, even small
stochastic uctuations may cause wide divergences in state space trajectories thus
precluding reliable action [Kauman (1969); Aldana et al. (2007)].
Previous studies have also demonstrated with simulated transient or permanent
mutations that biological networks are robust to noisy perturbations [von Dassow
et al. (2000); Albert and Othmer (2003); Espinosa-Soto et al. (2004)]. However,
besides recent studies demonstrating that a wide array of biological GRN are critical,
no study has shown that biological networks are robust to noisy perturbations and at
the same time optimize their capacity to explore alternative states in time and space.
Little is known as well as to which structural traits (topologies of interconnections
and structure of logical functions) are linked to the optimized capacity of GRN to
explore alternatives, and at the same time are able to retain previous attractors.
In this contribution we rst review our own work on simple model networks in
which we have particularly searched for generic dynamical behaviors in the face of
noise. Then we review our work on an experimentally grounded GRN, that consti-
tutes a model for the logic of necessary and sucient gene regulatory interactions
required for primordial cell fate determination during early stages of ower devel-
opment. We specically address if such network has a peculiar behavior in response
to noise in comparison to the simple model networks analyzed, and if such behavior
maximizes its capacity to explore the state space at the same time that it retains
the observed steady states under dierent noisy regimes.
5.2. Noisy Dynamics of Boolean Networks
Both model and real networks are constituted by units that process information
and are subject to some noise regime. Concerning the unit's dynamics, we assume
that the state of the units are Boolean variables. Boolean variables, which can take
one of two values, 0 (OFF) or 1 (ON), and Boolean functions have been extensively
used to model the state and dynamics of complex systems | see [Kaplan and Glass
(1997)] for an introduction. Even though Boolean networks are oversimplications
of real systems, it has been shown that Boolean functions are good approximations
to the nonlinear functions encountered in many control systems [Kauman (1993);
Weng et al. (1999); Aldana et al. (2003); Wolfram (2002)]. For instance, Random
Boolean networks (RBNs) were proposed by Kauman (1993) as models of genetic
regulatory networks, and have also been studied in a number of other contexts
[Weng et al. (1999); Aldana et al. (2003)]. Wolfram (2002), in contrast, proposed
that cellular automata (CA) models | a class of ordered Boolean networks with
identical units | may explain the real-world's complexity.
In general, it is assumed that the state
i
(t + 1) of unit i at time t + 1 depends
on the state of the set of its neighbors | including itself | at time t as
h
i
1
(t);
i
2
(t); : : : ;
i
k
i
(t)
i
i
(t + 1) =F
i
;
(5.1)
