Genetic Regulatory Networks: A Nonlinear Signal Processing Perspective - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 512

13.3.1 Attractors, Nonlinear Digital Filters, Associative Memory,

and Root Signals

Real genetic regulatory networks are highly stable in the presence of pertur-
bations of the genes. Within the Boolean network formalism, this means that
when a minimal number of genes change value (say, by means of some ex-
ternal stimulus), the system transitions into states that reside in the same
basin of attraction from which the network eventually flows back to the
same attractor. Generally speaking, large basins of attraction correspond to
higher stability. Such stability of networks in living organisms allows the cells
to maintain their functional state within the tissue environment. 22
Although in developmental biology, epigenetic, heritable changes in cell
determination have been well established, it is now becoming evident that
the same type of mechanisms may be also responsible in carcinogenesis and
that gene expression patterns can be inherited without the need for mutational
changes in DNA. 33 In the Boolean network framework, this can be explained
by hysteresis , which is a change in the system's state caused by a stimulus that
is not changed back when the stimulus is withdrawn. 22 Thus, if the change of
some particular gene does in fact cause a transition to a different attractor, the
network will often remain in the new attractor even if that gene is switched
off. Thus, the attractors of a Boolean network also represent a type of memory
of the dynamical system. 22
Virtually the same type of behavior is exhibited by many nonlinear digital
filters. Let us consider a popular class of nonlinear filters called stack filters ,
first introduced by Wendt et al. 34 1

is sliding across a binary-valued one-dimensional (1D) signal of arbitrary
length. At every location of the window, the contents inside the window are
used as input variables to some fixed Boolean function f . That is,

Suppose a window of length n = 2 m +

y i = f ( x i − m , ... ,x i , ... ,x i + m ) (13.2)

represents the output of the Boolean function corresponding to the window
centered on the i th value of the input signal. The sequence of outputs y i can be
thought of as an output signal of the filter. If this Boolean function is monotone
(positive), meaning that it can be written without complemented variables in
its disjunctive normal form,* then the filter defined by such a Boolean function
is a stack filter. It also corresponds to a neural network in which the weights of
all the threshold logic gates are nonnegative. In the real-valued domain, stack
filters can be defined as follows. Let X

be a real-valued
vector of observations comprising the contents of the filter window. Then a
stack filter S

= ( X 1 ,X 2 , ... ,X n )

( X ) is defined as

max min P K } ,

S ( X ) = { X j : j ∈ P 1 } , ... , min { X j : j ∈

* Thus, the disjunctive normal form contains only disjunctions and conjunctions. A monotone
Boolean function can also be defined as follows: f

( x 1 , ... ,x n ) is called monotone if for any two

, ˜

n

˜

α β ∈ { } α i ≤ β i for every i (1 ≤ ≤ ( α) ≤ ( β)

vectors ˜ 0 , 1 such that i n ), we have f ˜ f .