Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 52

x 2

f B (x) − γ x

F(x) − γ x

A

q 2

x 1

B

f θ 2 (x) − γ x

f A (x) − γ x

Fig. 2.3 Sliding mode solution. These type of solutions may occur when the vector fields in
regions A and B point in opposite directions ( f A , f B ). At the boundary of A and B (the segment
x 2 = θ 2 ), the vector field may be defined as a convex combination of the two vector fields:
F ( x )= αf A ( x )+(1 − α ) f B ( x ) . The values of α range between [0 , 1] , forming the convex
hull cone. A sliding mode solution, with vector field f θ 2 ( x ) , can be found by setting x 2 = θ 2 and
F 2 ( x 1 ,θ 2 ) − γθ 2 =0 , and computing the appropriate value for α

where x =( x 1 ,...,x n ) t is a non-negative vector of variables. The non-negative
quantities f i ( x ) and γ i x i represent production and loss (or transformation) rates
for each variable x i . The functions f i : R

+ → R + will be constant in rectangular
regions of the state space whose boundaries will be called thresholds .The( n

n

− 1)-
dimensional hyperplanes defined by these thresholds partition the state space into
hyper-rectangular regions which are called domains or boxes (see an example
in Sect. 2.2.3.3 ). For any domain D , the function f ( x )=( f 1 ( x ) ,...,f n ( x )) is
constant for all x

D , and it follows that the PWA system can be written as an
affine vector field x = f D

∈

− γx, x ∈ D where f D is constant in D .

Ω is called the focal point for the flow in D ,
and globally attracts the trajectories until they reach the boundaries of the domain.
The focal points define the possible transitions associated with the domain D ; the
transition graph describes these transitions and gives the qualitative behavior of
the system. This graph can be efficiently computed, and its properties analyzed
(see the example in Sect. 2.2.3.3 ).
On the thresholds, the solutions have to be appropriately defined, typically
through a construction due to Filippov. This construction considers all the solutions,
as if the step function could take all the values of the interval [0 , 1] on the threshold.
To be more explicit, let n =2and consider two re gular domains, A and B , separated
by one threshold ( x 2 = θ 2 ), as in Fig. 2.3 .Letco denote the closed convex hull of a
set of vector fields. We define the differential inclusion

The point φ ( D )= γ − 1 f D ∈

x ∈ H ( x ) , (2.5)

with

H ( x )=co f D ( x ) − ∂D , if x γx : D s D s , a switching domain

∈ ∈