Biomedical Engineering Reference
In-Depth Information
a
b
c
Fig. 2.11 ( a ) The interaction graph of Fig. 2.9 together with bounds, logical thresholds and logical
parameters. ( b ) The table gives the focal point of i =1 , 2 according to the state of the system. This
table results from the one of Fig. 2.10 and the parameter values given in ( a ). ( c ) The asynchronous
state graph resulting from the data given in ( a ). This asynchronous state graph can be easily built
from the table given in ( b )
oscillations. It is easy to show that from any initial state, there always exists a path
leading to an attractor (and so, there always exists at least one attractor). It is in
this weak sense that one can consider that attractors perform an attraction. The state
graph of Fig. 2.11 contains a unique attractor, the stable state (2 , 1), and indeed,
from every initial state, there exists a path leading to this unique attractor.
2.3.2.2
Relationships with the Continuous Approaches
The dynamics of a network whose interaction graph is G may be described, using
the piecewise affine model (see Sects. 2.2.3.3 and 2.2.2.4 ), by the following system:
x i = κ i +
j∈G i
s ( s ji ) ( x j ji )
κ ji ·
γ i ·
x i
( i =1 ,...,n ) ,
(2.25)
where: κ i and γ i are the “basal” synthesis rate and the degradation rate of i ; G i is
the set of regulators j of i ; κ ji , θ ji and s ji
are the synthesis rate, the
quantitative threshold and the sign associated with the interaction j
∈{ + ,
−}
i ; s + and s
are the step functions defined in Sect. 2.2.2.4 . We will now describe how to obtain,
from the quantitative parameters κ i ij i and θ ij , the qualitative parameters b i ,t ij
and K i,Ω describing an asynchronous state graph abstracting the system ( 2.25 ).
First, for all i ,let Θ i =
{
θ ij |
i
G j }
be the set of “out-going” quantitative
thresholds of i ,andset
b i =c ard ( Θ i )
(first step) .
(2.26)
Then, consider the resulting set of discrete states X = i =1 { 0 , 1 ...,b i }
,andthe
discretization mapping
n
d : R
→ X,
d ( x )=( d 1 ( x 1 ) ,...,d n ( x n )) ,
i ( x i )=card( {θ ∈ Θ i |x i ≥ θ i } ) .
Search WWH ::




Custom Search