Biomedical Engineering Reference
In-Depth Information
P
2
x
2
P
3
θ
2
x
(
t
)
P
1
θ
1
x
1
Fig. 2.6
Phase plane for the piecewise linear system of Eq. (
2.22
), with all parameters as in
Fig.
2.5
a (except
κ
10
=
κ
20
=0
and
m
i
=
∞
). The nullclines cannot be defined as in
the continuous model (
2.17
) but, instead, the threshold values
θ
1
,
θ
2
divide the plane into four
rectangular regions, where the vector field is constant. There are still two stable steady states
(
P
1
,
P
2
), but the unstable steady state is now defined as an unstable Filippov equilibrium (
P
3
).
One solution is shown in black, which may be compared to that shown in Fig.
2.5
a
P
1
and
P
2
belong to the boundary of their respective domains (
B
10
and
B
01
), so that
any trajectory entering one of these domains remains there. In contrast, trajectories
starting in
B
00
or
B
11
will switch to another domain. This leads to the following
transition graph for the bistable switch:
01
←−
11
00
−→
10
where
P
1
is represented by
,and
P
3
is not represented in this diagram,
as it is located in the middle, at the boundary of the four regular domains. This
discrete abstraction (in the sense of hybrid systems) is a qualitative description of
the behavior of the dynamical system. It can be used to check some qualitative
properties of the system. Software exist that are able to compute the graph and
check some of its properties, with
model checking
techniques.
10
,
P
2
by
01
2.2.3.4
Towards Control of Genetic Networks
An important problem is to be able to lead the system to a prescribed behavior. In
control theory, the input represents the actions that a user (here a biologist) is able to
exert on the system. From an experimental point of view, one common manipulation
is to change the synthesis rate of messenger RNA by addition of a plasmid (a small
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