Biology Reference
In-Depth Information
b.
Use the result of (a) to find the maximum length of any transient and any attractor,
the number of different attractors, and the size of each basin of attraction.
=
1, things become more interesting. The result of Exercise
6.11
(a) generalizes as follows:
Proposition 6.10.
When we allow for
p
,
1
Let N
=
D
max
(
n
),
p
for some n. Then along any trajectory
we always have Act
t
+
1
={
i
∈[
n
]:
s
i
(
t
)
=
p
i
}
if Act
t
=∅
,
and Act
t
+
1
=∅
whenever Act
t
=∅
.
Proposition
6.10
allows for an easy characterization of all features of the dynamics
when
th
=
1.
p
is constant and
,
1
Exercise 6.12.
Let
N
=
D
max
(
n
),
p
for some
n
, where
p
=
(
p
,...,
p
)
is
constant.
a.
Show that the basin of attraction of the steady state consists of all vectors
s
such
that for at least one
q
∈{
0
,...
p
}
we have
{
i
∈[
n
]:
s
i
=
q
}=∅
.
b.
Show that every periodic attractor has length
p
+
1 and forms its own basin of
attraction.
c.
Find a formula that relates the number of different attractors to the number of
functions that map
.
d.
Find a formula for the size of the basin of attraction of the steady state attrac-
tor in the spirit of (c) and also find a formula for the maximum length of any
transient.
[
n
]
onto
{
0
,
1
,...,
p
}
Let us introduce here some terminology that we will use later in this and the next
section.
Definition 6.11.
,
th
Let
N
=
D
,
p
be a network, let
i
∈
V
D
be a node, and let
s
(
0
)
∈
St
N
be an initial state. We say that
i
is
minimally cycling (in the trajectory of
s
0 holds.
Let
AT
be an attractor in
N
. We say that node
i
is
minimally cycling in AT
if it
is minimally cycling in every trajectory that starts in
AT
. A node
i
is
active in AT
if
it fires in at least one state of
AT
. We say that the attractor
AT
is
minimal
if every
active node in
AT
is minimally cycling, and we call
AT fully active
if every node of
the network is active in
AT
.
The following exercise gives a nice and useful condition on initial states whose
trajectories contain at least one minimally cycling node.
Exercise 6.13.
Let
N
(
0
)
)
if for every
t
0 the implication
s
i
(
t
)
=
p
i
⇒
s
i
(
t
+
1
)
=
,
1
,
1
=
D
and let
s
(
0
)
∈
St
N
.
a.
Prove that if there exists a directed cycle
C
=
(
i
1
,
i
2
,...,
i
2
k
,
i
1
)
in
D
such that
s
i
j
(
0
)
=
j
mod 2 for all
j
∈[
2
k
]
, then there exists a minimally cycling node in
the trajectory of
s
(
0
)
.
b.
Is the condition in point (a) also necessary for the existence of aminimally cycling
node?
Search WWH ::
Custom Search