Biology Reference
In-Depth Information
,
th
Theorem 6.6.
Let N
=
D
,
p
be a network whose connectivity D is acyclic.
Then
a.
the maximum length of each attractor in N is 1,
b.
the maximum number of different attractors in N is 1,
c.
the size of the basin of attraction of the steady state attractor is
|
St
N
|=
i
=
1
(
p
i
+
)
1
,
p
∗
−
d.
+
the transients have length at most n
1
.
Exercise 6.7.
Prove part (d).
Hint:
Use arguments similar to the proof of Lemma
6.5
and the solution of Exercise
6.2
.
The wording of Theorem
6.6
certainly looks overly formal for such a simple result,
but it provides a nice template for the kinds of results we might hope to be able to
prove for certain classes of digraphs. The bound in point (d) is sharp in the class of
all networks with acyclic connectivities, which can be easily seen by considering the
trivial case of a digraph with just one node and no arcs. However, this bound can be
improved under some additional assumptions on the connectivity. Project 1 of the
online supplement [
1
] gives some guidelines for readers who wish to prove stricter
bounds under additional assumptions.
Let us take another look at Lemma
6.5
. It tells us that the existence of a directed
cycle in the connectivity
D
is a
necessary
condition for the existence of a periodic
attractor in a network
N
,
th
, but it does not answer the question whether
the existence of a directed cycle in
D
is a
sufficient
condition. This type of question
is best explored by looking first at the simplest possible connectivities that contain a
directed cycle. Our next subsection will investigate the possible dynamics for these
types of connectivities.
=
D
,
p
6.4.2
Directed Cycle Graphs
Let
n
be a positive integer. For convenience, we will write
i
%
n
for
((
i
−
1
)
mod
n
)
+
1.
Let
D
c
(
n
)
be the
directed cycle graph
on
[
n
]
with arc set
A
c
(
n
)
={
i
,(
i
+
1
)
%
n
:
i
∈
[
. Throughout this subsection we will tacitly assume that we are given a neuronal
network
N
n
]}
,
th
for some positive integer
n
.
In such networks, node
i
gets input
only
from node
=
D
,
p
with
D
=
D
c
(
n
)
(
i
−
1
)
%
n
.Thus,if
i
∈[
n
]
is such that
th
i
>
1, then
s
i
(
t
)
=
0 for all
t
>
0 and each trajectory will eventually
th
=
1
reach the steady state
p
. Therefore, we will focus here exclusively on the case
which allows for more interesting dynamics.
Let
s
(
0
)
be a given initial state. For all
t
0let
Act
t
={
∈[
]:
s
i
(
)
=
}
i
n
t
0
(6.2)
be the set of neurons that fire at time
t
. We call
Act
t
the
active set at time t.
Note that for cyclic connectivities, a firing of node
i
at time
t can
induce at most
one firing at time
t
+
1, and it follows that
|
Act
t
| |
Act
t
+
1
|
along any trajectory,
which in turn implies that
|
Act
t
|
is constant on any attractor. Thus throughout any
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