Biology Reference
In-Depth Information
How about the lengths of the transients? Recall that
|
Act
t
|
is a nonincreasing
function of
t
0. Thus, for any trajectory there must be a smallest time
t
0
such that
|
t
0
. We can get a bound for the lengths of transients by
carefully considering how
t
0
is related to the refractory periods.
Exercise 6.9.
Let
N
Act
t
|=|
Act
t
0
|
for all
t
,
1
=
D
,
p
be a neuronal network with
D
=
D
c
(
n
)
. Consider
a trajectory and let
t
0
be defined as above. Prove that:
a.
The length of the transient is at most
t
0
+
p
∗
−
1.
b.
If
t
0
>
0, then there exists a node
i
with
s
i
(
t
0
−
1
)
=
0 and
s
(
i
+
1
)
%
n
(
t
0
)>
0.
p
∗
, then for all nodes
i
as in (b) we must have
p
i
c.
If
t
0
>
<
p
(
i
+
1
)
%
n
.
If all nodes have the same refractory period, then the scenario of point (c) of
Exercise
6.9
cannot occur, and point (b) implies that
t
0
p
∗
. In view of point (a),
this in turn implies the following result of [
7
].
Theorem 6.9.
,
1
Let N
=
D
,
p
be a neuronal network with D
=
D
c
(
n
)
, and
p
=
(
p
,...,
p
)
is constant. Then the length of any transient is at most
2
p
−
1
.
p
is not a constant vector, then the scenario of point (c) of Exercise
6.9
can
occur, which adds significant complications. Exercise 6.19 [
1
] invites the reader to
explore one such example and Theorem 3 of [
7
] gives a precise upper bound of the
length of transients for all networks
N
If
,
1
.
How about the sizes of the basins of attraction? At the time of this writing no
complete characterization for all cases was known. We invite our readers to explore
this problem on their own. Project 2 [
1
] gives some guidance.
=
D
c
(
n
),
p
6.4.3
Complete Loop-Free Digraphs
Both directed cycles and acyclic digraphs are characterized by “small” arc sets in
the sense that directed cycle graphs contain just enough arcs to allow for a directed
cycle that visits all nodes, and acyclic digraphs are missing the arcs that would close
a directed cycle. However, the word “small” here does not mean the same thing as
“few in number.”
Exercise 6.10.
Show that for every
n
>
1 there exists an acyclic digraph
D
=
2
, which is exactly half of the maximum number of
arcs in any loop-free digraph with
n
vertices.
with arc set of size
n
[
n
]
,
A
D
On the other extreme are
complete loop-free digraphs D
max
(
n
)
=[
n
]
,
A
D
whose
arc set
A
D
contains every pair
i
,
j
with
i
,
j
∈[
n
]
such that
i
=
j
. In this subsection
,
th
wewill explore the possible dynamics for networks of the form
N
=
D
max
(
n
),
p
.
=
th
=
1.
As before, let us first consider the simplest case when
p
),
1
,
1
Exercise 6.11.
Let
N
=
D
max
(
n
for some
n
.
a.
Show that along any trajectory we always have
Act
t
+
1
=[
n
]\
Act
t
, unless
Act
t
=∅
, in which case the system has reached the steady state.
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