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derive bounds on these numbers for networks whose connectivities
D
belong to one
of several important classes of digraphs. These bounds will be expressed as functions
of the number
n
of neurons. We will usually need to make some assumptions on
p
p
and
th
∗
denotes max
th
.Weare
primarily interested in bounds that hold for all
n
and fixed values of
p
∗
and
th
∗
.
th
. Throughout this chapter
p
∗
denotes max
and
6.4.1
Acyclic Digraphs
A digraph
D
is
acyclic
if
D
does not contain any directed cycle. While none of the
digraphs in Figures
6.2
and
6.3
is acyclic, Figure
6.4
gives an example of an acyclic
digraph.
Lemma 6.5.
,
th
Let N
=
D
,
p
be a neuronal network whose connectivity D is
acyclic. Then
{
p
}
is the only attractor in N.
Proof.
Let
N
,
D
be as in the assumptions and assume
D
=[
n
]
,
A
D
. Suppose
there exists a periodic attractor
AT
in
N
, and let
s
(
0
)
be a state in this attractor. Move
the system forward to time
t
=
n
.ByExercise
6.2
, some node fires in
s
(
t
)
, that
is, there must exist
i
∈[
n
]
with
s
i
(
t
)
=
0. Fix such
i
=
i
(
t
)
. By the rules of the
network dynamics, there must exist another node
i
(
t
−
1
)
with
i
(
t
−
1
),
i
(
t
)
∈
A
D
such that
s
i
(
t
−
1
)
(
t
−
1
)
=
0. By recursion we can now construct a sequence of nodes
i
(
t
),
i
(
t
−
1
),...,
i
(
0
)
∈[
n
]
such that
s
i
(
t
−
k
)
(
t
−
k
)
=
0 for all
k
∈{
0
,...,
n
}
and
i
(
t
−
k
),
i
(
t
−
k
+
1
)
∈
A
D
for all
k
∈[
n
]
. Since there are only
n
nodes total, there
must exist
<
k
such that
i
()
=
i
(
k
)
and
(
i
(),
i
(
+
1
),...,
i
(
k
))
is a directed
cycle in
D
, which contradicts the assumption on
D
.
Lemma
6.5
makes networks with acyclic connectivities rather uninteresting from
our point of view, but investigating the class of acyclic digraphs will allow us to lay
some groundwork for subsequent explorations. First of all, notice that Lemma
6.5
completely characterizes all four features of the dynamics that we are investigating
here in that it implies points (a)-(d) of the following result.
FIGURE 6.4
An acyclic digraph on [7]. Arcs
i
,
j
∈
A
D
are represented by arrows
i
→
j
.
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