Biology Reference
In-Depth Information
Example
6.3
(c) shows that the condition on
s
(
0
)
in Exercise
6.2
is sufficient, but
not necessary, for
s
(
0
)
to be in the basin of attraction of the steady state attractor.
Definition 6.4.
Let
D
=
V
D
,
A
D
be a digraph. A
directed path
in
D
from
v
1
to
v
k
is
a sequence
(v
1
,v
2
,...,v
k
)
of vertices
v
∈
V
D
such that
v
,v
+
1
∈
A
D
for all
∈
[
k
−
1
]
, and all vertices in the sequence, except possibly
v
1
and
v
k
, are pairwise distinct.
A
directed cycle
in
D
is a directed path
v
1
=
v
k
.The
length
of
a directed path (or cycle) is the number of arcs that connect its successive vertices.
For example, in the digraph of Figure
6.2
b, the sequence (1, 5, 2, 7, 3) is a directed
path of length 4 and the sequence (2, 7, 6, 4, 2) is a directed cycle of length 4. Two
directed cycles will be considered disjoint if their set of vertices are disjoint. In particu-
lar, the directed cycles (7, 3, 5, 7) and (2, 7, 6, 4, 2) in this digraph are not disjoint despite
the fact that they do not use any common arcs. We will usually consider directed
cycles that differ by a cyclic shift as identical; for example, (7, 3, 5, 7), (3, 5, 7, 3), and
(5, 7, 3, 5) would be considered three representations of the same directed cycle.
Exercise 6.3.
Find all directed cycles in the digraph of Figure
6.2
a. What is the
maximum length of a directed path in this digraph?
(v
1
,v
2
,...,v
k
)
with
,
th
=
,
we can define another digraph
D
N
whose vertices are the states and whose arcs indicate the
successor state
for each state,
that is,
For any neuronal network
N
D
p
s
∗
s
∗
(
.
This digraph will be called the
state transition digraph
of the network. Note that
if
s
,
is an arc in this digraph if and only if
s
(
t
)
=
s
implies
s
(
t
+
1
)
=
t
)
=
1
, then we can represent each state of a network
N
very
conveniently by the set of all nodes that fire in this state. If the number of nodes is
not too large, this allows us to actually draw
D
N
. Figure
6.3
gives an example.
Notice that the state transition digraph
D
N
of a network
N
p
=
(
1
,
1
,...,
1
)
,
th
is different
from the
network connectivity D.
For example,
D
N
has many more nodes than
D
(2
7
=
D
,
p
128 vs. 7 in the example of Figures
6.3
and
6.2
b). Moreover, while we are
always assuming that the network connectivity
D
is loop-free, the state transition
digraph contains the loop
=
p
,
p
.
=
1 the state transition digraph
D
N
must be drawn in such a way that each
state is actually represented by its state vector. This may still be possible for very
small
n
,asinExercise
6.5
.
State transition digraphs, as long as we can draw them, give a complete visual
picture of the network dynamics. Let us take a closer look at Figure
6.3
. The steady
state is represented by the empty set of its firing nodes and forms a loop
If
p
,thatis,a
directed cycle of length one. There are six directed cycles of length 2 each, and there is
one directed cycle of length 5. These directed cycles of lengths larger than 1 correspond
to periodic attractors of the network. All nodes outside of the union of these directed
paths are transient states. For each transient state
∅
,
∅
s
there exists a unique directed path
from
s
to a state in some attractor that visits no other persistent states. The basin of
attraction of each attractor consists of all states fromwhich the attractor can be reached
via a directed path. The attractors {(124), (3567)}, {(147), (2356)}, {(157), (2346)}
and the steady state attractor
{∅}
form their own basins of attraction; the basins of
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