Biology Reference
In-Depth Information
attractor a firing of node
i
at time
t always will
induce a firing of node
(
i
+
1
)
%
n
at
time
t
+
1. If
t
0
is such that
|
Act
t
+
t
0
|=|
Act
t
0
|
for all
t
0, then for all
t
0wehave
Act
t
+
t
0
={
(
i
+
t
)
%
n
:
i
∈
Act
t
0
}
.
(6.3)
Exercise 6.8.
Prove that there exists a divisor
m
of
n
such that
Act
t
0
+
m
=
For node
i
to fire at time
t
, the rules of the dynamics require that it must be in state
p
i
at time
t
Act
t
0
.
−
1. After node
i
has fired at time
t
(formally:
s
i
(
t
)
=
0), its state will increase
until it reaches the end of its refractory period:
s
i
(
t
+
1
)
=
1
,...,
s
i
(
t
+
p
i
)
=
p
i
.
This implies that
m
1for
m
as in Exercise
6.8
, that is, the distance between
two firing cells is at least
p
i
+
p
i
+
1. In particular, this will be true for
i
of maximum
p
∗
. These observations prove the following.
refractory period
p
i
=
,
1
Proposition 6.7.
Let N
=
D
,
p
be a neuronal network with D
=
D
c
(
n
)
. Then
n
p
∗
+
the length
|
AT
|
of any attractor is a divisor of n and
|
Act
t
|
1
for all t
>
0
.
p
∗
+
Moreover
1
.
How many
different
attractors are there in
N?
It follows from our proof of Proposition
6.7
that along any attractor the state
s
i
(
,
if
|
AT
|
>
1
,
then
|
AT
|
)
of any node will be uniquely determined by its refractory period and the set
Act
t
.
We may consider each attractor with fixed
t
|
Act
t
|=
k
>
0 as a necklace with
k
red
p
∗
+
beads and
n
−
k
(
1
)
black beads. Pick a state
s
in the attractor. Each red bead
corresponds to a block of
p
∗
+
1 positions of
s
of the form
(
∗
,...,
∗
,
0
)
, where the
wildcards
correspond to the stages in the refractory period of the corresponding
neuron, but not to firing. This type of block represents one firing together with the
mandatory waiting period that must precede the next firing. Black beads correspond to
optional extensions of the minimal waiting period between two consecutive elements
of
Act
t
. There is a one-to-one correspondence between periodic attractors in
N
and
this type of necklace. Now the well-known formula for the number of different such
necklaces (e.g., Theorem 1 in [
6
]) plus the fact that there exists exactly one steady
state attractor implies the following result .
Theorem 6.8.
∗
,
1
Let N
=
D
,
p
be a neuronal network with D
=
D
c
(
n
)
. Then
the number of different attractors is
n
−
kp
∗
a
k
a
n
p
∗
+
1
1
1
+
φ(
a
)
,
(6.4)
kp
∗
n
−
p
∗
+
k
=
1
a
∈{
di
v
isors of gcd
(
k
,
n
−
k
(
1
))
}
where
is Euler's phi function.
In particular, the number of attractors in neuronal networks
N
φ
,
1
grows exponentially in
n
. (See Corollary 2 in [
7
].) In order to get a feel for the depen-
dence of this number on
n
, the reader may want to do some numerical explorations
as suggested in Exercise 6.18 [
1
].
=
D
c
(
n
),
p
Search WWH ::
Custom Search