Biology Reference
In-Depth Information
)
of each neuron needs to specify whether or not the neuron fires at time t and whether
it has rested long enough to fire again. We code this information as follows: s i (
p
= (
p 1 ,...,
p n )
which specifies the refractory periods of all neurons. The state s i (
t
t
) =
0
means that the neuron fires at time t ; more generally, a state s i (
t
)<
p i signifies that
neuron i fired s i (
t
)
time steps earlier, that is, in episode number t
s i (
t
)
and is not
yet ready to fire in episode t
p i signifies that neuron i has reached
the end of its refractory period and is ready to fire in episode t
+
1; and s i (
t
) =
+
1.
When will a neuron i that is ready to fire
(
s i (
t
) =
p i )
actually fire in episode t
+
1?
At time t , neuron i receives firing inputs from all neurons j with
j
,
i
A D and
s j (
1 if and only if there are sufficiently many such
neurons. The exact meaning of “sufficiently many” may again differ from neuron
to neuron and is conceptualized by a positive integer th i that we call the i th firing
threshold. The vector
t
) =
0, and it will fire at time t
+
th
will be another parameter of our models.
Now we are ready for a formal definition.
Definition 6.2.
= (
th 1 ,...,
th n )
, th
A neuronal network is a triple N
=
D
,
p
, where D
=
th
[
] ,
A D
= (
p 1 ,...,
p n )
= (
th 1 ,...,
th n )
n
is a loop-free digraph and
p
and
are vectors of positive integers. The state
s of the system at time t is a vector
s
. The state
space of N will be denoted by St N . The dynamics of N is defined as follows:
(
t
) = (
s 1 (
t
),...,
s n (
t
))
, where s i (
t
) ∈{
0
,
1
,...,
p i }
for all i
∈[
n
]
￿f s i (
t
)<
p i , then s i (
t
+
1
) =
s i (
t
) +
1.
￿f s i (
t
) =
p i and there exist at least th i different j
∈[
n
]
with s j (
t
) =
0 and
j
,
i
A D , then s i (
t
+
1
) =
0.
￿f s i (
t
) =
p i and there are fewer than th i different j
∈[
n
]
with s j (
t
) =
0 and
j
,
i
A D , then s i (
t
+
1
) =
p i .
at time 0 will be called an initial state. Each initial state uniquely deter-
mines a trajectory
A state
s
(
0
)
s
(
0
),
s
(
1
),
s
(
2
),...,
s
(
t
),...
of successive states for all times t .
, th
Example 6.3.
Consider the neuronal network N
=
D
,
p
, where D is the
), th
digraph of Figure 6.2 a,
p
= (
1
,
1
,
1
,
2
= (
2
,
1
,
1
,
1
)
.Find
s
(
1
),
s
(
2
),
s
(
3
)
for
the following initial states: (a)
s
(
0
) = (
0
,
1
,
1
,
1
)
,(b)
s
(
0
) = (
1
,
0
,
0
,
0
)
,(c)
s
(
0
) =
(
0
,
1
,
1
,
2
)
.
Solution: a.
If
, node 1 fires and thus cannot fire in the next time
step, node 2will fire in the next step since it is at the end of its refractory
period and receives firing input from th 2 =
s
(
0
) = (
0
,
1
,
1
,
1
)
1 node, node 3 does not
receive any firing input and will stay in state p 3 =
1, and node 4 will
reach the end of its refractory period p 4
=
2. Thus we get
s
(
1
) =
(
. Now nodes 3 and 4 receive firing input from the required
number of nodes, while nodes 1 and 2 do not receive any such input.
Thus
1
,
0
,
1
,
2
)
. Now node 1 receives firing input from th 1 =
2 nodes, while node 2 does not receive such input and nodes 3 and 4
must enter their refractory periods. It follows that
s
(
2
) = (
1
,
1
,
0
,
0
)
s
(
3
) = (
0
,
1
,
1
,
1
)
.
 
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