Biology Reference
In-Depth Information
b.
If
s
(
0
)
=
(
1
,
0
,
0
,
0
)
, then in the next step node 1 receives firing input
from 2
th
1
nodes and will fire in step 1. Thus
s
(
1
)
=
(
0
,
1
,
1
,
1
)
,
and it follows from our calculations for part (a) that
s
(
2
)
=
(
1
,
0
,
1
,
2
)
and
s
(
3
)
=
(
1
,
1
,
0
,
0
)
.
c.
If
s
(
0
)
=
(
0
,
1
,
1
,
2
)
, then
s
(
1
)
=
(
1
,
0
,
1
,
0
)
. Now node 1 receives
firing input from only 1
th
1
node and will not fire in the next step,
but node 3 receives firing input from 1
<
th
3
node and will fire. Thus
s
(
2
)
=
(
1
,
1
,
0
,
1
)
. Similarly
s
(
3
)
=
(
1
,
1
,
1
,
2
)
.
Example
6.3
illustrates a number of important phenomena in the dynamics of our
networks. First of all, notice that
in part (a) we got
s
(
3
)
=
s
(
0
)
. Thus
s
(
4
)
=
s
(
1
),
s
(
5
)
=
s
(
2
)
, and so on. The trajectory will cycle indefinitely through
the set of states
AT
. The states in
AT
are called
persistent states
because every trajectory that visits one of these states will return to it infinitely often,
and the set
AT
itself is called the
attractor
of
={
s
(
0
),
s
(
1
),
s
(
2
)
}
.
The meaning of the word “attractor” will become clear when we consider the solu-
tion to part (b). Notice that
s
(
0
)
is in the same set
AT
as in the previous paragraph,
so the trajectory will again cycle indefinitely through this set. In particular, the initial
state
s
(
1
)
will never be visited again along the trajectory; it is a
transient
state.
By Dirichlet's Pigeonhole Principle, this kind of thing must always happen in
a discrete dynamical system with a finite-state space: Every trajectory will eventu-
ally return to a state it has already visited and will from then on cycle indefinitely
through its attractor. The states that a trajectory visits before reaching its attractor occur
only once. They form the
transient
(part) of the trajectory. Note that in part (b) of
Example
6.3
the
length of the transient,
i.e., the number of its transient states, is 1;
whereas in part (a) the length of the transient is 0.
The trajectories of the initial states in parts (a) and (b) will be out of sync by one
time step; nevertheless, they both end up in the same attractor
AT
that has a length of 3.
In part (c) of Example
6.3
, we get
s
(
0
)
=
(
1
,
0
,
0
,
0
)
(
)
=
(
,
,
,
)
=
p
. In this state, every
neuron has reached the end of its refractory period and no neuron fires. Thus no
neuron fires at any subsequent time, the length of the transient is 3, and we get
s
3
1
1
1
2
s
is an attractor of length 1. Attractors of
length 1 are called
steady state attractors
and their unique elements are called
steady
states.
In contrast, attractors of length
(
3
)
=
s
(
4
)
=
...
It follows that the set
{
p
}
>
1are
periodic attractors.
Exercise 6.1.
Prove that
p
is the only steady state in any network
N
=
,
th
The set of all initial states whose trajectories will eventually reach a given attractor
AT
is called the
basin of attraction of AT.
The initial states in parts (a) and (b) of
Example
6.3
belong to the same basin of attraction, while the initial state in part (c)
belongs to a different basin of attraction, namely the basin of the steady state.
Exercise 6.2.
D
,
p
.
,
th
Let
N
=
D
,
p
be a neuronal network and let
s
(
0
)
∈
St
N
be a
state such that
s
i
is in the basin of attraction of the
steady state attractor and determine the length of the transient.
>
0 for all
i
∈[
n
]
. Prove that
s
(
0
)
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