Biology Reference
In-Depth Information
FIGURE 6.2
(a) A digraph with
V
D
=[
4
]
and (b) a digraph with
V
D
=[7].Arcs
i
,
j
∈
A
D
are represented
by arrows
i
→
j
.
by
gcd
. For a positive integer
n
the symbol
[
n
]
will be used as shorthand for the set
{
.
Definition 6.1.
1
,...,
n
}
A
directed graph
or
digraph
is a pair
D
=
V
D
,
A
D
, where
V
D
.Theset
V
D
is called the set of
vertices
or
nodes
of
D
, and the set
A
D
is
called the set of
arcs
of
D
.
Digraphs are convenient models for all sorts of networks of interacting agents.
The agents are elements of
V
D
, while an arc
A
D
⊆
A
D
signifies that agent
v
may
influence the state of agent
w
. Note that the interactions may not be symmetric; there-
fore directed graphs are usually more accurate models of networks than undirected
graphs. Figure
6.2
gives a visual representation of two digraphs.
An arc of the form
v, w
∈
v, v
is called a
loop
; a digraph that does not contain such arcs is
called
loop-free.
In our intended interpretation, the set
V
D
stands for the set of neurons
in the network, while an arc
A
D
signifies that neuron
v
may send synaptic
input to neuron
w
. Since we are not considering neuronal networks in which a neuron
may send synaptic input to itself, the corresponding digraphs will be loop-free.
Moreover, it will be convenient for our purposes to consider only digraphs such
that
V
D
=[
v, w
∈
for some positive integer
n
.
Each of our network models will have an associated digraph
D
that specifies the
connectivity
of the network. While the connectivity is assumed here to remain fixed at
all times, we are interested in the
dynamics
of our networks, that is, how the
state
of the
network changes over time. Time
t
is assumed to progress in discrete steps or episodes,
so that we will have a state
n
]
s
(
t
)
of the network for each
t
=
0
,
1
,
2
,...
The
state
s
(
t
)
of
the network at time t
will simply be the vector
s
(
t
)
=
(
s
1
(
t
),
s
2
(
t
),...,
s
n
(
t
))
, where
s
i
(
V
D
at time
t
. According to what was said in Section
6.2
,
at any given time step, a neuron
i
may either fire, or be at rest. We will assume that
firing lasts exactly one episode, and neuron
i
needs a
refractory period
of
p
i
time steps
before it can fire again, where
p
i
is a positive integer. We will allow the refractory
periods
p
i
to differ from neuron to neuron. Each of our models will have a parameter
t
)
is the state of node
i
∈
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