Biomedical Engineering Reference
In-Depth Information
Fig. 8.8
A two-dimensional
cellular automata model
In the usual case of CA realised on a
D
-dimensional grid,
N
consists of
D
-tuples
of indices from a coordinate set:
I
D
I
:
N
⊆
The 2D cellular model therefore becomes
I
2
N
⊆
S
|
N
|
→
T
:
S
To consider an automaton specified as a CA, let
λ
and
α
be the global state and
the global transition function of the CA, respectively. Then,
λ
={
τ
|
τ
:
I
2
→
S
}
and
α(λ(i,j))
=
T(τ
|
N
+
(i, j ))
for all
τ
in
λ
and
(i, j)
in
I
2
.
Definition 3
(State transition of a cell) The heart muscle system is composed of
heterogeneous cells, the CA model of the muscle system,
CAM
CA
, is characterised
by having no dependency on the type of cells.
CAM
CA
is defined as follows:
CAM
CA
=
S,N,T
S
={
Active, Passive, Refractory
}
N
={
(
0
,
0
), (
1
,
0
), (
−
1
,
0
), (
0
,
1
), (
0
,
−
1
)
}
s
m,n
=
s
m,n
(t
+
1
)
s
m,n
=
T(s
m,n
,s
m
+
1
,n
,s
m
−
1
,n
,s
m,n
+
1
,s
m,n
−
1
)
where,
s
m,n
denotes the state of the cell located at
(m, n)
and
T
is a transition func-
tion for
CAM
CA
that specifies the next state, as shown in Fig.
8.9
.
Each cell in the heart muscle should be in one of the states
Active
,
Passive
or
Refractory
. Initially, all cells are
Passive
. In this state, the cell is discharged elec-
trically and has no influence on its neighbouring cells. When an electrical impulse
propagates, the cell becomes charged and eventually activated (
Active
state). The
cell then transmits an electrical impulse to its neighbour cells. The electrical im-
pulse is propagated to all the cells in the heart muscle. After activation, the cell
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