Biomedical Engineering Reference
In-Depth Information
Table 3.1.
Parameter values for modified MG model.
parameter value
k
1
0.036 min
−
1
0.666 min
−
1
k
2
L
1
10
L
2
0.005
0.958 min
−
1
k
i
3.58 min
−
1
k
e
h
5
0.9 min
−
1
k
t
0.108
0.57 min
−
1
σ
λ
0.01
θ
0.0l
q
4000
ν
12
k
4
c
10
0.024 mm
2
min
−
1
D
δ
1
the volume ratio
h
and the extracellular cAMP decay rate via
h
→
hδ
and
k
e
→
k
e
/δ
; none of the other parameters are affected.
It is important to understand how an excitable system can propagate
waves. In the above model, the cAMP secreted by a given cell diffuses to
that cell's neighbors and hence can excite them into emitting cAMP. Each
cell in turn relaxes back to the quiescent state, but there are always down-
stream cells that are just then being excited. The net effect is that a single
localized emission of cAMP can lead to a pulse propagating throughout the
system. Alternatively, a region in which the cAMP system is in the sponta-
neously oscillating range of parameters can serve as an emitter that sends
periodic pulse trains. The dispersion relationship for the waves (i.e., the rela-
tionship between the temporal forcing period and the spatial wavelength) can
be calculated and compared to direct measurements [11].
Often, the visualized field contains rotating spiral patterns. The general
theory of excitable media predicts that spirals are a stable, self-sustaining
pattern that often arises via the breaking of wavefronts. Specifically, we can
imagine that a propagating pulse is disturbed, perhaps by passing through
a regime of reduced excitability. The wavefront will then break, leading to
two points, at which the phase of the wave wraps around 2
π
as we go around
them. Each of the phase singularities can potentially form the core of a spiral;
hence, a typical such event will lead to a counter-rotating pair. This type of
generic occurrence is shown in Figure 3.5. A spiral formed from our basic MG
model equations is shown in Figure 3.6.
