Environmental Engineering Reference
In-Depth Information
B.3.1 Biharmonic approximation of neural models
In Chapter 6, Section
6.2
, we presented an approximated representation of Eq. (
B.12
),
based on a Taylor expansion of the integral term in the neighborhood of
φ
=
φ
0
(i.e.,
for small values of
z
). If the kernel function has axial symmetry and only the terms
up to the fourth order are retained, Eq. (
B.12
) can be approximated as (
Murray
,
2002
)
∂φ
∂
2
4
t
≈
f
(
φ
)
+
ω
0
(
φ
−
φ
0
)
+
ω
2
∇
φ
+
ω
4
∇
φ,
(B.19)
where the terms on the right-hand-side are the same as in Eq. (
6.2
)and
w
m
is the
m
th-order moment of the kernel function defined in Eq. (
6.3
). In the case of axial
symmetric kernel functions, the odd-order moments of
(
z
) are zero.
It can be shown that a second-order expansion (i.e., a Fisher equation) is unable to
lead to persistent patterns (e.g.,
Murray
,
2002
) and that the biharmonic term
ω
4
φ
is needed in the series expansion to obtain [with approximation (
B.19
)] deterministic
patterns that do not vanish with time. In fact, linear-stability analysis of the state
φ
=
φ
ω
∇
4
0
with respect to a perturbation
ˆ
e
γ
t
+
i
k
·
r
leads to the dispersion relation:
φ
(
r
,
t
)
∝
f
(
ω
2
k
2
ω
4
k
4
γ
(
k
)
=
φ
0
)
+
ω
0
−
2
+
4
.
(B.20)
0), the most unstable mode
k
max
is zero and no patterns emerge. In the case of biharmonic approximation (
B.19
)
(i.e., when
In the absence of the biharmonic term (i.e., when
ω
=
4
ω
=
0), th
e most unstable mode can be easily obtained from Eq. (
B.20
)
4
5
√
ω
=
.
/ω
as
k
max
0
4
. Patterns emerge when
k
max
is real and different from zero (i.e.,
2
ω
2
and
ω
4
need to have the same sign), and
γ
>
(
k
max
)
0:
2
2
ω
f
(
γ
(
k
max
)
=
φ
0
)
+
ω
0
−
ω
4
>
0
.
(B.21)
4
φ
=
φ
0
in the absence of spatial dynamics requires
f
(
In addition, the stability of
φ
0
)
to be negative, as in the first of conditions (
B.16
). Moreover, in many applications
φ
is
always nonnegative. This condition is met when
ω
0
<
0. Because in this case
ω
0
and
f
(
φ
0
) are both negative, Eq. (
B.21
), combined with the requirement that
ω
2
and
ω
4
have the same sign, implies that pattern formation occurs only if
ω
2
and
ω
4
are also
negative. However, the condition that
ω
4
are negative is only necessary
and not sufficient for pattern formation as condition (
B.21
) would still need to be met
for the instability to emerge.
ω
0
,
ω
2
,and
B.4 Patterns emerging from differential-flow instability
The third major deterministic model of self-organized pattern formation associ-
ated with symmetry-breaking instability is due to differential flow. This mechanism
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