Environmental Engineering Reference
In-Depth Information
In some cases, the spatial interactions modulated by the kernel function act only
within short distances. Therefore
ω
(
z
) quickly tends to zero for increasing values of
z
.
Thus, depending on the shape of
(
z
), conditions leading to pattern formation in
neural models can be formalized through a Taylor expansion of the integral term of
Eq. (
6.1
) for small values of
z
(
Murray
,
2002
):
ω
∂φ
∂
2
4
t
−
f
(
φ
)
+
ω
0
(
φ
−
φ
0
)
+
ω
2
∇
φ
+
ω
4
∇
φ
+···
,
(6.2)
4
is the biharmonic operator (see Subsection
5.1.2.2
)and
where
∇
w
m
are the
m
th-order
moments of the kernel function
1
m
!
z
m
ω
=
ω
(
z
)
d
z
,
m
=
0
,
2
,
4
,....
(6.3)
m
In Eq. (
6.2
) we assume that the dynamics are isotropic, i.e., that the kernel function
has axial symmetry. Thus, because in this case the odd-order moments of
ω
(
z
)are
zero, we do not include the odd-order terms in the Taylor expansion.
When
4
is “sufficiently small” the biharmonic term in Eq. (
6.2
) can be negligible
and the equationmay be rewritten in the formof a classical reaction-diffusion equation
(or
Fisher's equation
):
ω
∂φ
∂
2
t
f
1
(
φ
)
+
ω
2
∇
φ,
(6.4)
with
f
1
(
0
). Because in Eq. (
6.4
) only the low-order terms of
the Taylor expansion are retained, these dynamics are driven only by short-range
interactions in the neighborhood of (
x
φ
)
=
f
(
φ
)
+
ω
0
(
φ
−
φ
,
y
).
ω
4
is not “small” the biharmonic term cannot be neglected and
the dynamics are expressed by Eq. (
6.2
). If the moments of order higher than
Conversely, when
ω
4
are negligible, Eq. (
6.2
) can be truncated to the fourth order (
long-range diffusion
equation
). Thus
ω
4
multiplies the biharmonic term, which accounts for long-range interactions. Notice
how the Swift-Hohenberg spatial coupling presented in Chapter 5 and in Section
6.6
can always be written in the same form as truncated equation (
6.2
) with a suitable
kernel function.
ω
2
modulates the effect of short-range interactions, and moment
6.3 Examples of pattern-forming processes
A common feature of deterministic models of symmetry-breaking instability (see
Appendix B) is that the emergence of periodic patterns arises from the balance
between positive (activation) and negative (inhibition) interactions (e.g.,
Shnerb et al.
,
2003
). For example, in the neural model both pattern emergence and pattern geometry
are determined by the interplay between short-range facilitation (or cooperation) and
long-range competition (or inhibition).
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