Digital Signal Processing Reference
In-Depth Information
interaction kernel
9
exp
2
s
2
2
−
ξ
κ(ξ)
=
γ
−
β
,
(11.3)
where the first term corresponds to an excitatory short-range mechanism with
width
s
and the second term corresponds to a generic inhibitory mechanism
of relative strength
.Itcan be shown that such an interaction kernel activates
only one connected region at the equilibrium, the
active bubble
(described in
more detail below).
8
At each iteration of the algorithm, the active bubbles are simultaneously
developed in layers
β
according to a procedure described by the set of
coupled differential equations:
9
X
and
Y
I
(
x
)
a
x
a
=−
α
˙
x
a
+
(κ
∗
X
)
a
+
,
X
a
=
ϕ(
x
a
)
(11.4)
I
(
y
)
b
y
b
˙
=−
α
x
b
+
(κ
∗
Y
)
+
,
Y
b
=
ϕ(
y
b
)
(11.5)
b
x
a
(
0
)
=
y
b
(
0
)
=
0
,
˙
ϕ(ξ)
=
/
+
(
−
λξ)
where
x
denotes a time derivative,
1
1
exp
is the sigmoidal
function, and
is the 2D spatial convolution operator. In the presynaptic layer,
I
(
x
a
is a noise random variable that is uncorrelated for two different neurons.
It represents a random excitation of the presynaptic neurons that is slowly
varying compared to the dynamics of
∗
X
and
Y
.Inthe postsynaptic layer, we
have
I
(
y
)
b
=
υ
J
ba
T
ba
X
a
,
(11.6)
a
where
is the coupling coefficient between the two layers. Equation 11.6
implies that the activity flow from one active presynaptic layer to the postsy-
naptic layers
b
υ
is proportional to the weight of the dynamic link
J
ba
and
the similarity function
T
ba
of the local features. When the activation in the two
layers is led to equilibrium, then the dynamic links are amplified according to
∈ Y
J
ba
=
υ
J
ba
T
ba
Y
b
X
a
.
(11.7)
The constraint
a
(
is externally enforced and the
iteration concludes by resetting the active zones, i.e.,
x
a
=
J
ba
+
J
ba
)
=
1,
∀
b
∈ Y
a
,
b
.
The next iteration begins with a new noise amplitude
I
(
x
a
that produces
a different active bubble in the presynaptic layer. The mechanism of active
bubble formation in the postsynaptic layer is affected by the previous itera-
tions, as these are reflected in the present state of the dynamic link weights.
Aproperty of this NN is that a presynaptic neuron
a
is not activated in iso-
lation, but always together with some of its neighbors. If the neighbors of
a
have strong links with a region in the postsynaptic layer
y
b
=
0,
∀
, the dynamic links
that emanate from neuron
a
and abut on this region will be further amplified.
Y
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