Biomedical Engineering Reference
In-Depth Information
*
=
(a)
(c)
v
0
|
v
|
Limit cycles
v
d
Initial
point
v
(b)
FIGURE 2.9
Mathematical concepts of
ΣΔ
learning: (a) a one-dimensional
L
1
regularizer and (b) its derivative that leads
to a binary (Boolean) function, (c) limit cycles due to
ΣΔ
learning about the minima that leads to spike generation.
minimum
v
*, as shown in
Figure 2.9
c. This limit-
cycle behavior will capture the spiking dyna-
mics of the system. It can be verified that, if
||
W
||
∞
≤ 1, then
v
*
=
0 is the location of the minimum
with
f
(
v*
,
W
)
=
0.
The link between the optimization criterion
of Eq.
(2.13)
and
ΣΔ
modulation (or integrate-
and-fire neuron) is through a stochastic gradient
minimization
[43]
of the cost function given by
Eq.
(2.14)
. If the input random vector
x
is
assumed to be stationary and if the probability
density function of
x
is assumed to be well
behaved (i.e., the gradient of the expectation
operator is equal to the expectation of the gradi-
ent operator), the stochastic gradient step with
respect to
v
for each iteration
n
yields
The bounded property of
w
[
n
] then leads to the
asymptotic property:
−→
1
E
n
{
d
[
n
]}
n
→∞
λ
E
n
{
W
[
n
]
T
X
[
n
]},
(2.18)
where
C
n
{
.
} denotes an empirical expectation
with respect to time index
n
. Thus, the recursion
Eq.
(2.16)
produces a binary (spike) sequence,
the mean of which asymptotically encodes the
transformed input at infinite resolution. This is
illustrated in
Figure 2.9
c, which shows a two-
dimensional optimization contour. The objec-
tive of the
ΣΔ
learning is to follow the trajectory
from an initial condition to the minimum and
induce limit cycles about the minimum
v*
. The
dynamics of the limit cycles then encode the
shape of the optimization contour and hence
also encode the estimation parameters.
The maximization step (decorrelation) in Eq.
(2.13)
yields updates for matrix
W
according to:
V
[
n
]=
V
[
n
− 1]−
∂
f
(
V
,
W
)
∂
V
,
(2.15)
(
n
−1)
W
[
n
]=
W
[
n
− 1]+ξ
∂
f
(
V
,
W
)
∂
W
,
V
[
n
]=
V
[
n
− 1]+
W
[
n
− 1]
T
X
[
n
− 1]−
d
[
n
],
(2.19)
n
−1
(2.16)
which leads to
where
d
[
n
]
=
sgn(
v
[
n
−
1]) denotes a Boolean
function indicating whether a spike is gener-
ated. Solution of the discrete-time recursion Eq.
(2.16)
leads to
W
[
n
]=
W
[
n
− 1]−
ξ
V
[
n
− 1]
X
[
n
− 1]
T
;
W
[
n
]∈
C
.
(2.20)
The adaptation step in Eq.
(2.20)
can be expressed
by its digital equivalent as
N
−1
N
1
N
d
[
n
]=
1
N
W
[
n
]
T
X
[
n
]+
1
N
(
V
[
n
]−
V
[0]
)
.
W
[
n
]=
W
[
n
− 1]−ξ [
n
]sgn(
X
[
n
− 1])
T
;
W
[
n
]∈
C
,
n
=1
n
=0
(2.21)
(2.17)
