Biomedical Engineering Reference
In-Depth Information
2
1
3
5
4
W
45
Figure 7.6:
Weighted connections between perceptrons.
g
N
.
g
a
j
=
=
w(i, j)a
i
(7.4)
i
=
1
Often, the function
g()
is a threshold function as in the left panel of Fig. 7.7 and defined by
1
if
≥
g(
)
θ
0
≤
a
j
=
=
θ
but
g()
may also take on a more smooth function such as a sigmoid
Threshold
Sigmoid
1
1
0
0
In
In
Figure 7.7:
Threshold and sigmoid activation functions.
Using the threshold
g()
function and appropriate choice of weights, McCulloch and Pitts were
able to show how to create
NOT
,
AND
, and
OR
logical functions as shown in the top panel of Fig. 7.8.
The input to the gates (e.g.,
x
1
and
x
2
) are always either 0 or 1.
As shown in the bottom panel of Fig. 7.8, the logic function can also be considered graphically.
Inputs are graphed on the axes and the squares represent the desired outputs, filled for a logical 1 and open
for a logical 0. The dotted line represents a
decision line
that is determined by the weights and threshold.
For a two-input perceptron, consider a group of inputs and weights that when summed are exactly at the
threshold
