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N
'
(
1
-
x
)
x
(
)
i
(25)
P
=
p
1
-
p
i
X
i
i
1
with
N
X
=-
Â
21
(26)
P X
=
1
,
X
=
0
which simply arises from the consideration that the simultaneous presence
or absence of various events with probabilities p i or (1 - p i ) respectively is
just the product of these probabilities. The presence or absence of events is
governed by the exponents x i and (1 - x i ) in the Bernoulli product which are
1 and (1 - 1) = 0 in presence, and 0 and (1 - 0) = 1 in absence of the vent of
interest, namely the activation of the synaptic junctions of the i th fiber.
Having established the probability of a particular input state, we have
simply to find out under which conditions the element fires in order to
establish its probability of firing. This, however, we know from our earlier
considerations (eqs. (15), (16)) which define those input states that activate
the output fiber. As we may recall, an activated output ( y = 1) is obtained
when the internal state Z e q uals, or exceeds, zero:
Â
Zn ii
=
- > -
qe
with n i representing the number of (positive or negative) synaptic junctions
of the i th fiber and q being, of course, the threshold. Hence, for a given
threshold and a certain input state X ( x 1 , x 2 ,..., x N ) output state y q, X is
defined by
1
0
for
for
S
S
nx
nx
->-
-<-
qe
qe
,
.
Ó
ii
y
=
(26)
q
,
X
ii
Since whenever the output is activated y will assume a value of unity, the
probability p of its activation is the sum of all probabilities of those input
states that give y a value of one”:
N
X
=-
 q,
21
(27)
p
=
y
XX
.
X
=
0
Since all terms in the above expression will automatically disappear when-
ever an input state is present that fails to activate the output ( y = 0), eq.
(27) represents indeed the activation probability of the output fiber. If we
again assume that the ON state of the output fiber conforms with the
universal pulse duration D t , which holds for all pulses traveling along the
input fibers, we are in a position to associate with the probability of output
excitation a frequency f acco r ding to eqs. (26), (27):
1
D
= Â
(28)
f
y XX
,
q,
t
which we will call—for reasons to be given in a moment—the “internal
frequency”.
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