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Plus Phase
Minus Phase
CHL
CPCA
Combo
CHL
CPCA
Combo
0
0
0
+
+
+
-
0
-
0
+
+
Tab le 5 . 1 : Directions of weight change according to the CHL and CPCA Hebbian learning rules for four qualitative conditions
of minus and plus phase activation coproduct values. CPCA, which doesn't care about phases, is assumed to take place in the plus
phase. A plausible biological mechanism produces the combination of the two rules, shown in the Combo column.
that were discussed in section 4.2 are capable of imple-
menting the critical aspect of CHL. However, recall that
we already showed in section 4.5.3 that these biological
mechanisms were consistent with the CPCA Hebbian
learning rule. Thus, to the extent that CHL is inconsis-
tent with CPCA, we cannot possibly argue that the bi-
ology supports both. However, what we can (and will)
argue is that the biology is consistent with the combina-
tion of CPCA and CHL.
Table 5.1 shows how the CHL and CPCA learning
rules agree and differ on the direction of weight change
for four different qualitative values of the minus and
plus phase coproducts of sending unit x i and receiving
unit y j . Note that CPCA does not have phase-based
variables, so we further specify that it occurs in the plus
phase, because it makes sense to learn the correlation
structure of the plus phase, not the minus phase. As
you can see, the two learning rules agree for the first
row of the table, corresponding to the case where there
was no activity in the expectation (minus) phase. In the
second row, the rules differ, but the conflict is not as
bad as it could be, because they do not predict different
signs of weight change — one rule predicts no weight
change while the other predicts a weight change. Thus,
the combination of the two rules is not so drastically in
conflict with either one. Further, we will see in chap-
ter 6 that the combination of error-driven and Hebbian
associative learning can be generally beneficial for solv-
ing many different kinds of tasks.
Three of the four cells in table 5.1 are consistent with
the CPCA Hebbian learning rule, which we have al-
ready shown can be accounted for by the biology of
LTP/D (section 4.5.3). Thus, what remains to be ex-
plained is the lower left-hand cell of the table, where
the coproduct was active in the minus phase, but inac-
tive in the plus phase. CHL requires a weight decrease
here (i.e., LTD). We refer to this cell as the error cor-
rection case, because it represents an incorrect expec-
tation that should be suppressed through error-driven
learning. In other words, this cell represents the situ-
ation where there was a strong expectation associated
with these units (in the minus phase) that was not ac-
tually experienced in the outcome (plus phase). This
is the most important contribution of error-driven learn-
ing, because it enables the network to correct a faulty
expectation or output. Otherwise, Hebbian learning is
capable of doing the right things.
To explain this error correction case, we appeal to the
relationship between intracellular calcium ion concen-
tration and the direction of synaptic modification pro-
posed by Artola et al. (1990), which was discussed
in section 4.2 and shown in figure 4.2. To refresh, the
idea is that there are two thresholds for synaptic modi-
fication, ￿ + and ￿ ￿ . A level of intracellular calcium
( [Ca ++ ] i ) that is higher than the high threshold ￿ +
leads to LTP, while one lower than this high threshold
but above the lower ￿ ￿ threshold leads to LTD.
Under this two-threshold mechanism, it seems plau-
sible that minus phase synaptic activity ( x i y j )thatis
not followed by similar or greater levels of plus phase
synaptic activity ( xi yj ) will lead to a level of [Ca ++ ] i
that is above the ￿ ￿ threshold but below the ￿ + thresh-
old, thus resulting in the LTD required for error correc-
tion. In short, minus phase activations are not persistent
enough on their own to build up the LTP level of cal-
cium, and thus only lead to LTD if not maintained into
the plus phase, which is presumably longer-lasting and
thus capable of accumulating LTP-levels of calcium.
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