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ω
) (8)
Notice that, consistently with the property of synaptic directionality (A.1), if
“
I
” is zero, neither potentiation, nor depression takes place, so term “I” can
be regarded as an “allowance term”. According to
Methods
, “a” and
O
can be
made proportional in the linear part of the logistic function so that, according
to equation 4,
I
can be replaced by
ω
ω
=
ξI
(
O
−
or
ω
in the homosynaptic case, so that:
ω
)=
ξ
[(
a
2
ω
=
ξ
(
a
ω
)(
a
−
ω
)
−
a
]
(9)
which is represented in Fig. 3.a, where the potentiation and depression regions
described by Artola et al [1] are present. The curves also exhibit metaplasticity
of the LTP threshold.
In the case of heterosynaptic plasticity, the postsynaptic activation a is the
sum of the contribution of the synapse being evaluated,
ωI
, and the potential
from nearby synapses,
a
n
. Recalling property B.1.
a
=
ωI
+
a
n
(10)
Isolating
I
and taking into account that this term is always positive or zero, the
incremental presynaptic rule (Eq. 8) becomes:
ω
=
ξ
[
a
−
a
n
]
+
ω
(
a
−
ω
)
(11)
where [
a
a
n
is negative. Setting
a
n
to a positive value, as for example 0.3, last expression
gives rise to the more realistic curve of figure 3.b in which for low postsynaptic
activations (
a<a
n
) neither potentiation nor depression takes place. The curve
also exhibits a LTD threshold for
a
=
a
n
.
This same equation is consistent with the ABS rule in which the horizontal
axis represents the presynaptic activity,
I
, instead of
a
. Substituting equation
10 in equation 9, equation 9 is expressed in terms of
I
as:
−
a
n
]
+
indicates the positive component of
a
−
a
n
, or zero if
a
−
ω
=
ξI
(
ωI
+
a
n
−
ω
)
(12)
which is the curve represented in Fig. 3.c showing a leftward shift of the LTP
threshold for greater values of
a
n
. At the same time, the curve exhibits the
uncommon properties described by Artola and Singer [8] in which synaptic po-
tentiation occurs with negligible values of
I
in the case of very large values
of
a
n
.
Finally, it is possible to take property A.3 into account in the case of het-
erosynaptic plasticity (see also property B.2).
According to this, the greater the number of AMPA and NMDA channels,
that is, the greater the synaptic weight, the greater is also the leakage of
K
+
and
Mg
2+
ions through these channels. Representing this leakage by a subtractive
term proportional to the weight, the postsynaptic activation is calculated as:
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