Probabilistic versus Incremental Presynaptic Learning in Biologically Plausible Synapses - Foundations on Natural and Artificial Computation - page 82

Fig. 2. Different models of synaptic plasticity and metaplasticity. (a) Hebb rule and
Sejnowski's covariance rule. (b) The ABS qualitative model by Artola, Brocher and
Singer.

a n , represents the postsynaptic activation due to neighbouring synapses to the
one under consideration. For higher values of a n , the curve is shifted to the left.

data. In the ABS model, LTP and LTD thresholds shift to lower values for higher
levels of the activation of neighbouring synapses a n . For example, if the presy-
naptic activity (action potential frequency) of a specific synapse is negligible,
and activations from neighbouring synapses are high, as in curve a n, 1 ,synaptic
potentiation occurs in that synapse.
In the following sections, the incremental and probabilistic versions of the
presynaptic rule [9] will be analyzed in order to assess their biological plausibility
in the light of the mentioned synaptic properties (i.e. metaplasticity of both LTP
and LTD thresholds, and the ABS rule).
In the incremental version of the presynaptic rule, the variation of the synaptic
weight ω is calculated as:

ω = ξI ( O − ω ) (1)

where I and O (in uppercase) are the presynaptic and postsynaptic action po-
tential frequencies or, if normalized, the presynaptic and postsynaptic action
potential probabilities respectively. ξ is a small positive constant. According to
Grossberg[9] and Minai [10] this incremental version of the presynaptic rule is
asymptotically equivalent to the following probabilistic expression:
ω = P ( o

i )

(2)

which is regarded to be “the probabilistic version of the presynaptic rule”. P ( i )

is the conditional probability of a postsynaptic action potential, o ,givenapresy-
naptic action potential i (notice that binary action potentials are represented in
lowercase while action potential frequencies or probabilities are in uppercase).

2 Methods

The two versions of the presynaptic rule correspond respectively to the following
mathematical models: