Probabilistic versus Incremental Presynaptic Learning in Biologically Plausible Synapses - Foundations on Natural and Artificial Computation - page 83

In the first model a rate-code neuron is used, in which the output of the
neuron is a number expressing its firing frequency or probability. In this model
the incremental rule of synaptic plasticity is applied.
In the latter, a spiking neuron model is used, in which the output is binary,
and triggers according to the calculated firing probability. In this case, the prob-
abilistic version of the presynaptic rule is applied.
In the first rate-code model, the action potential probabilities of presynaptic
and postsynaptic neurons are respectively written I and O (in upper case, as ex-
plained before). The synaptic weight ω relates I and the excitatory postsynaptic
potential, e , at a specific synapse j .

e j = ω j I j (3)

The neuron's activation, a , is obtained after summing the postsynaptic potentials
of all synapses:

n n

a = e j = ω j I j (4)

j =1 j =1

Afterwards, the action potential probability is calculated as a non-linear (sig-
moidal) function of a : O = f ( a ).
The rate code model is simplified by working in the linear part of the sigmoidal
curve.
In the second mathematical model, the so called spiking output model, synap-
tic weights are calculated taking into account the degree of correlation between
presynaptic and postsynaptic action potentials considered as binary outputs.
When an action potential is triggered it is represented by a binary one, whereas
the absence of an action potential is represented by a binary zero. The output
of the presynaptic neuron ( bit =1or bit = 0) is denoted by i (lowercase) while
the postsynaptic output ( bit =1or bit = 0) is denoted by o . The probabilistic
version of the presynaptic rule for obtaining synaptic weights is:

ω = P ( o

i )= n ( o

∩ i )

(5)

n ( i )

where n( ) in the numerator counts the number of coincidences between presy-
naptic and postsynaptic binary outputs.
The calculus of e and a is the same as for the rate code output, being ω a
conditional probability:

e j = P ( o
i j

) I j (6)

Note that I j (uppercase) is also a probability and not a binary output. In the
spiking output model, o is generated according to O , which is obtained by ap-
plying to “a” a logistic function f ( a )ofthetype:

1
1+ e − ( a − s )

O = (7)

T