The influence of the place-value structure of the Arabic number ...

The output function of the hidden layer was the identity function making the output
equal to the activation. The output of the output layer was determined as follows: The two
nodes of the output layer reflect two distinct decisions. When the activation of the left node
exceeds a threshold level θ, this indicates that the first number was the larger one of the pair.
Contrarily, when the threshold θ is first exceeded by the right node, this means that the second
number was larger. The threshold was set to θ = 0.5 in all of the current networks (cf. Verguts
et al., 2005; for a similar approach). Because activations approach a certain level in an
asymptotic manner, there is the possibility that none of both activations exceeds θ and the
loop would run forever. To prevent overly long “responses”, a time limit was introduced after
which the decision process was terminated. The actual choice of a time limit is dependent on
the choice of the parameter τ. For smaller/larger values of τ the loop takes longer/faster to
reach maximum activation. A τ-value of 0.01 and a time limit of rt max = 50 loops were found
to maximize the range of the simulated reaction time distribution while at the same time
minimizing the number of time limits actually reached.
After the network model solved the comparison task for one pair of numbers within
the given time limit, activation of the output nodes is compared to the correct output. For
instance, when comparing 43 to 78, activation of the left node may be 0.12 and 0.67 for the
right node. However, in this case, the correct output t j for the left node would be 0 and 1 for
the right node. Nevertheless, it is not mandatory that the obtained activations equal the correct
output (cf. Verguts et al., 2005). For a correct decision the actual activation just needs to be
closer to the true value for that particular output node than to the true value for the other
output node, or put differently: in case the difference between the actual activation and the
correct output does not exceed a difference of d > 0.5 the correct output is returned by the
model. For our example, the output for the left node would be 0, because 0 .12 − 0 < 0. 5 and
246