Appendix for the APA Website
The Rescorla-Wagner model of classical conditioning (Rescorla and Wagner 1972; for a review, see
Miller, Barnet and Grahame 1995) assumes that the gain in associative strength due to the pairing of
a CS with a US is proportional to the difference between the maximum strength of association that
the US can support, and the current associative strength of all the CSs present on this trial. As a
consequence, the rate of change in conditioned responding to a CS decreases as the number of
training trials increases, resulting in a negatively accelerated curve of learning across training.
More formally, two equations describe how associative strength of a given CS (CSx) change on each
trial:
∆𝑉𝑋𝑛+1 = 𝑎𝑋 𝛽(𝜆 − 𝑉𝑡𝑜𝑡 )
(1)
and
𝑉𝑋𝑛+1 = 𝑉𝑋𝑛 + ∆𝑉𝑋𝑛+1
(2)
where ∆𝑉𝑋𝑛+1 is the change in the associative strength (V) of CSx on Trial n + 1; αx is the associability
of CSx (range: 0 to 1), β is the associability of the US (range: 0 to 1); λ is the maximum associative
strength that the US can support; and Vtot is the sum of associative strengths of all CSs (including X )
present on Trial n. In Equation 2 , 𝑉𝑋𝑛+1 is the associative strength of CSx after Trial n + 1; 𝑉𝑋𝑛 is the
associative strength of CSx immediately before Trial n + 1 (i.e. , after trial n); and ∆𝑉𝑋𝑛+1 is the change
in the associate strength of CSx as a result of Trial n + 1, as assessed by Equation 1.
The training sequences were constructed by drawing randomly among a set of runs from which
number and length were previously determined. For each type of trials (i.e., CS-US and CS-alone),
there were one run of five trials, two runs of four trials, four runs of three trials, eight runs of two
trials and 16 runs of a single trial. The resulting sequences comprised 114 trials [i.e.,
2(1*5+2*4+4*3+8*2+16*1)] and conformed exactly to a binomial distribution of two equally
probable events.
In the Perruchet's paradigm, there is a single CS and a single US. In the simulation below, αβ was
given the function of a learning rate parameter. The paradigm comprises both CS-US pairs
(acquisition) and CS alone trials (extinction). λ was set to 1 for acquisition trials and zero for
extinction trials, and αβ took different values for the different type of trials. The Rescorla-Wagner
model assumes that αβ for extinction is a number smaller than αβ for acquisition but larger than
zero. αβ was set to 0.04 for CS-US pairs and 0.03 for CS-alone trials in Simulation (A), and these
values were increased exponentially from Simulation (A) to simulation (D). These values were
empirically selected to cover cases in which the learning curve does not reach asymptote before the
end of the 114 trials (A), to cases in which asymptote is reached within the very first trials (D).
Simulation (E) was aimed to test whether the assumption that αβ for extinction is a number smaller
than αβ for acquisition is crucial, by inverting the relation. Finally, Simulation (F) was aimed at
exploring whether the ratio between the learning rate parameters for acquisition and extinction was
important, by using much more distant values (see Table 1).
Values of αβ
Simulations E1-E2 trials E1-alone trials
0,04
0,03
(A)
0,08
0,06
(B)
0,16
0,12
(C)
0,32
0,24
(D)
0,06
0,08
(E)
0,16
0,03
(F)
TABLE 1.
The results shown in Figure 1 have been averaged over 10.000 randomly generated sequences for
each simulation. Unsurprisingly, associative strength (upper panels) increased following a negatively
accelerated function across the 114 trials for all simulations. The value of the asymptote and the
number of trials needed to reach this asymptote differed in a predictive way as a function of learning
parameters. Except for Simulation (F) in which there was a strong asymmetry between the leaning
rates for acquisition and extinction, the asymptotic value is far from 1 (λ), which is typical of the
prediction of the model with a partial reinforcement paradigm.
The variation of associative strength as a function of the length and type of the preceding run of trials
was more important for our concern. An overall look on the lower panels of Figure 1 suggests the
prevalence of an upward linear trend. However, there were two minor exceptions. First, there was a
slight, but consistent departure from linearity in the middle part of the curves. Second, the curve for
Simulation (D) tended to be sigmoïdally shaped. Recall that the learning rates were very strong,
leading to reach asymptote within the very first trials (around eight trials, i.e. four E1-E2 trials on the
average). Neither of these exceptions violates monotonicity. These exceptions could turn out to be
relevant for further fine-grained analyses of the data, but for any practical purpose, it appears that a
linear function provides a satisfactory approximation for making predictions.
Figure 2