The mathematics behind PKH1982
Another departure from global error term models such as RW1972 (Rescorla & Wagner, 1972), the PKH1982 model (Pearce et al., 1982) does not use an error term for learning excitatory associations (but does for inhibitory associations), and ties stimulus associability (\(\alpha\)) to absolute global prediction error.
note: The implementation of this model closely follows the technical note from the CAL-R group where possible. Divergences are noted.
1 - Generating expectations
Let \(v_{k,j}\) denote the excitatory strength from stimulus \(k\) to stimulus \(j\), and \(v_{k,\overline j}\) the inhibitory strength from stimulus \(k\) to stimulus \(j\) (effectively, a “no j” representation). On any given trial, the net expectation of stimulus \(j\), \(e_j\), is given by:
\[ \tag{Eq.1} e_j = \sum_{k}^{K}x_k v_{k,j} - \sum_{k}^{K}x_k v_{k,\overline j} \]
where \(x_k\) denotes the presence (1) or absence (0) of stimulus \(k\), and the set \(K\) represents all stimuli in the design.
2 - Learning associations
Changes to the excitatory and inhibitory associations between stimuli are given by:
\[ \tag{Eq.2a} \Delta v_{i,j} = \delta_jx_i \beta_{ex,j} \alpha_i \lambda_j \] \[ \tag{Eq.2b} \Delta v_{i,\overline j} = x_i \beta_{in,j} \alpha_i |\overline{\lambda_j}| \]
where \(\beta_{ex,j}\) and \(\beta_{in,j}\) represent learning rates for excitatory and inhibitory associations, respectively, as determined by stimulus \(j\), \(\alpha_i\) is the associability of stimulus \(i\), respectively, and \(\lambda_j\) and \(\overline {\lambda_j}\) are the excitatory asymptote and the overexpectation of stimulus \(j\), respectively.
Importantly, \(\delta_j\) in Eq.2a is a parameter that is equal to 1 if the expectation of stimulus \(j\), is lower than its excitatory asymptote (i.e., \(e_j < \lambda_j\)), but 0 if not. This implies that the model stops strengthening \(v_{i,j}\) if the expectation of \(j\) is higher than its excitatory assymptote.
As mentioned in the introductory note, the PKH1982 model does not learn excitatory associations via correction error. However, the model does learn inhibitory associations via correction error, as the overexpectation term above, \(\overline {\lambda_j}\) is equal to \(min(\lambda_j - e_j, 0)\), where \(min\) is the minimum function. This implies \(\overline {\lambda_j}\) only takes non-zero values when the expectation of \(j\) is higher than its intensity on the trial (\(\lambda_j\)).
3 - Learning to attend
The associability parameter \(\alpha_i\) changes completely from trial to trial as a function of learning (note the lack of \(\Delta\) below), with the change being equal to the difference of the absolute global error, via:
\[ \tag{Eq.3} \Delta \alpha_{i} = x_i \sum_{j}^{K}\gamma_j(|\lambda_j - e_j|) \] where \(\gamma_j\) denotes the contribution of the prediction error based on the jth stimulus. To this regard, it is important to note that Pearce et al. (1982) did not extend their model to account for the predictive power of within-compound associations, yet the implementation of the model in this package does. This can sometimes result in unexpected behaviour, and as such, Eq. 3 above includes the extra parameter \(\gamma_j\) (defaulting to 1/K) that denotes whether the expectation of stimulus \(j\) contributes to attentional learning. As such, the user can set these parameters manually in order to reflect contribution of the different experimental stimuli. For example, in a simple “AB>(US)” design, setting \(\gamma_{US}\) = 1 and \(\gamma_{A} = \gamma_{B} = 0\) leads to the behaviour of the original model.
The PKH1982 model improves upon the Pearce & Hall (1980) model by adding an extra parameter that controls the rate at which associability changes. If we qualify the changes in associability determined by Eq.3 via \(\Delta\alpha_{i}^{n}\) (meaning they happened after trial \(n\)), then we can quantify the total associability of stimulus \(i\) after trial \(n\) via:
\[ \tag{Eq.4} \alpha_{i}^{n} = \begin{cases} (1-\theta_i) \alpha_{i}^{n-1} + \theta_i\Delta\alpha_{j}^n &\text{, if } x_i = 1\\ \alpha_{i}^{n} & \text{, otherwise} \end{cases} \] where \(\theta_i\) is a parameter determining both the rate at which associability decays (via \(1-\theta_i\)), and the rate at which increments in attention occur. Note that changes in associability only apply to stimuli presented on the trial (i.e., \(x_i = 1\)); attention to absent stimuli remains unchanged.
4 - Generating responses
There is no specification of response-generating mechanisms in PKH1982. However, the simplest response function that can be adopted is the identity function on stimulus expectations. If so, the responses reflecting the nature of \(j\), \(r_j\), are given by:
\[ \tag{Eq.5} r_j = e_j \]