# The mathematics behind MAC1975

A grand departure from global error term models such as RW1972 (Rescorla & Wagner, 1972), the MAC1975 model (Mackintosh, 1975) uses local error terms and changes stimulus associability ($$\alpha$$) via an error comparison mechanism that promotes learning about uncertain stimuli:

## 1 - Generating expectations

Let $$v_{k,j}$$ denote the associative strength from stimulus $$k$$ to stimulus $$j$$. On any given trial, the expectation of stimulus $$j$$, $$e_j$$, is given by:

$\tag{Eq.1} e_j = \sum_{k}^{K}x_k v_{k,j}$

$$x_k$$ 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 association from stimulus $$i$$ to $$j$$, $$v_{i,j}$$, are given by:

$\tag{Eq.2} \Delta v_{i,j} = x_i \alpha_i \beta_j (\lambda_j - v_{i,j})$

where $$\alpha_i$$ is the associability of (or attention devoted to) stimulus $$i$$, $$\beta_j$$ is a learning rate parameter determined by the properties of $$j$$, and $$\lambda_j$$ is a the maximum association strength supported by $$j$$ (the asymptote).

## 3 - Learning to attend

The parameter $$\alpha_i$$ changes as a function of learning, proportionally to the difference between the absolute errors conveyed by $$i$$ and all the other predictors1, via:

$\tag{Eq.3} \Delta \alpha_{i} = x_i\theta_i \sum_{j}^{K}\gamma_j(|\lambda_j - \sum_{k \ne i}^{K}v_{k,j}|-|\lambda_j - v_{i,j}|)$ where $$\theta_i$$ is an attentional learning rate parameter for stimulus $$i$$ (usually fixed across all stimuli). Although Mackintosh (1975) did not extend their model to account for the predictive power of within-compound associations, the implementation of the model in this package does. This can sometimes result in unexpected behavior, and as such, Eq. 3 above includes an 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 to reflect the 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 behavior of the original model.

## 4 - Generating responses

There is no specification of response-generating mechanisms in MAC1975. 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:

$\tag{Eq.4} r_j = e_j$

### References

Le Pelley, M. E., Mitchell, C. J., Beesley, T., George, D. N., & Wills, A. J. (2016). Attention and associative learning in humans: An integrative review. Psychological Bulletin, 142, 1111–1140. https://doi.org/10.1037/bul0000064
Mackintosh, N. J. (1975). A theory of attention: Variations in the associability of stimuli with reinforcement. Psychological Review, 82, 276–298. https://doi.org/10.1037/h0076778
Rescorla, R. A., & Wagner, A. R. (1972). A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement. In A. H. Black & W. F. Prokasy (Eds.), Classical conditioning II: Current research and theory. (pp. 64–69). Appleton-Century-Crofts.

1. Mackintosh (1975) did not fully specify the equations governing the change in stimulus associability. Instead, we adopt here the equation Le Pelley et al. (2016) used in their implementation of the model.↩︎