Mackintosh's (1975) attentional learning model, as implemented by Le Pelley et al. (2016).
List of model parameters
Matrix of training items
Boolean specifying whether to include extended information in the output (see below)
The function operates as a stateful list processor (slp; see Wills et al., 2017). Specifically, it takes a matrix (tr) as an argument, where each row represents a single training trial, while each column represents the different types of information required by the model, such as the elemental representation of the training stimuli, and the presence or absence of an outcome. It returns the output activation on each trial (a.k.a. sum of associative strengths of cues present on that trial), as a vector. The slpMack75 function also returns the final state of the model - a vector of associative and attentional strengths between each stimulus and the outcome representation.
st must be a list containing the following items:
lr - the associative learning rate (fixed for a given
simulation), as denoted by
theta in Equation 1 of Mackintosh
alr - the attentional learning rate parameter. It can be set without
limit (see alpha below), but we recommend setting this parameter to somewhere
between 0.1 and 1.
w - a vector of initial associative strengths. If you are not
sure what to use here, set all values to zero.
alpha - a vector of initial attentional strengths. If the
updated value is above 1 or below 0.1, it is capped to 1 and 0.1
colskip - the number of optional columns to be skipped in the tr
matrix. colskip should be set to the number of optional columns you have
added to the tr matrix, PLUS ONE. So, if you have added no optional
columns, colskip=1. This is because the first (non-optional) column
contains the control values (details below).
tr must be a matrix, where each row is one trial
presented to the model. Trials are always presented in the order
specified. The columns must be as described below, in the order
ctrl - a vector of control codes. Available codes are:
0 = normal trial 1 = reset model (i.e. set associative strengths back to their initial values as specified in w) 2 = Freeze learning 3 = Reset associative weights to initial state, but keep attentional strengths in alpha 4 = Reset attentional strengths to initial state, but keep association weights.
Control codes are actioned before the trial is processed.
opt1, opt2, ... - any number of preferred optional columns, the
names of which can be chosen by the user. It is important that these
columns are placed after the control column, and before the remaining
columns (see below). These optional columns are ignored by the
function, but you may wish to use them for readability. For example, you
might choose to include columns such as block number, trial number and
condition. The argument colskip (see above) must be set to the number of
optional columns plus one.
x1, x2, ... - activation of any number of input elements. There
must be one column for each input element. Each row is one trial. In
simple applications, one element is used for each stimulus (e.g. a
simulation of blocking (Kamin, 1969), A+, AX+, would have two inputs,
one for A and one for X). In simple applications, all present elements
have an activation of 1 and all absence elements have an activation of
0. However, slpMack75 supports any real number for activations, e.g. one
might use values between 0 and 1 to represent differing cue saliences.
t - Teaching signal (a.k.a. lambda). Traditionally, 1 is used to
represent the presence of the outcome, and 0 is used to represent the
absence of the outcome, although slpMack75 supports any real values for lambda.
If you are planning to use multiple outcomes, see Note 2.
xtdo (eXTenDed Output) - if set to TRUE, function will
additionally return trial-level data including attentional strengths and
the updated associative strengths after each trial (see Value).
Returns a list containing three components (if xtdo = FALSE) or five components (if xtdo = TRUE, xoutw and xouta is also returned):
Vector of summed associative strength for each trial.
Vector of final associative strengths.
Vector of final attentional weights.
Matrix of trial-level data of the associative strengths at the end of the trial, after each has been updated.
Matrix of trial-level data of the attentional strengths at the end of the trial, after each has been updated.
1. Mackintosh (1975) did not formalise how to update the cues' associability, but described when associability increases or decreases in Equation 4 and 5. He assumed that the change in alpha would reflect the difference between the prediction error generated by the current cue and the combined influence (a sum) of all other cues. Le Pelley et al. (2016) provided a linear function in Equation 2 that adheres to this description. This expression is probably the simplest way to express Mackintosh's somewhat vague description in mathematical terms. A linear function is also easier to computationally implement. So we decided to use Equation 2 from Le Pelley et al. (2016) for updating attentional strengths.
2. At present, only single-outcome experiments are officially supported. If you want to simulate a two-outcome study, consider using +1 for one outcome, and -1 for the other outcome. Alternatively, run a separate simulation for each outcome.
Lenard Dome, Andy Wills, Tom Beesley
Kamin, L.J. (1969). Predictability, surprise, attention and conditioning. In Campbell, B.A. & Church, R.M. (eds.), Punishment and Aversive Behaviour. New York: Appleton-Century-Crofts, 1969, pp.279-296.
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(10), 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.
Wills, A.J., O'Connell, G., Edmunds, C.E.R., & Inkster, A.B.(2017). Progress in modeling through distributed collaboration: Concepts, tools, and category-learning examples. Psychology of Learning and Motivation, 66, 79-115.
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