Author: Victor Navarro

# directional_models

#### Victor Navarro

# The behaviour of directional models

Imagine a procedure in which the offset of stimulus A is immediately followed by the onset of stimulus B. Though it would be reasonable to assume that the association from A to B might increase, it is not immediately clear what would happen to the “backward” association (from B to A).

Though some models propose that in most instances the backward association will increase (e.g., Honey, Dwyer, and Iliescu 2020), some models propose that the backward association does not automatically increase. In `calmr` those models are called “directional models”. Currently, `calmr` deems the following models as directional:

- RW1972
- MAC1975
- PKH1982

# Denoting directional trials

In `calmr`, we denote directional trials by using the `>` symbol between any two (groups) of stimuli. Doing so tells `calmr` to parse the trial into trial periods, one followed after the other. As such, specifying a trial via `1A>B` denotes the delivery of one trial in which stimulus A appears first, and stimulus B appears second. A few variations on that theme:

- `1AB` denotes a trial in which A and B are simultaneously presented (note there is no `>` symbol).
- `1AB>B` denotes a trial in which A and B are simultaneously presented in the first period, but only B remains in the second period.
- `1A>B>A` denotes a trial in which A is presented first, followed by B, and then A again.

There is technically no limitation in the number of periods one could specify, but keep in mind that the number of periods will increase model runtime.

# The behaviour of directional models

With the generalization of trials into trial periods, `calmr` must choose how associations should be expressed and learned across periods.

## Expression

For expression, `calmr` assumes that all of the stimuli in the trial will simultaneously contribute to the net responding. Thus, both stimuli A and B in a `1A>B` trial will contribute to responding, even though they occur at different times. Similarly, in a `1AB>B` trial, both A and B will contribute to responding, but B will contribute only once.

## Learning

For learning, `calmr` takes no shortcuts, updating associations between adjacent periods in the predetermined order these periods appear. Take for example the `1A>B` trial above. When using the RW1972 model (Rescorla and Wagner 1972), we expect A to associate with B, because A is followed by B. Mechanistically speaking, the association from A to B strengthens whenever the presentation of A does not generate a strong enough expectation of the B that immediately follows it.1

It is important to note that the second period of the `A>B` trial also undergoes updating. As B is presented, it might generate expectations if it has any associations with other stimuli, but, because B is followed by nothing, all of those associations will undergo extinction2.

# A simple example

To exemplify all of the above, let’s consider how the RW model behaves under directional and simultaneous trials. We begin by defining a simple design:

    library(calmr)
    design <- data.frame(
      group = c("Directional", "Simultaneous"),
      phase1 = c("10A>B", "10AB")
    )
    parsed <- parse_design(design)
    print(parsed)
    # note: periods are recognized internally; see parsed@mapping

Let’s run the model with default parameters and look at the associations in each group.

    pars <- get_parameters(parsed, model = "RW1972")
    exp_res <- run_experiment(design, parameters = pars, model = "RW1972")
    patch_graphs(graph(exp_res))

As the graphs above show, only the A to B association is positive in the Directional group, but both the A to B and B to A associations are positive in the Simultaneous group.

# A more complex example

Let’s now consider a design in which stimulus A is followed by a US in phase 1, but is itself preceded by stimulus B in phase 2 (with no US). A second group will have the same treatment in phase 1, but B and A will be presented simultaneously in phase 2. We will also add probe trials of A (`#A`) to check how strongly A is associated with the US in both groups.

    design <- data.frame(
      group = c("Directional", "Simultaneous"),
      phase1 = "10A>(US)",
      phase2 = c("10B>A/10#A", "10BA/10#A")
    )
    pars <- get_parameters(design, model = "RW1972")
    exp_res <- run_experiment(design, parameters = pars, model = "RW1972")
    plots <- plot(exp_res, type = "responses")
    plots <- lapply(plots, function(x) x + ggplot2::ggtitle(unique(x$data$group)))
    patch_plots(plots)

To appreciate the key difference between the two groups, focus on the middle, topmost panel in each group. Those panels show how much responding A produces on the probe trials (`#A`) during phase 2. Each line represents a different target stimulus.

When it comes to US-oriented responses, A in the Directional group undergoes simple extinction. In `B>A` trials, A is not followed by the US. This is not the case for the Simultaneous group. For that group, the simultaneous nature of AB trials partially protects the A-US association from extinguishing, by having B acquire an inhibitory association with the US (see the center, rightmost panel for the Simultaneous group above).

# Conclusion

Directional models in `calmr` can have unexpected behaviours, especially when compared to non-directional models. As such, you must be careful when specifying the trial periods in your design. Tinkering is your friend, but if you see any erratic behaviour, please consider dropping me a line or opening an issue on GitHub.

# References

Honey, Robert C., Dominic M. Dwyer, and Adela F. Iliescu. 2020. “HeiDI: A Model for Pavlovian Learning and Performance with Reciprocal Associations.” Psychological Review 127 (5): 829–52. https://doi.org/10.1037/rev0000196.
Rescorla, Robert A., and A. R. Wagner. 1972. “A Theory of Pavlovian Conditioning: Variations in the Effectiveness of Reinforcement and Nonreinforcement.” In Classical Conditioning II: Current Research and Theory., edited by A. H. Black and W. F. Prokasy, 64–69. New York: Appleton-Century-Crofts.

1. Internally, we use the stimuli in period p to generate expectations, and use the `union` between the stimuli in periods p and p+1 as the teaching signal.↩︎

2. Although not covered here, you safely can assume that every “learning” step in directional models behaves similarly, regardless of whether we update association or other (e.g., attentional) weights.↩︎