An Associative Model of Geometry Learning
Noam Miller & Sara Shettleworth
University of Toronto, Toronto, Ontario, Canada
The puzzle of geometry learning
Animals can learn to use the shape or geometry of an enclosure to locate a hidden goal
(review in Cheng & Newcombe, 2005). Remarkably, more informative features in the
enclosure don’t seem to block or overshadow geometry learning and may even potentiate
it, supporting Cheng’s idea of a “geometric module” impenetrable to other spatial
information. But spatial learning in an arena or watermaze is an operant task. By taking
this into account, our model shows how underlying competitive learning between geometry
and other cues, as in the Rescorla-Wagner (RW) model, determines observed spatial
choices in geometry and other instrumental learning tasks.
How the model solves the puzzle
In this example of a geometry learning task, disoriented subjects are required to locate a
reward hidden in one corner of a rectangular enclosure marked by a prominent feature. The
best predictor of the reward’s location is obviously the feature, but animals also learn the
geometry, and such learning is not blocked by prior training with the feature, e.g. in a square
enclosure (Wall et al., 2004). This happens because the animal’s behavior, not the a priori
predictiveness of the cues, determines their frequency of presentation. Subjects only learn
about cues when they visit corners containing those cues. Choice of what corner to visit is
determined by the associative strength of cues at each corner relative to the total of
associative strengths at all corners.
In the situation shown above, subjects quickly develop a preference for the corner with
the predictive feature. But when they visit that corner they also experience a pairing of its
geometry with reward, leading the associative strength of the correct geometry to increase
more than it would have without a feature at the correct corner. Thus, the feature enhances
learning about geometry rather than overshadowing it. Because strong preferences in the
model can develop long before associative strengths are asymptotic, prior training does not
block geometry, as shown below in a comparison of model and data from Wall et all. 2004.
F
C
R
N
How the model works
There are several cues that the subject can learn to use to locate the reward: the
geometry of the corners, the feature, and other contextual cues. Each of these is a cue
in the model: G(eometry), F(eature), C(ontext), W(rong geometry).
When a subject visits a corner and experiences reward or nonreward there, the cues at
that corner change in strength as in RW:
(1)
ΔV = α β (λ – ΣV),
If subjects’ choices of search locations are based on what they have already learned
about the various cues at each corner, the probability of choosing a location (PL)
should be dependent on the associative strengths of the cues at that location (VL). We
define:
(2)
PL = VL / ΣVL,
Other predictions
 Pearce et al. (2006, Experiment 1) found potentiation of
geometry learning by a feature in a rectangular enclosure, and
overshadowing of geometry by the feature in a kite-shaped
enclosure. Both groups had an innately attractive feature at an
incorrect corner. This experiment indirectly shows the
important difference between geometrically ambiguous and
unambiguous enclosures. In an ambiguous enclosure, such as
a rectangle, attractive features at any corner increase the
perceived reward contingency of the correct geometry by
increasing the probability of choosing the correct over the
rotational corner. This leads to potentiation.
and modify the original equation so that subjects only learn about a cue when they
visit a location that contains it:
(3)
ΔV = α β (λ – ΣV) PL.
When visiting a rewarded corner, λ = 1; when visiting an unrewarded corner, λ = 0.
An additional term is added to the above equation for each corner that a cue is present
at. So, in our example, for the correct geometry (G), which is present at both the
correct and rotational corners, the equation would become:
(4)
ΔVG = αG β (1 – VGFC) PCorr + αG β (0 – VGC) PRot.
Blocking and potentiation in the watermaze
In a watermaze, subjects are usually permitted to swim until they locate the
platform. Thus, the probability of visiting the correct corner on a given trial is
always 1 but there is also some non-zero probability of visiting each of the other
corners along the way. The model accounts for multiple-choice paradigms by taking
into account the cumulative probability of each of the possible paths the subject can
take to the platform.
Pearce et al. (2001) trained rats in an unambiguous triangular watermaze with the
platform at one of the corners along the base. Group Beacon had a beacon attached
to the platform, which was
always in the same corner;
group None had no beacon;
group Random had a beacon
and platform that moved
randomly between the correct
and incorrect corners from
trial to trial. Here we show
how the model correctly
predicts the results of a test
trial with no beacon present
for any of the groups. For such
a test, predicted choice
proportions are based on
relative associative strengths
of the cues other than the
beacon.
 In an unambiguous enclosure like the kite, an attractive
feature at an incorrect corner increases the number of errors,
leading to overshadowing of the correct geometry.
 The interactions between features and geometry depend on
the shape of the enclosure and the locations of the features.
Features at the same locations enhance learning about each
other, whether this learning is excitatory or inhibitory.
 The model predicts that subjects trained in a rectangular
enclosure with four distinct features at the corners (as in
Cheng, 1986), will prefer, if tested in a square enclosure (no
geometric information), the feature that was at the near
corner to the feature that was at the rotational corner, even
though neither was paired with a reward during training.
 The model also predicts the results of experiments
comparing features of differing sizes (e.g. Goutex et al., 2001)
and ones that span whole walls (Graham et al., 2006; Sovrano
et al., 2003), multiple features (Cheng, 1986), the effects of
enclosure size (e.g. Vallortigara et al., 2005), and varying
shape (Tommasi & Polli, 2004), and touchscreen versions of
geometry tasks (Kelly & Spetch, 2004).
 This model is a general model of operant discrimination
learning and choice. It predicts opposite results in an operant
task to those found by Wagner et al. (1968) in a Pavlovian
discrimination vs. pseudo-discrimination task.
References
Miller, N. Y., & Shettleworth, S. J.(in press). Learning about
environmental geometry: An associative model. JEP:ABP.
(pdf available from noam.miller@utoronto.ca).