â Pe=
arce
et al. (2006, Experiment 1) found potentiation of geo=
metry
learning by a feature in a rectangular enclosure, and ove=
rshadowing
of geometry by the feature in a kite-shaped enc=
losure.
Both groups had an innately attractive feature at an inc=
orrect
corner. This experiment indirectly shows the imp=
ortant
difference between geometrically ambiguous and una=
mbiguous
enclosures. In an ambiguous enclosure, such as a
rectangle, attractive features at any corner increase the per=
ceived
reward contingency of the correct geometry by inc=
reasing
the probability of choosing the correct over the rot=
ational
corner. This leads to potentiation.
â In=
an
unambiguous enclosure like the kite, an attractive fea=
ture
at an incorrect corner increases the number of errors, lea=
ding
to overshadowing of the correct geometry.
â 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 =
span>other, whether this learning is excitatory or inhibitory.
â =
span>The model predicts that subjects trained in a rectangular enclosure with four distinct features at the corners (as in <=
/span>Cheng, 1986), will prefer, if tested in a square enclosure (no geometric information), the
feature that was at the near cor=
ner to
the feature that was at the rotational corner, even though neither was paired with a reward during training.
=
â =
span>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).
â =
span>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 =
span>discrimination vs. pseudo-discrimination task.
=
b>
<=
b>An
Associative Model of Geometry Learning
Noam Miller &am=
p;
Sara Shettleworth
University of
Toronto, Toronto, Ontario, Canada
=
=
How the model solves the puzzle
=
=
=
Blocking
and potentiation in the watermaze
=
=
=
Animals can learn to use the shape or geometry of an enclosure to
locate a hidden goal (review in Cheng & Newcomb=
e,
2005). Remarkably, more informative features in the enc=
losure
don’t seem to block or overshadow geometry learning and may even
potentiate it, supporting Cheng’s idea of a “geomet=
ric
module” impenetrable to other spatial inf=
ormation.
But spatial learning in an arena or watermaze is an operant task. By takin=
g thi=
s into
account, our model shows how underlying competitive learning between geome=
try and=
other
cues, as in the Rescorla-Wagner (RW) model, determines observed spatial =
span>cho=
ices
in geometry and other instrumental learning tasks.
In this example=
of a
geometry learning task, disoriented subjects are required to locate a rew=
ard
hidden in one corner of a rectangular enclosure marked by a prominent feat=
ure.
The best predictor of the reward’s location is
obviously the feature, but animals also learn the geo=
metry,
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 vis=
it
corners containing those cues. Choice of what corner to vis=
it is
determined by the associative strength of cues at each corner relative to =
the
total of associative strengths at all corners.
=
span> 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 mor=
e 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 blo=
ck
geometry, as shown below in a comparison of model and data from Wall et al=
l.
2004.
=
=
The puzzle of geometry learning
Mil=
ler,
N. Y., & Shettleworth, S. J.(in press). Learning about <=
span
style=3D'position:absolute;top:90.55%;left:75.45%;width:26.35%;height:1.69=
%'>env=
ironmental
geometry: An associative model. JEP:ABP. &=
#13;
(pdf
available from noam.miller@utoronto.ca).<=
/span>
There are sever=
al
cues that the subject can learn to use to locate the reward: the =
span>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(ro=
ng
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) Δ=
V =3D 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 probabili=
ty
of choosing a location (PL) should be dependent on the associative strengths of =
the
cues at that location (VL). We define:
(2) PL =3D VL / ΣVL,
and modify the original equati=
on
so that subjects only learn about a cue when they visit a location that contains it:
(3) ΔV =3D α β (λ – ΣV) PL.
When visiting a rewarded corne=
r, λ =
=3D 1;
when visiting an unrewarded corner, &=
#955; =3D 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)  =
; &n=
bsp;
ΔVG =3D αG β=
(1
– VGFC) PCorr + αG β=
(0
– VGC) PRot.
&=
#13;
In a watermaze, subjects are usually permitted to swim until they
locate the platform. Thus, the probability of visiting the corr=
ect
corner on a given trial is always 1 but there is al=
so
some non-zero probability of visiting each of the other cor=
ners
along the way. The model accounts for multiple-choice paradigms by taking =
into
account the cumulative probability of each of the possible paths the subje=
ct
can take to the platform.
Pearce et al. (=
2001)
trained rats in an unambiguous triangular watermaze with the pla=
tform
at one of the corners along the base. Group Beacon had a beacon attached=
span>
to the platform, which was alw=
ays in
the same corner; group None had no beacon; gro=
up
Random had a beacon and platform that moved ran=
domly
between the correct and incorrect corners from tri=
al to
trial. Here we show how the model correctly pre=
dicts
the results of a test trial with no beacon present <=
/span>for=
any
of the groups. For such a test, predicted choice=
pro=
portions
are based on relative associative strengths=
of =
the
cues other than the beacon.
C
=
N
=
F
=
R
=