![](3D"slide0001_image001.gif")
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.
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 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 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.
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 other, whether this learning is excitatory or inhibitory.
â The model predicts that subjects trained in a rectangular
<=
b>An
Associative Model of Geometry Learning
Noam Miller &am=
p;
Sara Shettleworth
University of
Toronto, Toronto, Ontario, Canada
![](3D"slide0001_image003.gif")
![](3D"slide0001_image005.gif")
![](3D"slide0001_image006.gif")
=
=
How the model solves the puzzle
=
=
References
=
=
=
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 cho=
ices
in geometry and other instrumental learning tasks.
=
=
The puzzle of geometry learning
(pdf
available from noam.miller@utoronto.ca).<=
/span>
=
=
How the model works
There are sever=
al
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(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;
![](3D"slide0001_image008.gif")
![](3D"slide0001_image009.gif")
![](3D"slide0001_image010.gif")
![](3D"slide0001_image011.gif")
![](3D"slide0001_image012.gif")
![](3D"slide0001_image013.gif")
![](3D"slide0001_image014.gif")
![](3D"slide0001_image016.jpg")
![](3D"slide0001_image018.jpg")
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
=
=
Other predictions
![](3D"slide0001_image020.jpg")
![](3D"slide0001_image022.jpg")
![](3D"slide0001_image023.gif")
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
=
![](3D"slide0001_image024.gif")
N
=
F
=
R
=