An Associative Model of Geometry Learning Noam Miller & Sara Shettleworth

1. An Associative Model of Geometry Learning Noam Miller & Sara Shettleworth University of Toronto, Toronto, Ontario, Canada The puzzle of geometry learning How the model works 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.  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. • 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, • 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. 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. 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. F C R N 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).