Original ArticlesSpontaneous, modality-general abstraction of a ratio scale
Introduction
Magnitudes such as size, duration, and number share similar psychophysical signatures, appear to use overlapping neural resources, and can influence each other in dual tasks. These observations are consistent with the existence of a shared analog code or generalized magnitude representation (Gallistel and Gelman, 2000, Holyoak and Glass, 1978, Pinel et al., 2004, Walsh, 2003; see Bonn and Cantlon, 2012, Bueti and Walsh, 2009, Cantlon et al., 2009, Cohen Kadosh et al., 2008; and Lourenco, 2015 for extensive reviews). However, the shared-code hypothesis remains underspecified because the existing data has not revealed much about the code’s internal structure. In addition, it remains unclear whether the many studies demonstrating interactions between magnitude dimensions are tapping into more than one possible mechanism.
At a minimum, a shared code should be inherently meaningful across many magnitude domains and across sensory modalities. Two types of relative-magnitude representation—ratios and ranks—automatically offer such generality at different levels of granularity. Ratios and ranks are dimensionless quantities that abstract away from original metrics; for example, the ratio of 1:2 is meaningful on any intensity scale such as loudness or size. Some recent evidence suggests that ratios, represented by pairs of lines of different length or subsets of dot arrays painted in a particular color, are spontaneously represented in a fronto-parietal network in adult humans and macaques (Vallentin and Nieder, 2008, Vallentin and Nieder, 2010, Jacob and Nieder, 2009, Jacob et al., 2012), but it is unclear whether these representations are restricted to their particular dimensions. In principle, ratios could support cross-dimension mapping between pairs of structurally similar analog magnitudes (Srinivasan & Carey, 2010), but current evidence for such transfer is limited.
We explore the possibility that humans spontaneously represent fine-grained information about ratios and ranks in a format that can be compared across modalities and dimensions, providing a candidate for a generalized magnitude representation.
An abstract representation of relative magnitude should allow observers to transfer information about a set of two or more stimuli from one dimension to another without presenting both dimensions simultaneously. Evidence for the transfer of representations of relative magnitude across dimensions or across sensory modalities is scattered across several literatures; here we review a selection of representative examples.
Magnitude-estimation experiments designed to measure subjective sensation demonstrated that after observing a change in magnitude in one dimension relative to an anchor stimulus, subjects are able to generate equivalent proportional changes in other dimensions, given explicit instruction (Luce, 1990, Luce, 2002, Shepard, 1981, Stevens, 1975, Stevens et al., 1960). However, more ambiguous instructions in this task could elicit a wider range of magnitude estimates from unconstrained, heterogeneous transformation rules. Using a more constrained bisection task, Balci and Gallistel (2006) found that within-dimension calculation of proportions likely explained the transfer of duration discrimination to numerical discrimination behavior in humans, but it is unknown how generalizable this result is across multiple dimensions, and whether subjects spontaneously represent rank or proportion relations from sequences.
Cross-dimension transfer of relative magnitudes has been shown in infants for more imprecise representations resembling the concepts of more or less. For example, Lourenco and Longo (2010) showed that when infants learned to associate arbitrary features with large and small object sizes, they expected a similar association between those same features and large and small numerosities or durations. In another study, de Hevia and Spelke (2010) showed that after exposure to a series of stimuli with increasing or decreasing numerosities, 8-month-olds failed to dishabituate to sequences of lines changing length in the same direction, but dishabituated to sequences proceeding in the opposite direction. These studies leave open the question of whether infants generate more precise representations such as ratios and multi-item ranks.
In audition, humans and macaques retain representations of pitch-height changes in sequences of tones (‘melodic contour’; Brosch et al., 2004, Dowling and Fujitani, 1971, Marvin, 1997, Marvin and Laprade, 1987, Trehub et al., 1987). One study found that these representations can be constructed from and transferred across other auditory continua such as brightness and loudness (McDermott, Lehr, & Oxenham, 2008). Other studies found that adults can compare melodies to line drawings that represent long sequences of pitch-height changes (Prince, Schmuckler, & Thompson, 2009), suggesting a modality-independent representation of height. However, the granularity of these abstract representations of pitch and other auditory contours remains unknown.
In summary, previous studies are consistent with the existence of precise, spontaneous, relative-magnitude representations that can be transferred or mapped across diverse dimensions, but no series of experiments has demonstrated precision, spontaneity, and generality of these underlying representations simultaneously while keeping the behavioral methodology constant. Moreover, to our knowledge, no study has yet explicitly distinguished between ratio-based and rank-order-based representations of magnitude sets as potential candidates for generalized magnitude representations.
We provide evidence that human adults use precise, relative-magnitude information to compare sequences within and across sensory modalities and dimensions. Using a sequence-comparison method, we tested the specific hypotheses that subjects (1) can automatically extract ratio information within visual and auditory modalities and (2) can use it to compare sequences across sensory modalities and across the dimensions of space, time and number.
We created pairs of stimulus sequences containing a randomly generated, standard sequence and a comparison sequence that preserved the standard’s abstract structure with varying levels of precision. The comparison could be the same sequence (Same sequences, for within-dimension comparisons only), a sequence in which between-item ratios were preserved (Ratio sequences), a sequence in which only the between-item ranks was preserved (Rank sequences), and a pseudorandom sequence that violated the rank-ordering of the standard (Different sequences); see Fig. 1 for an illustration. We predicted that perceived similarity of patterns would decrease as a function of increased information loss from standard to comparison: Same > Ratio > Rank > Different.
Section snippets
Experiment 1: within-dimension sequence comparisons
In this experiment, sequence pairs were presented in the same stimulus dimension, with separate groups of subjects tested in each. Visual sequences consisted of three squares varying in the dimensions of height or surface area. Auditory sequences consisted of three, band-pass-filtered samples of white noise varying in the dimensions of brightness (center frequency) or loudness (bandwidth and gain). These particular dimensions were chosen for the following reasons: (1) loudness and brightness
Experiment 2: cross-dimension sequence comparisons
A domain-general code, whether shared as a common resource or simply a common code generated by all systems representing magnitudes, should be able to support comparisons across dimensions. In Experiment 2a, we demonstrate that fine-grained representations of relative magnitudes extracted in Experiment 1 can be compared across vision and audition. In Experiment 2b, we show that similar cross-dimension comparison behavior extends to time (interval duration) and number (Arabic numerals).
Omnibus analysis
To compare within-dimension and across-dimension ratings, we ran an omnibus analysis in which we combined all data, excluding Same trials from Experiment 1. As with previous analyses, sequence type was difference-coded. We simple-coded for 3 condition types: Within conditions that included Same trials (labeled Within(+Same) in the coefficient names), the Within condition that excluded Same trials (labeled Within(ØSame) in the coefficient names), and Across conditions. In addition to the random
General discussion
We have demonstrated that subjects can compare sequences on the basis of the precision of relative-magnitude information preserved between sequence pairs. Specifically, patterns that preserved inter-item ratio information were rated as more similar than patterns that only preserved inter-item rank information, a distinction not required by the task but spontaneously imposed by subjects. This is consistent with the hypothesis that a dimensionless representation of a ratio scale supports the
Acknowledgments
The authors wish to thank Richard Aslin, Steven Piantadosi, and Véronique Izard for their comments. An earlier version of this work was submitted to the University of Rochester as a portion of C.D.B.’s dissertation. The research was partially supported by an NSF-GRFP, NSF-GROW, and STEM-Chateaubriand to C.D.B. and NSF 1459625 and NIH R01HD064636 to J.F.C.
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