A Bayesian Perspective on Accumulation in the Magnitude System

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A common metric for time, space and number

Walsh proposed that the seemingly distinct domains of space, time and number may be processed by a single cross-domain magnitude system in the brain, a proposal that he named ATOM, for A Theory Of Magnitudes (Walsh, 2003). This domain-general magnitude system is thought to be involved in processing temporal, spatial, and numerical magnitudes, including various dimensions such as size, area, length, density for example. ATOM addresses domains that we experience in terms of “more than” or “less than”. ATOM argues that this shared neural substrate confers benefits because it supports the coordination of magnitudes that are relevant for action (Walsh, 2003; Bueti and Walsh, 2009). For example, when human and non-human animals want to grasp an object, magnitude is relevant to perceive the size of the object, how distant the object is and when we should close our hand to grasp it. Two different schemas can be drawn for processing time, space, number and other magnitude dimensions. In the first case, the different magnitudes can be independently analyzed, processed and compared, according to each individual metric (Figure 1.3 A). The second possibility is to consider a generalized magnitude system (ATOM) in which all the different magnitudes are similarly processed, according to a common metric (Figure 1.3 B).

Investigating the directional symmetry imposed by a common metric

One prediction from ATOM and the possible existence of a common neural code is that we should observe bi-directional interactions between all the different magnitude dimensions (Winter, Marghetis, and Matlock, 2015). In other words, time, space and number should equally interfere between each other. One known effect in the literature is the size-congruency effect. When instructed to judge the magnitude of a digit, participants tend to respond slower if the physical size of the digit is incongruent with its magnitude (Pinel et al., 2004; Henik and Tzelgov, 1982). In other words, if the size of the digit is congruent with its value (e.g. “1” and “9”), the numerical judgment will be facilitated compared to the condition in which the size of the digit is incongruent with its value (e.g. “1” and “9”). Xuan et al. (2007) used a congruent vs. incongruent paradigm and asked the participants to perform a temporal judgments task between two stimuli. They had to judge if the second stimulus was presented for a shorter or a longer duration than the first one. Stimulus consisted in an open square that could vary in size. Results showed that temporal estimations were influenced by variations of the physical size of the stimulus. Similarly, Dormal & Pesenti (2007) designed a Stroop task in which participants were required to compare the length or the numerosity of two linear arrays of dots. Results showed a significant main effect on response latencies: responses were provided faster in the congruent condition than in the incongruent one, when participants performed numerical judgments. However, in the spatial task, the number of dots did not interfere with the processing of spatial information. Such asymmetry has also been reported by Hurewitz et al. (2006). In their study, participants were presented with pairs of arrays of dots with varying circle sizes and were required to make numerosity judgments. The authors also investigated if varying the number of circles interfered with judgments of the cumulative filled area. They found an interference effect of the size of the circles when participants had to judge the number of circles such that reaction times and error rates were larger in the incongruent condition than in the congruent one. However, the effect of numerosity on the area comparison was weaker. In these reports of interference between magnitude dimensions, behavioral effects were concluded on the basis of increased reaction times and error rates in incongruent conditions (e.g., small number presented with a long duration) which prevented the direct evaluation of participants’ magnitude perception per se. As such, no clear direction of interference effects could be concluded from the studies beyond the existence of an interaction.
To further investigate interferences across dimensions and the possible existence of bi-directional interactions, Casasanto and Boroditsky (2008) conducted a series of experiment in which participants had to perform spatial and temporal reproduction task. Such design allowed the authors to quantify the size of the interaction (e.g. reproduced duration compared to the objective duration) and to test for possible asymmetry in the interference. In each task participants saw lines or dots on the screen and had to reproduce the spatial displacement or the duration of the trial. The results showed that for a given duration, participants over- (under) estimated the duration when the line traveled a long (short) distance on the screen. No effect of duration on spatial reproduction was found. In another study (Bottini and Casasanto, 2010), duration judgments have been found to be biased by the semantic of words: the estimation of the duration increased as a function of the implicit spatial length of the word. For a given duration, participants judged that the word “Highway” stayed longer on the screen than the word “Pencil”, for example. However, the implicit duration of a word did not interfere with spatial judgments. Here again this finding highlights directional asymmetries between magnitude dimensions, which is not consistent with one of our prediction, according to which bidirectional interactions should be observed across magnitudes. Similar asymmetries have been reported between number and time, with numerical information interfering with duration judgments but not the reverse (Dormal, Seron, and Pesenti, 2006). Dormal et al. (2006) used a Stroop task in which participants were presented with visually flashing dots and had to compare either the numerosity or the duration of each trial. In the duration task, results showed that congruent condition was answered faster than the incongruent one. No effect was observed in the numerosity comparison task. Droit-Volet et al. (2003) also investigated the interferences in the processing of time and number information. Results showed that an increase of the number of stimuli induced an increase of the “long” responses in the duration discrimination task. In the numerical bisection task, no time interference on the processing of number was found. In this experiment, the effect of number on duration was stronger in 5-years-old children than in 8-years-old children and in adults, suggesting that these asymmetries appear early in development. Supporting this idea, de Hevia & Spelke (2009) found that non-symbolic numerical displays affected the subjective midpoint of a horizontal line in both adults, 3- and 5-year-old children. While number is often reported to affect duration judgments manipulating the duration of events has seldom been reported to affect numerical and spatial magnitudes (Javadi and Aichelburg, 2012; Lambrechts, Walsh, and van Wassenhove, 2013; Cai and Connell, 2015; Martin, Wiener, and van Wassenhove, 2017).

A Bayesian Perspective on Magnitude Estimations

Recent discussions in the field suggest that the combination and evaluation of quantities in a common representational system would be realized on the basis of Bayesian computations (Petzschner, Glasauer, and Stephan, 2015; Shi, Church, and Meck, 2013). When performing temporal, spatial or numerical judgements, the information that we receive comes from a noisy environment. One concept from the signal detection theory (Green and Swets, 1989; Peterson, Birdsall, and Fox, 1954) is that when observers are instructed to detect the presence of a signal, observers will be correct in some cases (hit) and incorrect in others (false alarm). Errors may be due to an uncertainty in the decision process, coming from the noisy sensory input but also coming from the previous situations an observer has experienced. This is where Bayes’ theorem becomes useful in magnitude estimations. In a Bayesian framework, decision is made by combining a priori information (prior) with noisy sensory input (likelihood), weighing the two information sources by their relative uncertainty. This can be summarized with the following equation (Petzschner, Glasauer, and Stephan, 2015): ( | ) ∝ ( | ) ∙ ( ).
Where ( | ) represents the noisy likelihood, ( ) represents the a priori knowledge and ( | ) correspond to the posterior. Figure 1.4 compares a classical model of magnitude estimation and a generative model based on Bayesian probabilities.

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A Common Metric for Time, Space and Number and the issue of scaling

While ATOM predicts scaling across magnitudes (i.e. a mapping factor across magnitude dimensions should exist), an alternative and more parsimonious interpretation of existing number-time interference effects in behavioral data may be a task-driven categorization of magnitudes. In fact, according to ATOM, perceived duration should increase as a function of the tested digit (Figure 3.1, panel A), and the difference in perceived duration between two digits should also increase as a function of the tested duration (Figure 3.1, panel B). However, results in several studies (Rammsayer and Verner, 2016; Chang et al., 2011) suggested that perceived duration did not increase as a function of digit value but rather increased along a dichotomical categorization of numerosity (small or large numerical value). I argue in this section that the number-time interaction may depend on the instructions given to the participants.
Rammsayer and Verner (2016) recently investigated the effect of the value on reproduced duration. Participants were instructed to reproduce the duration of presentation of a digit, and to judge its numerical value. The mean reproduced durations were significantly longer when large digits (8 and 9) were presented as compared to small digits (1 and 2). However, visual inspection and additional analysis of their results (see Figure 3.2) did not reveal any significant increase of the reproduced duration as a function of the numerical value. These findings go against the previous consideration (see Figure 1A) and do not support the hypothesis of a general system in which different magnitudes share a common metric.

Static vs. Dynamic displays and the direction of interference effects

The directionality of the interactions between non-temporal and temporal magnitude dimensions reported in the literature may largely depend on the experimental paradigm being used. There is a growing body of evidence showing that the numerical value of a digit interferes with temporal judgments (Rammsayer and Verner 2016; Oliveri et al. 2008; Chang et al. 2011), and similar effects are reported when numerical information is presented in non-symbolic form: the larger the number of items in a set, the longer the perceived the duration (Dormal, Seron, and Pesenti, 2006; Mo, 1971, 1974, 1975; Xuan et al., 2007; Javadi and Aichelburg 2012). Yet, while duration necessarily accumulates over time, numerical information does not. Indeed, numerical information can be statically provided to the participants (all the items of a set are presented at the same time, during the entire trial) or dynamically (the number of items increases in time, to reach its maximum value at the end of the trial). Investigating the number-time interaction with a static or a dynamic design provide opposite results: whereas it is often reported that large numerosities lengthen the perceived duration when a static design is used (Dormal, Seron, and Pesenti, 2006; Mo, 1971; Javadi and Aichelburg 2012), perceived duration is resilient to non-temporal manipulation when using a dynamic design (Lambrechts, Walsh, and van Wassenhove, 2013; Martin, Wiener, and van Wassenhove, 2017). The present section investigates this specific point.
Javadi and Aichelburg (2012) instructed participants to judge which of two successive sets of items was presented longer (duration task) or which was more numerous (number task). Their results revealed a positive correlation between time and number, with more numerous sets being judged to last longer. These results are in line with a seminal series of experiments showing that temporal estimation increased as a function of numerosity. Mo (1971) initially reported that the proportion of “long” judgments in a duration task significantly increased as a function of the number of dots presented to the participants; in a second study (Mo, 1974), participants were instructed to judge the duration of the second stimuli in a pair, and the proportion of “longer” responses was shown to decrease when the number of dots of the first stimulus increased. These observations indicate that participants perceived the first stimulus as being longer than its actual physical duration; when the numerosity of the second stimulus was manipulated, the proportion of “longer” responses increased as a function of the number magnitude. In a third study (Mo, 1975), participants were presented with sets of dots that varied in numerosity and were instructed to reproduce the duration. Once again, results showed a general tendency for temporal reproduction to increase as numerosity increased. More recently, Dormal and colleagues (2006) used a Stroop task in which participants were presented with visually flashing dots and had to compare either the numerosity or the duration of each trial. In the duration task, results showed that neutral and congruent conditions (in which larger (smaller) numerosity were matched with larger (shorter) durations) were answered faster than the incongruent ones. Overall, these findings suggest an asymmetry in the number-time interaction, with numerical magnitude interfering with duration judgments. The same is also true in the case of space-time interactions: most studies using a static design revealed that the larger the physical size of a stimulus, the longer the perceived duration (Xuan et al., 2007; Rammsayer and Verner, 2014, 2015). For instance, in a reproduction task, Casasanto and Boroditsky (2008) showed that participants (under-) over- estimated the duration when a line covered a (short) long distance. In another experiment (Xuan et al., 2007), manipulating the size of stimuli interfered with temporal estimations such that reaction times and error rates increased in incongruent conditions compared to congruent ones. Altogether, the pattern of results supports the general idea of a generalized magnitude system with perceived duration increasing as a function of non-temporal (i.e. spatial and numerical) magnitude in dynamic designs. However, duration seems to be resilient to numerical and spatial manipulations when sensory evidence accumulates over time (Lambrechts, Walsh, and van Wassenhove, 2013; Martin, Wiener, and van Wassenhove, 2017). The next paragraph investigates this specific point.

Table of contents :

1| Introduction
1.1 Magnitudes in the Brain
1.1.1 The Weber-Fechner law
1.1.3 Regression effect
1.2 Same mechanisms for the processing of time, space and number?
1.2.1 The parietal lobe in fMRI studies
1.2.2 The parietal lobe in clinical studies
1.3 A Theory Of Magnitudes (ATOM)
1.3.1 A common metric for time, space and number
1.3.2 Investigating the directional symmetry imposed by a common metric
1.3.3 A common metric implies a scaling effect between magnitudes
1.4 A Bayesian Perspective on Magnitude Estimations
1.5 Aim of this thesis
2| A Bayesian Perspective on Accumulation in the Magnitude System
2.1 Summary
2.2 Reference
3| A Theory Of Magnitudes
3.1 A Common Metric for Time, Space and Number and the issue of scaling
3.2 Static vs. Dynamic displays and the direction of interference effects
3.3 Numerical magnitude affects temporal encoding, not temporal reproduction
3.4 Non symbolic magnitude is automatically processed, not the symbolic one
4| The larger the longer, but not all the time
4.1 Summary
4.2 Materials & Methods
4.3 Results
4.4 Discussion & Conclusion
5| Discussion and Conclusion
5.1 A common neural code for Time, Space and Number?
5.2 From a Bayesian perspective, time, space and number do not share the same priors
5.3 The rate of accumulation of sensory evidence interferes with Numerical and Spatial estimates
5.4 Perceived duration does not always increase as a function of the tested numerosity: no scaling effect
5.5 Non-temporal magnitudes do not interfere with duration at a perceptual level
5.6 The number-time interaction is modulated by attention and numerical format
5.7 Conclusions


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