Beyond the SNARC effect: distance–number mapping occurs in the peripersonal space

Effect of egocentric and allocentric reference frames on spatial-numerical associations

From an embodied view of cognition, sensorimotor mechanisms are strongly involved in abstract processing, such as Arabic number meanings. For example, spatial cognition can influence number processing. These spatial-numerical associations (SNAs) have been deeply explored since the seminal SNAs of response code (SNARC) effect (i.e., faster left/right sided responses to small/large magnitude numbers, respectively). Although these SNAs along the transverse plane (left-to-right axis) have been extensively studied in cognitive sciences, no systematic assessment of other planes of the tridimensional space has been afforded. Moreover, there is no evidence of how SNAs organise themselves throughout the changes in spatial body-reference frames (egocentric and allocentric). Hence, this study aimed to explore how SNAs organise themselves along the transverse and sagittal planes when egocentric and allocentric changes are processed during body displacements in the environment. In the first experiment, the results revealed that, when the participants used an egocentric reference, SNAs were observed only along the sagittal plane. In a second experiment that used an allocentric reference, the reversed pattern of results was observed: SNAs were present only along the transverse plane of the body. Overall, these findings suggest that, depending on the spatial reference frames of the body, SNAs are strongly flexible.

Spatio-Numerical Mapping in 3D

The close link between number and space is illustrated by the Spatial Numerical Association of Response Codes (SNARC) effect. The current research focuses on the flexibility of the SNARC across three dimensions. Shaki and Fischer (2018) pointed out that spatial attributes of stimuli and response effectors can favor an ad hoc spatial representation. In this paper, we aimed to broaden this perspective using two Go/NoGo experiments with digits being presented at two spatial locations while a central response was required. In Experiment 1, stimuli appeared either to the left or right (horizontal) and below or above fixation (vertical). In Experiment 2, as the monitor was laying down flat on the desk, stimuli appeared either to the left or right (horizontal) and either close or far from the observer (midsagittal). The results of Experiment 1 show significant effects for the two dimensions (horizontal, vertical), while in Experiment 2, we observe only a barely significant effect for the sagittal axis. We interpret these findings as showing (1) the importance of motor response spatialization in eliciting the SNAs and (2) the dominance of the vertical axis over the horizontal when the spatial component of the motor response is removed.

The SNARC effect: a preregistered study on the interaction of horizontal, vertical, and sagittal spatial-numerical associations

Small numbers are processed faster through left-sided than right-sided responses, whereas large numbers are processed faster through right-sided than left-sided responses [i.e., the Spatial-Numerical Association of Response Codes (SNARC) effect]. This effect suggests that small numbers are mentally represented on the left side of space, whereas large numbers are mentally represented on the right side of space, along a mental number line. The SNARC effect has been widely investigated along the horizontal Cartesian axis (i.e., left-right). Aleotti et al. (Cognition 195:104111, 2020), however, have shown that the SNARC effect could also be observed along the vertical (i.e., small numbers-down side vs. large numbers-up side) and the sagittal axis (i.e., small numbers-near side vs. large numbers-far side). Here, we investigated whether the three Cartesian axes could interact to elicit the SNARC effect. Participants were asked to decide whether a centrally presented Arabic digit was odd or even. Responses were collected through an ad hoc-made response box on which the SNARC effect could be compatible for one, two, or three Cartesian axes. The results showed that the higher the number of SNARC-compatible Cartesian axes, the stronger the SNARC effect. We suggest that numbers are represented in a three-dimensional number space defined by interacting Cartesian axes.

Early is left and up: Saccadic responses reveal horizontal and vertical spatial associations of serial order in working memory

Maintaining serial order in working memory is crucial for cognition. Recent theories propose that serial information is achieved by positional coding of items on a spatial frame of reference. In line with this, an early-left and late-right spatial-positional association of response code (SPoARC) effect has been established. Various theoretical accounts have been put forward to explain the SPoARC effect (the mental whiteboard hypothesis, conceptual metaphor theory, polarity correspondence, or the indirect spatial-numerical association effect). Crucially, while all these accounts predict a left-to-right orientation of the SPoARC effect, they make different predictions regarding the direction of a possible vertical SPoARC effect. In this study, we therefore investigated SPoARC effects along the horizontal and vertical spatial dimension by means of saccadic responses. We replicated the left-to-right horizontal SPoARC effect and established for the first time an up-to-down vertical SPoARC effect. The direction of the vertical SPoARC effect was in contrast to that predicted by metaphor theory, polarity correspondence, or by the indirect spatial-numerical association effect. Rather, our results support the mental whiteboard-hypothesis, according to which positions can be flexibly coded on an internal space depending on the task demands. We also found that the strengths of the horizontal and vertical SPoARC effects were correlated, showing that some people are more prone than others to use spatial references for position coding. Our results therefore suggest that context templates used for position marking are not necessarily spatial in nature but depend on individual strategy preferences.

Children's spatial-numerical associations on horizontal, vertical, and sagittal axes

There is substantial evidence linking numerical magnitude to the physical properties of space. The most influential support for this connection comes from the SNARC effect (spatial-numerical association of response codes), in which responses to small/large numbers are faster on the left/right side of space, respectively. The SNARC effect has been extensively replicated, and is understood as horizontal mapping of numerical magnitude. However, much less is known about how numbers are represented on the vertical and sagittal axes, and whether spatial-numerical associations on different axes emerge during childhood. To that end, we tested two groups of children, aged 5-7 years and 8 and 9 years, on a single-digit magnitude comparison task with response buttons positioned either upper/lower (vertical), left/right (horizontal) or near/far (sagittal). Our results provide evidence of spatial-numerical mapping on all three axes for both age groups that are similar in strength. This indicates that, even at an early stage of formal education, children can flexibly assign numerical magnitude to any spatial dimension. To examine the contribution of extracorporeal space and spatio-anatomical mapping to the SNARC effect across axes, these sources were pitted against each other by swapping the position of the response hands in Experiment 1b. Switching hand position did not reveal convincing evidence for SNARC effects on any axis. Results are discussed with respect to the utility of three-dimensional mental number lines, and potential avenues for future research are outlined.

The spatial representation of number, time, and serial order following sensory deprivation: A systematic review

The spatial representation of numerical and temporal information is thought to be rooted in our multisensory experiences. Accordingly, we may expect visual or auditory deprivation to affect the way we represent numerical magnitude and time spatially. Here, we systematically review recent findings on how blind and deaf individuals represent abstract concepts such as magnitude and time (e.g., past/future, serial order of events) in a spatial format. Interestingly, available evidence suggests that sensory deprivation does not prevent the spatial "re-mapping" of abstract information, but differences compared to normally sighted and hearing individuals may emerge depending on the specific dimension considered (i.e., numerical magnitude, time as past/future, serial order). Herein we discuss how the study of sensory deprived populations may shed light on the specific, and possibly distinct, mechanisms subserving the spatial representation of these concepts. Furthermore, we pinpoint unresolved issues that need to be addressed by future studies to grasp a full understanding of the spatial representation of abstract information associated with visual and auditory deprivation.

The SNARC effect in two dimensions: Evidence for a frontoparallel mental number plane

The existence of an association between numbers and space is known for a long time. The most prominent demonstration of this relationship is the spatial numerical association of response codes (SNARC) effect, describing the fact that participants' reaction times are shorter with the left hand for small numbers and with the right hand for large numbers, when being asked to judge the parity of a number (Dehaene et al., J. Exp. Psychol., 122, 371-396, 1993). The SNARC effect is commonly seen as support for the concept of a mental number line, i.e. a mentally conceived line where small numbers are represented more on the left and large numbers are represented more on the right. The SNARC effect has been demonstrated for all three cardinal axes and recently a transverse SNARC plane has been reported (Chen et al., Exp. Brain Res., 233(5), 1519-1528, 2015). Here, by employing saccadic responses induced by auditory or visual stimuli, we measured the SNARC effect within the same subjects along the horizontal (HM) and vertical meridian (VM) and along the two interspersed diagonals. We found a SNARC effect along HM and VM, which allowed predicting the occurrence of a SNARC effect along the two diagonals by means of linear regression. Importantly, significant differences in SNARC strength were found between modalities. Our results suggest the existence of a frontoparallel mental number plane, where small numbers are represented left and down, while large numbers are represented right and up. Together with the recently described transverse mental number plane our findings provide further evidence for the existence of a three-dimensional mental number space.