Qi Chen

Verbal-spatial and visuospatial coding of number-space interactions.

A tight correspondence has been postulated between the representations of number and space. The spatial numerical association of response codes (SNARC) effect, which reflects the observation that people respond faster with the left-hand side to small numbers and with the right-hand side to large numbers, is regarded as strong evidence for this correspondence. The dominant explanation of the SNARC effect is that it results from visuospatial coding of magnitude (e.g., the mental number line hypothesis). In a series of experiments, we demonstrated that this is only part of the story and that verbal-spatial coding influences processes and representations that have been believed to be purely visuospatial. Additionally, when both accounts were directly contrasted, verbal-spatial coding was observed in absence of visuospatial coding. Relations to other number-space interactions and implications for other tasks are discussed.

Beyond the mental number line: A neural network model of number–space interactions

It is commonly assumed that there is an interaction between the representations of number and space (e.g., Dehaene, Bossini, & Giraux, 1993; Walsh, 2003), typically ascribed to a mental number line. The exact nature of this interaction has remained elusive, however. Here we propose that spatial aspects are not inherent to number representations, but that instead spatial and numerical representations are separate. However, cultural factors establish ties between them. By extending earlier models (Gevers, Verguts, Reynvoet, Caessens, & Fias, 2006; Verguts, Fias, & Stevens, 2005) based on this hypothesis, the authors present computer simulations showing that a model incorporating this idea can account for data from a series of studies. These results suggest that number-space interactions are emergent properties resulting from the interaction between different brain areas.

Spatial Intuition in Elementary Arithmetic: A Neurocomputational Account

Elementary arithmetic (e.g., addition, subtraction) in humans has been shown to exhibit spatial properties. Its exact nature has remained elusive, however. To address this issue, we combine two earlier models for parietal cortex: A model we recently proposed on number-space interactions and a modeling framework of parietal cortex that implements radial basis functions for performing spatial transformations. Together, they provide us with a framework in which elementary arithmetic is based on evolutionarily more basic spatial transformations, thus providing the first implemented instance of Dehaene and Cohens recycling hypothesis.

Spontaneous summation or numerosity-selective coding?

A key debate in numerical cognition concerns the neural code for number representation (e.g., Nieder and Merten, 2007; Roggeman et al., 2007; Viswanathan and Nieder, 2013). One idea is that individual neurons are tuned to individual numbers, with decreasing response to numbers with increasing distance (numerosity-selective coding or labeled-line coding). An alternative, more implicit way of representing number is by summation coding. Here, individual neurons fire either monotonically stronger or weaker to increasing number. The number can then be decoded from the pooled cell activity. n nThe computational properties of both coding types have been studied. A summation code but not a numerosity-selective code was extracted without number-related training from a visual display in a recent modeling study (Stoianov and Zorzi, 2012). Also, the summation code serves as a precursor for a numerosity-selective code in such models (Dehaene and Changeux, 1993; Verguts and Fias, 2004). Furthermore, each coding type has distinct advantages; summation coding is more suited for smaller-larger (i.e., magnitude) processing, numerosity-selective coding is more efficient for same-different number discrimination (Verguts, 2007). n nIn a number of papers, Nieder and colleagues demonstrated numerosity-selective coding in macaque monkeys (e.g., Nieder et al., 2002; Nieder and Miller, 2004). However, number was always relevant for the task; in other words, animals were trained on number (e.g., Nieder et al., 2002). The computational modeling work jointly predicts that summation coding is primary and foundational to numerosity-selective coding, and that in the absence of number-relevant training, only summation coding would be observed. Consistently, Roitman et al. (2007) showed that only summation coding was observed in a single-unit recording study in which number was not relevant for solving the task. However, number was relevant for computing the reward at trial offset, so it may still be that number-relevant learning took place during training. n nTo determine the natural numerical coding system (i.e., without number learning), Viswanathan and Nieder (2013) recorded cells from ventral intraparietal area [VIP, in intraparietal sulcus (IPS)] and from prefrontal cortex (PFC) in two monkeys in a task without number relevance (and hence number learning). They found that neurons in both brain areas responded maximally to a given number (e.g., one neuron responded maximally to 1, another neuron maximally to 2, and so on). They interpret their data as suggesting numerosity-selective coding. They also found that the most frequently preferred numbers for these neurons were numbers 1 and 5, whereas a relatively small set of neurons were classified as tuned to intermediate numbers 2, 3, and 4. However, given the computational primacy of summation coding, we consider the possibility that the authors instead sampled summation coding neurons. Here, we show that the data are consistent with summation coding, and that summation coding can account for subtle and unexplained aspects of the data.

Biasing the organism for novelty: A pervasive property of the attention system

Although the functional and anatomical independences between the orienting and the executive attention networks have been well established, surprisingly little is known about the potential neural interaction between them. Recent studies point out that spatial inhibition of return (IOR), a mechanism associated with the orienting network, and nonspatial inhibition of return, a mechanism associated with the executive network, might bias the organism for novel locations and objects, respectively. By orthogonally combining the spatial and the nonspatial IOR paradigms in this fMRI study, we demonstrate that the orienting and the executive networks interact and compensate each other in biasing the attention system for novelty. Behaviorally, participants responded slower to the target at the old location only when the color of the target was novel, and participants responded slower to the old color representation only when the target appeared at a novel spatial location. Neurally, the orienting network was involved in slowing down responses to the old location only when the nonspatial IOR mechanism in the executive network was not operative (i.e., when the color of the target was novel); the prefrontal executive network was involved in slowing down responses to the old color representation only when the spatial IOR mechanism in the orienting network was not functioning (i.e., when the target appeared at a novel location). Hum Brain Mapp, 2010.

The Time Course of Spatial Attention Shifts in Elementary Arithmetic

It has been proposed that elementary arithmetic induces spatial shifts of attention. However, the timing of this arithmetic-space association remains unknown. Here we investigate this issue with a target detection paradigm. Detecting targets in the right visual field was faster than in the left visual field when preceded by an addition operation, while detecting targets in the left visual field was faster than in the right visual field when preceded by a subtraction operation. The arithmetic-space association was found both at the end of the arithmetic operation and during calculation. In contrast, the processing of operators themselves did not induce spatial biases. Our results suggest that the arithmetic-space association resides in the mental arithmetic operation rather than in the individual numbers or the operators. Moreover, the temporal course of this effect was different in addition and subtraction.

Behavioral and neural interaction between spatial inhibition of return and the Simon effect

It has been well documented that the anatomically independent attention networks in the human brain interact functionally to achieve goal-directed behaviors. By combining spatial inhibition of return (IOR) which implicates the orienting network with some executive function tasks (e.g., the Stroop and the flanker tasks) which implicate the executive network, researchers consistently found that the interference effects are significantly reduced at cued compared to uncued locations, indicating the functional interaction between the two attention networks. However, a unique, but consistent effect is observed when spatial IOR is combined with the Simon effect: the Simon effect is significantly larger at the cued than uncued locations. To investigate the neural substrates underlying this phenomenon, we orthogonally combined the spatial IOR with the Simon effect in the present event-related fMRI study. Our behavioral data replicated previous results by showing larger Simon effect at the cued location. At the neural level, we found shared spatial representation system between spatial IOR and the Simon effect in bilateral posterior parietal cortex (PPC); spatial IOR specifically activated bilateral superior parietal cortex while the Simon effect specifically activated bilateral middle frontal cortex. Moreover, left precentral gyrus was involved in the neural interaction between spatial IOR and the Simon effect by showing significantly higher neural activity in the “Cued_Congruent” condition. Taken together, our results suggest that due to the shared spatial representation system in the PPC, responses were significantly facilitated when spatial IOR and the Simon effect relied on the same spatial representations, i.e., in the “Cued_Congruent” condition. Correspondingly, the sensorimotor system was significantly involved in the “Cued_Congruent” condition to fasten the responses, which indirectly resulted in the enhanced Simon effect at the cued location.

Addition and Subtraction but Not Multiplication and Division Cause Shifts of Spatial Attention

Many studies have shown that solving addition and subtraction problems can induce overt shifts of spatial attention. In particular, right-side targets are detected faster than left-side targets when preceded by an addition operation, while left-side targets are detected faster than right-side targets when preceded by a subtraction operation. However, the interaction between space and arithmetic in multiplication or division is hardly studied and remains controversial. In order to make a strong case for the interaction between space and mental arithmetic, we attempted to replicate the spatial-arithmetic association in addition and subtraction (Experiment 1), and at the same time investigated whether shift of spatial attention would also be induced by multiplication or division operations (Experiment 2). We found that solving addition problems facilitated the detection of right-side targets, whereas left-side targets were detected faster after solving subtraction problems. However, no interaction between space and arithmetic operation was observed in multiplication or division. The implication of these findings is discussed.

Neural Correlates underlying Size Constancy in Virtual Three-Dimensional Space

The perceived size of an object remains relatively constant although its retinal size keeps decreasing as the object moves away along the depth dimension of the 3D space, i.e. size constancy. Neural mechanisms generating size constancy in virtual 3D space, however, remain poorly understood. By constructing a virtual 3D world in the MR scanner, we positioned the same 3D objects either near or far from the observers so that the near and far objects were perceived as having the same physical size despite their differences in retinal size. To control for the effect of differential retinal size, an additional 2D condition was introduced: a large and a small object, with matched retinal images as the near and far objects in the 3D condition, respectively, were presented on a 2D screen. Differences in retinal size activated overlapped areas in bilateral inferior occipital gyrus (IOG) in both experiments. The overlapped areas in IOG, however, showed different patterns of functional connectivity with different neural networks, depending on the perceived size of objects. In particular, IOG showed enhanced connectivity with bilateral superior parietal cortex in the 2D condition, but with inferior temporal and prefrontal cortex in the virtual 3D condition, i.e., size constancy.

Numerical Proportion Representation: A Neurocomputational Account

Proportion representation is an emerging subdomain in numerical cognition. However, its nature and its correlation with simple number representation remain elusive, especially at the theoretical level. To fill this gap, we propose a gain-field model of proportion representation to shed light on the neural and computational basis of proportion representation. The model is based on two well-supported neuroscientific findings. The first, gain modulation, is a general mechanism for information integration in the brain; the second relevant finding is how simple quantity is neurally represented. Based on these principles, the model accounts for recent relevant proportion representation data at both behavioral and neural levels. The model further addresses two key computational problems for the cognitive processing of proportions: invariance and generalization. Finally, the model provides pointers for future empirical testing.