数量加工单方向干扰时间知觉的认知与神经机制

近年来,数量加工对时间知觉的单方向干扰效应受到广泛关注,然而关于该效应的认知和神经机制的研究却缺乏深度的文献梳理和理论分析。本文从抽象数量、数字符号等认知加工如何干扰时间知觉的角度,系统总结了近期数量加工单方向影响时间知觉的研究进展;并在对相关文献加以梳理的基础上,详细分析了该效应可能的认知和神经机制,并提出若干未来研究的设想。     

Subitizing: Magical numbers or mere superstition?

Summary It is widely believed that humans are endowed with a specialized numerical process, calledsubitizing, which enables them to apprehend rapidly and accurately the numerosity of small sets of objects. A major part of the evidence for this process is a purported discontinuity in the mean response time (RT) versus numerosity curves at about 4 elements, when subjects enumerate up to 7 or more elements in a visual display. In this article, RT data collected in a speeded enumeration experiment are subjected to a variety of statistical analyses, including several tests on the RT distributions. None of these tests reveals a significant discontinuity as numerosity increases. The data do suggest a strong stochastic dominance in RT by display numerosity, indicating that the mental effort required to enumerate does increase with each additional element in the display, both within and beyond the putative subitizing range.     

Perceived numerosity of spatiotemporal events

Numerosity discrimination was examined when items were varied in space-time position rather than in space only. Observers were instructed to indicate which of two adjacent streamsof visual events contained more items. The precision of numerosity discrimination of dynamic events was not remarkably different from that of static patterns. Two basic numerosity biases previously found for static dot patterns—inhibitory overestimation and satellite underestimation—were demonstrated for items distributed randomly over a spatiotemporal interval. It was also demonstrated that two streams, equated in the number and luminous energy of items, are not judged equal in their visible number if items in one of these two streams have longer duration than items in the second stream. These findings can be accounted for by the occupancy model of perceived numerosity (Allik & Tuulmets, 1991a) if it is supposed that the impact that each element has on its neighborhood is spread along both spatial and temporal coordinates. Perceived numerosity decreases with both spatial and temporal proximity between the-visual-items. Space and time have interchangeable effects on perceived numerosity: the amount of numerosity bias caused by the spatial proximity of items can also be produced by the properly chosen temporal proximity of items     

Temporal information affects the performance of numerosity discrimination: Behavioral evidence for a shared system for numerosity and temporal processing

In this study, we tested whether and how temporal information would affect the performance of numerosity discrimination in sequential events, in an attempt to make the relationship clear between temporal and numerosity processing. We manipulated the duration of event presentation (i.e., the stimulus duration) and the duration of the sequence (i.e., the total interval). We also employed three levels of standard event numbers (i.e., 5, 10, and 20) to test whether and how the effect would differ among event numbers. The results showed that temporal information affected the performance of numerosity discrimination; precision deteriorated when stimulus duration and the total interval were manipulated, and the number of events in the longer total interval were judged as less numerous than those in the shorter total interval across standard numbers. Accordingly, our results suggested the existence of a shared system between time and numerosity processing.     

Numerosity-duration interference: a Stroop experiment

The existence of a possible common mechanism for duration and numerosity processing was tested with a Stroop task. Participants had to compare either the duration or the numerosity of sequences of flashing dots for which the duration and numerosity were independently manipulated to create congruent, incongruent or neutral pairs. Results show that the numerical cues interfered with duration processing, whereas the temporal cues did not interfere with numerosity processing. These findings extend the idea of an automatic access to magnitude to non-symbolic sequentially presented material, and reflect a probable difference in the mandatory processing of numerosity and duration.     

Audiotactile interactions in temporal perception

In the present review, we focus on how commonalities in the ontogenetic development of the auditory and tactile sensory systems may inform the interplay between these signals in the temporal domain. In particular, we describe the results of behavioral studies that have investigated temporal resolution (in temporal order, synchrony/asynchrony, and simultaneity judgment tasks), as well as temporal numerosity perception, and similarities in the perception of frequency across touch and hearing. The evidence reviewed here highlights features of audiotactile temporal perception that are distinctive from those seen for other pairings of sensory modalities. For instance, audiotactile interactions are characterized in certain tasks (e.g., temporal numerosity judgments) by a more balanced reciprocal influence than are other modality pairings. Moreover, relative spatial position plays a different role in the temporal order and temporal recalibration processes for audiotactile stimulus pairings than for other modality pairings. The effect exerted by both the spatial arrangement of stimuli and attention on temporal order judgments is described. Moreover, a number of audiotactile interactions occurring during sensory-motor synchronization are highlighted. We also look at the audiotactile perception of rhythm and how it may be affected by musical training. The differences emerging from this body of research highlight the need for more extensive investigation into audiotactile temporal interactions. We conclude with a brief overview of some of the key issues deserving of further research in this area.     

The Ornstein-Uhlenbeck model for decision time in cognitive tasks: An example of control of nonlinear network dynamics

The Ornstein-Uhlenbeck (OU) model for human decision-making has been successfully applied to account for response accuracy and response time (RT) data in recent two-choice decision models. A variant of the OU model is shown to arise from the response dynamics of a nonlinear network consisting of randomly connected neural processing units. When feedback control of the network is effected by the stimulus onset, the average network response is an autocorrelated random signal satisfying the stochastic differential equation for the OU process. An alternative, more general, stimulus detection procedure is proposed which involves the use of an adaptive Kalman filter process to track any temporal change in autoregressive parameters. The predicted decision time distributions suggest that both the OU and the Kalman filter processes can serve as alternative models for RT data in experimental tasks. Received: 15 October 1998 / Accepted: 27 May 1999     

Numerical reproduction in pigeons

Pigeons were trained to match the number of responses made during a production phase to the number of keylight flashes (2, 4, or 6) in a previous sample phase. In Experiment 1, there were 2 conditions in which the flashes were programmed to occur at a constant rate or within a constant overall duration. For both conditions, although accuracy was relatively low, responding increased linearly with flash number and coefficients of variation decreased. Positive transfer to novel numbers was obtained only when test and baseline trials had the same temporal organization, but multiple regressions revealed significant control by number independently of temporal cues. In Experiment 2, flashes were programmed to occur pseudorandomly to degrade the validity of temporal cues. Results were similar to in Experiment 1. A prototype response class model accounted for the major features of the data. According to the model, responses during the production phase are shaped into higher order units that are associated with different stimulus numbers and function as a rough category scale of numerosity.     

Behavioral evidence for format-dependent processes in approximate numerosity representation

A genuinely abstract number representation is thought to be capable of representing the numerosity of any set of discrete elements, whether they are sequentially or simultaneously presented. Recent neuroimaging studies, however, have demonstrated that different areas of intraparietal sulcus play a role in extracting numerosity across simultaneous or sequential presentation during a quantification process, suggesting the existence of a format-dependent numerical system. To test whether behavioral evidence exists for format-dependent numerical processing in adult humans, we measured the Weber fractions of numerosity discrimination for sequential stimuli, simultaneous stimuli, and cross-format stimuli with a carefully controlled experimental procedure. The results showed distinct differences between the performance in the simultaneous and sequential conditions, supporting the existence of format-dependent processes for numerosity representation. Moreover, the performance on cross-format trials differed among participants, with the exception that performance was always worse than in the simultaneous condition. Taken together, our findings suggest that numerical representation may involve a complex set of multiple stages.     

非数量视觉特征对数量感知的影响

以往许多研究探讨了各种视觉特征对数量判断任务的影响,但得出了不一致的结论.本研究依据信号检测论原理对刺激特征影响数量判断的方式进行考察.结果发现:刺激大小、形状和分布均匀度等因素可影响数量判断的反应标准,但不会影响辨别力;刺激聚集或组块会直接对数量判断的辨别力产生影响.结果表明,不同刺激特征对数量判断的影响方式存在差异.一方面,人类具有从刺激特征中直接抽取数量信息进行认知比较的能力;另一方面,刺激点聚集或组块可以直接影响被试的数量感知.此外,我们还推测,组块是通过改变客体表征的方式来影响数量感知的.     

The impact of stimulus and response variability on S-R correspondence effects

Six experiments investigated how variability on irrelevant stimulus dimensions and variability on response dimensions contribute to spatial and nonspatial stimulus-response (S-R) correspondence effects. Experiments 1-3 showed that, when stimuli varied in location and number, S-R correspondence effects for location or numerosity occurred when responses varied on these dimensions but not when responses were invariant on these dimensions. These results are consistent with the response-discrimination account, according to which S-R correspondence effects should only arise for a dimension that is used for discriminating between responses in working memory. Experiments 4-6 showed that, when responses varied in location and number, both invariant and variable stimulus number produced correspondence effects in S-R numerosity. In summary, the present results indicate that the usefulness of a particular dimension for response discrimination can be sufficient for producing S-R correspondence effects, whereas variability of a stimulus dimension is not sufficient for producing such effects.     

Bidirectional examination of the interaction between time and numerosity corroborates that numerosity influences time estimation but not vice versa

There has been great interest in the idea that time, number, and space share a common magnitude system. However, only a handful of studies examined bidirectional interaction between time and number and the results varied depending on the specifics of the methods and stimulus properties of each study. The present study investigated bidirectional interaction between time and number using estimation tasks. We used duration (Experiment 1) and numerosity (Experiment 2) estimation tasks to investigate the effect of numerosity‐on‐duration and duration‐on‐numerosity estimation. The results from the two experiments demonstrated that numerosity influences duration processing but not vice versa; that is, there was unidirectional interaction between numerosity and time. The duration of stimulus presentation was overestimated for stimuli larger in (task‐irrelevant) numerosity. Possible mechanisms underlying the unidirectional interaction between time and number are discussed.     

Numerosity discrimination in deep neural networks: Initial competence,developmental refinement and experience statistics

Both humans and non‐human animals exhibit sensitivity to the approximate number of items in a visual array, as indexed by their performance in numerosity discrimination tasks, and even neonates can detect changes in numerosity. These findings are often interpreted as evidence for an innate ‘number sense’. However, recent simulation work has challenged this view by showing that human‐like sensitivity to numerosity can emerge in deep neural networks that build an internal model of the sensory data. This emergentist perspective posits a central role for experience in shaping our number sense and might explain why numerical acuity progressively increases over the course of development. Here we substantiate this hypothesis by introducing a progressive unsupervised deep learning algorithm, which allows us to model the development of numerical acuity through experience. We also investigate how the statistical distribution of numerical and non‐numerical features in natural environments affects the emergence of numerosity representations in the computational model. Our simulations show that deep networks can exhibit numerosity sensitivity prior to any training, as well as a progressive developmental refinement that is modulated by the statistical structure of the learning environment. To validate our simulations, we offer a refinement to the quantitative characterization of the developmental patterns observed in human children. Overall, our findings suggest that it may not be necessary to assume that animals are endowed with a dedicated system for processing numerosity, since domain‐general learning mechanisms can capture key characteristics others have attributed to an evolutionarily specialized number system.     

Response and encoding factors in “ignoring” irrelevant information

Subjects classified either the numerosity or numeric value of elements in successive stimulus displays. In separate experiments, responses were indicated by oral naming, card sorting, manual tapping, and oral “tapping.” Incongruent levels of numeric value slowed naming and sorting, but not tapping, when numerosity was the cue for responding. Incongruent numerosity slowed tapping, but not naming and sorting, when numeric value was the cue. Changes in stimulus response mapping may thus critically alter the ability to ignore an irrelevant stimulus dimension.     

A probabilistic model for the discrimination of visual number

This paper proposes a probabilistic model of how humans identify the number of dots within a briefly presented visual display. The model is an application of Thurstone’s law of comparative judgment, and it is assumed that the internal representation of numerosity consists of log-spaced random variables. The discrimination between any two different numerosities is consequently described as a function of max/rain, where max and rain are the larger and smaller numbers, respectively. The model was tested in two experiments in which the Weber fraction for numerosity, corresponding with the critical ratio of max and min, was found to have the value of 162. It was concluded that the classical span of subitizing numerosity is but a special case of the span of discrimination.     

Incidental encoding of numerosity in visual long-term memory

The visual system can readily extract numerosity information from brief experiences. This numerical perception is characterized by diminishing accuracy as numerosity increases, and impaired discrimination for similar quantities and large magnitudes. Here we assess whether these properties apply more broadly to numerosity in visual long-term memory. In surprise memory tests, we observed: Remarkable accuracy in estimating the number of repetitions of an exemplar image (Experiment 1a), that this accuracy decreased but remained high when estimating over categories (Experiments 1b and 1c), that numerical discrimination from memory exhibited psychophysical distance and size effects (Experiment 2), that these effects may derive from stored representations rather than post hoc approximation (Experiment 3a), and that they can reflect total elapsed experience in addition to discrete counts (Experiment 3b). Similar to how numerosity is readily extracted during visual perception, our results suggest that numerosity is encoded incidentally in visual long-term memory.     

Calibrating the mental number line

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.     

Single-parameter power law psychophysics of auditory numerosity and the psychological moment hypothesis.

In the present study, numerosity estimation was investigated. A two-parameter Stevens power law analysis was performed on a total of 944 subjects in six experiments. Two pulse ranges (2-17 or 17-253 pulses) and six pulse rates (either constant or randomly varied within trial blocks) were used, variously, in an unsuccessful attempt to find evidence for a psychological moment, under the supposition that the exponent (or, possibly, the measure constant) would become smaller as increasing numbers of pulses fell within the interval determined by each psychological moment. A single-parameter reanalysis of these six experiments under the initial value condition that a (standard) stimulus of one pulse be assigned a theoretical response (modulus) of one yielded single-parameter equations whose exponents were reliably less varied than those for conventional two-parameter equations in Experiments 1-4 (with randomly varying pulse rates from trial to trial) but not less varied in Experiments 5 and 6 (in which pulse rates were constant within trial blocks). It was concluded that the variable pulse rate condition, with its reduced exponent variability and presumed reduced temporal confounding, provides a more valid estimate of the single-parameter power law exponent for numerosity, which was found to be 0.80.     

Spontaneous non‐verbal counting in toddlers

A wealth of studies have investigated numerical abilities in infants and in children aged 3 or above, but research on pre‐counting toddlers is sparse. Here we devised a novel version of an imitation task that was previously used to assess spontaneous focusing on numerosity (i.e. the predisposition to grasp numerical properties of the environment) to assess whether pre‐counters would spontaneously deploy sequential (item‐by‐item) enumeration and whether this ability would rely on the object tracking system (OTS) or on the approximate number system (ANS). Two‐and‐a‐half‐year‐olds watched the experimenter performing one‐by‐one insertion of ‘food tokens’ into an opaque animal puppet and then were asked to imitate the puppet‐feeding behavior. The number of tokens varied between 1 and 6 and each numerosity was presented many times to obtain a distribution of responses during imitation. Many children demonstrated attention to the numerosity of the food tokens despite the lack of any explicit cueing to the number dimension. Most notably, the response distributions centered on the target numerosities and showed the classic variability signature that is attributed to the ANS. These results are consistent with previous studies on sequential enumeration in non‐human primates and suggest that pre‐counting children are capable of sequentially updating the numerosity of non‐visible sets through additive operations and hold it in memory for reproducing the observed behavior.