Analogue Magnitude System Research Articles

Evidence for magnitude representations of social hierarchies: Size and distance effects.

Social status is often metaphorically construed in terms of spatial relations such as height, size, and numerosity. This has led to the idea that social status might partially be represented by an analogue magnitude system, responsible for processing the magnitude of various physical and abstract dimensions. Accordingly, processing of social status should obey Weber’s law. We conducted three studies to investigate whether social status comparisons would indicate behavioral outcomes derived from Weber’s law: the distance effect and the size effect. Dependent variable was the latency of status comparisons for a variety of both learned and familiar hierarchies. As predicted and in line with previous findings, we observed a clear distance effect. However, the effect of size variation differed from the size effect hypothesized a priori, and an unexpected interaction between the two effects was observed. In conclusion, we provide a robust confirmation of previous observations of the distance effect in social status comparisons, but the shape of the size effect requires new theorizing.

Intuitive statistical inferences in chimpanzees and humans follow Weber’s law

Humans and nonhuman great apes share a sense for intuitive statistical reasoning, making intuitive probability judgments based on proportional information. This ability is of fundamental importance, in particular for inferring general regularities from finite numbers of observations and, vice versa, for predicting the outcome of single events using prior information. To date it remains unclear which cognitive mechanism underlies and enables this capacity. The aim of the present study was to gain deeper insights into the cognitive structure of intuitive statistics by probing its signatures in chimpanzees and humans. We tested 24 sanctuary-living chimpanzees in a previously established paradigm which required them to reason from populations of food items with different ratios of preferred (peanuts) and non-preferred items (carrot pieces) to randomly drawn samples. In a series of eight test conditions, the ratio between the two ratios to be discriminated (ROR) was systematically varied ranging from 1 (same proportions in both populations) to 16 (high magnitude of difference between populations). One hundred and forty-four human adults were tested in a computerized version of the same task. The main result was that both chimpanzee and human performance varied as a function of the log(ROR) and thus followed Weber’s law. This suggests that intuitive statistical reasoning relies on the same cognitive mechanism that is used for comparing absolute quantities, namely the analogue magnitude system.

Interference Between Number Magnitude and Parity.

Interference between number magnitude and other properties can be explained by either an analogue magnitude system interfering with a continuous representation of the other properties or by discrete, categorical representations in which the corresponding number and property categories interfere. In this study, we investigated whether parity, a discrete property which supposedly cannot be stored on an analogue representation, could interfere with number magnitude. We found that in a parity decision task the magnitude interfered with the parity, highlighting the role of discrete representations in numerical interference. Additionally, some participants associated evenness with large values, while others associated evenness with small values, therefore, a new interference index, the dual index was introduced to detect this heterogeneous interference. The dual index can be used to reveal any heterogeneous interference that were missed in previous studies. Finally, the magnitude-parity interference did not correlate with the magnitude-response side interference (Spatial-Numerical Association of Response Codes [SNARC] effect) or with the parity-response side interference (Markedness Association of Response Codes [MARC] effect), suggesting that at least some of the interference effects are not the result of the stimulus property markedness.

Quantity discrimination in angelfish (Pterophyllum scalare) is maintained after a 30-s retention interval in the large but not in the small number range.

The ability to discriminate between sets that differ in the number of elements can be useful in different contexts and may have survival and fitness consequences. As such, numerical/quantity discrimination has been demonstrated in a diversity of animal species. In the laboratory, this ability has been analyzed, for example, using binary choice tests. Furthermore, when the different number of items first presented to the subjects are subsequently obscured, i.e., are not visible at the moment of making a choice, the task requires memory for the size of the sets. In previous work, angelfish (Pterophyllum scalare) have been found to be able to discriminate shoals differing in the number of shoal members both in the small (less than 4) and the large (4 or more) number range, and they were able to perform well even when a short memory retention interval (2-15s) was imposed. In the current study, we increased the retention interval to 30s during which the shoals to choose between were obscured, and investigated whether angelfish could show preference for the larger shoal they saw before this interval. Subjects were faced with a discrimination between numerically small shoals (≤4 fish) and also between numerically large (≥4 fish) shoals of conspecifics. We found angelfish not to be able to remember the location of larger versus smaller shoals in the small number range, but to exhibit significant memory for the larger shoal in the large number range as long as the ratio between these shoals was at least 2:1. These results, together with prior findings, suggest the existence of two separate quantity estimation systems, the object file system for small number of items that does not work with the longer retention interval and the analogue magnitude system for larger number of items that does.

Short-Term Memory Effects on Crossing the Boundary: Discrimination between Large and Small Quantities in Angelfish (Pterophyllum scalare).

Rudimentary quantification abilities are found in numerous animal species and in human infants all demonstrating the ability to discriminate between quantities differing in numerical size. An open question is whether individuals rely on different underlying systems to discriminate between large (analogue magnitude system (AMS) for number of items exceeding 3) and small quantities (object-file system (OFS) for number of items below 4), or they use only one system (AMS) for the entire number range. The two-system hypothesis has been supported by finding reduced ability to discriminate between quantities that cross the large-small boundary in several species. Recently, the role of cognitive representation, i.e., memory, in quantity discrimination has also been recognized. Here, we investigated whether angelfish can discriminate quantities across the boundary under two memory conditions. In a binary choice test, single angelfish were allowed to see groups (shoals) of conspecifics of different numerical size on the two sides of their test tank. In Experiment 1, their choice was recorded after a 2-sec retention interval during which shoal size information was unavailable. Angelfish were able to discriminate the larger shoal across the boundary when the shoals differed by a 2:1 or higher ratio, but not when the ratio was lower. In Experiment 2, however, with a 15-sec retention interval, angelfish could only detect a four-fold difference in ratio but failed to detect a three- or a two-fold difference across the boundary. These results suggest that angelfish can remember smaller differences for a short (2 sec) but not for a longer (15 sec) period. Together with previous findings, the current results support the idea that angelfish use two distinct systems for representing quantity, but they may recruit the AMS even for the small number range under some circumstances, e.g., when higher memory demand is imposed by a greater retention interval.

What counts in estimation? The nature of the preverbal system.

It has been proposed that the development of verbal counting is supported by a more ancient preverbal system of estimation, the most widely canvassed candidates being the accumulator originally proposed by Gibbon and colleagues and the analogue magnitude system proposed by Dehaene and colleagues. The aim of this chapter is to assess the strengths and weaknesses of these models in terms of their capacity to emulate the statistical properties of verbal counting. The emphasis is put on the emergence of exact representations, autoscaling, and commensurability of noise characteristics. We also outline the modified architectures that may help improve models' power to meet these criteria. We propose that architectures considered in this chapter can be used to generate predictions for experimental testing and provide an example where we test the hypothesis whether the visual sense of number, ie, ability to discriminate numerosity without counting, entails enumeration of objects.

Quantification acuity in spontaneous shoaling decisions of three-spined sticklebacks.

The ability to discriminate between different quantities is widespread throughout the animal kingdom, and the underlying mechanisms of quantity discrimination are currently intensely discussed. In contrast, questions elucidating the limits of quantity estimation received rather little attention so far. Here, we examined fine-tuned quantity estimation in the three-spined stickleback (Gasterosteus aculeatus) in a natural context, i.e. during shoaling decisions. Wild-caught focal fish were given the spontaneous choice between two shoals which differed in group size by 1 fish (0 vs. 1, 1 vs. 2, 2 vs. 3, 3 vs. 4, 4 vs. 5, 5 vs. 6 and 6 vs. 7), based on visual assessment. The results show that sticklebacks generally prefer to shoal with the larger group. They discriminated numerical contrasts up to 6 versus 7, equalling a numerical ratio of 0.86. Preference patterns followed Weber's law, i.e. decreased with increasing numerical ratio. This pattern was found across all numerical conditions as well as within the small number range (ranging from 1 vs. 2 to 3 vs. 4). The results suggest that wild-caught three-spined sticklebacks are spontaneously able (i.e. without prior learning) to detect subtle differences in shoal sizes. Further, they confirm findings of previous studies highlighting the contribution of the analogue magnitude system to quantity estimation in fishes.

Binding space and time through action.

Space and time are intimately coupled dimensions in the human brain. Several lines of evidence suggest that space and time are processed by a shared analogue magnitude system. It has been proposed that actions are instrumental in establishing this shared magnitude system. Here we provide evidence in support of this hypothesis, by showing that the interaction between space and time is enhanced when magnitude information is acquired through action. Participants observed increases or decreases in the height of a visual bar (spatial magnitude) while judging whether a simultaneously presented sequence of acoustic tones had accelerated or decelerated (temporal magnitude). In one condition (Action), participants directly controlled the changes in bar height with a hand grip device, whereas in the other (No Action), changes in bar height were externally controlled but matched the spatial/temporal profile of the Action condition. The sign of changes in bar height biased the perceived rate of the tone sequences, where increases in bar height produced apparent increases in tone rate. This effect was amplified when the visual bar was actively controlled in the Action condition, and the strength of the interaction was scaled by the magnitude of the action. Subsequent experiments ruled out that this was simply explained by attentional factors, and additionally showed that a monotonic mapping is also required between grip force and bar height in order to bias the perception of the tones. These data provide support for an instrumental role of action in interfacing spatial and temporal quantities in the brain.

From small to large: Numerical discrimination by young domestic chicks (Gallus gallus).

Human adults and nonhuman primates share a subset of nonverbal numerical skills that are considered the evolutionary foundation of more complex numerical reasoning. Intriguing experiments have shown that 10- to 12-month-old infants are able to distinguish between large (8 vs. 12) and small (1 vs. 2, 1 vs. 3, 2 vs. 3) sets of objects but seem incapable of comparing quantities that fall in the middle area between large and small numerosities, such as 1 versus 4. This finding suggests that there are two separate nonverbal numerical systems. Other researchers argue that there is continuity in the representation of numbers. Experimental evidence demonstrating that newborn chicks are able to process addition and subtraction such as (4-1) versus (1 + 1) lends support to the latter hypothesis. Here, using an experimental paradigm to test numerical discrimination, we demonstrated that newborn chicks are able to distinguish between some numerical comparisons, such as 2 vs. 3, 2 vs. 8, 6 vs. 9, 8 vs. 14, 4 vs. 6, and 4 vs. 8. These findings support the hypothesis that a single system processes both small and large numerosities. The results of these experiments demonstrate that small and large numbers can be discriminated via "analogue magnitude" system (AMS). Those data can be accounted for in terms of a select mechanism prompting the functioning of either system and, therefore, a different processing of the stimuli. When the modality of presentation of the stimuli focuses the attention on the whole collection, the elaboration would be carried out by the AMS.

TO BE OR NOT TO BE: AN INFORMATIVE NON-SYMBOLIC NUMERICAL MAGNITUDE PROCESSING STUDY ABOUT SMALL VERSUS LARGE NUMBERS IN INFANTS

Many studies tested the association between numerical magnitude processing and mathematical achievement with conflicting findings reported for individuals with mathematical learning disorders. Some of the inconsistencies might be explained by the number of non-symbolic stimuli or dot collections used in studies. It has been hypothesized that there is an object-file system for ‘small’ and an analogue magnitude system for ‘large’ numbers. This two-system account has been supported by the set size limit of the object-file system (three items). A boundary was defined, accordingly, categorizing numbers below four as ‘small’ and from four and above as ‘large’. However, data on ‘small’ number processing and on the ‘boundary’ between small and large numbers are missing. In this contribution we provide data from infants discriminating between the number sets 4 vs. 8 and 1 vs. 4, both containing the number four combined with a small and a large number respectively. Participants were 25 and 26 full term 9- month-olds for 4 vs. 8 and 1 vs. 4 respectively. The stimuli (dots) were controlled for continuous variables. Eyetracking was combined with the habituation paradigm. The results showed that the infants were successful in discriminating 1 from 4, but failed to discriminate 4 from 8 dots. This finding supports the assumption of the number four as a ‘small’ number and enlarges the object-file system’s limit. This study might help to explain inconsistencies in studies. Moreover, the information may be useful in answering parent’s questions about challenges that vulnerable children with number processing problems, such as children with mathematical learning disorders, might encounter. In addition, the study might give some information on the stimuli that can be used to effectively foster children’s magnitude processing skills.

Relative quantity judgments in the beluga whale (Delphinapterus leucas) and the bottlenose dolphin (Tursiops truncatus)

Numerous studies have documented the ability of many species to make relative quantity judgments using an analogue magnitude system. We investigated whether one beluga whale, Delphinapterus leucas, and three bottlenose dolphins, Tursiops truncatus, were capable of selecting the larger of two sets of quantities, and analyzed if their performance matched predictions from the object file model versus the analog accumulator model. In Experiment 1, the two sets were presented simultaneously, under water, and they were visually (condition 1) or echoically (condition 2) available at the time of choice. In experiment 2, the two sets were presented above the water, successively (condition 1) or sequentially, item-by-item (condition 2), so that they were not visually available at the time of choice (condition 1) or at any time throughout the experiment (condition 2). We analyzed the effect of the ratio between quantities, the difference between quantities, and the total number of items presented on the subjects’ choices. All subjects selected the larger of the two sets of quantities above chance levels in all conditions. However, unlike most previous studies, the subjects’ choices did not match the predictions from the accumulator model. Whether these findings reflect interspecies differences in the mechanisms which underpin relative quantity judgments remains to be determined.

One, two, three, four, or is there something more? Numerical discrimination in day-old domestic chicks

Human adults master sophisticated, abstract numerical calculations that are mostly based on symbolic language and thus inimitably human. Humans may nonetheless share a subset of non-verbal numerical skills, available soon after birth and considered the evolutionary foundation of more complex numerical reasoning, with other animals. These skills are thought to be based on the two systems: the object file system which processes small values (<3) and the analogue magnitude system which processes large magnitudes (>4). Infants' ability to discriminate 1 vs. 2, 1 vs. 3, 2 vs. 3, but not 1 vs. 4, seems to indicate that the two systems are independent, implying that the conception of a continuous number processing system is based on precursors that appear to be interrupted at or about the number four. The findings from the study being presented here indicating that chicks are able to make a series of discriminations regarding that borderline number (1 vs. 4, 1 vs. 5, 2 vs. 4) support the hypothesis that there is continuity in the number system which processes both small and large numerousness.

Processing negative numbers by transforming negatives to positive range and by sign shortcut

Numerals are processed by a phylogenetically old analogue magnitude system. Can culturally new negative numerals be processed using this same representation? To find out whether magnitude representation could be used, we contrasted three possible processing mechanisms: an extended magnitude system for both positive and negative numbers, a mirroring mechanism that could transform negative values to the positive range to be processed on the positive magnitude system, and a sign shortcut strategy that can process the signs of numbers independently of the absolute values of numerals. To test these three hypotheses, a comparison task was used and the reaction time pattern, numerical distance, and Spatial-Numerical Association of Response Codes (SNARC) effect was analysed. The results revealed a mirroring process along with a sign shortcut mechanism. The SNARC effect was observed only when positive numbers were compared.

Cognition, evolution, and behavior

Preface Acknowledgments Chapter 1. Cognition and the study of behavior 1.1 What is comparative cognition about? 1.2 Kinds of explanation for behavior 1.3 Approaches to comparative cognition 1.4 Summary Chapter 2. Evolution, behavior, and cognition: A primer 2.1 Testing adaptation 2.2 Mapping phylogeny 2.3 Evolution, cognition, and the structure of behavior 2.4 Evolution and the brain 2.5 What does all this have to do with comparative psychology? 2.6 Summarizing and looking ahead Part I. Fundamental Mechanisms Chapter 3. Perception and attention 3.1 Specialized sensory systems 3.2 How can we find out what animals perceive? 3.3 Some psychophysical principles 3.4 Signal detection theory 3.5 Perception and evolution: Sensory ecology 3.6 Search and attention 3.7 Attention and foraging: The behavioral ecology of attention 3.8 Summary Chapter 4. Learning: Introduction and Pavlovian conditioning 4.1 General processes and constraints on 4.2 A framework for thinking about learning 4.3 When and how will learning evolve? 4.4 Pavlovian conditioning: Conditions for learning 4.5 What is learned? 4.6 Conditional control of behavior: Occasion setting and modulation 4.7 Effects of learning on behavior 4.8 Concluding remarks Chapter 5. Recognition learning 5.1 Habituation 5.2 Perceptual learning 5.3 Imprinting 5.4 The behavioral ecology of social recognition: Recognizing kin 5.5. Forms of recognition learning compared Chapter 6. Discrimination, classification, and concepts 6.1 Three examples 6.2 Untrained responses to natural stimuli 6.3 Classifying complex natural stimuli 6.4 Discrimination learning 6.5 Category discrimination and concepts 6.6 Summary and conclusions Chapter 7. Memory 7.1 Functions and properties of memory 7.2 Methods for studying memory in animals 7.3 Conditions for memory 7.4 Species differences in memory? 7.5 Mechanisms: What is remembered and why is it forgotten? 7.6 Memory and consciousness 7.7 Summary and conclusions Part II. Physical Cognition Chapter 8. Getting around: Spatial cognition 8.1 Mechanisms for spatial orientation 8.2 Modularity and integration 8.3 Acquiring spatial knowledge: The conditions for learning 8.4 Do animals have cognitive maps? 8.5 Summary Chapter 9. Timing 9.1 Circadian rhythms 9.2 Interval timing: Data 9.3 Interval timing: Theories 9.4 Two timing systems? Chapter 10. Numerical competence 10.1 Numerosity discrimination and the analogue magnitude system 10.2 The object tracking system 10.3. Ordinal comparison: Numerosity, serial position, and transitive inference 10.4 Labels and language 10.5 Numerical cognition and comparative psychology Chapter 11. Cognition and the consequences of behavior: Foraging, planning, instrumental learning and using tools 11.1 Foraging 11.2 Long term or short term maximizing: Do animals plan ahead? 11.3 Causal learning and instrumental behavior 11.4 Using tools 11.5 On causal learning and killjoy explanations Part III. Social Cognition Chapter 12. Social intelligence 12.1 The social intelligence hypothesis 12.2 The nature of social knowledge 12.3 Intentionality and social understanding 12.4 Theory of mind 12.5 Cooperation 12.6 Summary Chapter 13. Social learning 13.1 Social learning in context 13.2 Mechanisms : Social learning without imitation 13.3 Mechanisms: Imitation 13.4 Do nonhuman animals teach? 13.5 Animal cultures? 13.6 Conclusions Chapter 14. Communication and language 14.1 Basic issues 14.2 Natural communication systems 14.3 Trying to teach human language to other species 14.4 Language evolution and animal communication: Current directions 14.5 Conclusions Chapter 15. Summing up and looking ahead 15.1 Modularity and the animal mind 15.2 Theory and method in comparative cognition 15.3 Humans vs. other species: Different in degree or kind? 15.4 The future: Tinbergen's four questions, and a fifth one References Index

Numerical abilities in Williams syndrome: Dissociating the analogue magnitude system and verbal retrieval

Two numerical systems—the analogue magnitude system and verbal retrieval—were investigated in Williams syndrome (WS) with three numerical tasks: simple addition, simple multiplication, and number comparison. A new matching technique was introduced in selecting the proper control groups. The WS group was relatively fast in the addition and multiplication tasks, but was slow in number comparison. No reverse numerical effect was observed in the comparison task, and the distance effect was stronger than that in the control groups. The findings indicate a profile with an impaired analogue magnitude system and less impaired verbal retrieval in Williams syndrome.

Food and token quantity discrimination in capuchin monkeys (Cebus apella)

Quantity discrimination is adaptive in a variety of ecological contexts and different taxa discriminate stimuli differing in numerousness, both in the wild and in laboratory settings. Quantity discrimination between object arrays has been suggested to be more demanding than between food arrays but, to our knowledge, the same paradigm has never been used to directly compare them. We investigated to what extent capuchin monkeys' relative numerousness judgments (RNJs) with food and token are alike. Tokens are inherently non-valuable objects that acquire an associative value upon exchange with the experimenter. Our aims were (1) to assess capuchins' RNJs with food (Experiment 1) and with tokens (Experiment 2) by presenting all the possible pair-wise choices between one to five items, and (2) to evaluate on which of the two proposed non-verbal mechanisms underlying quantity discrimination (analogue magnitude and object file system) capuchins relied upon. In both conditions capuchins reliably selected the larger amount of items, although their performance was higher with food than with tokens. The influence of the ratio between arrays on performance indicates that capuchins relied on the same system for numerical representation, namely analogue magnitude, regardless of the type of stimuli (food or tokens) and across both the small and large number ranges.