Is there an innate geometric module? Effects of experience with angular geometric cues on spatial re-orientation based on the shape of the environment

Environment geometry alters subiculum boundary vector cell receptive fields in adulthood and early development

Boundaries to movement form a specific class of landmark information used for navigation: Boundary Vector Cells (BVCs) are neurons which encode an animal's location as a vector displacement from boundaries. Here we characterise the prevalence and spatial tuning of subiculum BVCs in adult and developing male rats, and investigate the relationship between BVC spatial firing and boundary geometry. BVC directional tunings align with environment walls in squares, but are uniformly distributed in circles, demonstrating that environmental geometry alters BVC receptive fields. Inserted barriers uncover both excitatory and inhibitory components to BVC receptive fields, demonstrating that inhibitory inputs contribute to BVC field formation. During post-natal development, subiculum BVCs mature slowly, contrasting with the earlier maturation of boundary-responsive cells in upstream Entorhinal Cortex. However, Subiculum and Entorhinal BVC receptive fields are altered by boundary geometry as early as tested, suggesting this is an inherent feature of the hippocampal representation of space.

Graded human sensitivity to geometric and topological concepts

In a seminal study, Dehaene et al. (2006) found evidence that adults and children are sensitive to geometric and topological (GT) concepts using a novel odd-one-out task. However, performance on this task could reflect more general cognitive abilities than intuitive knowledge of GT concepts. Here, we developed a new 2-alternative forced choice (2-AFC) version of the original task where chance represents a higher bar to clear (50% vs. 16.67%) and where the role of general cognitive abilities is minimized. Replicating the original finding, American adult participants showed above-chance sensitivity to 41 of the 43 GT concepts tested. Moreover, their performance was not strongly driven by two general cognitive abilities, fluid intelligence and mental rotation, nor was it strongly associated with mathematical achievement as measured by ACT/SAT scores. The performance profile across the 43 concepts as measured by the new 2-AFC task was found to be highly correlated with the profiles as measured using the original odd-one-out task, as an analysis of data sets spanning populations and ages revealed. Most significantly, an aggregation of the 43 concepts into seven classes of GT concepts found evidence for graded sensitivity. Some classes, such as Euclidean geometry and Topology, were found to be more domain-specific: they "popped out" for participants and were judged very quickly and highly accurately. Others, notably Symmetry and Geometric transformations, were found to be more domain-general: better predicted by participants' general cognitive abilities and mathematical achievement. These results shed light on the graded nature of GT concepts in humans and challenge computational models that emphasize the role of induction.

Geometry intuitions without vision? A study in blind children and adults

Geometry intuitions seem to be rooted in a non-verbal system that humans possess since early age. However, the mechanisms underlying the comprehension of basic geometric concepts remain elusive. Some authors have suggested that the starting point of geometry development could be found in the visual perception of specific features in our environment, thus conferring to vision a foundational role in the acquisition of geometric skills. To examine this assumption, a test probing intuitive understanding of basic geometric concepts was presented to congenitally blind children and adults. Participants had to detect the intruder among four different shapes, from which three instantiated a specific geometrical concept and one (the intruder) violated it. Although they performed above the chance level, the blind presented poorer performance than the sighted participants who did the task in the visual modality (i.e., with the eyes open), but performed equally well than the sighted who did the task in the tactile modality (i.e., with a blindfold). We therefore provide evidence that geometric abilities are impacted by the lack of vision.

Comparative spatial memory and cue use: The contributions of Marcia L. Spetch to the study of small-scale spatial cognition

Dr. Marcia Spetch is a Canadian experimental psychologist who specializes in the study of comparative cognition. Her research over the past four decades has covered many diverse topics, but focused primarily on the comparative study of small-scale spatial cognition, navigation, decision making, and risky choice. Over the course of her career Dr. Spetch has had a profound influence on the study of these topics, and for her work she was named a Fellow of the Association for Psychological Science in 2012, and a Fellow of the Royal Society of Canada in 2017. In this review, I provide a biographical sketch of Dr. Spetch's academic career, and revisit her contributions to the study of small-scale spatial cognition in two broad areas: the use of environmental geometric cues, and how animals cope with cue conflict. The goal of this review is to highlight the contributions of Dr. Spetch, her students, and her collaborators to the field of comparative cognition and the study of small-scale spatial cognition. As such, this review stands to serve as a tribute and testament to Dr. Spetch's scientific legacy.

Extra-Visual Systems in the Spatial Reorientation of Cavefish

Disoriented humans and animals are able to reorient themselves using environmental geometry ("metric properties" and "sense") and local features, also relating geometric to non-geometric information. Here we investigated the presence of these reorientation spatial skills in two species of blind cavefish (Astyanax mexicanus and Phreatichthys andruzzii), in order to understand the possible role of extra-visual senses in similar spatial tasks. In a rectangular apparatus, with all homogeneous walls (geometric condition) or in presence of a tactilely different wall (feature condition), cavefish were required to reorient themselves after passive disorientation. We provided the first evidence that blind cavefish, using extra-visual systems, were able i) to use geometric cues, provided by the shape of the tank, in order to recognize two geometric equivalent corners on the diagonal, and ii) to integrate the geometric information with the salient cue (wall with a different surface structure), in order to recover a specific corner. These findings suggest the ecological salience of the environmental geometry for spatial orientation in animals and, despite the different niches of adaptation, a potential shared background for spatial navigation. The geometric spatial encoding seems to constitute a common cognitive tool needed when the environment poses similar requirements to living organisms.

The orientation of homing pigeons (Columba livia f.d.) with and without navigational experience in a two-dimensional environment

Homing pigeons are known for their excellent homing ability, and their brains seem to be functionally adapted to homing. It is known that pigeons with navigational experience show a larger hippocampus and also a more lateralised brain than pigeons without navigational experience. So we hypothesized that experience may have an influence also on orientation ability. We examined two groups of pigeons (11 with navigational experience and 17 without) in a standard operant chamber with a touch screen monitor showing a 2-D schematic of a rectangular environment (as “geometric” information) and one uniquely shaped and colored feature in each corner (as “landmark” information). Pigeons were trained first for pecking on one of these features and then we examined their ability to encode geometric and landmark information in four tests by modifying the rectangular environment. All tests were done under binocular and monocular viewing to test hemispheric dominance. The number of pecks was counted for analysis. Results show that generally both groups orientate on the basis of landmarks and the geometry of environment, but landmark information was preferred. Pigeons with navigational experience did not perform better on the tests but showed a better conjunction of the different kinds of information. Significant differences between monocular and binocular viewing were detected particularly in pigeons without navigational experience on two tests with reduced information. Our data suggest that the conjunction of geometric and landmark information might be integrated after processing separately in each hemisphere and that this process is influenced by experience.

First Direct Evidence of Cue Integration in Reorientation: A New Paradigm

There are several models of the use of geometric and feature cues in reorientation (Cheng, Huttenlocher, & Newcombe, ). The adaptive combination approach posits that people integrate cues with weights that depend on cue salience and learning, or, when discrepancies are large, they choose between cues based on these variables (Cheng, Shettleworth, Huttenlocher, & Rieser, ; Newcombe & Huttenlocher, ). In a new paradigm designed to evaluate integration and choice, disoriented participants attempted to return to a heading direction, in a trapezoidal enclosure in which feature and geometric cues both unambiguously specified a heading, but later the feature was moved. With discrepancies greater than 90 degrees, participants choose geometry. With smaller discrepancies, integration appeared in three of five situations; otherwise, participants used geometry alone. Variation depended on direction of feature movement and whether the nearest corner was acute or obtuse. The results have implications for contrasting adaptive combination and modularity theory, and for future research, offering a new paradigm for reorientation research, and for testing cue integration more broadly.

An adaptive cue combination model of human spatial reorientation

Previous research has proposed an adaptive cue combination view of the development of human spatial reorientation (Newcombe & Huttenlocher, 2006), whereby information from multiple sources is combined in a weighted fashion in localizing a target, as opposed to being modular and encapsulated (Hermer & Spelke, 1996). However, no prior work has formalized this proposal and tested it against existing empirical data. We propose a computational model of human spatial reorientation that is motivated by probabilistic approaches to optimal perceptual cue integration (e.g. Ernst & Banks, 2002) and to spatial location coding (Huttenlocher, Hedges, & Duncan, 1991). We show that this model accounts for data from a variety of human reorientation experiments, providing support for the adaptive combination view of reorientation.

Comparative cognition of number and space: the case of geometry and of the mental number line

Evidence is discussed about the use of geometric information for spatial orientation and the association between space and numbers in non-human animals. A variety of vertebrate species can reorient using simple Euclidian geometry of the environmental surface layout, i.e. in accord with metric and sense (right/left) relationships among extended surfaces. There seems to be a primacy of geometric over non-geometric information in spatial reorientation and, possibly, innate encoding of the sense of direction. The hippocampal formation plays a key role in geometry-based reorientation in mammals, birds, amphibians and fish. Although some invertebrate species show similar behaviours, it is unclear whether the underlying mechanisms are the same as in vertebrates. As to the links between space and number representations, a disposition to associate numerical magnitudes onto a left-to-right-oriented mental number line appears to exist independently of socio-cultural factors, and can be observed in animals with very little numerical experience, such as newborn chicks and human infants. Such evidence supports a nativistic foundation of number-space association. Some speculation about the possible underlying mechanisms is provided together with consideration on the difficulties inherent to any comparison among species of different taxonomic groups.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Spatial Impairment and Memory in Genetic Disorders: Insights from Mouse Models

Research across the cognitive and brain sciences has begun to elucidate some of the processes that guide navigation and spatial memory. Boundary geometry and featural landmarks are two distinct classes of environmental cues that have dissociable neural correlates in spatial representation and follow different patterns of learning. Consequently, spatial navigation depends both on the type of cue available and on the type of learning provided. We investigated this interaction between spatial representation and memory by administering two different tasks (working memory, reference memory) using two different environmental cues (rectangular geometry, striped landmark) in mouse models of human genetic disorders: Prader-Willi syndrome (PWScrm+/p- mice, n = 12) and Beta-catenin mutation (Thr653Lys-substituted mice, n = 12). This exploratory study provides suggestive evidence that these models exhibit different abilities and impairments in navigating by boundary geometry and featural landmarks, depending on the type of memory task administered. We discuss these data in light of the specific deficits in cognitive and brain function in these human syndromes and their animal model counterparts.

Wild hummingbirds can use the geometry of a flower array

Animals use cues from their environment to orient in space and to navigate their surroundings. Geometry is a cue whose informational content may originate from the metric properties of a given environment, and its use has been demonstrated in the laboratory in nearly every species of animal tested. However, it is not clear whether geometric information, used by animals typically tested in small, rectangular boxes, is directly relevant to animals in their natural environment. Here we present the first data that confirm the use of geometric cues by a free-living animal in the wild. We trained rufous hummingbirds to visit a rectangular array of four artificial flowers, one of which was rewarded. In some trials a conspicuous landmark cued the reward. Following array translocation and rotation, we presented hummingbirds with three tests. When trained and tested with the landmark, or when trained and tested without it, hummingbirds failed to show geometric learning. However, when trained with a landmark but tested without it, hummingbirds produced the classic geometric response, showing that they had learned the geometric relationships (distance and direction) of several non-reward visual elements of the environment. While it remains that the use of geometry to relocate a reward may be an experimental artefact, its use is not confined to the laboratory.

The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers

During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a "geometrical language" with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to predict future locations. The sequences vary in complexity according to a well-defined language comprising elementary primitives and recursive rules. A detailed analysis of error patterns indicates that primitives of symmetry and rotation are spontaneously detected and used by adults, preschoolers, and adult members of an indigene group in the Amazon, the Munduruku, who have a restricted numerical and geometrical lexicon and limited access to schooling. Furthermore, subjects readily combine these geometrical primitives into hierarchically organized expressions. By evaluating a large set of such combinations, we obtained a first view of the language needed to account for the representation of visuospatial sequences in humans, and conclude that they encode visuospatial sequences by minimizing the complexity of the structured expressions that capture them.

Origins of Knowledge: Insights from Precocial Species

Behavioral responses are influenced by knowledge acquired during the lifetime of an individual and by predispositions transmitted across generations. Establishing the origin of knowledge and the role of the unlearned component is a challenging task, given that both learned and unlearned knowledge can orient perception, learning, and the encoding of environmental features since the first stages of life. Ethical and practical issues constrain the investigation of unlearned knowledge in altricial species, including human beings. On the contrary, precocial animals can be tested on a wide range of tasks and capabilities immediately after birth and in controlled rearing conditions. Insects and precocial avian species are very convenient models to dissect the knowledge systems that enable young individuals to cope with their environment in the absence of specific previous experience. We present the state of the art of research on the origins of knowledge that comes from different models and disciplines. Insects have been mainly used to investigate unlearned sensory preferences and prepared learning mechanisms. The relative simplicity of the neural system and fast life cycle of insects make them ideal models to investigate the neural circuitry and evolutionary dynamics of unlearned traits. Among avian species, chicks of the domestic fowl have been the focus of many studies, and showed to possess unlearned knowledge in the sensory, physical, spatial, numerical and social domains. Solid evidence shows the existence of unlearned knowledge in different domains in several species, from sensory and social preferences to the left-right representation of the mental number line. We show how non-mammalian models of cognition, and in particular precocial species, can shed light into the adaptive value and evolutionary history of unlearned knowledge.

Core geometry in perspective

Research on animals, infants, children, and adults provides evidence that distinct cognitive systems underlie navigation and object recognition. Here we examine whether and how these systems interact when children interpret 2D edge-based perspectival line drawings of scenes and objects. Such drawings serve as symbols early in development, and they preserve scene and object geometry from canonical points of view. Young children show limits when using geometry both in non-symbolic tasks and in symbolic map tasks that present 3D contexts from unusual, unfamiliar points of view. When presented with the familiar viewpoints in perspectival line drawings, however, do children engage more integrated geometric representations? In three experiments, children successfully interpreted line drawings with respect to their depicted scene or object. Nevertheless, children recruited distinct processes when navigating based on the information in these drawings, and these processes depended on the context in which the drawings were presented. These results suggest that children are flexible but limited in using geometric information to form integrated representations of scenes and objects, even when interpreting spatial symbols that are highly familiar and faithful renditions of the visual world.

Avian cognition: examples of sophisticated capabilities in space and song

Although birds have traditionally and colloquially been considered less cognitively complex than mammals, and especially primates, more recent research has consistently refuted these assumptions. We argue that the impressive abilities of birds to navigate and communicate require considerable information-processing capabilities. These capacities include collecting, organizing, and selecting from a wide variety of navigational cues to orient toward and find a goal location in the spatial domain, and utilizing open-ended categorization and possibly even abstract reasoning to discriminate species-specific acoustic features of songs and calls. Furthermore, these abilities may be present across many avian species, providing evidence for domain-general cognitive facilities. We provide examples of processes in spatial learning and communication in birds, and locate them within the general literature, as evidence that the term 'bird-brain' should not be considered a pejorative.

Geometric and featural systems, separable and combined: Evidence from reorientation in people with Williams syndrome

Spatial reorientation by humans and other animals engages geometric representations of surface layouts as well as featural landmarks; however, the two types of information are thought to be behaviorally and neurally separable. In this paper, we examine the use of these two types of information during reorientation among children and adults with Williams syndrome (WS), a genetic disorder accompanied by abnormalities in brain regions that support use of both geometry and landmarks. Previous studies of reorientation in adolescents and adults with WS have shown deficits in the ability to use geometry for reorientation, but intact ability to use features, suggesting that the two systems can be differentially impaired by genetic disorder. Using a slightly modified layout, we found that many WS participants could use geometry, and most could use features along with geometry. However, the developmental trajectories for the two systems were quite different from one other, and different from those found in typical development. Purely geometric responding was not correlated with age in WS, and search processes appeared similar to those in typically developing (TD) children. In contrast, use of features in combination with geometry was correlated with age in WS, and search processes were distinctly different from TD children. The results support the view that use of geometry and features stem from different underlying mechanisms, that the developmental trajectories and operation of each are altered in WS, and that combination of information from the two systems is atypical. Given brain abnormalities in regions supporting the two kinds of information, our findings suggest that the co-operation of the two systems is functionally altered in this genetic syndrome.

25 years of research on the use of geometry in spatial reorientation: a current theoretical perspective

The purpose of this article is to review and evaluate the range of theories proposed to explain findings on the use of geometry in reorientation. We consider five key approaches and models associated with them and, in the course of reviewing each approach, five key issues. First, we take up modularity theory itself, as recently revised by Lee and Spelke (Cognitive Psychology, 61, 152-176, 2010a; Experimental Brain Research, 206, 179-188, 2010b). In this context, we discuss issues concerning the basic distinction between geometry and features. Second, we review the view-matching approach (Stürzl, Cheung, Cheng, & Zeil, Journal of Experimental Psychology: Animal Behavior Processes, 34, 1-14, 2008). In this context, we highlight the possibility of cross-species differences, as well as commonalities. Third, we review an associative theory (Miller & Shettleworth, Journal of Experimental Psychology: Animal Behavior Processes, 33, 191-212, 2007; Journal of Experimental Psychology: Animal Behavior Processes, 34, 419-422, 2008). In this context, we focus on phenomena of cue competition. Fourth, we take up adaptive combination theory (Newcombe & Huttenlocher, 2006). In this context, we focus on discussing development and the effects of experience. Fifth, we examine various neurally based approaches, including frameworks proposed by Doeller and Burgess (Proceedings of the National Academy of Sciences of the United States of America, 105, 5909-5914, 2008; Doeller, King, & Burgess, Proceedings of the National Academy of Sciences of the United States of America, 105, 5915-5920, 2008) and by Sheynikhovich, Chavarriaga, Strösslin, Arleo, and Gerstner (Psychological Review, 116, 540-566, 2009). In this context, we examine the issue of the neural substrates of spatial navigation. We conclude that none of these approaches can account for all of the known phenomena concerning the use of geometry in reorientation and clarify what the challenges are for each approach.

Wild hummingbirds rely on landmarks not geometry when learning an array of flowers

Rats, birds or fish trained to find a reward in one corner of a small enclosure tend to learn the location of the reward using both nearby visual features and the geometric relationships of corners and walls. Because these studies are conducted under laboratory and thereby unnatural conditions, we sought to determine whether wild, free-living rufous hummingbirds (Selasphorus rufus) learning a single reward location within a rectangular array of flowers would similarly employ both nearby visual landmarks and the geometric relationships of the array. Once subjects had learned the location of the reward, we used test probes in which one or two experimental landmarks were moved or removed in order to reveal how the birds remembered the reward location. The hummingbirds showed no evidence that they used the geometry of the rectangular array of flowers to remember the reward. Rather, they used our experimental landmarks, and possibly nearby, natural landmarks, to orient and navigate to the reward. We believe this to be the first test of the use of rectangular geometry by wild animals, and we recommend further studies be conducted in ecologically relevant conditions in order to help determine how and when animals form complex geometric representations of their local environments.

Asymmetrical interference effects between two-dimensional geometric shapes and their corresponding shape words

Nativists have postulated fundamental geometric knowledge that predates linguistic and symbolic thought. Central to these claims is the proposal for an isolated cognitive system dedicated to processing geometric information. Testing such hypotheses presents challenges due to difficulties in eliminating the combination of geometric and non-geometric information through language. We present evidence using a modified matching interference paradigm that an incongruent shape word interferes with identifying a two-dimensional geometric shape, but an incongruent two-dimensional geometric shape does not interfere with identifying a shape word. This asymmetry in interference effects between two-dimensional geometric shapes and their corresponding shape words suggests that shape words activate spatial representations of shapes but shapes do not activate linguistic representations of shape words. These results appear consistent with hypotheses concerning a cognitive system dedicated to processing geometric information isolated from linguistic processing and provide evidence consistent with hypotheses concerning knowledge of geometric properties of space that predates linguistic and symbolic thought.

Core foundations of abstract geometry

Human adults from diverse cultures share intuitions about the points, lines, and figures of Euclidean geometry. Do children develop these intuitions by drawing on phylogenetically ancient and developmentally precocious geometric representations that guide their navigation and their analysis of object shape? In what way might these early-arising representations support later-developing Euclidean intuitions? To approach these questions, we investigated the relations among young children's use of geometry in tasks assessing: navigation; visual form analysis; and the interpretation of symbolic, purely geometric maps. Children's navigation depended on the distance and directional relations of the surface layout and predicted their use of a symbolic map with targets designated by surface distances. In contrast, children's analysis of visual forms depended on the size-invariant shape relations of objects and predicted their use of the same map but with targets designated by corner angles. Even though the two map tasks used identical instructions and map displays, children's performance on these tasks showed no evidence of integrated representations of distance and angle. Instead, young children flexibly recruited geometric representations of either navigable layouts or objects to interpret the same spatial symbols. These findings reveal a link between the early-arising geometric representations that humans share with diverse animals and the flexible geometric intuitions that give rise to human knowledge at its highest reaches. Although young children do not appear to integrate core geometric representations, children's use of the abstract geometry in spatial symbols such as maps may provide the earliest clues to the later construction of Euclidean geometry.

Spontaneous reorientation is guided by perceived surface distance, not by image matching or comparison

Humans and animals recover their sense of position and orientation using properties of the surface layout, but the processes underlying this ability are disputed. Although behavioral and neurophysiological experiments on animals long have suggested that reorientation depends on representations of surface distance, recent experiments on young children join experimental studies and computational models of animal navigation to suggest that reorientation depends either on processing of any continuous perceptual variables or on matching of 2D, depthless images of the landscape. We tested the surface distance hypothesis against these alternatives through studies of children, using environments whose 3D shape and 2D image properties were arranged to enhance or cancel impressions of depth. In the absence of training, children reoriented by subtle differences in perceived surface distance under conditions that challenge current models of 2D-image matching or comparison processes. We provide evidence that children's spontaneous navigation depends on representations of 3D layout geometry.

Spatial reorientation by geometry in bumblebees

Human and non-human animals are capable of using basic geometric information to reorient in an environment. Geometric information includes metric properties associated with spatial surfaces (e.g., short vs. long wall) and left-right directionality or 'sense' (e.g. a long wall to the left of a short wall). However, it remains unclear whether geometric information is encoded by explicitly computing the layout of surface geometry or by matching images of the environment. View-based spatial encoding is generally thought to hold for insect navigation and, very recently, evidence for navigation by geometry has been reported in ants but only in a condition which does not allow the animals to use features located far from the goal. In this study we tested the spatial reorientation abilities of bumblebees (Bombus terrestris). After spatial disorientation, by passive rotation both clockwise and anticlockwise, bumblebees had to find one of the four exit holes located in the corners of a rectangular enclosure. Bumblebees systematically confused geometrically equivalent exit corners (i.e. corners with the same geometric arrangement of metric properties and sense, for example a short wall to the left of a long wall). However, when one wall of the enclosure was a different colour, bumblebees appeared to combine this featural information (either near or far from the goal) with geometric information to find the correct exit corner. Our results show that bumblebees are able to use both geometric and featural information to reorient themselves, even when features are located far from the goal.

Navigation as a source of geometric knowledge: young children's use of length, angle, distance, and direction in a reorientation task

Geometry is one of the highest achievements of our species, but its foundations are obscure. Consistent with longstanding suggestions that geometrical knowledge is rooted in processes guiding navigation, the present study examines potential sources of geometrical knowledge in the navigation processes by which young children establish their sense of orientation. Past research reveals that children reorient both by the shape of the surface layout and the shapes of distinctive landmarks, but it fails to clarify what shape properties children use. The present study explores 2-year-old children's sensitivity to angle, length, distance and direction by testing disoriented children's search in a variety of fragmented rhombic and rectangular environments. Children reoriented themselves in accord with surface distances and directions, but they failed to use surface lengths or corner angles either for directional reorientation or as local landmarks. Thus, navigating children navigate by some but not all of the abstract properties captured by formal Euclidean geometry. While navigation systems may contribute to children's developing geometric understanding, they likely are not the sole source of abstract geometric intuitions.

Chicks, like children, spontaneously reorient by three-dimensional environmental geometry, not by image matching

Spatial reorientation by layout geometry occurs in numerous species, but its underlying mechanisms are debated. While some argue that navigating animals' sense of place is based on geometric computations over three-dimensional representations, others claim it depends on panoramic image-matching processes. Because children reorient by subtle three-dimensional perturbations of the terrain and not by salient two-dimensional brightness contours on surfaces or freestanding columns, children's sense of place cannot be explained by image matching. To test image-matching theories in a different species, the present experiment investigates the reorientation performance of domestic chicks (Gallus gallus) in environments similar to those used with children. Chicks, like children, spontaneously reoriented by geometric relationships of subtle three-dimensional terrains, and not by salient two-dimensional brightness contours on surfaces or columns. These findings add to the evidence for homologous navigation systems in humans and other vertebrates, and they cast doubt on image-matching theories of reorientation in these species.

Core knowledge of object, number, and geometry: a comparative and neural approach

Studies on the ontogenetic origins of human knowledge provide evidence for a small set of separable systems of core knowledge dealing with the representation of inanimate and animate objects, number, and geometry. Because core knowledge systems are evolutionarily ancient, they can be investigated from a comparative perspective, making use of various animal models. In this review, I discuss evidence showing precocious abilities in nonhuman species to represent (a) objects that move partly or fully out of view and their basic mechanical properties such as solidity, (b) the cardinal and ordinal/sequential aspects of numerical cognition and rudimentary arithmetic with small numerosities, and (c) the geometrical relationships among extended surfaces in the surrounding layout. Controlled rearing studies suggest that the abilities associated with core knowledge systems of objects, number, and geometry are observed in animals in the absence (or with very reduced) experience, supporting a nativistic foundation of such cognitive mechanisms. Animal models also promise a fresh approach to the issue of the neurobiological and genetic mechanisms underlying the expression of core knowledge systems.

Spatial reorientation by geometry with freestanding objects and extended surfaces: a unifying view

The macroscopic, three-dimensional surface layout geometry of an enclosure apparently provides a different contribution for spatial reorientation than the geometric cues associated with freestanding objects arranged in arrays with similar geometric shape. Here, we showed that a unitary spatial representation can account for the capability of animals to reorient both by extended surfaces and discrete objects in a small-scale spatial task. We trained domestic chicks to locate a food-reward from an opening on isolated cylinders arranged either in a geometrically uninformative (square-shaped) or informative (rectangular-shaped) arrays. The arrays were located centrally within a rectangular-shaped enclosure. Chicks trained to access the reward from a fixed position of openings proved able to reorient according to the geometric cues specified by the shape of the enclosure in all conditions. Chicks trained in a fixed position of opening with geometric cues provided both by the arena and the array proved able to reorient according to each shape separately. However, chicks trained to access the reward from a variable position of openings failed to reorient. The results suggest that the physical constrains associated with the presence of obstacles in a scene, rather than their apparent visual extension, are crucial for spatial reorientation.

Arena geometry and path shape: when rats travel in straight or in circuitous paths?

We show here that the global geometry of the environment affects the shape of the paths of travel in rats. To examine this, individual rats were introduced into an unfamiliar arena. One group of rats (n=8) was tested in a square arena (2 m × 2 m), and the other group (n=8) in a round arena (2 m diameter). Testing was in a total darkness, since in the absence of visual information the geometry is not perceived immediately and the extraction of environment shape is slower. We found that while the level of the rats' activity did not seem to differ between both arenas, path shape differed significantly. When traveling along the perimeter, path shape basically followed the arena walls, with perimeter paths curving along the walls of the round arena, while being straight along the walls of the square arena. A similar impact of arena geometry was observed for travel away from the arena walls. Indeed, when the rats abandoned the arena walls to crosscut through the center of the arena, their center paths were circuitous in the round arena and relatively straight in the square arena. We suggest that the shapes of these paths are exploited for the same spatial task: returning back to a familiar location in the unsighted environment.

Large number discrimination by mosquitofish

Background

Recent studies have demonstrated that fish display rudimentary numerical abilities similar to those observed in mammals and birds. The mechanisms underlying the discrimination of small quantities (<4) were recently investigated while, to date, no study has examined the discrimination of large numerosities in fish.

Methodology/Principal Findings

Subjects were trained to discriminate between two sets of small geometric figures using social reinforcement. In the first experiment mosquitofish were required to discriminate 4 from 8 objects with or without experimental control of the continuous variables that co-vary with number (area, space, density, total luminance). Results showed that fish can use the sole numerical information to compare quantities but that they preferentially use cumulative surface area as a proxy of the number when this information is available. A second experiment investigated the influence of the total number of elements to discriminate large quantities. Fish proved to be able to discriminate up to 100 vs. 200 objects, without showing any significant decrease in accuracy compared with the 4 vs. 8 discrimination. The third experiment investigated the influence of the ratio between the numerosities. Performance was found to decrease when decreasing the numerical distance. Fish were able to discriminate numbers when ratios were 1∶2 or 2∶3 but not when the ratio was 3∶4. The performance of a sample of undergraduate students, tested non-verbally using the same sets of stimuli, largely overlapped that of fish.

Conclusions/Significance

Fish are able to use pure numerical information when discriminating between quantities larger than 4 units. As observed in human and non-human primates, the numerical system of fish appears to have virtually no upper limit while the numerical ratio has a clear effect on performance. These similarities further reinforce the view of a common origin of non-verbal numerical systems in all vertebrates.

View-based strategy for reorientation by geometry

Human and non-human animals can use geometric information (metric information and left-right discrimination sense) to reorient themselves in an environment. The hypothesis that in so doing they rely on allocentric (map-like) representations has received wide consensus. However, theoretical models suggest that egocentric representations may represent efficient strategies for visuo-spatial navigation. Here, we provide, for the first time, evidence that a view-based strategy is effectively used by animals to reorient themselves in an array of landmarks. Domestic chicks were trained to locate a food-reward in a rectangular array of either four indistinguishable or distinctive pipes. In the key experimental series, the pipes had four openings, only one of which allowed the chicks to access the reward. The direction of the open access relative to the array was either maintained stable or it was changed throughout training. The relative position of the pipes in the array was maintained stable in both training conditions. Chicks reoriented according to configural geometry as long as the open access pointed in the same direction during training but failed when the positions of the openings was changed throughout training. When the correct pipe was characterized by a distinctive featural cue, chicks learnt to locate the reward irrespective of the stability of the direction to openings, indicating that place-navigation was dissociated from non-spatial learning. These findings provide evidence that view-based strategies to reorient by geometry could be used by animals.

Origins of spatial, temporal and numerical cognition: Insights from comparative psychology

Contemporary comparative cognition has a large repertoire of animal models and methods, with concurrent theoretical advances that are providing initial answers to crucial questions about human cognition. What cognitive traits are uniquely human? What are the species-typical inherited predispositions of the human mind? What is the human mind capable of without certain types of specific experiences with the surrounding environment? Here, we review recent findings from the domains of space, time and number cognition. These findings are produced using different comparative methodologies relying on different animal species, namely birds and non-human great apes. The study of these species not only reveals the range of cognitive abilities across vertebrates, but also increases our understanding of human cognition in crucial ways.

Use of geometry for spatial reorientation in children applies only to symmetric spaces

Proponents of the geometric module hypothesis argue that following disorientation, many species reorient by use of macro-environment geometry. It is suggested that attention to the surface layout geometry of natural terrain features may have been selected for over evolutionary time due to the enduring and unambiguous location information it provides. Paradoxically, however, tests of the hypothesis have been exclusively conducted in symmetric (hence 'unnatural' and geometrically ambiguous) environments. The present series of studies examines reorientation by 18-month-3-year-old children in a rectangular versus irregular quadrilateral enclosure (Study 1), a rectangular versus irregular quadrilateral array (Study 2) and an isosceles versus irregular triangular array (Study 3). Children were successful in symmetric but not asymmetric environments, casting doubt on the functional argument for an empirical basis of the geometric module hypothesis.

Two systems of spatial representation underlying navigation

We review evidence for two distinct cognitive processes by which humans and animals represent the navigable environment. One process uses the shape of the extended 3D surface layout to specify the navigator's position and orientation. A second process uses objects and patterns as beacons to specify the locations of significant objects. Although much of the evidence for these processes comes from neurophysiological studies of navigating animals and neuroimaging studies of human adults, behavioral studies of navigating children shed light both on the nature of these systems and on their interactions.

A modular geometric mechanism for reorientation in children

Although disoriented young children reorient themselves in relation to the shape of the surrounding surface layout, cognitive accounts of this ability vary. The present paper tests three theories of reorientation: a snapshot theory based on visual image-matching computations, an adaptive combination theory proposing that diverse environmental cues to orientation are weighted according to their experienced reliability, and a modular theory centering on encapsulated computations of the shape of the extended surface layout. Seven experiments test these theories by manipulating four properties of objects placed within a cylindrical space: their size, motion, dimensionality, and distance from the space's borders. Their findings support the modular theory and suggest that disoriented search behavior centers on two processes: a reorientation process based on the geometry of the 3D surface layout, and a beacon-guidance process based on the local features of objects and surface markings.

Beyond core knowledge: Natural geometry

For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive systems for representing the shapes of large-scale, navigable surface layouts and of small-scale, movable forms and objects. Each of these systems applies to some but not all perceptible arrays and captures some but not all of the three fundamental Euclidean relationships of distance (or length), angle, and direction (or sense). Like natural number (Carey, 2009), Euclidean geometry may be constructed through the productive combination of representations from these core systems, through the use of uniquely human symbolic systems.