Which coordinate system for modelling path integration?

Path integration is a navigation strategy widely observed in nature where an animal maintains a running estimate, called the home vector, of its location during an excursion. Evidence suggests it is both ancient and ubiquitous in nature, and has been studied for over a century. In that time, canonical and neural network models have flourished, based on a wide range of assumptions, justifications and supporting data. Despite the importance of the phenomenon, consensus and unifying principles appear lacking. A fundamental issue is the neural representation of space needed for biological path integration. This paper presents a scheme to classify path integration systems on the basis of the way the home vector records and updates the spatial relationship between the animal and its home location. Four extended classes of coordinate systems are used to unify and review both canonical and neural network models of path integration, from the arthropod and mammalian literature. This scheme demonstrates analytical equivalence between models which may otherwise appear unrelated, and distinguishes between models which may superficially appear similar. A thorough analysis is carried out of the equational forms of important facets of path integration including updating, steering, searching and systematic errors, using each of the four coordinate systems. The type of available directional cue, namely allothetic or idiothetic, is also considered. It is shown that on balance, the class of home vectors which includes the geocentric Cartesian coordinate system, appears to be the most robust for biological systems. A key conclusion is that deducing computational structure from behavioural data alone will be difficult or impossible, at least in the absence of an analysis of random errors. Consequently it is likely that further theoretical insights into path integration will require an in-depth study of the effect of noise on the four classes of home vectors.     

Sensory feedback in a bump attractor model of path integration

Mammalian spatial navigation systems utilize several different sensory information channels. This information is converted into a neural code that represents the animal’s current position in space by engaging place cell, grid cell, and head direction cell networks. In particular, sensory landmark (allothetic) cues can be utilized in concert with an animal’s knowledge of its own velocity (idiothetic) cues to generate a more accurate representation of position than path integration provides on its own (Battaglia et al. The Journal of Neuroscience 24(19):4541–4550 (2004)). We develop a computational model that merges path integration with feedback from external sensory cues that provide a reliable representation of spatial position along an annular track. Starting with a continuous bump attractor model, we explore the impact of synaptic spatial asymmetry and heterogeneity, which disrupt the position code of the path integration process. We use asymptotic analysis to reduce the bump attractor model to a single scalar equation whose potential represents the impact of asymmetry and heterogeneity. Such imperfections cause errors to build up when the network performs path integration, but these errors can be corrected by an external control signal representing the effects of sensory cues. We demonstrate that there is an optimal strength and decay rate of the control signal when cues appear either periodically or randomly. A similar analysis is performed when errors in path integration arise from dynamic noise fluctuations. Again, there is an optimal strength and decay of discrete control that minimizes the path integration error.     

Path integration — a network model

Path integration is a primary means of navigation for a number of animals. We present a model which performs path integration with a neural network. This model is based on a neural structure called a sinusoidal array, which allows an efficient representation of vector information with neurons. We show that exact path integration can easily be achieved by a neural network. Thus deviations from the direct home trajectory, found previously in experiments with ants, can not be explained by computational limitations of the nervous system. Instead we suggest that the observed deviations are caused by a strategy to simplify landmark navigation.     

The ant’s path integration system: a neural architecture

 A model is developed by which path integration as observed in many animal species could be implemented neurobiologically. The proposed architecture is able to describe the navigation behaviour of Cataglyphis ants, and that of other social insects, at the level of interacting neurons. The basic idea of this architecture is the concept of activity patterns travelling along neural chains. Although experimental evidence has yet to be provided, this concept seems biologically plausible and not limited to the navigation problem. Neural chains are able to represent variables by activity patterns with high accuracy and temporal stability. Moreover, they are able to integrate incremental signals with high precision. Cyclical chains of neurons show superior performance as soon as cyclical variables are to be represented and integrated. Finally, representation of cyclical variables by travelling activity peaks allows simple approximations of goniometric functions as they are used in path integration systems. Received: 15 November 1994/Accepted in revised form: 30 May 1995     

No need for a cognitive map: decentralized memory for insect navigation

In many animals the ability to navigate over long distances is an important prerequisite for foraging. For example, it is widely accepted that desert ants and honey bees, but also mammals, use path integration for finding the way back to their home site. It is however a matter of a long standing debate whether animals in addition are able to acquire and use so called cognitive maps. Such a 'map', a global spatial representation of the foraging area, is generally assumed to allow the animal to find shortcuts between two sites although the direct connection has never been travelled before. Using the artificial neural network approach, here we develop an artificial memory system which is based on path integration and various landmark guidance mechanisms (a bank of individual and independent landmark-defined memory elements). Activation of the individual memory elements depends on a separate motivation network and an, in part, asymmetrical lateral inhibition network. The information concerning the absolute position of the agent is present, but resides in a separate memory that can only be used by the path integration subsystem to control the behaviour, but cannot be used for computational purposes with other memory elements of the system. Thus, in this simulation there is no neural basis of a cognitive map. Nevertheless, an agent controlled by this network is able to accomplish various navigational tasks known from ants and bees and often discussed as being dependent on a cognitive map. For example, map-like behaviour as observed in honey bees arises as an emergent property from a decentralized system. This behaviour thus can be explained without referring to the assumption that a cognitive map, a coherent representation of foraging space, must exist. We hypothesize that the proposed network essentially resides in the mushroom bodies of the insect brain.     

The Preferred Directions of Conjunctive Grid X Head Direction Cells in the Medial Entorhinal Cortex Are Periodically Organized

The discovery of speed-modulated grid, head direction, and conjunctive grid x head direction cells in the medial entorhinal cortex has led to the hypothesis that path integration, the updating of one’s spatial representation based on movement, may be carried out within this region. This hypothesis has been formalized by many computational models, including a class known as attractor network models. While many of these models propose specific mechanisms by which path integration might occur, predictions of these specific mechanisms have not been tested. Here I derive and test a key prediction of one attractor network path integration mechanism. Specifically, I first demonstrate that this mechanism predicts a periodic distribution of conjunctive cell preferred directions in order to minimize drift. Next, I test whether conjunctive cell preferred directions are in fact periodically organized. Results indicate that conjunctive cells are preferentially tuned to increments of 36°, consistent with drift minimization in this path integration mechanism. By contrast, no periodicity was observed in the preferred directions of either pure grid or pure head direction cells. These results provide the first neural evidence of a nonuniform structure in the directional preferences of any head direction representation found in the brain.     

Multimodal integration for the representation of space in the posterior parietal cortex.

The posterior parietal cortex has long been considered an ''association'' area that combines information from different sensory modalities to form a cognitive representation of space. However, until recently little has been known about the neural mechanisms responsible for this important cognitive process. Recent experiments from the author''s laboratory indicate that visual, somatosensory, auditory and vestibular signals are combined in areas LIP and 7a of the posterior parietal cortex. The integration of these signals can represent the locations of stimuli with respect to the observer and within the environment. Area MSTd combines visual motion signals, similar to those generated during an observer''s movement through the environment, with eye-movement and vestibular signals. This integration appears to play a role in specifying the path on which the observer is moving. All three cortical areas combine different modalities into common spatial frames by using a gain-field mechanism. The spatial representations in areas LIP and 7a appear to be important for specifying the locations of targets for actions such as eye movements or reaching; the spatial representation within area MSTd appears to be important for navigation and the perceptual stability of motion signals.     

Spatial cognition and neuro-mimetic navigation: a model of hippocampal place cell activity

 A computational model of hippocampal activity during spatial cognition and navigation tasks is presented. The spatial representation in our model of the rat hippocampus is built on-line during exploration via two processing streams. An allothetic vision-based representation is built by unsupervised Hebbian learning extracting spatio-temporal properties of the environment from visual input. An idiothetic representation is learned based on internal movement-related information provided by path integration. On the level of the hippocampus, allothetic and idiothetic representations are integrated to yield a stable representation of the environment by a population of localized overlapping CA3-CA1 place fields. The hippocampal spatial representation is used as a basis for goal-oriented spatial behavior. We focus on the neural pathway connecting the hippocampus to the nucleus accumbens. Place cells drive a population of locomotor action neurons in the nucleus accumbens. Reward-based learning is applied to map place cell activity into action cell activity. The ensemble action cell activity provides navigational maps to support spatial behavior. We present experimental results obtained with a mobile Khepera robot. Received: 02 July 1999 / Accepted in revised form: 20 March 2000     

How do insects use path integration for their navigation?

 We combine experimental findings on ants and bees, and build on earlier models, to give an account of how these insects navigate using path integration, and how path integration interacts with other modes of navigation. At the core of path integration is an accumulator. This is set to an initial state at the nest and is updated as the insect moves so that it always reports the insect's current position relative to the nest. Navigation that uses path integration requires, in addition, a way of storing states of the accumulator at significant places for subsequent recall as goals, and a means of computing the direction to such goals. We discuss three models of how path integration might be used for this process, which we call vector navigation. Vector navigation is the principal means of navigating over unfamiliar terrain, or when landmarks are unavailable. Under other conditions, insects often navigate by landmarks, and ignore the output of the vector navigation system. Landmark navigation does not interfere with the updating of the accumulator. There is an interesting symmetry in the use of landmarks and path integration. In the short term, vector navigation can be independent of landmarks, and landmark navigation needs no assistance from path integration. In the longer term, visual landmarks help keep path vector navigation calibrated, and the learning of visual landmarks is guided by path integration. Received: 6 June 1999 / Accepted in revised form: 20 March 2000     

An optical flow-based integrated navigation system inspired by insect vision

Some insects use optic flow (OF) to perform their navigational tasks perfectly. Learning from insects' OF navigation strategies, this article proposes a bio-inspired integrated navigation system based on OF. The integrated navigation system is composed of an OF navigation system (OFNS) and an OF aided navigation system (OFAN). The OFNS uses a simple OF method to measure motion at each step along a path. The position information is then obtained by path integration. However, path integration leads to cumulative position errors which increase rapidly with time. To overcome this problem, the OFAN is employed to assist the OFNS in estimating and correcting these cumulative errors. The OFAN adopts an OF-based Kalman filter (KF) to continuously estimate the position errors. Moreover, based on the OF technique used in the OFNS, we develop a new OF method employed by the OFAN to generate the measurement input of the OF-based KF. As a result, both the OFNS and the OFAN in our integrated navigation system are derived from the same OF method so that they share input signals and some operations. The proposed integrated navigation system can provide accurate position information without interference from cumulative errors yet doing so with low computational effort. Simulations and comparisons have demonstrated its efficiency.     

Maintaining a cognitive map in darkness: the need to fuse boundary knowledge with path integration

Spatial navigation requires the processing of complex, disparate and often ambiguous sensory data. The neurocomputations underpinning this vital ability remain poorly understood. Controversy remains as to whether multimodal sensory information must be combined into a unified representation, consistent with Tolman's "cognitive map", or whether differential activation of independent navigation modules suffice to explain observed navigation behaviour. Here we demonstrate that key neural correlates of spatial navigation in darkness cannot be explained if the path integration system acted independently of boundary (landmark) information. In vivo recordings demonstrate that the rodent head direction (HD) system becomes unstable within three minutes without vision. In contrast, rodents maintain stable place fields and grid fields for over half an hour without vision. Using a simple HD error model, we show analytically that idiothetic path integration (iPI) alone cannot be used to maintain any stable place representation beyond two to three minutes. We then use a measure of place stability based on information theoretic principles to prove that featureless boundaries alone cannot be used to improve localization above chance level. Having shown that neither iPI nor boundaries alone are sufficient, we then address the question of whether their combination is sufficient and - we conjecture - necessary to maintain place stability for prolonged periods without vision. We addressed this question in simulations and robot experiments using a navigation model comprising of a particle filter and boundary map. The model replicates published experimental results on place field and grid field stability without vision, and makes testable predictions including place field splitting and grid field rescaling if the true arena geometry differs from the acquired boundary map. We discuss our findings in light of current theories of animal navigation and neuronal computation, and elaborate on their implications and significance for the design, analysis and interpretation of experiments.     

Triple-loop model of path control by head direction and place cells

 Arthropods as well as mammals are able to return straight home after a random search excursion under conditions that are designed to exclude all external cues. After a brief clarification of the terminology, two principal systems of information processing that can achieve this performance are introduced and analysed: Polar versus Cartesian path integration. The different demands and achievements of the two systems are confronted with neurophysiological findings on the functioning of the hippocampus, and with a recent comprehensive model of how the hippocampal place cells perform path integration. To connect the neurophysiological findings with the behavior of the animal, a new model is developed. It achieves three functionally diverse performances: maintenance and control of a compass direction, navigation by path integration, and formation of goals by connecting non-spatial features with their location. This is done by three interconnected feedback loops, set by a common reference variable. Their information-processing structure enables the animal not only to home but also to go straight from any stored goal to any other, without explicit representation of the distance between them, and without a topological arrangement of the store. The model explains behaviors not yet understood and predicts still undiscovered performances. Because it allows the isolation of orienting from storing functions yet also shows how they can be connected, the model may help to reconcile conflicting views on the function of the hippocampus. Received: 17 June 1999 / Accepted in revised form: 20 March 2000     

A Unified Model of Heading and Path Perception in Primate MSTd

Self-motion, steering, and obstacle avoidance during navigation in the real world require humans to travel along curved paths. Many perceptual models have been proposed that focus on heading, which specifies the direction of travel along straight paths, but not on path curvature, which humans accurately perceive and is critical to everyday locomotion. In primates, including humans, dorsal medial superior temporal area (MSTd) has been implicated in heading perception. However, the majority of MSTd neurons respond optimally to spiral patterns, rather than to the radial expansion patterns associated with heading. No existing theory of curved path perception explains the neural mechanisms by which humans accurately assess path and no functional role for spiral-tuned cells has yet been proposed. Here we present a computational model that demonstrates how the continuum of observed cells (radial to circular) in MSTd can simultaneously code curvature and heading across the neural population. Curvature is encoded through the spirality of the most active cell, and heading is encoded through the visuotopic location of the center of the most active cell''s receptive field. Model curvature and heading errors fit those made by humans. Our model challenges the view that the function of MSTd is heading estimation, based on our analysis we claim that it is primarily concerned with trajectory estimation and the simultaneous representation of both curvature and heading. In our model, temporal dynamics afford time-history in the neural representation of optic flow, which may modulate its structure. This has far-reaching implications for the interpretation of studies that assume that optic flow is, and should be, represented as an instantaneous vector field. Our results suggest that spiral motion patterns that emerge in spatio-temporal optic flow are essential for guiding self-motion along complex trajectories, and that cells in MSTd are specifically tuned to extract complex trajectory estimation from flow.     

Path integration of head direction: updating a packet of neural activity at the correct speed using neuronal time constants

A key question in understanding the neural basis of path integration is how individual, spatially responsive, neurons may self-organize into networks that can, through learning, integrate velocity signals to update a continuous representation of location within an environment. It is of vital importance that this internal representation of position is updated at the correct speed, and in real time, to accurately reflect the motion of the animal. In this article, we present a biologically plausible model of velocity path integration of head direction that can solve this problem using neuronal time constants to effect natural time delays, over which associations can be learned through associative Hebbian learning rules. The model comprises a linked continuous attractor network and competitive network. In simulation, we show that the same model is able to learn two different speeds of rotation when implemented with two different values for the time constant, and without the need to alter any other model parameters. The proposed model could be extended to path integration of place in the environment, and path integration of spatial view.     

Accurate Path Integration in Continuous Attractor Network Models of Grid Cells

Grid cells in the rat entorhinal cortex display strikingly regular firing responses to the animal''s position in 2-D space and have been hypothesized to form the neural substrate for dead-reckoning. However, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat''s velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of ∼10–100 meters and ∼1–10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.     

Orientation and navigation in Amphibia

Aquatic and terrestrial amphibians integrate acoustic, magnetic, mechanical, olfactory and visual directional information into a redundant-multisensory orientation system. The sensory information is processed to accomplish homing following active or passive displacement by either path integration, beaconing, pilotage, compass orientation or true navigation. There is evidence for two independent compass systems, a time-compensated compass based on celestial cues and a light-dependent magnetic inclination compass. Beaconing along acoustic or olfactory gradients emanating from the home site, as well as pilotage along fixed visual landmarks also form an important part in the behaviour of many species. True navigation has been shown in only one species, the aquatic salamander Notophthalmus viridescens. Evidence on the nature of the navigational map obtained so far is compatible with the magnetic map hypothesis.     

Memory use in insect visual navigation

The navigational strategies that are used by foraging ants and bees to reach a goal are similar to those of birds and mammals. Species from all these groups use path integration and memories of visual landmarks to navigate through familiar terrain. Insects have far fewer neural resources than vertebrates, so data from insects might be useful in revealing the essential components of efficient navigation. Recent work on ants and bees has uncovered a major role for associative links between long-term memories. We emphasize the roles of these associations in the reliable recognition of visual landmarks and the reliable performance of learnt routes. It is unknown whether such associations also provide insects with a map-like representation of familiar terrain. We suggest, however, that landmarks act primarily as signposts that tell insects what particular action they need to perform, rather than telling them where they are.     

Desert ant navigation: how miniature brains solve complex tasks

This essay presents and discusses the state of the art in studies of desert ant (Cataglyphis) navigation. In dealing with behavioural performances, neural mechanisms, and ecological functions these studies ultimately aim at an evolutionary understanding of the insect's navigational toolkit: its skylight (polarization) compass, its path integrator, its view-dependent ways of recognizing places and following landmark routes, and its strategies of flexibly interlinking these modes of navigation to generate amazingly rich behavioural outputs. The general message is that Cataglyphis uses path integration as an egocentric guideline to acquire continually updated spatial information about places and routes. Hence, it relies on procedural knowledge, and largely context-dependent retrieval of such knowledge, rather than on all-embracing geocentred representations of space.This revised version was published online in July 2003 with corrections to the text of the sections "Skylight compass" and "Path integration".     

Solving navigational uncertainty using grid cells on robots

To successfully navigate their habitats, many mammals use a combination of two mechanisms, path integration and calibration using landmarks, which together enable them to estimate their location and orientation, or pose. In large natural environments, both these mechanisms are characterized by uncertainty: the path integration process is subject to the accumulation of error, while landmark calibration is limited by perceptual ambiguity. It remains unclear how animals form coherent spatial representations in the presence of such uncertainty. Navigation research using robots has determined that uncertainty can be effectively addressed by maintaining multiple probabilistic estimates of a robot's pose. Here we show how conjunctive grid cells in dorsocaudal medial entorhinal cortex (dMEC) may maintain multiple estimates of pose using a brain-based robot navigation system known as RatSLAM. Based both on rodent spatially-responsive cells and functional engineering principles, the cells at the core of the RatSLAM computational model have similar characteristics to rodent grid cells, which we demonstrate by replicating the seminal Moser experiments. We apply the RatSLAM model to a new experimental paradigm designed to examine the responses of a robot or animal in the presence of perceptual ambiguity. Our computational approach enables us to observe short-term population coding of multiple location hypotheses, a phenomenon which would not be easily observable in rodent recordings. We present behavioral and neural evidence demonstrating that the conjunctive grid cells maintain and propagate multiple estimates of pose, enabling the correct pose estimate to be resolved over time even without uniquely identifying cues. While recent research has focused on the grid-like firing characteristics, accuracy and representational capacity of grid cells, our results identify a possible critical and unique role for conjunctive grid cells in filtering sensory uncertainty. We anticipate our study to be a starting point for animal experiments that test navigation in perceptually ambiguous environments.     

Multisensory integration in spatial orientation

Recent psychophysical studies on normal subjects, as well as brain imaging studies, have revised the concepts concerning the mechanisms underlying spatial orientation during navigation tasks. The emphasis has been put on internal models that allow the prediction of a planned trajectory and are essential in the steering of locomotion. Cognitive factors such as strategies and emotional parameters are now starting to be included in the research on spatial orientation. It is obvious that important individual and gender differences exist in the brain operations underlying spatial orientation in humans, which may help to understand the construction of a coherent perception and the organic neural disorders related to the internal representation of space.