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.     

Encoding certainty in bump attractors

Persistent activity in neuronal populations has been shown to represent the spatial position of remembered stimuli. Networks that support bump attractors are often used to model such persistent activity. Such models usually exhibit translational symmetry. Thus activity bumps are neutrally stable, and perturbations in position do not decay away. We extend previous work on bump attractors by constructing model networks capable of encoding the certainty or salience of a stimulus stored in memory. Such networks support bumps that are not only neutrally stable to perturbations in position, but also perturbations in amplitude. Possible bump solutions then lie on a two-dimensional attractor, determined by a continuum of positions and amplitudes. Such an attractor requires precisely balancing the strength of recurrent synaptic connections. The amplitude of activity bumps represents certainty, and is determined by the initial input to the system. Moreover, bumps with larger amplitudes are more robust to noise, and over time provide a more faithful representation of the stored stimulus. In networks with separate excitatory and inhibitory populations, generating bumps with a continuum of possible amplitudes, requires tuning the strength of inhibition to precisely cancel background excitation.     

Robust path integration in the entorhinal grid cell system with hippocampal feed-back

Animals are able to update their knowledge about their current position solely by integrating the speed and the direction of their movement, which is known as path integration. Recent discoveries suggest that grid cells in the medial entorhinal cortex might perform some of the essential underlying computations of path integration. However, a major concern over path integration is that as the measurement of speed and direction is inaccurate, the representation of the position will become increasingly unreliable. In this paper, we study how allothetic inputs can be used to continually correct the accumulating error in the path integrator system. We set up the model of a mobile agent equipped with the entorhinal representation of idiothetic (grid cell) and allothetic (visual cells) information and simulated its place learning in a virtual environment. Due to competitive learning, a robust hippocampal place code emerges rapidly in the model. At the same time, the hippocampo-entorhinal feed-back connections are modified via Hebbian learning in order to allow hippocampal place cells to influence the attractor dynamics in the entorhinal cortex. We show that the continuous feed-back from the integrated hippocampal place representation is able to stabilize the grid cell code. This research was supported by the EU Framework 6 ICEA project (IST-4-027819-IP).     

Continuous Attractor Network Model for Conjunctive Position-by-Velocity Tuning of Grid Cells

The spatial responses of many of the cells recorded in layer II of rodent medial entorhinal cortex (MEC) show a triangular grid pattern, which appears to provide an accurate population code for animal spatial position. In layer III, V and VI of the rat MEC, grid cells are also selective to head-direction and are modulated by the speed of the animal. Several putative mechanisms of grid-like maps were proposed, including attractor network dynamics, interactions with theta oscillations or single-unit mechanisms such as firing rate adaptation. In this paper, we present a new attractor network model that accounts for the conjunctive position-by-velocity selectivity of grid cells. Our network model is able to perform robust path integration even when the recurrent connections are subject to random perturbations.     

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.     

Evidence for Resetting the Directional Component of Path Integration in the House Mouse (Mus musculus)

Path integration is the animal's running computation of its position relative to a starting point based on recording all displacements. This process is known to be prone to cumulative errors and hence must somehow be corrected with the help of local spatial cues. In the reported experiments, the relative role of local spatial cues at two target locations was appraised. These two targets were the peripheral box of a female house mouse (Mus musculus) and a central box from where she retrieved her pups over a distance of 50 cm. The experimental conditions required the mouse to navigate between the two targets solely by means of path integration. To find out whether the mice continued integrating at the two target locations, directional misinformation was fed into their path integrating system. Hence, passive rotation of 90° proved to be sufficient to decide unambiguously whether the mice were misdirected. Such experimental interference consistently documented ongoing path integration at the target location outside the residential nest. In contrast, the same interference when the mouse was in the residential nest documented discontinuation of path integration in most cases. It is inferred that mice, when departing from residential nests, initially direct themselves by means of local guiding stimuli.     

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.     

Oscillatory neurocomputing with ring attractors: a network architecture for mapping locations in space onto patterns of neural synchrony

Theories of neural coding seek to explain how states of the world are mapped onto states of the brain. Here, we compare how an animal''s location in space can be encoded by two different kinds of brain states: population vectors stored by patterns of neural firing rates, versus synchronization vectors stored by patterns of synchrony among neural oscillators. It has previously been shown that a population code stored by spatially tuned ‘grid cells’ can exhibit desirable properties such as high storage capacity and strong fault tolerance; here it is shown that similar properties are attainable with a synchronization code stored by rhythmically bursting ‘theta cells’ that lack spatial tuning. Simulations of a ring attractor network composed from theta cells suggest how a synchronization code might be implemented using fewer neurons and synapses than a population code with similar storage capacity. It is conjectured that reciprocal connections between grid and theta cells might control phase noise to correct two kinds of errors that can arise in the code: path integration and teleportation errors. Based upon these analyses, it is proposed that a primary function of spatially tuned neurons might be to couple the phases of neural oscillators in a manner that allows them to encode spatial locations as patterns of neural synchrony.     

Weighted cue integration in the rodent head direction system

How the brain combines information from different sensory modalities and of differing reliability is an important and still-unanswered question. Using the head direction (HD) system as a model, we explored the resolution of conflicts between landmarks and background cues. Sensory cue integration models predict averaging of the two cues, whereas attractor models predict capture of the signal by the dominant cue. We found that a visual landmark mostly captured the HD signal at low conflicts: however, there was an increasing propensity for the cells to integrate the cues thereafter. A large conflict presented to naive rats resulted in greater visual cue capture (less integration) than in experienced rats, revealing an effect of experience. We propose that weighted cue integration in HD cells arises from dynamic plasticity of the feed-forward inputs to the network, causing within-trial spatial redistribution of the visual inputs onto the ring. This suggests that an attractor network can implement decision processes about cue reliability using simple architecture and learning rules, thus providing a potential neural substrate for weighted cue integration.     

Robust spatial working memory through homeostatic synaptic scaling in heterogeneous cortical networks

The concept of bell-shaped persistent neural activity represents a cornerstone of the theory for the internal representation of analog quantities, such as spatial location or head direction. Previous models, however, relied on the unrealistic assumption of network homogeneity. We investigate this issue in a network model where fine tuning of parameters is destroyed by heterogeneities in cellular and synaptic properties. Heterogeneities result in the loss of stored spatial information in a few seconds. Accurate encoding is recovered when a homeostatic mechanism scales the excitatory synapses to each cell to compensate for the heterogeneity in cellular excitability and synaptic inputs. Moreover, the more realistic model produces a wide diversity of tuning curves, as commonly observed in recordings from prefrontal neurons. We conclude that recurrent attractor networks in conjunction with appropriate homeostatic mechanisms provide a robust, biologically plausible theoretical framework for understanding the neural circuit basis of spatial working memory.     

Motor-maps, navigation and implicit space representation in the hippocampus

Multiple sensory-motor maps located in the brainstem and the cortex are involved in spatial orientation. Guiding movements of eyes, head, neck and arms they provide an approximately linear relation between target distance and motor response. This involves especially the superior colliculus in the brainstem and the parietal cortex. There, the natural frame of reference follows from the retinal representation of the environment. A model of navigation is presented that is based on the modulation of activity in those sensory-motor maps. The actual mechanism chosen was gain-field modulation, a process of multimodal integration that has been demonstrated in the parietal cortex and superior colliculus, and was implemented as attraction to visual cues (colour). Dependent on the metric of the sensory-motor map, the relative attraction to these cues implemented as gain field modulation and their position define a fixed point attractor on the plane for locomotive behaviour. The actual implementation used Kohonen-networks in a variant of reinforcement learning that are well suited to generate such topographically organized sensory-motor maps with roughly linear visuo-motor response characteristics. In the following, it was investigated how such an implicit coding of target positions by gain-field parameters might be represented in the hippocampus formation and under what conditions a direction-invariant space representation can arise from such retinotopic representations of multiple cues. Information about the orientation in the plane—as could be provided by head direction cells—appeared to be necessary for unambiguous space representation in our model in agreement with physiological experiments. With this information, Gauss-shaped “place-cells” could be generated, however, the representation of the spatial environment was repetitive and clustered and single cells were always tuned to the gain-field parameters as well     

A Controlled Attractor Network Model of Path Integration in the Rat

Cells in several areas of the hippocampal formation show place specific firing patterns, and are thought to form a distributed representation of an animals current location in an environment. Experimental results suggest that this representation is continually updated even in complete darkness, indicating the presence of a path integration mechanism in the rat. Adopting the Neural Engineering Framework (NEF) presented by Eliasmith and Anderson (2003) we derive a novel attractor network model of path integration, using heterogeneous spiking neurons. The network we derive incorporates representation and updating of position into a single layer of neurons, eliminating the need for a large external control population, and without making use of multiplicative synapses. An efficient and biologically plausible control mechanism results directly from applying the principles of the NEF. We simulate the network for a variety of inputs, analyze its performance, and give three testable predictions of our model.     

Path Integration of Head Direction: Updating a Packet of Neural Activity at the Correct Speed Using Axonal Conduction Delays

The head direction cell system is capable of accurately updating its current representation of head direction in the absence of visual input. This is known as the path integration of head direction. An important question is how the head direction cell system learns to perform accurate path integration of head direction. In this paper we propose a model of velocity path integration of head direction in which the natural time delay of axonal transmission between a linked continuous attractor network and competitive network acts as a timing mechanism to facilitate the correct speed of path integration. The model effectively learns a “look-up” table for the correct speed of path integration. In simulation, we show that the model is able to successfully learn two different speeds of path integration across two different axonal conduction delays, and without the need to alter any other model parameters. An implication of this model is that, by learning look-up tables for each speed of path integration, the model should exhibit a degree of robustness to damage. In simulations, we show that the speed of path integration is not significantly affected by degrading the network through removing a proportion of the cells that signal rotational velocity.     

Spatial Orientation in Blattella germanica L. Larvae

The aim of these experiments was to investigate the type of cues used in homing processes by young Blattella germanica L. larvae. Several types of stimuli were tested: path integration with kinesthetic cues and visual orientation with landmark cues. Tests measured the escape direction of larvae from the food box after disturbance. Either type of cue alone, path integration or visual landmarks, was sufficient to allow larvae to orient towards their shelters, but they oriented more precisély when both types of cue were used. When several landmark cues (proximal and distal) were present, their relative angular position seemed important in the orientation process. Macroscopic shapes in the environment appeared to be used as a global image, memorized to reach the shelter.     

Homing in German Cockroaches, Blattella germanica (L.) (Insecta: Dictyoptera): Multi-Channelled Orientation Cues

Cockroaches use navigational cues to elaborate their return path to the shelter. Our experiments investigated how individuals weighted information to choose where to search for the shelter in situations where path integration, visual and olfactory cues were conflicting. We showed that homing relied on a complex set of environmental stimuli, each playing a particular part. Path integration cues give cockroaches an estimation of the position of their goal, visual landmarks guide them to that position from a distance, while olfactory cues indicate the end of the path. Cockroaches gave the greatest importance to the first cues they encountered along their return path. Nevertheless, visual cues placed beyond aggregation pheromone deposits reduced their arrest efficiency and induced search in the area near the visual cues.     

MPI CyberMotion Simulator: implementation of a novel motion simulator to investigate multisensory path integration in three dimensions

Path integration is a process in which self-motion is integrated over time to obtain an estimate of one's current position relative to a starting point (1). Humans can do path integration based exclusively on visual (2-3), auditory (4), or inertial cues (5). However, with multiple cues present, inertial cues - particularly kinaesthetic - seem to dominate (6-7). In the absence of vision, humans tend to overestimate short distances (<5 m) and turning angles (<30°), but underestimate longer ones (5). Movement through physical space therefore does not seem to be accurately represented by the brain. Extensive work has been done on evaluating path integration in the horizontal plane, but little is known about vertical movement (see (3) for virtual movement from vision alone). One reason for this is that traditional motion simulators have a small range of motion restricted mainly to the horizontal plane. Here we take advantage of a motion simulator (8-9) with a large range of motion to assess whether path integration is similar between horizontal and vertical planes. The relative contributions of inertial and visual cues for path navigation were also assessed. 16 observers sat upright in a seat mounted to the flange of a modified KUKA anthropomorphic robot arm. Sensory information was manipulated by providing visual (optic flow, limited lifetime star field), vestibular-kinaesthetic (passive self motion with eyes closed), or visual and vestibular-kinaesthetic motion cues. Movement trajectories in the horizontal, sagittal and frontal planes consisted of two segment lengths (1st: 0.4 m, 2nd: 1 m; ±0.24 m/s(2) peak acceleration). The angle of the two segments was either 45° or 90°. Observers pointed back to their origin by moving an arrow that was superimposed on an avatar presented on the screen. Observers were more likely to underestimate angle size for movement in the horizontal plane compared to the vertical planes. In the frontal plane observers were more likely to overestimate angle size while there was no such bias in the sagittal plane. Finally, observers responded slower when answering based on vestibular-kinaesthetic information alone. Human path integration based on vestibular-kinaesthetic information alone thus takes longer than when visual information is present. That pointing is consistent with underestimating and overestimating the angle one has moved through in the horizontal and vertical planes respectively, suggests that the neural representation of self-motion through space is non-symmetrical which may relate to the fact that humans experience movement mostly within the horizontal plane.     

A gain-field encoding of limb position and velocity in the internal model of arm dynamics

Adaptability of reaching movements depends on a computation in the brain that transforms sensory cues, such as those that indicate the position and velocity of the arm, into motor commands. Theoretical consideration shows that the encoding properties of neural elements implementing this transformation dictate how errors should generalize from one limb position and velocity to another. To estimate how sensory cues are encoded by these neural elements, we designed experiments that quantified spatial generalization in environments where forces depended on both position and velocity of the limb. The patterns of error generalization suggest that the neural elements that compute the transformation encode limb position and velocity in intrinsic coordinates via a gain-field; i.e., the elements have directionally dependent tuning that is modulated monotonically with limb position. The gain-field encoding makes the counterintuitive prediction of hypergeneralization: there should be growing extrapolation beyond the trained workspace. Furthermore, nonmonotonic force patterns should be more difficult to learn than monotonic ones. We confirmed these predictions experimentally.     

Multisensory contributions to the 3-D representation of visuotactile peripersonal space in humans: evidence from the crossmodal congruency task.

In order to determine precisely the location of a tactile stimulus presented to the hand it is necessary to know not only which part of the body has been stimulated, but also where that part of the body lies in space. This involves the multisensory integration of visual, tactile, proprioceptive, and even auditory cues regarding limb position. In recent years, researchers have become increasingly interested in the question of how these various sensory cues are weighted and integrated in order to enable people to localize tactile stimuli, as well as to give rise to the 'felt' position of our limbs, and ultimately the multisensory representation of 3-D peripersonal space. We highlight recent research on this topic using the crossmodal congruency task, in which participants make speeded elevation discrimination responses to vibrotactile targets presented to the thumb or index finger, while simultaneously trying to ignore irrelevant visual distractors presented from either the same (i.e., congruent) or a different (i.e., incongruent) elevation. Crossmodal congruency effects (calculated as performance on incongruent-congruent trials) are greatest when visual and vibrotactile stimuli are presented from the same azimuthal location, thus providing an index of common position across different sensory modalities. The crossmodal congruency task has been used to investigate a number of questions related to the representation of space in both normal participants and brain-damaged patients. In this review, we detail the major findings from this research, and highlight areas of convergence with other cognitive neuroscience disciplines.     

Optimal cue integration in ants

In situations with redundant or competing sensory information, humans have been shown to perform cue integration, weighting different cues according to their certainty in a quantifiably optimal manner. Ants have been shown to merge the directional information available from their path integration (PI) and visual memory, but as yet it is not clear that they do so in a way that reflects the relative certainty of the cues. In this study, we manipulate the variance of the PI home vector by allowing ants (Cataglyphis velox) to run different distances and testing their directional choice when the PI vector direction is put in competition with visual memory. Ants show progressively stronger weighting of their PI direction as PI length increases. The weighting is quantitatively predicted by modelling the expected directional variance of home vectors of different lengths and assuming optimal cue integration. However, a subsequent experiment suggests ants may not actually compute an internal estimate of the PI certainty, but are using the PI home vector length as a proxy.     

Ontogenetic Development of Spatial Orientation in First- and Second-instar Cockroach Larvae (Blattella germanica (L.), Dictyoptera)

Changes in spatial orientation related to aging were investigated during the first two instars in Blattella germanica (L). Spatial orientation was tested through shelter return under various experimental conditions in order to understand how kinesthetic path integration and visual cues intervene in the construction of this behaviour. We were able to distinguish two types of changes: experience during the first instar and maturation between instars. Young larvae need 2–3 d to develop the appropriate shelter-return behaviour when they can only use visual cues. By contrast, they can use kinesthetic path-integration cues efficiently to return to the shelter as soon as they hatch.