Accurate Path Integration Research Articles

A spatial transformation-based CAN model for information integration within grid cell modules

The hippocampal-entorhinal circuit is considered to play an important role in the spatial cognition of animals. However, the mechanism of the information flow within the circuit and its contribution to the function of the grid-cell module are still topics of discussion. Prevailing theories suggest that grid cells are primarily influenced by self-motion inputs from the Medial Entorhinal Cortex, with place cells serving a secondary role by contributing to the visual calibration of grid cells. However, recent evidence suggests that both self-motion inputs and visual cues may collaboratively contribute to the formation of grid-like patterns. In this paper, we introduce a novel Continuous Attractor Network model based on a spatial transformation mechanism. This mechanism enables the integration of self-motion inputs and visual cues within grid-cell modules, synergistically driving the formation of grid-like patterns. From the perspective of individual neurons within the network, our model successfully replicates grid firing patterns. From the view of neural population activity within the network, the network can form and drive the activated bump, which describes the characteristic feature of grid-cell modules, namely, path integration. Through further exploration and experimentation, our model can exhibit significant performance in path integration. This study provides a new insight into understanding the mechanism of how the self-motion and visual inputs contribute to the neural activity within grid-cell modules. Furthermore, it provides theoretical support for achieving accurate path integration, which holds substantial implications for various applications requiring spatial navigation and mapping.

Learning accurate path integration in ring attractor models of the head direction system.

Ring attractor models for angular path integration have received strong experimental support. To function as integrators, head direction circuits require precisely tuned connectivity, but it is currently unknown how such tuning could be achieved. Here, we propose a network model in which a local, biologically plausible learning rule adjusts synaptic efficacies during development, guided by supervisory allothetic cues. Applied to the Drosophila head direction system, the model learns to path-integrate accurately and develops a connectivity strikingly similar to the one reported in experiments. The mature network is a quasi-continuous attractor and reproduces key experiments in which optogenetic stimulation controls the internal representation of heading in flies, and where the network remaps to integrate with different gains in rodents. Our model predicts that path integration requires self-supervised learning during a developmental phase, and proposes a general framework to learn to path-integrate with gain-1 even in architectures that lack the physical topography of a ring.

Influence of sensory modality and control dynamics on human path integration.

Path integration is a sensorimotor computation that can be used to infer latent dynamical states by integrating self-motion cues. We studied the influence of sensory observation (visual/vestibular) and latent control dynamics (velocity/acceleration) on human path integration using a novel motion-cueing algorithm. Sensory modality and control dynamics were both varied randomly across trials, as participants controlled a joystick to steer to a memorized target location in virtual reality. Visual and vestibular steering cues allowed comparable accuracies only when participants controlled their acceleration, suggesting that vestibular signals, on their own, fail to support accurate path integration in the absence of sustained acceleration. Nevertheless, performance in all conditions reflected a failure to fully adapt to changes in the underlying control dynamics, a result that was well explained by a bias in the dynamics estimation. This work demonstrates how an incorrect internal model of control dynamics affects navigation in volatile environments in spite of continuous sensory feedback.

A speed-accurate self-sustaining head direction cell path integration model without recurrent excitation

ABSTRACTThe head direction (HD) system signals HD in an allocentric frame of reference. The system is able to update firing based on internally derived information about self-motion, a process known as path integration. Of particular interest is how path integration might maintain concordance between true HD and internally represented HD. Here we present a self-sustaining two-layer model, capable of self-organizing, which produces extremely accurate path integration. The implications of this work for future investigations of HD system path integration are discussed.

Absence of Visual Input Results in the Disruption of Grid Cell Firing in the Mouse

SummaryGrid cells are spatially modulated neurons within the medial entorhinal cortex whose firing fields are arranged at the vertices of tessellating equilateral triangles [1]. The exquisite periodicity of their firing has led to the suggestion that they represent a path integration signal, tracking the organism’s position by integrating speed and direction of movement [2, 3, 4, 5, 6, 7, 8, 9, 10]. External sensory inputs are required to reset any errors that the path integrator would inevitably accumulate. Here we probe the nature of the external sensory inputs required to sustain grid firing, by recording grid cells as mice explore familiar environments in complete darkness. The absence of visual cues results in a significant disruption of grid cell firing patterns, even when the quality of the directional information provided by head direction cells is largely preserved. Darkness alters the expression of velocity signaling within the entorhinal cortex, with changes evident in grid cell firing rate and the local field potential theta frequency. Short-term (<1.5 s) spike timing relationships between grid cell pairs are preserved in the dark, indicating that network patterns of excitatory and inhibitory coupling between grid cells exist independently of visual input and of spatially periodic firing. However, we find no evidence of preserved hexagonal symmetry in the spatial firing of single grid cells at comparable short timescales. Taken together, these results demonstrate that visual input is required to sustain grid cell periodicity and stability in mice and suggest that grid cells in mice cannot perform accurate path integration in the absence of reliable visual cues.

The retrosplenial cortex is necessary for path integration in the dark

An increasing amount of data indicates that the retrosplenial cortex (RSC) plays a role in navigation and spatial memory. Moreover, it has been suggested that the RSC integrates mnemonic spatial features of the environment with self-motion information therefore enabling accurate path integration in darkness. This hypothesis rests on data obtained from animals trained in spatial memory tasks involving the conjoint use of allothetic and idiothetic information [8]. We examined the contribution of the RSC when animals are submitted to a path integration task with minimal memory requirement in light and dark. We found that RSC-lesioned rats exhibited a path integration deficit in the dark but not in the light. This suggests that the RSC is important for path integration and incorporates visuospatial information to maintain path integration accuracy.

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.

The spatial representations acquired in CA3 by self-organizing recurrent connections

Neural computation models have hypothesized that the dentate gyrus (DG) drives the storage in the CA3 network of new memories including, e.g., in rodents, spatial memories. Can recurrent CA3 connections self-organize, during storage, and form what have been called continuous attractors, or charts—so that they express spatial information later, when aside from a partial cue the information may not be available in the inputs? We use a simplified mathematical network model to contrast the properties of spatial representations self-organized through simulated Hebbian plasticity with those of charts pre-wired in the synaptic matrix, a control case closer to the ideal notion of continuous attractors. Both models form granular quasi-attractors, characterized by drift, which approach continuous ones only in the limit of an infinitely large network. The two models are comparable in terms of precision, but not of accuracy: with self-organized connections, the metric of space remains distorted, ill-adequate for accurate path integration, even when scaled up to the real hippocampus. While prolonged self-organization makes charts somewhat more informative about position in the environment, some positional information is surprisingly present also about environments never learned, borrowed, as it were, from unrelated charts. In contrast, context discrimination decreases with more learning, as different charts tend to collapse onto each other. These observations challenge the feasibility of the idealized CA3 continuous chart concept, and are consistent with a CA3 specialization for episodic memory rather than path integration.

Path integration: Effect of curved path complexity and sensory system on blindfolded walking

Path integration refers to the ability to integrate continuous information of the direction and distance traveled by the system relative to the origin. Previous studies have investigated path integration through blindfolded walking along simple paths such as straight line and triangles. However, limited knowledge exists regarding the role of path complexity in path integration. Moreover, little is known about how information from different sensory input systems (like vision and proprioception) contributes to accurate path integration. The purpose of the current study was to investigate how sensory information and curved path complexity affect path integration. Forty blindfolded participants had to accurately reproduce a curved path and return to the origin. They were divided into four groups that differed in the curved path, circle (simple) or figure-eight (complex), and received either visual (previously seen) or proprioceptive (previously guided) information about the path before they reproduced it. The dependent variables used were average trajectory error, walking speed, and distance traveled. The results indicated that (a) both groups that walked on a circular path and both groups that received visual information produced greater accuracy in reproducing the path. Moreover, the performance of the group that received proprioceptive information and later walked on a figure-eight path was less accurate than their corresponding circular group. The groups that had the visual information also walked faster compared to the group that had proprioceptive information. Results of the current study highlight the roles of different sensory inputs while performing blindfolded walking for path integration.

Age differences in virtual environment and real world path integration

Accurate path integration (PI) requires the integration of visual, proprioceptive, and vestibular self-motion cues and age effects associated with alterations in processing information from these systems may contribute to declines in PI abilities. The present study investigated age-related differences in PI in conditions that varied as a function of available sources of sensory information. Twenty-two healthy, young (23.8 ± 3.0 years) and 16 older (70.1 ± 6.4 years) adults participated in distance reproduction and triangle completion tasks (TCTs) performed in a virtual environment (VE) and two “real world” conditions: guided walking and wheelchair propulsion. For walking and wheelchair propulsion conditions, participants wore a blindfold and wore noise-blocking headphones and were guided through the workspace by the experimenter. For the VE condition, participants viewed self-motion information on a computer monitor and used a joystick to navigate through the environment. For TCTs, older compared to younger individuals showed greater errors in rotation estimations performed in the wheelchair condition, and for rotation and distance estimations in the VE condition. Distance reproduction tasks (DRTs), in contrast, did not show any age effects. These findings demonstrate that age differences in PI vary as a function of the available sources of information and by the complexity of outbound pathway.

Stationary envelope synthesis (SES): A universal method for phase coding by neural oscillators

Stationary envelope synthesis (SES): A universal method for phase coding by neural oscillators

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.

Elastic image mapping for 4-D dose estimation in thoracic radiotherapy

Demonstrate the path integration of a four-dimensional (4-D) dose distribution onto the 3-D anatomy. A computer-generated 4-D thoracic phantom with a lung tumour was constructed. Eight respiratory phases were generated. A radiotherapy treatment plan was applied to all the phases resulting in a 4-D dose distribution. An elastic image registration algorithm was used to find the vector displacement between all the image elements and the end expiration phase. The path-integrated tissue dose distribution and each component dose distribution were compared with the planned dose distribution. Numerical path integration was performed to calculate the tissue dose distribution. Loss of tumour coverage was the predominant effect observed with tumour motion in this study. The loss was asymmetric and dependent on the tumour trajectory. The elastic image registration allowed an accurate path integration through a 4-D data set to produce an accurate 3-D tissue dose estimate.

Evidence of Separable Spatial Representations in a Virtual Navigation Task.

Three experiments investigated spatial orientation in a virtual navigation task. Subjects had to adjust a homing vector indicating their end position relative to the origin of the path. It was demonstrated that sparse visual flow was sufficient for accurate path integration. Moreover, subjects were found to prefer a distinct egocentric or allocentric reference frame to solve the task. "Turners" reacted as if they had taken on the new orientation during turns of the path by mentally rotating their sagittal axis (egocentric frame). "Nonturners," by contrast, tracked the new orientation without adopting it (allocentric frame). When instructed to use their nonpreferred reference frame, both groups displayed no decline in response accuracy relative to their preferred frame; even when presented with reaction formats based on either ego or allocentric coordinates, with format unpredictable on a trial, both groups responded highly accurately. These findings support the assumption of coexisting spatial representations during navigation.

A subterranean mammal uses the magnetic compass for path integration.

Path integration allows animals to navigate without landmarks by continuously processing signals generated through locomotion. Insects such as bees and ants have evolved an accurate path integration system, assessing and coding rotations with the help of a general directional reference, the sun azimuth. In mammals, by contrast, this process can take place through purely idiothetic (mainly proprioceptive and vestibular) signals. However, without any stable external reference for measuring direction, path integration is highly affected by cumulative errors and thus has been considered so far as valid only for short-distance navigation. Here we show through two path integration experiments (homing and shortcut finding) that the blind mole rat assesses direction both through internal signals and by estimating its heading in relation to the earth's magnetic field. Further, it is shown that the greater the circumvolution and length of the traveled path, the more the animal relies on the geomagnetic field. This path integration system strongly reduces the accumulation of errors due to inaccuracies in the estimation of rotations and thus allows the mole rat to navigate efficiently in darkness, without the help of any landmark, over both short and long distances.

Temporary Inactivation of the Retrosplenial Cortex Causes a Transient Reorganization of Spatial Coding in the Hippocampus

The ability to navigate accurately is dependent on the integration of visual and movement-related cues. Navigation based on metrics derived from movement is referred to as path integration. Recent theories of navigation have suggested that posterior cortical areas, the retrosplenial and posterior parietal cortex, are involved in path integration during navigation. In support of this hypothesis, we have found previously that temporary inactivation of retrosplenial cortex results in dark-selective impairments on the radial maze (Cooper and Mizumori, 1999). To understand further the role of the retrosplenial cortex in navigation, we combined temporary inactivation of retrosplenial cortex with recording of complex spike cells in the hippocampus. Thus, behavioral performance during spatial memory testing could be compared with place-field responses before, and during, inactivation of retrosplenial cortex. In the first experiment, behavioral results confirmed that inactivation of retrosplenial cortex only impairs radial maze performance in darkness when animals are at asymptote levels of performance. A second experiment revealed that retrosplenial cortex inactivation impaired spatial learning during initial light training. In both experiments, the normal location of hippocampal "place fields" was changed by temporary inactivation of retrosplenial cortex, whereas other electrophysiological properties of the cells were not affected. The changes in place coding occurred in the presence, and absence, of behavioral impairments. We suggest that the retrosplenial cortex provides mnemonic spatial information for updating location codes in the hippocampus, thereby facilitating accurate path integration. In this way, the retrosplenial cortex and hippocampus may be part of an interactive neural system that mediates navigation.

An Analysis of 3D Particle Path Integration Algorithms

Several techniques for the numerical integration of particle paths in steady and unsteady vector (velocity) fields are analyzed. Most of the analysis applies to unsteady vector fields, however, some results apply to steady vector field integration. Multistep, multistage, and some hybrid schemes are considered. It is shown that due to initialization errors, many unsteady particle path integration schemes are limited to third-order accuracy in time. Multistage schemes require at least three times more internal data storage than multistep schemes of equal order. However, for timesteps within the stability bounds, multistage schemes are generally more accurate. A linearized analysis shows that the stability of these integration algorithms are determined by the eigenvalues of the local velocity tensor. Thus, the accuracy and stability of the methods are interpreted with concepts typically used in critical point theory. This paper shows how integration schemes can lead to erroneous classification of critical points when the timestep is finite and fixed. For steady velocity fields, we demonstrate that timesteps outside of the relative stability region can lead to similar integration errors. From this analysis, guidelines for accurate timestep sizing are suggested for both steady and unsteady flows. In particular, using simulation data for the unsteady flow around a tapered cylinder, we show that accurate particle path integration requires timesteps which are at most on the order of the physical timescale of the flow.