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.     

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.     

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.     

A novel somatosensory spatial navigation system outside the hippocampal formation

Spatially selective firing of place cells, grid cells, boundary vector/border cells and head direction cells constitutes the basic building blocks of a canonical spatial navigation system centered on the hippocampal-entorhinal complex. While head direction cells can be found throughout the brain, spatial tuning outside the hippocampal formation is often non-specific or conjunctive to other representations such as a reward. Although the precise mechanism of spatially selective firing activity is not understood, various studies show sensory inputs, particularly vision, heavily modulate spatial representation in the hippocampal-entorhinal circuit. To better understand the contribution of other sensory inputs in shaping spatial representation in the brain, we performed recording from the primary somatosensory cortex in foraging rats. To our surprise, we were able to detect the full complement of spatially selective firing patterns similar to that reported in the hippocampal-entorhinal network, namely, place cells, head direction cells, boundary vector/border cells, grid cells and conjunctive cells, in the somatosensory cortex. These newly identified somatosensory spatial cells form a spatial map outside the hippocampal formation and support the hypothesis that location information modulates body representation in the somatosensory cortex. Our findings provide transformative insights into our understanding of how spatial information is processed and integrated in the brain, as well as functional operations of the somatosensory cortex in the context of rehabilitation with brain-machine interfaces.Subject terms: Biological techniques, Cell biology     

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.     

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).     

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.     

Grid alignment in entorhinal cortex

The spatial responses of many of the cells recorded in all layers of rodent medial entorhinal cortex (mEC) show mutually aligned grid patterns. Recent experimental findings have shown that grids can often be better described as elliptical rather than purely circular and that, beyond the mutual alignment of their grid axes, ellipses tend to also orient their long axis along preferred directions. Are grid alignment and ellipse orientation aspects of the same phenomenon? Does the grid alignment result from single-unit mechanisms or does it require network interactions? We address these issues by refining a single-unit adaptation model of grid formation, to describe specifically the spontaneous emergence of conjunctive grid-by-head-direction cells in layers III, V, and VI of mEC. We find that tight alignment can be produced by recurrent collateral interactions, but this requires head-direction (HD) modulation. Through a competitive learning process driven by spatial inputs, grid fields then form already aligned, and with randomly distributed spatial phases. In addition, we find that the self-organization process is influenced by any anisotropy in the behavior of the simulated rat. The common grid alignment often orients along preferred running directions (RDs), as induced in a square environment. When speed anisotropy is present in exploration behavior, the shape of individual grids is distorted toward an ellipsoid arrangement. Speed anisotropy orients the long ellipse axis along the fast direction. Speed anisotropy on its own also tends to align grids, even without collaterals, but the alignment is seen to be loose. Finally, the alignment of spatial grid fields in multiple environments shows that the network expresses the same set of grid fields across environments, modulo a coherent rotation and translation. Thus, an efficient metric encoding of space may emerge through spontaneous pattern formation at the single-unit level, but it is coherent, hence context-invariant, if aided by collateral interactions.     

A Continuous Attractor Network Model Without Recurrent Excitation: Maintenance and Integration in the Head Direction Cell System

Motivated by experimental observations of the head direction system, we study a three population network model that operates as a continuous attractor network. This network is able to store in a short-term memory an angular variable (the head direction) as a spatial profile of activity across neurons in the absence of selective external inputs, and to accurately update this variable on the basis of angular velocity inputs. The network is composed of one excitatory population and two inhibitory populations, with inter-connections between populations but no connections within the neurons of a same population. In particular, there are no excitatory-to-excitatory connections. Angular velocity signals are represented as inputs in one inhibitory population (clockwise turns) or the other (counterclockwise turns). The system is studied using a combination of analytical and numerical methods. Analysis of a simplified model composed of threshold-linear neurons gives the conditions on the connectivity for (i) the emergence of the spatially selective profile, (ii) reliable integration of angular velocity inputs, and (iii) the range of angular velocities that can be accurately integrated by the model. Numerical simulations allow us to study the proposed scenario in a large network of spiking neurons and compare their dynamics with that of head direction cells recorded in the rat limbic system. In particular, we show that the directional representation encoded by the attractor network can be rapidly updated by external cues, consistent with the very short update latencies observed experimentally by Zugaro et al. (2003) in thalamic head direction cells.     

Coordinated learning of grid cell and place cell spatial and temporal properties: multiple scales,attention and oscillations

A neural model proposes how entorhinal grid cells and hippocampal place cells may develop as spatial categories in a hierarchy of self-organizing maps (SOMs). The model responds to realistic rat navigational trajectories by learning both grid cells with hexagonal grid firing fields of multiple spatial scales, and place cells with one or more firing fields, that match neurophysiological data about their development in juvenile rats. Both grid and place cells can develop by detecting, learning and remembering the most frequent and energetic co-occurrences of their inputs. The model''s parsimonious properties include: similar ring attractor mechanisms process linear and angular path integration inputs that drive map learning; the same SOM mechanisms can learn grid cell and place cell receptive fields; and the learning of the dorsoventral organization of multiple spatial scale modules through medial entorhinal cortex to hippocampus (HC) may use mechanisms homologous to those for temporal learning through lateral entorhinal cortex to HC (‘neural relativity’). The model clarifies how top-down HC-to-entorhinal attentional mechanisms may stabilize map learning, simulates how hippocampal inactivation may disrupt grid cells, and explains data about theta, beta and gamma oscillations. The article also compares the three main types of grid cell models in the light of recent data.     

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.     

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.     

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.     

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.     

Neural population representation hypothesis of visual flow and its illusory after effect in the brain: psychophysics,neurophysiology and computational approaches

The neural representation of motion aftereffects induced by various visual flows (translational, rotational, motion-in-depth, and translational transparent flows) was studied under the hypothesis that the imbalances in discharge activities would occur in favor in the direction opposite to the adapting stimulation in the monkey MST cells (cells in the medial superior temporal area) which can discriminate the mode (i.e., translational, rotational, or motion-in-depth) of the given flow. In single-unit recording experiments conducted on anaesthetized monkeys, we found that the rate of spontaneous discharge and the sensitivity to a test stimulus moving in the preferred direction decreased after receiving an adapting stimulation moving in the preferred direction, whereas they increased after receiving an adapting stimulation moving in the null direction. To consistently explain the bidirectional perception of a transparent visual flow and its unidirectional motion aftereffect by the same hypothesis, we need to assume the existence of two subtypes of MST D cells which show directionally selective responses to a translational flow: component cells and integration cells. Our physiological investigation revealed that the MST D cells could be divided into two types: one responded to a transparent flow by two peaks at the instances when the direction of one of the component flow matched the preferred direction of the cell, and the other responded by a single peak at the instance when the direction of the integrated motion matched the preferred direction. In psychophysical experiments on human subjects, we found evidence for the existence of component and integration representations in the human brain. To explain the different motion perceptions, i.e., two transparent flows during presentation of the flows and a single flow in the opposite direction to the integrated flows after stopping the flow stimuli, we suggest that the pattern-discrimination system can select the motion representation that is consistent with the perception of the pattern from two motion representations. We discuss the computational aspects related to the integration of component motion fields.     

Calibration of the head direction network: a role for symmetric angular head velocity cells

Continuous attractor networks require calibration. Computational models of the head direction (HD) system of the rat usually assume that the connections that maintain HD neuron activity are pre-wired and static. Ongoing activity in these models relies on precise continuous attractor dynamics. It is currently unknown how such connections could be so precisely wired, and how accurate calibration is maintained in the face of ongoing noise and perturbation. Our adaptive attractor model of the HD system that uses symmetric angular head velocity (AHV) cells as a training signal shows that the HD system can learn to support stable firing patterns from poorly-performing, unstable starting conditions. The proposed calibration mechanism suggests a requirement for symmetric AHV cells, the existence of which has previously been unexplained, and predicts that symmetric and asymmetric AHV cells should be distinctly different (in morphology, synaptic targets and/or methods of action on postsynaptic HD cells) due to their distinctly different functions.     

Recurrent networks with short term synaptic depression

Cortical circuitry shows an abundance of recurrent connections. A widely used model that relies on recurrence is the ring attractor network, which has been used to describe phenomena as diverse as working memory, visual processing and head direction cells. Commonly, the synapses in these models are static. Here, we examine the behaviour of ring attractor networks when the recurrent connections are subject to short term synaptic depression, as observed in many brain regions. We find that in the presence of a uniform background current, the network activity can be in either of three states: a stationary attractor state, a uniform state, or a rotating attractor state. The rotation speed can be adjusted over a large range by changing the background current, opening the possibility to use the network as a variable frequency oscillator or pattern generator. Finally, using simulations we extend the network to two-dimensional fields and find a rich range of possible behaviours.     

Timing over Tuning: Overcoming the Shortcomings of a Line Attractor during a Working Memory Task

How the brain stores information about a sensory stimulus in working memory is not completely known. Clues about the mechanisms responsible for working memory can be gleaned by recording from neurons during the performance of a delayed response task. I focus on the data recorded during such an experiment, a classic tactile discrimination task. I describe how the observed variability in the firing rate during a trial suggests that the type of attractor that is responsible for holding the stimulus information is not a fixed-point type attractor. I propose an alternate mechanism to a line attractor that allows the network to hold the value of an analog stimulus variable for the duration of the delay period, but rather than maintain a constant level of activity, the cells'' firing rate varies throughout the delay period. I describe how my proposed mechanism offers a substantial advantage over a line attractor: The tuning requirements of cell to cell connections are greatly eased from that of a line attractor. To accommodate a change in the length of the delay period, I show that the network can be altered by changing a single parameter - the timing of an executive signal that originates outside of the network. To demonstrate the mechanism, as well as the tuning benefits, I use a well known model of propagation in neuronal networks.     

Neural mechanisms of tactile motion integration in somatosensory cortex

How are local motion signals integrated to form a global motion percept? We investigate the neural mechanisms of tactile motion integration by presenting tactile gratings and plaids to the fingertips of monkeys, using the tactile analogue of a visual monitor and recording the responses evoked in somatosensory cortical neurons. The perceived directions of the gratings and plaids are measured in parallel psychophysical experiments. We identify a population of somatosensory neurons that exhibit integration properties comparable to those induced by analogous visual stimuli in area MT and find that these neural responses account for the perceived direction of the stimuli across all stimulus conditions tested. The preferred direction of the neurons and the perceived direction of the stimuli can be predicted from the weighted average of the directions of the individual stimulus features, highlighting that the somatosensory system implements a vector average mechanism to compute tactile motion direction that bears striking similarities to its visual counterpart.     

Persistent activity in limbic system neurons: neurophysiological and modeling perspectives

Neural activity persisting for one to hundreds of seconds has been postulated to be a substrate of memory. This review article illustrates examples of such activity in limbic system structures including the hippocampus, postsubiculum, and the anterodorsal thalamus. These neuronal responses include better known correlates with the spatial position as well as with head direction of the animal relative to its environment as well as other lesser known examples. Since head direction responses are greater when the animal is actively moving than when passively rotated, it has been proposed that there might be a general mechanism where the behavioral state of the animal can provide modulatory gating of such persistent signals. This would regulate the relative influence of these signals on downstream structures. Neural network attractor models of the head direction cell system are presented to demonstrate how these responses might originate, as well as the dynamics by which they are updated during movements.