Place and Grid Cells in a Loop: Implications for Memory Function and Spatial Coding

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.

The chicken and egg problem of grid cells and place cells

Place cells and grid cells are major building blocks of the hippocampal cognitive map. The prominent forward model postulates that grid-cell modules are generated by a continuous attractor network; that a velocity signal evoked during locomotion moves entorhinal activity bumps; and that place-cell activity constitutes summation of entorhinal grid-cell modules. Experimental data support the first postulate, but not the latter two. Several families of solutions that depart from these postulates have been put forward. We suggest a modified model (spatial modulation continuous attractor network; SCAN), whereby place cells are generated from spatially selective nongrid cells. Locomotion causes these cells to move the hippocampal activity bump, leading to movement of the entorhinal manifolds. Such inversion accords with the shift of hippocampal thought from navigation to more abstract functions.

Place cells dynamically refine grid cell activities to reduce error accumulation during path integration in a continuous attractor model

Navigation is one of the most fundamental skills of animals. During spatial navigation, grid cells in the medial entorhinal cortex process speed and direction of the animal to map the environment. Hippocampal place cells, in turn, encode place using sensory signals and reduce the accumulated error of grid cells for path integration. Although both cell types are part of the path integration system, the dynamic relationship between place and grid cells and the error reduction mechanism is yet to be understood. We implemented a realistic model of grid cells based on a continuous attractor model. The grid cell model was coupled to a place cell model to address their dynamic relationship during a simulated animal's exploration of a square arena. The grid cell model processed the animal's velocity and place field information from place cells. Place cells incorporated salient visual features and proximity information with input from grid cells to define their place fields. Grid cells had similar spatial phases but a diversity of spacings and orientations. To determine the role of place cells in error reduction for path integration, the animal's position estimates were decoded from grid cell activities with and without the place field input. We found that the accumulated error was reduced as place fields emerged during the exploration. Place fields closer to the animal's current location contributed more to the error reduction than remote place fields. Place cells' fields encoding space could function as spatial anchoring signals for precise path integration by grid cells.

Connectivity concepts in neuronal network modeling

Sustainable research on computational models of neuronal networks requires published models to be understandable, reproducible, and extendable. Missing details or ambiguities about mathematical concepts and assumptions, algorithmic implementations, or parameterizations hinder progress. Such flaws are unfortunately frequent and one reason is a lack of readily applicable standards and tools for model description. Our work aims to advance complete and concise descriptions of network connectivity but also to guide the implementation of connection routines in simulation software and neuromorphic hardware systems. We first review models made available by the computational neuroscience community in the repositories ModelDB and Open Source Brain, and investigate the corresponding connectivity structures and their descriptions in both manuscript and code. The review comprises the connectivity of networks with diverse levels of neuroanatomical detail and exposes how connectivity is abstracted in existing description languages and simulator interfaces. We find that a substantial proportion of the published descriptions of connectivity is ambiguous. Based on this review, we derive a set of connectivity concepts for deterministically and probabilistically connected networks and also address networks embedded in metric space. Beside these mathematical and textual guidelines, we propose a unified graphical notation for network diagrams to facilitate an intuitive understanding of network properties. Examples of representative network models demonstrate the practical use of the ideas. We hope that the proposed standardizations will contribute to unambiguous descriptions and reproducible implementations of neuronal network connectivity in computational neuroscience.

Experience-driven rate modulation is reinstated during hippocampal replay

Replay, the sequential reactivation within a neuronal ensemble, is a central hippocampal mechanism postulated to drive memory processing. While both rate and place representations are used by hippocampal place cells to encode behavioral episodes, replay has been largely defined by only the latter – based on the fidelity of sequential activity across neighboring place fields. Here, we show that dorsal CA1 place cells in rats can modulate their firing rate between replay events of two different contexts. This experience-dependent phenomenon mirrors the same pattern of rate modulation observed during behavior and can be used independently from place information within replay sequences to discriminate between contexts. Our results reveal the existence of two complementary neural representations available for memory processes.

How do our brains store memories? We now know that this is a complex and dynamic process, involving multiple regions of the brain. A brain region, called the hippocampus, plays an important role in memory formation. While we sleep, the hippocampus works to consolidate information, and eventually creates stable, long-term memories that are then stored in other parts of the brain.

But how does the hippocampus do this? Neuroscientists believe that it can replay the patterns of brain activity that represent particular memories. By repeatedly doing this while we sleep, the hippocampus can then direct the transfer of this information to the rest of the brain for storage.

The behaviour of nerve cells in the brain underpins these patterns of brain activity. When a nerve cell is active, it fires tiny electrical impulses that can be detected experimentally. The brain thus represents information in two ways: which nerve cells are active and when (sequential patterns); and how active the nerve cells are (how fast they fire electrical impulses or firing rate). For example, when an animal moves from one location to another, special place cells in the hippocampus become active in a distinct sequence. Depending on the context, they will also fire faster or slower.

We know that the hippocampus can replay sequential patterns of nerve cell activity during memory consolidation, but whether it can also replay the firing rates associated with a particular experience is still unknown. Tirole, Huelin Gorriz et al. set out to determine if the hippocampus could also preserve the information encoded by firing rate during replay.

In the experiments, rats explored two different environments that they had not seen before. The activity of the rats’ place cells was recorded before and after they explored, and also later while they were sleeping. Analysis of the recordings revealed that during replay, the rats’ hippocampi could indeed reproduce both the sequential patterns of activity and the firing rate of the place cells. It also confirmed that each environment was associated with unique firing rates – in other words, the firing rates were memory-specific.

These results contribute to our understanding of how the hippocampus represents and processes information about our experiences. More broadly, they also shed new light on how the brain lays down memories, by revealing a key part of the mechanism that it uses to consolidate that information.

Nootropic effects of LSD: Behavioral, molecular and computational evidence

The therapeutic use of classical psychedelic substances such as d-lysergic acid diethylamide (LSD) surged in recent years. Studies in rodents suggest that these effects are produced by increased neural plasticity, including stimulation of the mTOR pathway, a key regulator of metabolism, plasticity, and aging. Could psychedelic-induced neural plasticity be harnessed to enhance cognition? Here we show that LSD treatment enhanced performance in a novel object recognition task in rats, and in a visuo-spatial memory task in humans. A proteomic analysis of human brain organoids showed that LSD affected metabolic pathways associated with neural plasticity, including mTOR. To gain insight into the relation of neural plasticity, aging and LSD-induced cognitive gains, we emulated the experiments in rats and humans with a neural network model of a cortico-hippocampal circuit. Using the baseline strength of plasticity as a proxy for age and assuming an increase in plasticity strength related to LSD dose, the simulations provided a good fit for the experimental data. Altogether, the results suggest that LSD has nootropic effects.

The Role of Idiothetic Signals, Landmarks, and Conjunctive Representations in the Development of Place and Head-Direction Cells: A Self-Organizing Neural Network Model

Place and head-direction (HD) cells are fundamental to maintaining accurate representations of location and heading in the mammalian brain across sensory conditions, and are thought to underlie path integration-the ability to maintain an accurate representation of location and heading during motion in the dark. Substantial evidence suggests that both populations of spatial cells function as attractor networks, but their developmental mechanisms are poorly understood. We present simulations of a fully self-organizing attractor network model of this process using well-established neural mechanisms. We show that the differential development of the two cell types can be explained by their different idiothetic inputs, even given identical visual signals: HD cells develop when the population receives angular head velocity input, whereas place cells develop when the idiothetic input encodes planar velocity. Our model explains the functional importance of conjunctive "state-action" cells, implying that signal propagation delays and a competitive learning mechanism are crucial for successful development. Consequently, we explain how insufficiently rich environments result in pathology: place cell development requires proximal landmarks; conversely, HD cells require distal landmarks. Finally, our results suggest that both networks are instantiations of general mechanisms, and we describe their implications for the neurobiology of spatial processing.

Age-related episodic memory decline and the role of amyloid-β: a systematic review

Aging has been associated with the functional decline of episodic memory (EM). Unanswered questions are whether the decline of EM occurs even during healthy aging and whether this decline is related to amyloid-β (Aβ) deposition in the hippocampus.

Objective

The main purpose of this study was to investigate data on the relationship between the age-related EM decline and Aβ deposition.

Methods

We searched the Cochrane, MEDLINE, Scopus, and Web of Science databases and reference lists of retrieved articles that were published in the past 10 years. The initial literature search identified 517 studies. After screening the title, abstract, key words, and reference lists, 56 studies met the inclusion criteria.

Results

The overall results revealed that increases in Aβ are related to lower hippocampal volume and worse performance on EM tests. The results of this systematic review revealed that high levels of Aβ may be related to EM deficits and the progression to Alzheimer's disease.

Conclusions

We discussed the strengths and pitfalls of various tests and techniques used for investigating EM and Aβ deposition, methodological issues, and potential directions for future research.

The grid code for ordered experience

Entorhinal cortical grid cells fire in a periodic pattern that tiles space, which is suggestive of a spatial coordinate system. However, irregularities in the grid pattern as well as responses of grid cells in contexts other than spatial navigation have presented a challenge to existing models of entorhinal function. In this Perspective, we propose that hippocampal input provides a key informative drive to the grid network in both spatial and non-spatial circumstances, particularly around salient events. We build on previous models in which neural activity propagates through the entorhinal-hippocampal network in time. This temporal contiguity in network activity points to temporal order as a necessary characteristic of representations generated by the hippocampal formation. We advocate that interactions in the entorhinal-hippocampal loop build a topological representation that is rooted in the temporal order of experience. In this way, the structure of grid cell firing supports a learned topology rather than a rigid coordinate frame that is bound to measurements of the physical world.

Learning an Efficient Hippocampal Place Map from Entorhinal Inputs Using Non-Negative Sparse Coding

Cells in the entorhinal cortex (EC) contain rich spatial information and project strongly to the hippocampus where a cognitive map is supposedly created. These cells range from cells with structured spatial selectivity, such as grid cells in the medial EC (MEC) that are selective to an array of spatial locations that form a hexagonal grid, to weakly spatial cells, such as non-grid cells in the MEC and lateral EC (LEC) that contain spatial information but have no structured spatial selectivity. However, in a small environment, place cells in the hippocampus are generally selective to a single location of the environment, while granule cells in the dentate gyrus of the hippocampus have multiple discrete firing locations but lack spatial periodicity. Given the anatomic connection from the EC to the hippocampus, how the hippocampus retrieves information from upstream EC remains unclear. Here, we propose a unified learning model that can describe the spatial tuning properties of both hippocampal place cells and dentate gyrus granule cells based on non-negative sparse coding from EC inputs. Sparse coding plays an important role in many cortical areas and is proposed here to have a key role in the hippocampus. Our results show that the hexagonal patterns of MEC grid cells with various orientations, grid spacings and phases are necessary for the model to learn different place cells that efficiently tile the entire spatial environment. However, if there is a lack of diversity in any grid parameters or a lack of hippocampal cells in the network, this will lead to the emergence of hippocampal cells that have multiple firing locations. More surprisingly, the model can also learn hippocampal place cells even when weakly spatial cells, instead of grid cells, are used as the input to the hippocampus. This work suggests that sparse coding may be one of the underlying organizing principles for the navigational system of the brain.

A computational grid-to-place-cell transformation model indicates a synaptic driver of place cell impairment in early-stage Alzheimer's Disease

Alzheimer's Disease (AD) is characterized by progressive neurodegeneration and cognitive impairment. Synaptic dysfunction is an established early symptom, which correlates strongly with cognitive decline, and is hypothesised to mediate the diverse neuronal network abnormalities observed in AD. However, how synaptic dysfunction contributes to network pathology and cognitive impairment in AD remains elusive. Here, we present a grid-cell-to-place-cell transformation model of long-term CA1 place cell dynamics to interrogate the effect of synaptic loss on network function and environmental representation. Synapse loss modelled after experimental observations in the APP/PS1 mouse model was found to induce firing rate alterations and place cell abnormalities that have previously been observed in AD mouse models, including enlarged place fields and lower across-session stability of place fields. Our results support the hypothesis that synaptic dysfunction underlies cognitive deficits, and demonstrate how impaired environmental representation may arise in the early stages of AD. We further propose that dysfunction of excitatory and inhibitory inputs to CA1 pyramidal cells may cause distinct impairments in place cell function, namely reduced stability and place map resolution.

A theory of joint attractor dynamics in the hippocampus and the entorhinal cortex accounts for artificial remapping and grid cell field-to-field variability

The representation of position in the mammalian brain is distributed across multiple neural populations. Grid cell modules in the medial entorhinal cortex (MEC) express activity patterns that span a low-dimensional manifold which remains stable across different environments. In contrast, the activity patterns of hippocampal place cells span distinct low-dimensional manifolds in different environments. It is unknown how these multiple representations of position are coordinated. Here, we develop a theory of joint attractor dynamics in the hippocampus and the MEC. We show that the system exhibits a coordinated, joint representation of position across multiple environments, consistent with global remapping in place cells and grid cells. In addition, our model accounts for recent experimental observations that lack a mechanistic explanation: variability in the firing rate of single grid cells across firing fields, and artificial remapping of place cells under depolarization, but not under hyperpolarization, of layer II stellate cells of the MEC.

Cognitive swarming in complex environments with attractor dynamics and oscillatory computing

Neurobiological theories of spatial cognition developed with respect to recording data from relatively small and/or simplistic environments compared to animals' natural habitats. It has been unclear how to extend theoretical models to large or complex spaces. Complementarily, in autonomous systems technology, applications have been growing for distributed control methods that scale to large numbers of low-footprint mobile platforms. Animals and many-robot groups must solve common problems of navigating complex and uncertain environments. Here, we introduce the NeuroSwarms control framework to investigate whether adaptive, autonomous swarm control of minimal artificial agents can be achieved by direct analogy to neural circuits of rodent spatial cognition. NeuroSwarms analogizes agents to neurons and swarming groups to recurrent networks. We implemented neuron-like agent interactions in which mutually visible agents operate as if they were reciprocally connected place cells in an attractor network. We attributed a phase state to agents to enable patterns of oscillatory synchronization similar to hippocampal models of theta-rhythmic (5-12 Hz) sequence generation. We demonstrate that multi-agent swarming and reward-approach dynamics can be expressed as a mobile form of Hebbian learning and that NeuroSwarms supports a single-entity paradigm that directly informs theoretical models of animal cognition. We present emergent behaviors including phase-organized rings and trajectory sequences that interact with environmental cues and geometry in large, fragmented mazes. Thus, NeuroSwarms is a model artificial spatial system that integrates autonomous control and theoretical neuroscience to potentially uncover common principles to advance both domains.

Hippocampal theta coordinates memory processing during visual exploration

The hippocampus supports memory encoding and retrieval, which may occur at distinct phases of the theta cycle. These processes dynamically interact over rapid timescales, especially when sensory information conflicts with memory. The ability to link hippocampal dynamics to memory-guided behaviors has been limited by experiments that lack the temporal resolution to segregate encoding and retrieval. Here, we simultaneously tracked eye movements and hippocampal field potentials while neurosurgical patients performed a spatial memory task. Phase-locking at the peak of theta preceded fixations to retrieved locations, indicating that the hippocampus coordinates memory-guided eye movements. In contrast, phase-locking at the trough of theta followed fixations to novel object-locations and predicted intact memory of the original location. Theta-gamma phase amplitude coupling increased during fixations to conflicting visual content, but predicted memory updating. Hippocampal theta thus supports learning through two interleaved processes: strengthening encoding of novel information and guiding exploration based on prior experience.

Regulation of gamma-frequency oscillation by feedforward inhibition: A computational modeling study

Throughout the brain, reciprocally connected excitatory and inhibitory neurons interact to produce gamma-frequency oscillations. The emergent gamma rhythm synchronizes local neural activity and helps to select which cells should fire in each cycle. We previously found that such excitation-inhibition microcircuits, however, have a potentially significant caveat: the frequency of the gamma oscillation and the level of selection (i.e., the percentage of cells that are allowed to fire) vary with the magnitude of the input signal. In networks with varying levels of brain activity, such a feature may produce undesirable instability on the time and spatial structure of the neural signal with a potential for adversely impacting important neural processing mechanisms. Here we propose that feedforward inhibition solves the latter instability problem of the excitation-inhibition microcircuit. Using computer simulations, we show that the feedforward inhibitory signal reduces the dependence of both the frequency of population oscillation and the level of selection on the magnitude of the input excitation. Such a mechanism can produce stable gamma oscillations with its frequency determined only by the properties of the feedforward network, as observed in the hippocampus. As feedforward and feedback inhibition motifs commonly appear together in the brain, we hypothesize that their interaction underlies a robust implementation of general computational principles of neural processing involved in several cognitive tasks, including the formation of cell assemblies and the routing of information between brain areas.

Development and topographical organization of projections from the hippocampus and parahippocampus to the retrosplenial cortex

The rat hippocampal formation (HF), parahippocampal region (PHR), and retrosplenial cortex (RSC) play critical roles in spatial processing. These regions are interconnected, and functionally dependent. The neuronal networks mediating this reciprocal dependency are largely unknown. Establishing the developmental timing of network formation will help to understand the emergence of this dependency. We questioned whether the long-range outputs from HF-PHR to RSC in Long Evans rats develop during the same time periods as previously reported for the intrinsic HF-PHR connectivity and the projections from RSC to HF-PHR. The results of a series of retrograde and anterograde tracing experiments in rats of different postnatal ages show that the postnatal projections from HF-PHR to RSC display low densities around birth, but develop during the first postnatal week, reaching adult-like densities around the time of eye-opening. Developing projections display a topographical organization similar to adult projections. We conclude that the long-range projections from HF-PHR to RSC develop in parallel with the intrinsic circuitry of HF-PHR and the projections of RSC to HF-PHR.

Grid-like Neural Representations Support Olfactory Navigation of a Two-Dimensional Odor Space

Searching for food, friends, and mates often begins with an airborne scent. Importantly, odor concentration rises with physical proximity to an odorous source, suggesting a framework for orienting within olfactory landscapes to optimize behavior. Here, we created a two-dimensional odor space composed purely of odor stimuli to model how a navigator encounters smells in a natural environment. We show that human subjects can learn to navigate in olfactory space and form predictions of to-be-encountered smells. During navigation, fMRI responses in entorhinal cortex and ventromedial prefrontal cortex take the form of grid-like representations with hexagonal periodicity and entorhinal grid strength scaled with behavioral performance across subjects. The identification of olfactory grid-like codes with 6-fold symmetry highlights a unique neural mechanism by which odor information can be assembled into spatially navigable cognitive maps, optimizing orientation, and path finding toward an odor source.

The entorhinal cognitive map is attracted to goals

Grid cells with their rigid hexagonal firing fields are thought to provide an invariant metric to the hippocampal cognitive map, yet environmental geometrical features have recently been shown to distort the grid structure. Given that the hippocampal role goes beyond space, we tested the influence of nonspatial information on the grid organization. We trained rats to daily learn three new reward locations on a cheeseboard maze while recording from the medial entorhinal cortex and the hippocampal CA1 region. Many grid fields moved toward goal location, leading to long-lasting deformations of the entorhinal map. Therefore, distortions in the grid structure contribute to goal representation during both learning and recall, which demonstrates that grid cells participate in mnemonic coding and do not merely provide a simple metric of space.

Spatial synchronization codes from coupled rate-phase neurons

During spatial navigation, the frequency and timing of spikes from spatial neurons including place cells in hippocampus and grid cells in medial entorhinal cortex are temporally organized by continuous theta oscillations (6-11 Hz). The theta rhythm is regulated by subcortical structures including the medial septum, but it is unclear how spatial information from place cells may reciprocally organize subcortical theta-rhythmic activity. Here we recorded single-unit spiking from a constellation of subcortical and hippocampal sites to study spatial modulation of rhythmic spike timing in rats freely exploring an open environment. Our analysis revealed a novel class of neurons that we termed 'phaser cells,' characterized by a symmetric coupling between firing rate and spike theta-phase. Phaser cells encoded space by assigning distinct phases to allocentric isocontour levels of each cell's spatial firing pattern. In our dataset, phaser cells were predominantly located in the lateral septum, but also the hippocampus, anteroventral thalamus, lateral hypothalamus, and nucleus accumbens. Unlike the unidirectional late-to-early phase precession of place cells, bidirectional phase modulation acted to return phaser cells to the same theta-phase along a given spatial isocontour, including cells that characteristically shifted to later phases at higher firing rates. Our dynamical models of intrinsic theta-bursting neurons demonstrated that experience-independent temporal coding mechanisms can qualitatively explain (1) the spatial rate-phase relationships of phaser cells and (2) the observed temporal segregation of phaser cells according to phase-shift direction. In open-field phaser cell simulations, competitive learning embedded phase-code entrainment maps into the weights of downstream targets, including path integration networks. Bayesian phase decoding revealed error correction capable of resetting path integration at subsecond timescales. Our findings suggest that phaser cells may instantiate a subcortical theta-rhythmic loop of spatial feedback. We outline a framework in which location-dependent synchrony reconciles internal idiothetic processes with the allothetic reference points of sensory experience.

Parsing Hippocampal Theta Oscillations by Nested Spectral Components during Spatial Exploration and Memory-Guided Behavior

Theta oscillations reflect rhythmic inputs that continuously converge to the hippocampus during exploratory and memory-guided behavior. The theta-nested operations that organize hippocampal spiking could either occur regularly from one cycle to the next or be tuned on a cycle-by-cycle basis. To resolve this, we identified spectral components nested in individual theta cycles recorded from the mouse CA1 hippocampus. Our single-cycle profiling revealed theta spectral components associated with different firing modulations and distinguishable ensembles of principal cells. Moreover, novel co-firing patterns of principal cells in theta cycles nesting mid-gamma oscillations were the most strongly reactivated in subsequent offline sharp-wave/ripple events. Finally, theta-nested spectral components were differentially altered by behavioral stages of a memory task; the 80-Hz mid-gamma component was strengthened during learning, whereas the 22-Hz beta, 35-Hz slow gamma, and 54-Hz mid-gamma components increased during retrieval. We conclude that cycle-to-cycle variability of theta-nested spectral components allows parsing of theta oscillations into transient operating modes with complementary mnemonic roles.

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Spectral profiling of single theta waves enables studying inter-cycle variability

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Theta spectral components feature different spiking patterns and ensembles

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Co-firing in theta cycles nesting mid-gamma undergo enhanced offline reactivation

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Theta components relate differently to learning and memory retrieval demands

Computational models of memory consolidation and long-term synaptic plasticity during sleep

The brain stores memories by persistently changing the connectivity between neurons. Sleep is known to be critical for these changes to endure. Research on the neurobiology of sleep and the mechanisms of long-term synaptic plasticity has provided data in support of various theories of how brain activity during sleep affects long-term synaptic plasticity. The experimental findings - and therefore the theories - are apparently quite contradictory, with some evidence pointing to a role of sleep in the forgetting of irrelevant memories, whereas other results indicate that sleep supports the reinforcement of the most valuable recollections. A unified theoretical framework is in need. Computational modeling and simulation provide grounds for the quantitative testing and comparison of theoretical predictions and observed data, and might serve as a strategy to organize the rather complicated and diverse pool of data and methodologies used in sleep research. This review article outlines the emerging progress in the computational modeling and simulation of the main theories on the role of sleep in memory consolidation.

Theta sequences of grid cell populations can provide a movement-direction signal

It has been proposed that path integration in mammals is performed by the convergence of internally generated speed and directional inputs onto grid cells. Although this hypothesis has been supported by the discovery that head direction, speed, and grid cells are intermixed within entorhinal cortex and by the recent finding that head-direction inputs are necessary for grid firing, many details on how grid cells are generated have remained elusive. For example, analysis of recording data suggests that substituting head direction for movement direction accrues errors that preclude the formation of grid patterns. To address this discrepancy, we propose that the organization of grid networks makes it plausible that movement-direction signals are an output from grid cells and that temporally precise grid cell sequences provide a robust directional signal to other spatial and directional cell types.