The self-organization of grid cells in 3D

Modeling hippocampal spatial cells in rodents navigating in 3D environments

Studies on the neural correlates of navigation in 3D environments are plagued by several issues that need to be solved. For example, experimental studies show markedly different place cell responses in rats and bats, both navigating in 3D environments. In this study, we focus on modelling the spatial cells in rodents in a 3D environment. We propose a deep autoencoder network to model the place and grid cells in a simulated agent navigating in a 3D environment. The input layer to the autoencoder network model is the HD layer, which encodes the agent's HD in terms of azimuth (θ) and pitch angles (ϕ). The output of this layer is given as input to the Path Integration (PI) layer, which computes displacement in all the preferred directions. The bottleneck layer of the autoencoder model encodes the spatial cell-like responses. Both grid cell and place cell-like responses are observed. The proposed model is verified using two experimental studies with two 3D environments. This model paves the way for a holistic approach using deep neural networks to model spatial cells in 3D navigation.

Spatial periodicity in grid cell firing is explained by a neural sequence code of 2-D trajectories

Spatial periodicity in grid cell firing has been interpreted as a neural metric for space providing animals with a coordinate system in navigating physical and mental spaces. However, the specific computational problem being solved by grid cells has remained elusive. Here, we provide mathematical proof that spatial periodicity in grid cell firing is the only possible solution to a neural sequence code of 2-D trajectories and that the hexagonal firing pattern of grid cells is the most parsimonious solution to such a sequence code. We thereby provide a teleological cause for the existence of grid cells and reveal the underlying nature of the global geometric organization in grid maps as a direct consequence of a simple local sequence code. A sequence code by grid cells provides intuitive explanations for many previously puzzling experimental observations and may transform our thinking about grid cells.

Advantages of Persistent Cohomology in Estimating Animal Location From Grid Cell Population Activity

Many cognitive functions are represented as cell assemblies. In the case of spatial navigation, the population activity of place cells in the hippocampus and grid cells in the entorhinal cortex represents self-location in the environment. The brain cannot directly observe self-location information in the environment. Instead, it relies on sensory information and memory to estimate self-location. Therefore, estimating low-dimensional dynamics, such as the movement trajectory of an animal exploring its environment, from only the high-dimensional neural activity is important in deciphering the information represented in the brain. Most previous studies have estimated the low-dimensional dynamics (i.e., latent variables) behind neural activity by unsupervised learning with Bayesian population decoding using artificial neural networks or gaussian processes. Recently, persistent cohomology has been used to estimate latent variables from the phase information (i.e., circular coordinates) of manifolds created by neural activity. However, the advantages of persistent cohomology over Bayesian population decoding are not well understood. We compared persistent cohomology and Bayesian population decoding in estimating the animal location from simulated and actual grid cell population activity. We found that persistent cohomology can estimate the animal location with fewer neurons than Bayesian population decoding and robustly estimate the animal location from actual noisy data.

Grid codes underlie multiple cognitive maps in the human brain

Grid cells fire at multiple positions that organize the vertices of equilateral triangles tiling a 2D space and are well studied in rodents. The last decade witnessed rapid progress in two other research lines on grid codes-empirical studies on distributed human grid-like representations in physical and multiple non-physical spaces, and cognitive computational models addressing the function of grid cells based on principles of efficient and predictive coding. Here, we review the progress in these fields and integrate these lines into a systematic organization. We also discuss the coordinate mechanisms of grid codes in the human entorhinal cortex and medial prefrontal cortex and their role in neurological and psychiatric diseases.

Grid cells, border cells, and discrete complex analysis

We propose a mechanism enabling the appearance of border cells-neurons firing at the boundaries of the navigated enclosures. The approach is based on the recent discovery of discrete complex analysis on a triangular lattice, which allows constructing discrete epitomes of complex-analytic functions and making use of their inherent ability to attain maximal values at the boundaries of generic lattice domains. As it turns out, certain elements of the discrete-complex framework readily appear in the oscillatory models of grid cells. We demonstrate that these models can extend further, producing cells that increase their activity toward the frontiers of the navigated environments. We also construct a network model of neurons with border-bound firing that conforms with the oscillatory models.

Grid Cells, Border Cells and Discrete Complex Analysis

We propose a mechanism enabling the appearance of border cells-neurons firing at the boundaries of the navigated enclosures. The approach is based on the recent discovery of discrete complex analysis on a triangular lattice, which allows constructing discrete epitomes of complex-analytic functions and making use of their inherent ability to attain maximal values at the boundaries of generic lattice domains. As it turns out, certain elements of the discrete-complex framework readily appear in the oscillatory models of grid cells. We demonstrate that these models can extend further, producing cells that increase their activity towards the frontiers of the navigated environments. We also construct a network model of neurons with border-bound firing that conforms with the oscillatory models.

New Approaches to 3D Vision

New approaches to 3D vision are enabling new advances in artificial intelligence and autonomous vehicles, a better understanding of how animals navigate the 3D world, and new insights into human perception in virtual and augmented reality. Whilst traditional approaches to 3D vision in computer vision (SLAM: simultaneous localization and mapping), animal navigation (cognitive maps), and human vision (optimal cue integration) start from the assumption that the aim of 3D vision is to provide an accurate 3D model of the world, the new approaches to 3D vision explored in this issue challenge this assumption. Instead, they investigate the possibility that computer vision, animal navigation, and human vision can rely on partial or distorted models or no model at all. This issue also highlights the implications for artificial intelligence, autonomous vehicles, human perception in virtual and augmented reality, and the treatment of visual disorders, all of which are explored by individual articles. This article is part of a discussion meeting issue 'New approaches to 3D vision'.

Transforming Social Perspectives with Cognitive Maps

Growing evidence suggests that cognitive maps represent relations between social knowledge similar to how spatial locations are represented in an environment. Notably, the extant human medial temporal lobe literature assumes associations between social stimuli follow a linear associative mapping from an egocentric viewpoint to a cognitive map. Yet, this form of associative social memory doesn't account for a core phenomenon of social interactions in which social knowledge learned via comparisons to the self, other individuals, or social networks are assimilated within a single frame of reference. We argue that hippocampal-entorhinal coordinate transformations, known to integrate egocentric and allocentric spatial cues, inform social perspective switching between the self and others. We present evidence that the hippocampal formation helps inform social interactions by relating self versus other social attribute comparisons to society in general, which can afford rapid and flexible assimilation of knowledge about the relationship between the self and social networks of varying proximities. We conclude by discussing the ramifications of cognitive maps in aiding this social perspective transformation process in states of health and disease.

A Plane-Dependent Model of 3D Grid Cells for Representing Both 2D and 3D Spaces Under Various Navigation Modes

Grid cells are crucial in path integration and representation of the external world. The spikes of grid cells spatially form clusters called grid fields, which encode important information about allocentric positions. To decode the information, studying the spatial structures of grid fields is a key task for both experimenters and theorists. Experiments reveal that grid fields form hexagonal lattice during planar navigation, and are anisotropic beyond planar navigation. During volumetric navigation, they lose global order but possess local order. How grid cells form different field structures behind these different navigation modes remains an open theoretical question. However, to date, few models connect to the latest discoveries and explain the formation of various grid field structures. To fill in this gap, we propose an interpretive plane-dependent model of three-dimensional (3D) grid cells for representing both two-dimensional (2D) and 3D space. The model first evaluates motion with respect to planes, such as the planes animals stand on and the tangent planes of the motion manifold. Projection of the motion onto the planes leads to anisotropy, and error in the perception of planes degrades grid field regularity. A training-free recurrent neural network (RNN) then maps the processed motion information to grid fields. We verify that our model can generate regular and anisotropic grid fields, as well as grid fields with merely local order; our model is also compatible with mode switching. Furthermore, simulations predict that the degradation of grid field regularity is inversely proportional to the interval between two consecutive perceptions of planes. In conclusion, our model is one of the few pioneers that address grid field structures in a general case. Compared to the other pioneer models, our theory argues that the anisotropy and loss of global order result from the uncertain perception of planes rather than insufficient training.

How environmental movement constraints shape the neural code for space

Study of the neural code for space in rodents has many insights to offer for how mammals, including humans, construct a mental representation of space. This code is centered on the hippocampal place cells, which are active in particular places in the environment. Place cells are informed by numerous other spatial cell types including grid cells, which provide a signal for distance and direction and are thought to help anchor the place cell signal. These neurons combine self-motion and environmental information to create and update their map-like representation. Study of their activity patterns in complex environments of varying structure has revealed that this "cognitive map" of space is not a fixed and rigid entity that permeates space, but rather is variably affected by the movement constraints of the environment. These findings are pointing toward a more flexible spatial code in which the map is adapted to the movement possibilities of the space. An as-yet-unanswered question is whether these different forms of representation have functional consequences, as suggested by an enactivist view of spatial cognition.

Locally ordered representation of 3D space in the entorhinal cortex

As animals navigate on a two-dimensional surface, neurons in the medial entorhinal cortex (MEC) known as grid cells are activated when the animal passes through multiple locations (firing fields) arranged in a hexagonal lattice that tiles the locomotion surface¹. However, although our world is three-dimensional, it is unclear how the MEC represents 3D space². Here we recorded from MEC cells in freely flying bats and identified several classes of spatial neurons, including 3D border cells, 3D head-direction cells, and neurons with multiple 3D firing fields. Many of these multifield neurons were 3D grid cells, whose neighbouring fields were separated by a characteristic distance-forming a local order-but lacked any global lattice arrangement of the fields. Thus, whereas 2D grid cells form a global lattice-characterized by both local and global order-3D grid cells exhibited only local order, creating a locally ordered metric for space. We modelled grid cells as emerging from pairwise interactions between fields, which yielded a hexagonal lattice in 2D and local order in 3D, thereby describing both 2D and 3D grid cells using one unifying model. Together, these data and model illuminate the fundamental differences and similarities between neural codes for 3D and 2D space in the mammalian brain.

Place cells and geometry lead to a flexible grid pattern

Place cells and grid cells are important neurons involved in spatial navigation in the mammalian brain. Grid cells are believed to play an important role in forming a cognitive map of the environment. Experimental observations in recent years showed that the grid pattern is not invariant but is influenced by the shape of the spatial environment. However, the cause of this deformation remains elusive. Here, we focused on the functional interactions between place cells and grid cells, utilizing the information of location relationships between the firing fields of place cells to optimize the previous grid cell feedforward generation model and expand its application to more complex environmental scenarios. Not only was the regular equilateral triangle periodic firing field structure of the grid cells reproduced, but the expected results were consistent with the experiment for the environment with various complex boundary shapes and environmental deformation. Even in the field of three-dimensional spatial grid patterns, forward-looking predictions have been made. This provides a possible model explanation for how the coupling of grid cells and place cells adapt to the diversity of the external environment to deepen our understanding of the neural basis for constructing cognitive maps.

Modeling the grid cell activity on non-horizontal surfaces based on oscillatory interference modulated by gravity

Internal representation of the space is a fundamental and crucial function of the animal's brain. Grid cells in the medial entorhinal cortex are thought to provide an environment-invariant metric system for the navigation of the animal. Most experimental and theoretical studies have focused on the horizontal planar codes of grid cell, while how this metric coordinate system is configured in the actual three-dimensional space remains unclear. Evidence has implied the spatial cognition may not be fully volumetric. We proposed an oscillatory interference model with a novel gravity and body plane modulation to simulate grid cell activity in complex space for rodents. The animal can perceive the rotation of its body plane along the local surface by sensing the gravity, causing the modulation to the dendritic oscillations. The results not only reproduce the firing patterns of the grid cell recorded from known experiments, but also predict the grid codes in novel environments. It further demonstrates that the gravity signal is indispensable for the animal's navigation, and supports the hypothesis that the periodic firing of the grid cell is intrinsically not a volumetric code in three-dimensional space. This will provide new insights to understand the spatial representation of the actual world in the brain.

Overview of computational models of hippocampus and related structures: Introduction to the special issue

Extensive computational modeling has focused on the hippocampal formation and associated cortical structures. This overview describes some of the factors that have motivated the strong focus on these structures, including major experimental findings and their impact on computational models. This overview provides a framework for describing the topics addressed by individual articles in this special issue of the journal Hippocampus.

Partial coherence and frustration in self-organizing spherical grids

Nearby grid cells have been observed to express a remarkable degree of long-range order, which is often idealized as extending potentially to infinity. Yet their strict periodic firing and ensemble coherence are theoretically possible only in flat environments, much unlike the burrows which rodents usually live in. Are the symmetrical, coherent grid maps inferred in the lab relevant to chart their way in their natural habitat? We consider spheres as simple models of curved environments and waiting for the appropriate experiments to be performed, we use our adaptation model to predict what grid maps would emerge in a network with the same type of recurrent connections, which on the plane produce coherence among the units. We find that on the sphere such connections distort the maps that single grid units would express on their own, and aggregate them into clusters. When remapping to a different spherical environment, units in each cluster maintain only partial coherence, similar to what is observed in disordered materials, such as spin glasses.

Modeling grid fields instead of modeling grid cells : An effective model at the macroscopic level and its relationship with the underlying microscopic neural system

A neuron's firing correlates are defined as the features of the external world to which its activity is correlated. In many parts of the brain, neurons have quite simple such firing correlates. A striking example are grid cells in the rodent medial entorhinal cortex: their activity correlates with the animal's position in space, defining 'grid fields' arranged with a remarkable periodicity. Here, we show that the organization and evolution of grid fields relate very simply to physical space. To do so, we use an effective model and consider grid fields as point objects (particles) moving around in space under the influence of forces. We reproduce several observations on the geometry of grid patterns. This particle-like behavior is particularly salient in a recent experiment in which two separate grid patterns merge. We discuss pattern formation in the light of known results from physics of two-dimensional colloidal systems. Notably, we study the limitations of the widely used 'gridness score' and show how physics of 2d systems could be a source of inspiration, both for data analysis and computational modeling. Finally, we draw the relationship between our 'macroscopic' model for grid fields and existing 'microscopic' models of grid cell activity and discuss how a description at the level of grid fields allows to put constraints on the underlying grid cell network.

Can we study 3D grid codes non-invasively in the human brain? Methodological considerations and fMRI findings

Recent human functional magnetic resonance imaging (fMRI) and animal electrophysiology studies suggest that grid cells in entorhinal cortex are an efficient neural mechanism for encoding knowledge about the world, not only for spatial location but also for more abstract cognitive information. The world, be it physical or abstract, is often high-dimensional, but grid cells have been mainly studied on a simple two-dimensional (2D) plane. Recent theoretical studies have proposed how grid cells encode three-dimensional (3D) physical space, but it is unknown whether grid codes can be examined non-invasively in humans. Here, we investigated whether it was feasible to test different 3D grid models using fMRI based on the direction-modulated property of grid signals. In doing so, we developed interactive software to help researchers visualize 3D grid fields and predict grid activity in 3D as a function of movement directions. We found that a direction-modulated grid analysis was sensitive to one type of 3D grid model - a face-centred cubic (FCC) lattice model. As a proof of concept, we searched for 3D grid-like signals in human entorhinal cortex using a novel 3D virtual reality paradigm and a new fMRI analysis method. We found that signals in the left entorhinal cortex were explained by the FCC model. This is preliminary evidence for 3D grid codes in the human brain, notwithstanding the inherent methodological limitations of fMRI. We believe that our findings and software serve as a useful initial stepping-stone for studying grid cells in realistic 3D worlds and also, potentially, for interrogating abstract high-dimensional cognitive processes.

A hierarchical anti-Hebbian network model for the formation of spatial cells in three-dimensional space

Three-dimensional (3D) spatial cells in the mammalian hippocampal formation are believed to support the existence of 3D cognitive maps. Modeling studies are crucial to comprehend the neural principles governing the formation of these maps, yet to date very few have addressed this topic in 3D space. Here we present a hierarchical network model for the formation of 3D spatial cells using anti-Hebbian network. Built on empirical data, the model accounts for the natural emergence of 3D place, border, and grid cells, as well as a new type of previously undescribed spatial cell type which we call plane cells. It further explains the plausible reason behind the place and grid-cell anisotropic coding that has been observed in rodents and the potential discrepancy with the predicted periodic coding during 3D volumetric navigation. Lastly, it provides evidence for the importance of unsupervised learning rules in guiding the formation of higher-dimensional cognitive maps.

Neurons in the hippocampal formation encode diverse spatial properties. Here, the authors present a hierarchical network model for 3D spatial navigation that accounts for the observed neuronal representations and predict as yet unreported cell types with planar selectivity.

3-D Maps and Compasses in the Brain

The world has a complex, three-dimensional (3-D) spatial structure, but until recently the neural representation of space was studied primarily in planar horizontal environments. Here we review the emerging literature on allocentric spatial representations in 3-D and discuss the relations between 3-D spatial perception and the underlying neural codes. We suggest that the statistics of movements through space determine the topology and the dimensionality of the neural representation, across species and different behavioral modes. We argue that hippocampal place-cell maps are metric in all three dimensions, and might be composed of 2-D and 3-D fragments that are stitched together into a global 3-D metric representation via the 3-D head-direction cells. Finally, we propose that the hippocampal formation might implement a neural analogue of a Kalman filter, a standard engineering algorithm used for 3-D navigation.

Cognitive Mapping Based on Conjunctive Representations of Space and Movement

It is a challenge to build robust simultaneous localization and mapping (SLAM) system in dynamical large-scale environments. Inspired by recent findings in the entorhinal–hippocampal neuronal circuits, we propose a cognitive mapping model that includes continuous attractor networks of head-direction cells and conjunctive grid cells to integrate velocity information by conjunctive encodings of space and movement. Visual inputs from the local view cells in the model provide feedback cues to correct drifting errors of the attractors caused by the noisy velocity inputs. We demonstrate the mapping performance of the proposed cognitive mapping model on an open-source dataset of 66 km car journey in a 3 km × 1.6 km urban area. Experimental results show that the proposed model is robust in building a coherent semi-metric topological map of the entire urban area using a monocular camera, even though the image inputs contain various changes caused by different light conditions and terrains. The results in this study could inspire both neuroscience and robotic research to better understand the neural computational mechanisms of spatial cognition and to build robust robotic navigation systems in large-scale environments.

Computations in the deep vs superficial layers of the cerebral cortex

A fundamental question is how the cerebral neocortex operates functionally, computationally. The cerebral neocortex with its superficial and deep layers and highly developed recurrent collateral systems that provide a basis for memory-related processing might perform somewhat different computations in the superficial and deep layers. Here we take into account the quantitative connectivity within and between laminae. Using integrate-and-fire neuronal network simulations that incorporate this connectivity, we first show that attractor networks implemented in the deep layers that are activated by the superficial layers could be partly independent in that the deep layers might have a different time course, which might because of adaptation be more transient and useful for outputs from the neocortex. In contrast the superficial layers could implement more prolonged firing, useful for slow learning and for short-term memory. Second, we show that a different type of computation could in principle be performed in the superficial and deep layers, by showing that the superficial layers could operate as a discrete attractor network useful for categorisation and feeding information forward up a cortical hierarchy, whereas the deep layers could operate as a continuous attractor network useful for providing a spatially and temporally smooth output to output systems in the brain. A key advance is that we draw attention to the functions of the recurrent collateral connections between cortical pyramidal cells, often omitted in canonical models of the neocortex, and address principles of operation of the neocortex by which the superficial and deep layers might be specialized for different types of attractor-related memory functions implemented by the recurrent collaterals.

A single-cell spiking model for the origin of grid-cell patterns

Spatial cognition in mammals is thought to rely on the activity of grid cells in the entorhinal cortex, yet the fundamental principles underlying the origin of grid-cell firing are still debated. Grid-like patterns could emerge via Hebbian learning and neuronal adaptation, but current computational models remained too abstract to allow direct confrontation with experimental data. Here, we propose a single-cell spiking model that generates grid firing fields via spike-rate adaptation and spike-timing dependent plasticity. Through rigorous mathematical analysis applicable in the linear limit, we quantitatively predict the requirements for grid-pattern formation, and we establish a direct link to classical pattern-forming systems of the Turing type. Our study lays the groundwork for biophysically-realistic models of grid-cell activity.

Periodic population codes: From a single circular variable to higher dimensions, multiple nested scales, and conceptual spaces

Across the nervous system, neurons often encode circular stimuli using tuning curves that are not sine or cosine functions, but that belong to the richer class of von Mises functions, which are periodic variants of Gaussians. For a population of neurons encoding a single circular variable with such canonical tuning curves, computing a simple population vector is the optimal read-out of the most likely stimulus. We argue that the advantages of population vector read-outs are so compelling that even the neural representation of the outside world's flat Euclidean geometry is curled up into a torus (a circle times a circle), creating the hexagonal activity patterns of mammalian grid cells. Here, the circular scale is not set a priori, so the nervous system can use multiple scales and gain fields to overcome the ambiguity inherent in periodic representations of linear variables. We review the experimental evidence for this framework and discuss its testable predictions and generalizations to more abstract grid-like neural representations.

Selforganization of modular activity of grid cells

A unique topographical representation of space is found in the concerted activity of grid cells in the rodent medial entorhinal cortex. Many among the principal cells in this region exhibit a hexagonal firing pattern, in which each cell expresses its own set of place fields (spatial phases) at the vertices of a triangular grid, the spacing and orientation of which are typically shared with neighboring cells. Grid spacing, in particular, has been found to increase along the dorso‐ventral axis of the entorhinal cortex but in discrete steps, that is, with a modular structure. In this study, we show that such a modular activity may result from the self‐organization of interacting units, which individually would not show discrete but rather continuously varying grid spacing. Within our “adaptation” network model, the effect of a continuously varying time constant, which determines grid spacing in the isolated cell model, is modulated by recurrent collateral connections, which tend to produce a few subnetworks, akin to magnetic domains, each with its own grid spacing. In agreement with experimental evidence, the modular structure is tightly defined by grid spacing, but also involves grid orientation and distortion, due to interactions across modules. Thus, our study sheds light onto a possible mechanism, other than simply assuming separate networks a priori, underlying the formation of modular grid representations.

Ten Years of Grid Cells

The medial entorhinal cortex (MEC) creates a neural representation of space through a set of functionally dedicated cell types: grid cells, border cells, head direction cells, and speed cells. Grid cells, the most abundant functional cell type in the MEC, have hexagonally arranged firing fields that tile the surface of the environment. These cells were discovered only in 2005, but after 10 years of investigation, we are beginning to understand how they are organized in the MEC network, how their periodic firing fields might be generated, how they are shaped by properties of the environment, and how they interact with the rest of the MEC network. The aim of this review is to summarize what we know about grid cells and point out where our knowledge is still incomplete.

Neural encoding of large-scale three-dimensional space-properties and constraints

How the brain represents represent large-scale, navigable space has been the topic of intensive investigation for several decades, resulting in the discovery that neurons in a complex network of cortical and subcortical brain regions co-operatively encode distance, direction, place, movement etc. using a variety of different sensory inputs. However, such studies have mainly been conducted in simple laboratory settings in which animals explore small, two-dimensional (i.e., flat) arenas. The real world, by contrast, is complex and three dimensional with hills, valleys, tunnels, branches, and—for species that can swim or fly—large volumetric spaces. Adding an additional dimension to space adds coding challenges, a primary reason for which is that several basic geometric properties are different in three dimensions. This article will explore the consequences of these challenges for the establishment of a functional three-dimensional metric map of space, one of which is that the brains of some species might have evolved to reduce the dimensionality of the representational space and thus sidestep some of these problems.

Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns

Lattices abound in nature-from the crystal structure of minerals to the honey-comb organization of ommatidia in the compound eye of insects. These arrangements provide solutions for optimal packings, efficient resource distribution, and cryptographic protocols. Do lattices also play a role in how the brain represents information? We focus on higher-dimensional stimulus domains, with particular emphasis on neural representations of physical space, and derive which neuronal lattice codes maximize spatial resolution. For mammals navigating on a surface, we show that the hexagonal activity patterns of grid cells are optimal. For species that move freely in three dimensions, a face-centered cubic lattice is best. This prediction could be tested experimentally in flying bats, arboreal monkeys, or marine mammals. More generally, our theory suggests that the brain encodes higher-dimensional sensory or cognitive variables with populations of grid-cell-like neurons whose activity patterns exhibit lattice structures at multiple, nested scales.