Connecting multiple spatial scales to decode the population activity of grid cells

Reinforcement learning as a robotics-inspired framework for insect navigation: from spatial representations to neural implementation

Bees are among the master navigators of the insect world. Despite impressive advances in robot navigation research, the performance of these insects is still unrivaled by any artificial system in terms of training efficiency and generalization capabilities, particularly considering the limited computational capacity. On the other hand, computational principles underlying these extraordinary feats are still only partially understood. The theoretical framework of reinforcement learning (RL) provides an ideal focal point to bring the two fields together for mutual benefit. In particular, we analyze and compare representations of space in robot and insect navigation models through the lens of RL, as the efficiency of insect navigation is likely rooted in an efficient and robust internal representation, linking retinotopic (egocentric) visual input with the geometry of the environment. While RL has long been at the core of robot navigation research, current computational theories of insect navigation are not commonly formulated within this framework, but largely as an associative learning process implemented in the insect brain, especially in the mushroom body (MB). Here we propose specific hypothetical components of the MB circuit that would enable the implementation of a certain class of relatively simple RL algorithms, capable of integrating distinct components of a navigation task, reminiscent of hierarchical RL models used in robot navigation. We discuss how current models of insect and robot navigation are exploring representations beyond classical, complete map-like representations, with spatial information being embedded in the respective latent representations to varying degrees.

A topological deep learning framework for neural spike decoding

The brain's spatial orientation system uses different neuron ensembles to aid in environment-based navigation. Two of the ways brains encode spatial information are through head direction cells and grid cells. Brains use head direction cells to determine orientation, whereas grid cells consist of layers of decked neurons that overlay to provide environment-based navigation. These neurons fire in ensembles where several neurons fire at once to activate a single head direction or grid. We want to capture this firing structure and use it to decode head direction and animal location from head direction and grid cell activity. Understanding, representing, and decoding these neural structures require models that encompass higher-order connectivity, more than the one-dimensional connectivity that traditional graph-based models provide. To that end, in this work, we develop a topological deep learning framework for neural spike train decoding. Our framework combines unsupervised simplicial complex discovery with the power of deep learning via a new architecture we develop herein called a simplicial convolutional recurrent neural network. Simplicial complexes, topological spaces that use not only vertices and edges but also higher-dimensional objects, naturally generalize graphs and capture more than just pairwise relationships. Additionally, this approach does not require prior knowledge of the neural activity beyond spike counts, which removes the need for similarity measurements. The effectiveness and versatility of the simplicial convolutional neural network is demonstrated on head direction and trajectory prediction via head direction and grid cell datasets.

Quantitative modeling of the emergence of macroscopic grid-like representations

When subjects navigate through spatial environments, grid cells exhibit firing fields that are arranged in a triangular grid pattern. Direct recordings of grid cells from the human brain are rare. Hence, functional magnetic resonance imaging (fMRI) studies proposed an indirect measure of entorhinal grid-cell activity, quantified as hexadirectional modulation of fMRI activity as a function of the subject's movement direction. However, it remains unclear how the activity of a population of grid cells may exhibit hexadirectional modulation. Here, we use numerical simulations and analytical calculations to suggest that this hexadirectional modulation is best explained by head-direction tuning aligned to the grid axes, whereas it is not clearly supported by a bias of grid cells toward a particular phase offset. Firing-rate adaptation can result in hexadirectional modulation, but the available cellular data is insufficient to clearly support or refute this option. The magnitude of hexadirectional modulation furthermore depends considerably on the subject's navigation pattern, indicating that future fMRI studies could be designed to test which hypothesis most likely accounts for the fMRI measure of grid cells. Our findings also underline the importance of quantifying the properties of human grid cells to further elucidate how hexadirectional modulations of fMRI activity may emerge.

Robust variability of grid cell properties within individual grid modules enhances encoding of local space

Although grid cells are one of the most well studied functional classes of neurons in the mammalian brain, the assumption that there is a single grid orientation and spacing per grid module has not been carefully tested. We investigate and analyze a recent large-scale recording of medial entorhinal cortex to characterize the presence and degree of heterogeneity of grid properties within individual modules. We find evidence for small, but robust, variability and hypothesize that this property of the grid code could enhance the ability of encoding local spatial information. Performing analysis on synthetic populations of grid cells, where we have complete control over the amount heterogeneity in grid properties, we demonstrate that variability, of a similar magnitude to the analyzed data, leads to significantly decreased decoding error, even when restricted to activity from a single module. Our results highlight how the heterogeneity of the neural response properties may benefit coding and opens new directions for theoretical and experimental analysis of grid cells.

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.

Coherently remapping toroidal cells but not Grid cells are responsible for path integration in virtual agents

It is widely believed that grid cells provide cues for path integration, with place cells encoding an animal's location and environmental identity. When entering a new environment, these cells remap concurrently, sparking debates about their causal relationship. Using a continuous attractor recurrent neural network, we study spatial cell dynamics in multiple environments. We investigate grid cell remapping as a function of global remapping in place-like units through random resampling of place cell centers. Dimensionality reduction techniques reveal that a subset of cells manifest a persistent torus across environments. Unexpectedly, these toroidal cells resemble band-like cells rather than high grid score units. Subsequent pruning studies reveal that toroidal cells are crucial for path integration while grid cells are not. As we extend the model to operate across many environments, we delineate its generalization boundaries, revealing challenges with modeling many environments in current models.

Dissociating effects of aging and genetic risk of sporadic Alzheimer's disease on path integration

Path integration is a spatial navigation ability that requires the integration of information derived from self-motion cues and stable landmarks, when available, to return to a previous location. Path integration declines with age and Alzheimer's disease (AD). Here, we sought to separate the effects of age and AD risk on path integration, with and without a landmark. Overall, 279 people participated, aged between 18 and 80 years old. Advanced age impaired the appropriate use of a landmark. Older participants furthermore remembered the location of the goal relative to their starting location and reproduced this initial view without considering that they had moved in the environment. This lack of adaptative behavior was not associated with AD risk. In contrast, participants at genetic risk of AD (apolipoprotein E ε4 carriers) exhibited a pure path integration deficit, corresponding to difficulty in performing path integration in the absence of a landmark. Our results show that advanced-age impacts landmark-supported path integration, and that this age effect is dissociable from the effects of AD risk impacting pure path integration.

A map of spatial navigation for neuroscience

Spatial navigation has received much attention from neuroscientists, leading to the identification of key brain areas and the discovery of numerous spatially selective cells. Despite this progress, our understanding of how the pieces fit together to drive behavior is generally lacking. We argue that this is partly caused by insufficient communication between behavioral and neuroscientific researchers. This has led the latter to under-appreciate the relevance and complexity of spatial behavior, and to focus too narrowly on characterizing neural representations of space-disconnected from the computations these representations are meant to enable. We therefore propose a taxonomy of navigation processes in mammals that can serve as a common framework for structuring and facilitating interdisciplinary research in the field. Using the taxonomy as a guide, we review behavioral and neural studies of spatial navigation. In doing so, we validate the taxonomy and showcase its usefulness in identifying potential issues with common experimental approaches, designing experiments that adequately target particular behaviors, correctly interpreting neural activity, and pointing to new avenues of research.

How Do You Build a Cognitive Map? The Development of Circuits and Computations for the Representation of Space in the Brain

In mammals, the activity of neurons in the entorhinal-hippocampal network is modulated by the animal's position and its movement through space. At multiple stages of this distributed circuit, distinct populations of neurons can represent a rich repertoire of navigation-related variables like the animal's location, the speed and direction of its movements, or the presence of borders and objects. Working together, spatially tuned neurons give rise to an internal representation of space, a cognitive map that supports an animal's ability to navigate the world and to encode and consolidate memories from experience. The mechanisms by which, during development, the brain acquires the ability to create an internal representation of space are just beginning to be elucidated. In this review, we examine recent work that has begun to investigate the ontogeny of circuitry, firing patterns, and computations underpinning the representation of space in the mammalian brain.

Are grid cells used for navigation? On local metrics, subjective spaces, and black holes

The symmetric, lattice-like spatial pattern of grid-cell activity is thought to provide a neuronal global metric for space. This view is compatible with grid cells recorded in empty boxes but inconsistent with data from more naturalistic settings. We review evidence arguing against the global-metric notion, including the distortion and disintegration of the grid pattern in complex and three-dimensional environments. We argue that deviations from lattice symmetry are key for understanding grid-cell function. We propose three possible functions for grid cells, which treat real-world grid distortions as a feature rather than a bug. First, grid cells may constitute a local metric for proximal space rather than a global metric for all space. Second, grid cells could form a metric for subjective action-relevant space rather than physical space. Third, distortions may represent salient locations. Finally, we discuss mechanisms that can underlie these functions. These ideas may transform our thinking about grid cells.

Multiple bumps can enhance robustness to noise in continuous attractor networks

A central function of continuous attractor networks is encoding coordinates and accurately updating their values through path integration. To do so, these networks produce localized bumps of activity that move coherently in response to velocity inputs. In the brain, continuous attractors are believed to underlie grid cells and head direction cells, which maintain periodic representations of position and orientation, respectively. These representations can be achieved with any number of activity bumps, and the consequences of having more or fewer bumps are unclear. We address this knowledge gap by constructing 1D ring attractor networks with different bump numbers and characterizing their responses to three types of noise: fluctuating inputs, spiking noise, and deviations in connectivity away from ideal attractor configurations. Across all three types, networks with more bumps experience less noise-driven deviations in bump motion. This translates to more robust encodings of linear coordinates, like position, assuming that each neuron represents a fixed length no matter the bump number. Alternatively, we consider encoding a circular coordinate, like orientation, such that the network distance between adjacent bumps always maps onto 360 degrees. Under this mapping, bump number does not significantly affect the amount of error in the coordinate readout. Our simulation results are intuitively explained and quantitatively matched by a unified theory for path integration and noise in multi-bump networks. Thus, to suppress the effects of biologically relevant noise, continuous attractor networks can employ more bumps when encoding linear coordinates; this advantage disappears when encoding circular coordinates. Our findings provide motivation for multiple bumps in the mammalian grid network.

How to build a cognitive map

Learning and interpreting the structure of the environment is an innate feature of biological systems, and is integral to guiding flexible behaviors for evolutionary viability. The concept of a cognitive map has emerged as one of the leading metaphors for these capacities, and unraveling the learning and neural representation of such a map has become a central focus of neuroscience. In recent years, many models have been developed to explain cellular responses in the hippocampus and other brain areas. Because it can be difficult to see how these models differ, how they relate and what each model can contribute, this Review aims to organize these models into a clear ontology. This ontology reveals parallels between existing empirical results, and implies new approaches to understand hippocampal-cortical interactions and beyond.

Spatial representation by ramping activity of neurons in the retrohippocampal cortex

Neurons in the retrohippocampal cortices play crucial roles in spatial memory. Many retrohippocampal neurons have firing fields that are selectively active at specific locations, with memory for rewarded locations associated with reorganization of these firing fields. Whether this is the sole strategy for representing spatial memories is unclear. Here, we demonstrate that during a spatial memory task retrohippocampal neurons encode location through ramping activity that extends across segments of a linear track approaching and following a reward, with the rewarded location represented by offsets or switches in the slope of the ramping activity. Ramping representations could be maintained independently of trial outcome and cues marking the reward location, indicating that they result from recall of the track structure. When recorded in an open arena, neurons that generated ramping activity during the spatial memory task were more numerous than grid or border cells, with a majority showing spatial firing that did not meet criteria for classification as grid or border representations. Encoding of rewarded locations through offsets and switches in the slope of ramping activity also emerged in recurrent neural network models trained to solve a similar spatial memory task. Impaired performance of model networks following disruption of outputs from ramping neurons is consistent with this coding strategy supporting navigation to recalled locations of behavioral significance. Our results suggest that encoding of learned spaces by retrohippocampal networks employs both discrete firing fields and continuous ramping representations. We hypothesize that retrohippocampal ramping activity mediates readout of learned models for goal-directed navigation.

Sense of self impacts spatial navigation and hexadirectional coding in human entorhinal cortex

Grid cells in entorhinal cortex (EC) encode an individual's location in space and rely on environmental cues and self-motion cues derived from the individual's body. Body-derived signals are also primary signals for the sense of self and based on integrated sensorimotor signals (proprioceptive, tactile, visual, motor) that have been shown to enhance self-centered processing. However, it is currently unknown whether such sensorimotor signals that modulate self-centered processing impact grid cells and spatial navigation. Integrating the online manipulation of bodily signals, to modulate self-centered processing, with a spatial navigation task and an fMRI measure to detect grid cell-like representation (GCLR) in humans, we report improved performance in spatial navigation and decreased GCLR in EC. This decrease in entorhinal GCLR was associated with an increase in retrosplenial cortex activity, which was correlated with participants' navigation performance. These data link self-centered processes during spatial navigation to entorhinal and retrosplenial activity and highlight the role of different bodily factors at play when navigating in VR.

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 Bionic Spatial Cognition Model and Method for Robots Based on the Hippocampus Mechanism

The hippocampus and its accessory are the main areas for spatial cognition. It can integrate paths and form environmental cognition based on motion information and then realize positioning and navigation. Learning from the hippocampus mechanism is a crucial way forward for research in robot perception, so it is crucial to building a calculation method that conforms to the biological principle. In addition, it should be easy to implement on a robot. This paper proposes a bionic cognition model and method for mobile robots, which can realize precise path integration and cognition of space. Our research can provide the basis for the cognition of the environment and autonomous navigation for bionic robots.

Spatial goal coding in the hippocampal formation

The mammalian hippocampal formation contains several distinct populations of neurons involved in representing self-position and orientation. These neurons, which include place, grid, head direction, and boundary-vector cells, are thought to collectively instantiate cognitive maps supporting flexible navigation. However, to flexibly navigate, it is necessary to also maintain internal representations of goal locations, such that goal-directed routes can be planned and executed. Although it has remained unclear how the mammalian brain represents goal locations, multiple neural candidates have recently been uncovered during different phases of navigation. For example, during planning, sequential activation of spatial cells may enable simulation of future routes toward the goal. During travel, modulation of spatial cells by the prospective route, or by distance and direction to the goal, may allow maintenance of route and goal-location information, supporting navigation on an ongoing basis. As the goal is approached, an increased activation of spatial cells may enable the goal location to become distinctly represented within cognitive maps, aiding goal localization. Lastly, after arrival at the goal, sequential activation of spatial cells may represent the just-taken route, enabling route learning and evaluation. Here, we review and synthesize these and other evidence for goal coding in mammalian brains, relate the experimental findings to predictions from computational models, and discuss outstanding questions and future challenges.

Effect of reward on electrophysiological signatures of grid cell population activity in human spatial navigation

The regular equilateral triangular periodic firing pattern of grid cells in the entorhinal cortex is considered a regular metric for the spatial world, and the grid-like representation correlates with hexadirectional modulation of theta (4-8 Hz) power in the entorhinal cortex relative to the moving direction. However, researchers have not clearly determined whether grid cells provide only simple spatial measures in human behavior-related navigation strategies or include other factors such as goal rewards to encode information in multiple patterns. By analysing the hexadirectional modulation of EEG signals in the theta band in the entorhinal cortex of patients with epilepsy performing spatial target navigation tasks, we found that this modulation presents a grid pattern that carries target-related reward information. This grid-like representation is influenced by explicit goals and is related to the local characteristics of the environment. This study provides evidence that human grid cell population activity is influenced by reward information at the level of neural oscillations.

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.

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.

Distance-tuned neurons drive specialized path integration calculations in medial entorhinal cortex

During navigation, animals estimate their position using path integration and landmarks, engaging many brain areas. Whether these areas follow specialized or universal cue integration principles remains incompletely understood. We combine electrophysiology with virtual reality to quantify cue integration across thousands of neurons in three navigation-relevant areas: primary visual cortex (V1), retrosplenial cortex (RSC), and medial entorhinal cortex (MEC). Compared with V1 and RSC, path integration influences position estimates more in MEC, and conflicts between path integration and landmarks trigger remapping more readily. Whereas MEC codes position prospectively, V1 codes position retrospectively, and RSC is intermediate between the two. Lowered visual contrast increases the influence of path integration on position estimates only in MEC. These properties are most pronounced in a population of MEC neurons, overlapping with grid cells, tuned to distance run in darkness. These results demonstrate the specialized role that path integration plays in MEC compared with other navigation-relevant cortical areas.

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.

Measuring and modeling the motor system with machine learning

The utility of machine learning in understanding the motor system is promising a revolution in how to collect, measure, and analyze data. The field of movement science already elegantly incorporates theory and engineering principles to guide experimental work, and in this review we discuss the growing use of machine learning: from pose estimation, kinematic analyses, dimensionality reduction, and closed-loop feedback, to its use in understanding neural correlates and untangling sensorimotor systems. We also give our perspective on new avenues, where markerless motion capture combined with biomechanical modeling and neural networks could be a new platform for hypothesis-driven research.

Evaluating State Space Discovery by Persistent Cohomology in the Spatial Representation System

Persistent cohomology is a powerful technique for discovering topological structure in data. Strategies for its use in neuroscience are still undergoing development. We comprehensively and rigorously assess its performance in simulated neural recordings of the brain's spatial representation system. Grid, head direction, and conjunctive cell populations each span low-dimensional topological structures embedded in high-dimensional neural activity space. We evaluate the ability for persistent cohomology to discover these structures for different dataset dimensions, variations in spatial tuning, and forms of noise. We quantify its ability to decode simulated animal trajectories contained within these topological structures. We also identify regimes under which mixtures of populations form product topologies that can be detected. Our results reveal how dataset parameters affect the success of topological discovery and suggest principles for applying persistent cohomology, as well as persistent homology, to experimental neural recordings.

Modularization of grid cells constrained by the pyramidal patch lattice

Grid cells provide a metric representation of self-location. They are organized into modules, showing discretized scales of grid spacing, but the underlying mechanism remains elusive. In this modeling study, we propose that the hexagonal lattice of pyramidal cell patches may underlie the discretization of grid spacing and orientation. In the continuous attractor network composed of interneurons, stellate and pyramidal cells, the hexagonal lattice of bump attractors is specifically aligned to the patch lattice under 22 conditions determined by the geometry of the patch lattice, while pyramidal cells exhibit synchrony to diverse extents. Given the bump attractor lattice in each module originates from those 22 scenarios, the experimental data on the grid spacing ratio and orientation difference between modules can be reproduced. This work recapitulates the patterns of grid spacing versus orientation in individual animals and reveals the correlation between microstructures and firing fields, providing a systems-level mechanism for grid modularity.

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Each module is modeled as a continuous attractor network with specific parameters

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The lattice of bump attractors is specifically aligned to the pyramidal patch lattice

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Twenty-two scenarios for the bump attractor lattice are proposed

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The grid spacing ratios and orientation differences are determined intrinsically

Biomimetic FPGA-based spatial navigation model with grid cells and place cells

The mammalian spatial navigation system is characterized by an initial divergence of internal representations, with disparate classes of neurons responding to distinct features including location, speed, borders and head direction; an ensuing convergence finally enables navigation and path integration. Here, we report the algorithmic and hardware implementation of biomimetic neural structures encompassing a feed-forward trimodular, multi-layer architecture representing grid-cell, place-cell and decoding modules for navigation. The grid-cell module comprised of neurons that fired in a grid-like pattern, and was built of distinct layers that constituted the dorsoventral span of the medial entorhinal cortex. Each layer was built as an independent continuous attractor network with distinct grid-field spatial scales. The place-cell module comprised of neurons that fired at one or few spatial locations, organized into different clusters based on convergent modular inputs from different grid-cell layers, replicating the gradient in place-field size along the hippocampal dorso-ventral axis. The decoding module, a two-layer neural network that constitutes the convergence of the divergent representations in preceding modules, received inputs from the place-cell module and provided specific coordinates of the navigating object. After vital design optimizations involving all modules, we implemented the tri-modular structure on Zynq Ultrascale+ field-programmable gate array silicon chip, and demonstrated its capacity in precisely estimating the navigational trajectory with minimal overall resource consumption involving a mere 2.92% Look Up Table utilization. Our implementation of a biomimetic, digital spatial navigation system is stable, reliable, reconfigurable, real-time with execution time of about 32 s for 100k input samples (in contrast to 40 minutes on Intel Core i7-7700 CPU with 8 cores clocking at 3.60 GHz) and thus can be deployed for autonomous-robotic navigation without requiring additional sensors.

Perineuronal nets stabilize the grid cell network

Grid cells are part of a widespread network which supports navigation and spatial memory. Stable grid patterns appear late in development, in concert with extracellular matrix aggregates termed perineuronal nets (PNNs) that condense around inhibitory neurons. It has been suggested that PNNs stabilize synaptic connections and long-term memories, but their role in the grid cell network remains elusive. We show that removal of PNNs leads to lower inhibitory spiking activity, and reduces grid cells' ability to create stable representations of a novel environment. Furthermore, in animals with disrupted PNNs, exposure to a novel arena corrupted the spatiotemporal relationships within grid cell modules, and the stored representations of a familiar arena. Finally, we show that PNN removal in entorhinal cortex distorted spatial representations in downstream hippocampal neurons. Together this work suggests that PNNs provide a key stabilizing element for the grid cell network.

The Tolman-Eichenbaum Machine: Unifying Space and Relational Memory through Generalization in the Hippocampal Formation

The hippocampal-entorhinal system is important for spatial and relational memory tasks. We formally link these domains, provide a mechanistic understanding of the hippocampal role in generalization, and offer unifying principles underlying many entorhinal and hippocampal cell types. We propose medial entorhinal cells form a basis describing structural knowledge, and hippocampal cells link this basis with sensory representations. Adopting these principles, we introduce the Tolman-Eichenbaum machine (TEM). After learning, TEM entorhinal cells display diverse properties resembling apparently bespoke spatial responses, such as grid, band, border, and object-vector cells. TEM hippocampal cells include place and landmark cells that remap between environments. Crucially, TEM also aligns with empirically recorded representations in complex non-spatial tasks. TEM also generates predictions that hippocampal remapping is not random as previously believed; rather, structural knowledge is preserved across environments. We confirm this structural transfer over remapping in simultaneously recorded place and grid cells.

Efficient sensory coding of multidimensional stimuli

According to the efficient coding hypothesis, sensory systems are adapted to maximize their ability to encode information about the environment. Sensory neurons play a key role in encoding by selectively modulating their firing rate for a subset of all possible stimuli. This pattern of modulation is often summarized via a tuning curve. The optimally efficient distribution of tuning curves has been calculated in variety of ways for one-dimensional (1-D) stimuli. However, many sensory neurons encode multiple stimulus dimensions simultaneously. It remains unclear how applicable existing models of 1-D tuning curves are for neurons tuned across multiple dimensions. We describe a mathematical generalization that builds on prior work in 1-D to predict optimally efficient multidimensional tuning curves. Our results have implications for interpreting observed properties of neuronal populations. For example, our results suggest that not all tuning curve attributes (such as gain and bandwidth) are equally useful for evaluating the encoding efficiency of a population.

Our brains are tasked with processing a wide range of sensory inputs from the world around us. Natural sensory inputs are often complex and composed of multiple distinctive features (for example, an object may be characterized by its size, shape, color, and weight). Many neurons in the brain play a role in encoding multiple features, or dimensions, of sensory stimuli. Here, we employ the computational technique of population modeling to examine how groups of neurons in the brain can optimally encode multiple dimensions of sensory stimuli. This work provides predictions for theory-driven experiments that can leverage emerging high-throughput neural recording tools to characterize the properties of neuronal populations in response to complex natural stimuli.

Exact and Robust Reconstructions of Integer Vectors Based on Multidimensional Chinese Remainder Theorem (MD-CRT)

The robust Chinese remainder theorem (CRT) has been recently proposed for robustly reconstructing a large nonnegative integer from erroneous remainders. It has found many applications in signal processing, including phase unwrapping and frequency estimation under sub-Nyquist sampling. Motivated by the applications in multidimensional (MD) signal processing, in this paper we propose the MD-CRT and robust MD-CRT for Integer vectors with respect to a general set of integer matrix moduli, which provides an algorithm to uniquely reconstruct integer vectors with respect to a general set of integer matrix moduli, which provides an algorithm to uniquely reconstruct an integer vector from its remainders, if it is in the fundamental parallelepiped of the lattice generated by a least common right multiple of all the moduli. For some special forms of moduli, we present explicit reconstruction formulae. Moreover, we derive the robust MD-CRT for integer vectors when the remaining integer matrices of all the moduli left divided by their greatest common left divisor (gcld) are pairwise commutative and coprime. Two different reconstruction algorithms are proposed, and accordingly, two different conditions on the remainder error bound for the reconstruction robustness are obtained, which are related to a quarter of the minimum distance of the lattice generated by the gcld of all the moduli or the Smith normal form of the gcld.

Fate of Duplicated Neural Structures

Statistical physics determines the abundance of different arrangements of matter depending on cost-benefit balances. Its formalism and phenomenology percolate throughout biological processes and set limits to effective computation. Under specific conditions, self-replicating and computationally complex patterns become favored, yielding life, cognition, and Darwinian evolution. Neurons and neural circuits sit at a crossroads between statistical physics, computation, and (through their role in cognition) natural selection. Can we establish a statistical physics of neural circuits? Such theory would tell what kinds of brains to expect under set energetic, evolutionary, and computational conditions. With this big picture in mind, we focus on the fate of duplicated neural circuits. We look at examples from central nervous systems, with stress on computational thresholds that might prompt this redundancy. We also study a naive cost-benefit balance for duplicated circuits implementing complex phenotypes. From this, we derive phase diagrams and (phase-like) transitions between single and duplicated circuits, which constrain evolutionary paths to complex cognition. Back to the big picture, similar phase diagrams and transitions might constrain I/O and internal connectivity patterns of neural circuits at large. The formalism of statistical physics seems to be a natural framework for this worthy line of research.

Grid cells are modulated by local head direction

Grid and head direction codes represent cognitive spaces for navigation and memory. Pure grid cells generate grid codes that have been assumed to be independent of head direction, whereas conjunctive cells generate grid representations that are tuned to a single head direction. Here, we demonstrate that pure grid cells also encode head direction, but through distinct mechanisms. We show that individual firing fields of pure grid cells are tuned to multiple head directions, with the preferred sets of directions differing between fields. This local directional modulation is not predicted by previous continuous attractor or oscillatory interference models of grid firing but is accounted for by models in which pure grid cells integrate inputs from co-aligned conjunctive cells with firing rates that differ between their fields. We suggest that local directional signals from grid cells may contribute to downstream computations by decorrelating different points of view from the same location.

Advantages and detection of phase coding in the absence of rhythmicity

The encoding of information in spike phase relative to local field potential (LFP) oscillations offers several theoretical advantages over equivalent firing rate codes. One notable example is provided by place and grid cells in the rodent hippocampal formation, which exhibit phase precession-firing at progressively earlier phases of the 6-12 Hz movement-related theta rhythm as their spatial firing fields are traversed. It is often assumed that such phase coding relies on a high amplitude baseline oscillation with relatively constant frequency. However, sustained oscillations with fixed frequency are generally absent in LFP and spike train recordings from the human brain. Hence, we examine phase coding relative to LFP signals with broadband low-frequency (2-20 Hz) power but without regular rhythmicity. We simulate a population of grid cells that exhibit phase precession against a baseline oscillation recorded from depth electrodes in human hippocampus. We show that this allows grid cell firing patterns to multiplex information about location, running speed and movement direction, alongside an arbitrary fourth variable encoded in LFP frequency. This is of particular importance given recent demonstrations that movement direction, which is essential for path integration, cannot be recovered from head direction cell firing rates. In addition, we investigate how firing phase might reduce errors in decoded location, including those arising from differences in firing rate across grid fields. Finally, we describe analytical methods that can identify phase coding in the absence of high amplitude LFP oscillations with approximately constant frequency, as in single unit recordings from the human brain and consistent with recent data from the flying bat. We note that these methods could also be used to detect phase coding outside of the spatial domain, and that multi-unit activity can substitute for the LFP signal. In summary, we demonstrate that the computational advantages offered by phase coding are not contingent on, and can be detected without, regular rhythmicity in neural activity.

Bio-inspired multi-scale fusion

We reveal how implementing the homogeneous, multi-scale mapping frameworks observed in the mammalian brain's mapping systems radically improves the performance of a range of current robotic localization techniques. Roboticists have developed a range of predominantly single- or dual-scale heterogeneous mapping approaches (typically locally metric and globally topological) that starkly contrast with neural encoding of space in mammalian brains: a multi-scale map underpinned by spatially responsive cells like the grid cells found in the rodent entorhinal cortex. Yet the full benefits of a homogeneous multi-scale mapping framework remain unknown in both robotics and biology: in robotics because of the focus on single- or two-scale systems and limits in the scalability and open-field nature of current test environments and benchmark datasets; in biology because of technical limitations when recording from rodents during movement over large areas. New global spatial databases with visual information varying over several orders of magnitude in scale enable us to investigate this question for the first time in real-world environments. In particular, we investigate and answer the following questions: why have multi-scale representations, how many scales should there be, what should the size ratio between consecutive scales be and how does the absolute scale size affect performance? We answer these questions by developing and evaluating a homogeneous, multi-scale mapping framework mimicking aspects of the rodent multi-scale map, but using current robotic place recognition techniques at each scale. Results in large-scale real-world environments demonstrate multi-faceted and significant benefits for mapping and localization performance and identify the key factors that determine performance.

Transition Scale-Spaces: A Computational Theory for the Discretized Entorhinal Cortex

Although hippocampal grid cells are thought to be crucial for spatial navigation, their computational purpose remains disputed. Recently, they were proposed to represent spatial transitions and convey this knowledge downstream to place cells. However, a single scale of transitions is insufficient to plan long goal-directed sequences in behaviorally acceptable time.

Here, a scale-space data structure is suggested to optimally accelerate retrievals from transition systems, called transition scale-space (TSS). Remaining exclusively on an algorithmic level, the scale increment is proved to be ideally [Formula: see text] for biologically plausible receptive fields. It is then argued that temporal buffering is necessary to learn the scale-space online. Next, two modes for retrieval of sequences from the TSS are presented: top down and bottom up. The two modes are evaluated in symbolic simulations (i.e., without biologically plausible spiking neurons). Additionally, a TSS is used for short-cut discovery in a simulated Morris water maze. Finally, the results are discussed in depth with respect to biological plausibility, and several testable predictions are derived. Moreover, relations to other grid cell models, multiresolution path planning, and scale-space theory are highlighted. Summarized, reward-free transition encoding is shown here, in a theoretical model, to be compatible with the observed discretization along the dorso-ventral axis of the medial entorhinal cortex. Because the theoretical model generalizes beyond navigation, the TSS is suggested to be a general-purpose cortical data structure for fast retrieval of sequences and relational knowledge.

Source code for all simulations presented in this paper can be found at https://github.com/rochus/transitionscalespace .

Deforming the metric of cognitive maps distorts memory

Environmental boundaries anchor cognitive maps that support memory. However, trapezoidal boundary geometry distorts the regular firing patterns of entorhinal grid cells, proposedly providing a metric for cognitive maps. Here we test the impact of trapezoidal boundary geometry on human spatial memory using immersive virtual reality. Consistent with reduced regularity of grid patterns in rodents and a grid-cell model based on the eigenvectors of the successor representation, human positional memory was degraded in a trapezoid environment compared with a square environment-an effect that was particularly pronounced in the narrow part of the trapezoid. Congruent with changes in the spatial frequency of eigenvector grid patterns, distance estimates between remembered positions were persistently biased, revealing distorted memory maps that explained behaviour better than the objective maps. Our findings demonstrate that environmental geometry affects human spatial memory in a similar manner to rodent grid-cell activity and, therefore, strengthen the putative link between grid cells and behaviour along with their cognitive functions beyond navigation.

Identifying Core Regions for Path Integration on Medial Entorhinal Cortex of Hippocampal Formation

Path integration is one of the functions that support the self-localization ability of animals. Path integration outputs position information after an animal’s movement when initial-position and movement information is input. The core region responsible for this function has been identified as the medial entorhinal cortex (MEC), which is part of the hippocampal formation that constitutes the limbic system. However, a more specific core region has not yet been identified. This research aims to clarify the detailed structure at the cell-firing level in the core region responsible for path integration from fragmentarily accumulated experimental and theoretical findings by reviewing 77 papers. This research draws a novel diagram that describes the MEC, the hippocampus, and their surrounding regions by focusing on the MEC’s input/output (I/O) information. The diagram was created by summarizing the results of exhaustively scrutinizing the papers that are relative to the I/O relationship, the connection relationship, and cell position and firing pattern. From additional investigations, we show function information related to path integration, such as I/O information and the relationship between multiple functions. Furthermore, we constructed an algorithmic hypothesis on I/O information and path-integration calculation method from the diagram and the information of functions related to path integration. The algorithmic hypothesis is composed of regions related to path integration, the I/O relations between them, the calculation performed there, and the information representations (cell-firing pattern) in them. Results of examining the hypothesis confirmed that the core region responsible for path integration was either stellate cells in layer II or pyramidal cells in layer III of the MEC.

Navigating with grid and place cells in cluttered environments

Hippocampal formation contains several classes of neurons thought to be involved in navigational processes, in particular place cells and grid cells. Place cells have been associated with a topological strategy for navigation, while grid cells have been suggested to support metric vector navigation. Grid cell‐based vector navigation can support novel shortcuts across unexplored territory by providing the direction toward the goal. However, this strategy is insufficient in natural environments cluttered with obstacles. Here, we show how navigation in complex environments can be supported by integrating a grid cell‐based vector navigation mechanism with local obstacle avoidance mediated by border cells and place cells whose interconnections form an experience‐dependent topological graph of the environment. When vector navigation and object avoidance fail (i.e., the agent gets stuck), place cell replay events set closer subgoals for vector navigation. We demonstrate that this combined navigation model can successfully traverse environments cluttered by obstacles and is particularly useful where the environment is underexplored. Finally, we show that the model enables the simulated agent to successfully navigate experimental maze environments from the animal literature on cognitive mapping. The proposed model is sufficiently flexible to support navigation in different environments, and may inform the design of experiments to relate different navigational abilities to place, grid, and border cell firing.

A geometric attractor mechanism for self-organization of entorhinal grid modules

Grid cells in the medial entorhinal cortex (MEC) respond when an animal occupies a periodic lattice of 'grid fields' in the environment. The grids are organized in modules with spatial periods, or scales, clustered around discrete values separated on average by ratios in the range 1.4-1.7. We propose a mechanism that produces this modular structure through dynamical self-organization in the MEC. In attractor network models of grid formation, the grid scale of a single module is set by the distance of recurrent inhibition between neurons. We show that the MEC forms a hierarchy of discrete modules if a smooth increase in inhibition distance along its dorso-ventral axis is accompanied by excitatory interactions along this axis. Moreover, constant scale ratios between successive modules arise through geometric relationships between triangular grids and have values that fall within the observed range. We discuss how interactions required by our model might be tested experimentally.

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.

Episodic memory: Neuronal codes for what, where, and when

Episodic memory is defined as the ability to recall events in a spatiotemporal context. Formation of such memories is critically dependent on the hippocampal formation and its inputs from the entorhinal cortex. To be able to support the formation of episodic memories, entorhinal cortex and hippocampal formation should contain a neuronal code that follows several requirements. First, the code should include information about position of the agent ("where"), sequence of events ("when"), and the content of the experience itself ("what"). Second, the code should arise instantly thereby being able to support memory formation of one-shot experiences. For successful encoding and to avoid interference between memories during recall, variations in location, time, or in content of experience should result in unique ensemble activity. Finally, the code should capture several different resolutions of experience so that the necessary details relevant for future memory-based predictions will be stored. We review how neuronal codes in entorhinal cortex and hippocampus follow these requirements and argue that during formation of episodic memories entorhinal cortex provides hippocampus with instant information about ongoing experience. Such information originates from (a) spatially modulated neurons in medial entorhinal cortex, including grid cells, which provide a stable and universal positional metric of the environment; (b) a continuously varying signal in lateral entorhinal cortex providing a code for the temporal progression of events; and (c) entorhinal neurons coding the content of experiences exemplified by object-coding and odor-selective neurons. During formation of episodic memories, information from these systems are thought to be encoded as unique sequential ensemble activity in hippocampus, thereby encoding associations between the content of an event and its spatial and temporal contexts. Upon exposure to parts of the encoded stimuli, activity in these ensembles can be reinstated, leading to reactivation of the encoded activity pattern and memory recollection.

On the Organization of Grid and Place Cells: Neural Denoising via Subspace Learning

Place cells in the hippocampus (HC) are active when an animal visits a certain location (referred to as a place field) within an environment. Grid cells in the medial entorhinal cortex (MEC) respond at multiple locations, with firing fields that form a periodic and hexagonal tiling of the environment. The joint activity of grid and place cell populations, as a function of location, forms a neural code for space. In this article, we develop an understanding of the relationships between coding theoretically relevant properties of the combined activity of these populations and how these properties limit the robustness of this representation to noise-induced interference. These relationships are revisited by measuring the performances of biologically realizable algorithms implemented by networks of place and grid cell populations, as well as constraint neurons, which perform denoising operations. Contributions of this work include the investigation of coding theoretic limitations of the mammalian neural code for location and how communication between grid and place cell networks may improve the accuracy of each population's representation. Simulations demonstrate that denoising mechanisms analyzed here can significantly improve the fidelity of this neural representation of space. Furthermore, patterns observed in connectivity of each population of simulated cells predict that anti-Hebbian learning drives decreases in inter-HC-MEC connectivity along the dorsoventral axis.

Stellate Cells in the Medial Entorhinal Cortex Are Required for Spatial Learning

Spatial learning requires estimates of location that may be obtained by path integration or from positional cues. Grid and other spatial firing patterns of neurons in the superficial medial entorhinal cortex (MEC) suggest roles in behavioral estimation of location. However, distinguishing the contributions of path integration and cue-based signals to spatial behaviors is challenging, and the roles of identified MEC neurons are unclear. We use virtual reality to dissociate linear path integration from other strategies for behavioral estimation of location. We find that mice learn to path integrate using motor-related self-motion signals, with accuracy that decreases steeply as a function of distance. We show that inactivation of stellate cells in superficial MEC impairs spatial learning in virtual reality and in a real world object location recognition task. Our results quantify contributions of path integration to behavior and corroborate key predictions of models in which stellate cells contribute to location estimation.

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Mice learn to estimate location by path integration and cue-based strategies

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Motor-related self-motion signals are used for path integration

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Accuracy of path integration decreases with distance

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Stellate cells in medial entorhinal cortex are required for spatial learning

Origin and role of path integration in the cognitive representations of the hippocampus: computational insights into open questions

Path integration is a straightforward concept with varied connotations that are important to different disciplines concerned with navigation, such as ethology, cognitive science, robotics and neuroscience. In studying the hippocampal formation, it is fruitful to think of path integration as a computation that transforms a sense of motion into a sense of location, continuously integrated with landmark perception. Here, we review experimental evidence that path integration is intimately involved in fundamental properties of place cells and other spatial cells that are thought to support a cognitive abstraction of space in this brain system. We discuss hypotheses about the anatomical and computational origin of path integration in the well-characterized circuits of the rodent limbic system. We highlight how computational frameworks for map-building in robotics and cognitive science alike suggest an essential role for path integration in the creation of a new map in unfamiliar territory, and how this very role can help us make sense of differences in neurophysiological data from novel versus familiar and small versus large environments. Similar computational principles could be at work when the hippocampus builds certain non-spatial representations, such as time intervals or trajectories defined in a sensory stimulus space.

Anticipatory Neural Activity Improves the Decoding Accuracy for Dynamic Head-Direction Signals

Insects and vertebrates harbor specific neurons that encode the animal's head direction (HD) and provide an internal compass for spatial navigation. Each HD cell fires most strongly in one preferred direction. As the animal turns its head, however, HD cells in rat anterodorsal thalamic nucleus (ADN) and other brain areas fire already before their preferred direction is reached, as if the neurons anticipated the future HD. This phenomenon has been explained at a mechanistic level, but a functional interpretation is still missing. To close this gap, we use a computational approach based on the movement statistics of male rats and a simple model for the neural responses within the ADN HD network. Network activity is read out using population vectors in a biologically plausible manner, so that only past spikes are taken into account. We find that anticipatory firing improves the representation of the present HD by reducing the motion-induced temporal bias inherent in causal decoding. The amount of anticipation observed in ADN enhances the precision of the HD compass read-out by up to 40%. More generally, our theoretical framework predicts that neural integration times not only reflect biophysical constraints, but also the statistics of behaviorally relevant stimuli; in particular, anticipatory tuning should be found wherever neurons encode sensory signals that change gradually in time.SIGNIFICANCE STATEMENT Across different brain regions, populations of noisy neurons encode dynamically changing stimuli. Decoding a time-varying stimulus from the population response involves a trade-off: For short read-out times, stimulus estimates are unreliable as the number of stochastic spikes is small; for long read-outs, estimates are biased because they lag behind the true stimulus. We show that optimal decoding of temporally correlated stimuli not only relies on finding the right read-out time window but requires neurons to anticipate future stimulus values. We apply this general framework to the rodent head-direction system and show that the experimentally observed anticipation of future head directions can be explained at a quantitative level from the neuronal tuning properties, network size, and the animal's head-movement statistics.

The multidimensional brain

Brain activity takes place in three spatial-plus time dimensions. This rather obvious claim has been recently questioned by papers that, taking into account the big data outburst and novel available computational tools, are starting to unveil a more intricate state of affairs. Indeed, various brain activities and their correlated mental functions can be assessed in terms of trajectories embedded in phase spaces of dimensions higher than the canonical ones. In this review, I show how further dimensions may not just represent a convenient methodological tool that allows a better mathematical treatment of otherwise elusive cortical activities, but may also reflect genuine functional or anatomical relationships among real nervous functions. I then describe how to extract hidden multidimensional information from real or artificial neurodata series, and make clear how our mind dilutes, rather than concentrates as currently believed, inputs coming from the environment. Finally, I argue that the principle "the higher the dimension, the greater the information" may explain the occurrence of mental activities and elucidate the mechanisms of human diseases associated with dimensionality reduction.

A neural-level model of spatial memory and imagery

We present a model of how neural representations of egocentric spatial experiences in parietal cortex interface with viewpoint-independent representations in medial temporal areas, via retrosplenial cortex, to enable many key aspects of spatial cognition. This account shows how previously reported neural responses (place, head-direction and grid cells, allocentric boundary- and object-vector cells, gain-field neurons) can map onto higher cognitive function in a modular way, and predicts new cell types (egocentric and head-direction-modulated boundary- and object-vector cells). The model predicts how these neural populations should interact across multiple brain regions to support spatial memory, scene construction, novelty-detection, 'trace cells', and mental navigation. Simulated behavior and firing rate maps are compared to experimental data, for example showing how object-vector cells allow items to be remembered within a contextual representation based on environmental boundaries, and how grid cells could update the viewpoint in imagery during planning and short-cutting by driving sequential place cell activity.