Framing the grid: effect of boundaries on grid cells and navigation

Modeling the grid cell activity based on cognitive space transformation

The grid cells in the medial entorhinal cortex are widely recognized as a critical component of spatial cognition within the entorhinal-hippocampal neuronal circuits. To account for the hexagonal patterns, several computational models have been proposed. However, there is still considerable debate regarding the interaction between grid cells and place cells. In response, we have developed a novel grid-cell computational model based on cognitive space transformation, which established a theoretical framework of the interaction between place cells and grid cells for encoding and transforming positions between the local frame and global frame. Our model not only can generate the firing patterns of the grid cells but also reproduces the biological experiment results about the grid-cell global representation of connected environments and supports the conjecture about the underlying reason. Moreover, our model provides new insights into how grid cells and place cells integrate external and self-motion cues.

Remapping revisited: how the hippocampus represents different spaces

The representation of distinct spaces by hippocampal place cells has been linked to changes in their place fields (the locations in the environment where the place cells discharge strongly), a phenomenon that has been termed 'remapping'. Remapping has been assumed to be accompanied by the reorganization of subsecond cofiring relationships among the place cells, potentially maximizing hippocampal information coding capacity. However, several observations challenge this standard view. For example, place cells exhibit mixed selectivity, encode non-positional variables, can have multiple place fields and exhibit unreliable discharge in fixed environments. Furthermore, recent evidence suggests that, when measured at subsecond timescales, the moment-to-moment cofiring of a pair of cells in one environment is remarkably similar in another environment, despite remapping. Here, I propose that remapping is a misnomer for the changes in place fields across environments and suggest instead that internally organized manifold representations of hippocampal activity are actively registered to different environments to enable navigation, promote memory and organize knowledge.

Grid cells: the missing link in understanding Parkinson's disease?

The mechanisms underlying Parkinson's disease (PD) are complex and not fully understood, and the box-and-arrow model among other current models present significant challenges. This paper explores the potential role of the allocentric brain and especially its grid cells in several PD motor symptoms, including bradykinesia, kinesia paradoxa, freezing of gait, the bottleneck phenomenon, and their dependency on cueing. It is argued that central hubs, like the locus coeruleus and the pedunculopontine nucleus, often narrowly interpreted in the context of PD, play an equally important role in governing the allocentric brain as the basal ganglia. Consequently, the motor and secondary motor (e.g., spatially related) symptoms of PD linked with dopamine depletion may be more closely tied to erroneous computation by grid cells than to the basal ganglia alone. Because grid cells and their associated central hubs introduce both spatial and temporal information to the brain influencing velocity perception they may cause bradykinesia or hyperkinesia as well. In summary, PD motor symptoms may primarily be an allocentric disturbance resulting from virtual faulty computation by grid cells revealed by dopamine depletion in PD.

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.

Simultaneous representation of multiple time horizons by entorhinal grid cells and CA1 place cells

Grid cells and place cells represent the spatiotemporal continuum of an animal's past, present, and future locations. However, their spatiotemporal relationship is unclear. Here, we co-record grid and place cells in freely foraging rats. We show that average time shifts in grid cells tend to be prospective and are proportional to their spatial scale, providing a nearly instantaneous readout of a spectrum of progressively increasing time horizons ranging hundreds of milliseconds. Average time shifts of place cells are generally larger compared to grid cells and also increase with place field sizes. Moreover, time horizons display nonlinear modulation by the animal's trajectories in relation to the local boundaries and locomotion cues. Finally, long and short time horizons occur at different parts of the theta cycle, which may facilitate their readout. Together, these findings suggest that population activity of grid and place cells may represent local trajectories essential for goal-directed navigation and planning.

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.

A sense of direction: spatial boundaries in a cognitive, cultural, and deep time perspective

This brief note points toward new potentials that lie at the interface between research on landscape archaeology and cognitive science. Recent advances in the cognitive and neural sciences have sharpened our understanding of spatial cognition, by providing new explanations for how the brain reduces the dimensionality of complex topography and geography for effective navigation. This research suggests that space is represented in grid-like structures in the brain, and that grid-like forms are a basic ingredient of spatial processing. At the same time, recent archaeological research shows that the organization of larger-scale space into linear forms, and in particular grid-like landscapes, is a relatively recent social invention, which suggests that these forms are historically and culturally contingent. Taken together, this research raises the question of how the dimensionality-reducing function of grid-like processing in the brain is related to higher-level conceptual and imaginative processing of space needed to plan and negotiate large-scale landscape structures. This brief note motivates this question and argues for further exploration of the relationships between biological, cognitive, and cultural processes related to space and its conceptualization between these fields of research.

The effects of spatial stability and cue type on spatial learning: Implications for theories of parallel memory systems

Some theories of spatial learning predict that associative rules apply under only limited circumstances. For example, learning based on a boundary has been claimed to be immune to cue competition effects because boundary information is the basis for the formation of a cognitive map, whilst landmark learning does not involve cognitive mapping. This is referred to as the cue type hypothesis. However, it has also been claimed that cue stability is a prerequisite for the formation of a cognitive map, meaning that whichever cue type was perceived as stable would enter a cognitive map and thus be immune to cue competition, while unstable cues will be subject to cue competition, regardless of cue type. In experiments 1 and 2 we manipulated the stability of boundary and landmark cues when learning the location of two hidden goals. One goal location was constant with respect to the boundary, and the other constant with respect to the landmark cues. For both cue types, the presence of distal orientation cues provided directional information. For half the participants the landmark cues were unstable relative to the boundary and orientation cues, whereas for the remainder of the participants the boundary was unstable relative to landmarks and orientation cues. In a second stage of training, all cues remained stable so that both goal locations could be learned with respect to both landmark and boundary information. According to the cue type hypothesis, boundary information should block learning about landmarks regardless of cue stability. According to the cue stability hypothesis, however, landmarks should block learning about the boundary when the landmarks appear stable relative to the boundary. Regardless of cue type or stability the results showed reciprocal blocking, contrary to both formulations of incidental cognitive mapping. Experiment 3 established that the results of Experiments 1 and 2 could not be explained in terms of difficulty in learning certain locations with respect to different cue types. In a final experiment, following training in which both landmarks and boundary cues signalled two goal locations, a new goal location was established with respect to the landmark cues, before testing with the boundary, which had never been used to define the new goal location. The results of this novel test of the interaction between boundary and landmark cues indicated that new learning with respect to the landmark had a profound effect on navigation with respect to the boundary, counter to the predictions of incidental cognitive mapping of boundaries.

Hippocampal place cells encode global location but not connectivity in a complex space

Flexible navigation relies on a cognitive map of space, thought to be implemented by hippocampal place cells: neurons that exhibit location-specific firing. In connected environments, optimal navigation requires keeping track of one's location and of the available connections between subspaces. We examined whether the dorsal CA1 place cells of rats encode environmental connectivity in four geometrically identical boxes arranged in a square. Rats moved between boxes by pushing saloon-type doors that could be locked in one or both directions. Although rats demonstrated knowledge of environmental connectivity, their place cells did not respond to connectivity changes, nor did they represent doorways differently from other locations. Place cells coded location in a global reference frame, with a different map for each box and minimal repetitive fields despite the repetitive geometry. These results suggest that CA1 place cells provide a spatial map that does not explicitly include connectivity.

Recurrent amplification of grid-cell activity

High-level cognitive abilities such as navigation and spatial memory are thought to rely on the activity of grid cells in the medial entorhinal cortex (MEC), which encode the animal's position in space with periodic triangular patterns. Yet the neural mechanisms that underlie grid-cell activity are still unknown. Recent in vitro and in vivo experiments indicate that grid cells are embedded in highly structured recurrent networks. But how could recurrent connectivity become structured during development? And what is the functional role of these connections? With mathematical modeling and simulations, we show that recurrent circuits in the MEC could emerge under the supervision of weakly grid-tuned feedforward inputs. We demonstrate that a learned excitatory connectivity could amplify grid patterns when the feedforward sensory inputs are available and sustain attractor states when the sensory cues are lost. Finally, we propose a Fourier-based measure to quantify the spatial periodicity of grid patterns: the grid-tuning index.

Grid coding, spatial representation, and navigation: Should we assume an isomorphism?

Grid cells provide a compelling example of a link between cellular activity and an abstract and difficult to define concept like space. Accordingly, a representational perspective on grid coding argues that neural grid coding underlies a fundamentally spatial metric. Recently, some theoretical proposals have suggested extending such a framework to nonspatial cognition as well, such as category learning. Here, we provide a critique of the frequently employed assumption of an isomorphism between patterns of neural activity (e.g., grid cells), mental representation, and behavior (e.g., navigation). Specifically, we question the strict isomorphism between these three levels and suggest that human spatial navigation is perhaps best characterized by a wide variety of both metric and nonmetric strategies. We offer an alternative perspective on how grid coding might relate to human spatial navigation, arguing that grid coding is part of a much larger conglomeration of neural activity patterns that dynamically tune to accomplish specific behavioral outputs.

Spatial Navigation

The hippocampus is critical for spatial navigation. In this review, we focus on the role of the hippocampus in three basic strategies used for spatial navigation: path integration, stimulus-response association, and map-based navigation. First, the hippocampus is not required for path integration unless the path of path integration is too long and complex. The hippocampus provides mnemonic support when involved in the process of path integration. Second, the hippocampus's involvement in stimulus-response association is dependent on how the strategy is conducted. The hippocampus is not required for the habit form of stimulus-response association. Third, while the hippocampus is fully engaged in map-based navigation, the shared characteristics of place cells, grid cells, head direction cells, and other spatial encoding cells, which are detected in the hippocampus and associated areas, offer a possibility that there is a stand-alone allocentric space perception (or mental representation) of the environment outside and independent of the hippocampus, and the spatially specific firing patterns of these spatial encoding cells are the unfolding of the intermediate stages of the processing of this allocentric spatial information when conveyed into the hippocampus for information storage or retrieval. Furthermore, the presence of all the spatially specific firing patterns in the hippocampus and the related neural circuits during the path integration and map-based navigation support such a notion that in essence, path integration is the same allocentric space perception provided with only idiothetic inputs. Taken together, the hippocampus plays a general mnemonic role in spatial navigation.

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.

Environmental Barriers Disrupt Grid-like Representations in Humans during Navigation

Environmental barriers fundamentally shape our behavior and conceptualization of space [1-5]. Evidence from rodents suggests that, in contrast to an open-field environment, where grid cells exhibit firing patterns with a 6-fold rotational symmetry [5, 6], barriers within the field abolish the 6-fold symmetry and fragment the grid firing fields into compartmentalized repeating "submaps" [5]. These results suggest that barriers may exert their influence on the cognitive map through organization of the metric representation of space provided by entorhinal neurons. We directly tested this hypothesis in humans, combining functional MRI with a virtual navigation paradigm in which we manipulated the local barrier structure. When participants performed a fixed-route foraging task in an open field, the functional MRI signal in right entorhinal cortex exhibited a 6-fold periodic modulation by movement direction associated with conjunctive grid cell firing [7]. However, when environments were compartmentalized by barriers, the grid-like 6-fold spatial metric was abolished. Instead, a 4-fold modulation of the entorhinal signal was observed, consistent with a vectorized organization of spatial metrics predicted by rodent models of navigation [5]. Collectively, these results provide mechanistic insight into why barriers compartmentalize our cognitive map, indicating that boundaries exert a powerful influence on the way environments are represented in human entorhinal cortex. Given that our daily environments are rarely wide open and are often segmented by barriers (e.g., the buildings of our home city), our findings have implications for applying models of cognitive mapping based on grid-like metrics [8] to naturalistic circumstances.

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.

Emergent elasticity in the neural code for space

Upon encountering a novel environment, an animal must construct a consistent environmental map, as well as an internal estimate of its position within that map, by combining information from two distinct sources: self-motion cues and sensory landmark cues. How do known aspects of neural circuit dynamics and synaptic plasticity conspire to accomplish this feat? Here we show analytically how a neural attractor model that combines path integration of self-motion cues with Hebbian plasticity in synaptic weights from landmark cells can self-organize a consistent map of space as the animal explores an environment. Intriguingly, the emergence of this map can be understood as an elastic relaxation process between landmark cells mediated by the attractor network. Moreover, our model makes several experimentally testable predictions, including (i) systematic path-dependent shifts in the firing fields of grid cells toward the most recently encountered landmark, even in a fully learned environment; (ii) systematic deformations in the firing fields of grid cells in irregular environments, akin to elastic deformations of solids forced into irregular containers; and (iii) the creation of topological defects in grid cell firing patterns through specific environmental manipulations. Taken together, our results conceptually link known aspects of neurons and synapses to an emergent solution of a fundamental computational problem in navigation, while providing a unified account of disparate experimental observations.

Environmental deformations dynamically shift the grid cell spatial metric

In familiar environments, the firing fields of entorhinal grid cells form regular triangular lattices. However, when the geometric shape of the environment is deformed, these time-averaged grid patterns are distorted in a grid scale-dependent and local manner. We hypothesized that this distortion in part reflects dynamic anchoring of the grid code to displaced boundaries, possibly through border cell-grid cell interactions. To test this hypothesis, we first reanalyzed two existing rodent grid rescaling datasets to identify previously unrecognized boundary-tethered shifts in grid phase that contribute to the appearance of rescaling. We then demonstrated in a computational model that boundary-tethered phase shifts, as well as scale-dependent and local distortions of the time-averaged grid pattern, could emerge from border-grid interactions without altering inherent grid scale. Together, these results demonstrate that environmental deformations induce history-dependent shifts in grid phase, and implicate border-grid interactions as a potential mechanism underlying these dynamics.

Hexagonal Grid Fields Optimally Encode Transitions in Spatiotemporal Sequences

Grid cells of the rodent entorhinal cortex are essential for spatial navigation. Although their function is commonly believed to be either path integration or localization, the origin or purpose of their hexagonal firing fields remains disputed. Here they are proposed to arise as an optimal encoding of transitions in sequences. First, storage requirements for transitions in general episodic sequences are examined using propositional logic and graph theory. Subsequently, transitions in complete metric spaces are considered under the assumption of an ideal sampling of an input space. It is shown that memory capacity of neurons that have to encode multiple feasible spatial transitions is maximized by a hexagonal pattern. Grid cells are proposed to encode spatial transitions in spatiotemporal sequences, with the entorhinal-hippocampal loop forming a multitransition system.

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.

Field repetition and local mapping in the hippocampus and the medial entorhinal cortex

Hippocampal place cells support spatial cognition and are thought to form the neural substrate of a global "cognitive map." A widely held view is that parts of the hippocampus also underlie the ability to separate patterns or to provide different neural codes for distinct environments. However, a number of studies have shown that in environments composed of multiple, repeating compartments, place cells and other spatially modulated neurons show the same activity in each local area. This repetition of firing fields may reflect pattern completion and may make it difficult for animals to distinguish similar local environments. In this review we 1) highlight some of the navigation difficulties encountered by humans in repetitive environments, 2) summarize literature demonstrating that place and grid cells represent local and not global space, and 3) attempt to explain the origin of these phenomena. We argue that the repetition of firing fields can be a useful tool for understanding the relationship between grid cells in the entorhinal cortex and place cells in the hippocampus, the spatial inputs shared by these cells, and the propagation of spatially related signals through these structures.

Framing of grid cells within and beyond navigation boundaries

Grid cells represent an ideal candidate to investigate the allocentric determinants of the brain's cognitive map. Most studies of grid cells emphasized the roles of geometric boundaries within the navigational range of the animal. Behaviors such as novel route-taking between local environments indicate the presence of additional inputs from remote cues beyond the navigational borders. To investigate these influences, we recorded grid cells as rats explored an open-field platform in a room with salient, remote cues. The platform was rotated or translated relative to the room frame of reference. Although the local, geometric frame of reference often exerted the strongest control over the grids, the remote cues demonstrated a consistent, sometimes dominant, countervailing influence. Thus, grid cells are controlled by both local geometric boundaries and remote spatial cues, consistent with prior studies of hippocampal place cells and providing a rich representational repertoire to support complex navigational (and perhaps mnemonic) processes.