Optimal configurations of spatial scale for grid cell firing under noise and uncertainty

Human navigation strategies and their errors result from dynamic interactions of spatial uncertainties

Goal-directed navigation requires continuously integrating uncertain self-motion and landmark cues into an internal sense of location and direction, concurrently planning future paths, and sequentially executing motor actions. Here, we provide a unified account of these processes with a computational model of probabilistic path planning in the framework of optimal feedback control under uncertainty. This model gives rise to diverse human navigational strategies previously believed to be distinct behaviors and predicts quantitatively both the errors and the variability of navigation across numerous experiments. This furthermore explains how sequential egocentric landmark observations form an uncertain allocentric cognitive map, how this internal map is used both in route planning and during execution of movements, and reconciles seemingly contradictory results about cue-integration behavior in navigation. Taken together, the present work provides a parsimonious explanation of how patterns of human goal-directed navigation behavior arise from the continuous and dynamic interactions of spatial uncertainties in perception, cognition, and action.

A Review of Brain-Inspired Cognition and Navigation Technology for Mobile Robots

Brain-inspired navigation technologies combine environmental perception, spatial cognition, and target navigation to create a comprehensive navigation research system. Researchers have used various sensors to gather environmental data and enhance environmental perception using multimodal information fusion. In spatial cognition, a neural network model is used to simulate the navigation mechanism of the animal brain and to construct an environmental cognition map. However, existing models face challenges in achieving high navigation success rate and efficiency. In addition, the limited incorporation of navigation mechanisms borrowed from animal brains necessitates further exploration. On the basis of the brain-inspired navigation process, this paper launched a systematic study on brain-inspired environment perception, brain-inspired spatial cognition, and goal-based navigation in brain-inspired navigation, which provides a new classification of brain-inspired cognition and navigation techniques and a theoretical basis for subsequent experimental studies. In the future, brain-inspired navigation technology should learn from more perfect brain-inspired mechanisms to improve its generalization ability and be simultaneously applied to large-scale distributed intelligent body cluster navigation. The multidisciplinary nature of brain-inspired navigation technology presents challenges, and multidisciplinary scholars must cooperate to promote the development of this technology.

Working memory and reward increase the accuracy of animal location encoding in the medial prefrontal cortex

The ability to perceive spatial environments and locate oneself during navigation is crucial for the survival of animals. Mounting evidence suggests a role of the medial prefrontal cortex (mPFC) in spatially related behaviors. However, the properties of mPFC spatial encoding and how it is influenced by animal behavior are poorly defined. Here, we train the mice to perform 3 tasks differing in working memory and reward-seeking: a delayed non-match to place (DNMTP) task, a passive alternation (PA) task, and a free-running task. Single-unit recording in the mPFC shows that although individual mPFC neurons exhibit spatially selective firing, they do not reliably represent the animal location. The population activity of mPFC neurons predicts the animal location. Notably, the population coding of animal locations by the mPFC is modulated by animal behavior in that the coding accuracy is higher in tasks involved in working memory and reward-seeking. This study reveals an approach whereby the mPFC encodes spatial positions and the behavioral variables affecting it.

State transitions in the statistically stable place cell population correspond to rate of perceptual change

The hippocampus occupies a central role in mammalian navigation and memory. Yet an understanding of the rules that govern the statistics and granularity of the spatial code, as well as its interactions with perceptual stimuli, is lacking. We analyzed CA1 place cell activity recorded while rats foraged in different large-scale environments. We found that place cell activity was subject to an unexpected but precise homeostasis—the distribution of activity in the population as a whole being constant at all locations within and between environments. Using a virtual reconstruction of the largest environment, we showed that the rate of transition through this statistically stable population matches the rate of change in the animals’ visual scene. Thus, place fields near boundaries were small but numerous, while in the environment’s interior, they were larger but more dispersed. These results indicate that hippocampal spatial activity is governed by a small number of simple laws and, in particular, suggest the presence of an information-theoretic bound imposed by perception on the fidelity of the spatial memory system.

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Neural activity in rodent CA1 place cell populations is homeostatically balanced

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Hippocampal place field size and frequency are governed by proximity to boundaries

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Transition rate through place cell population matches rate of change in visual scene

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.

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.

Combinatorial Neural Inhibition for Stimulus Selection across Space

The ability to select the most salient among competing stimuli is essential for animal behavior and operates no matter which spatial locations stimuli happen to occupy. We provide evidence that the brain employs a combinatorially optimized inhibition strategy for selection across all pairs of stimulus locations. With experiments in a key inhibitory nucleus in the vertebrate midbrain selection network, called isthmi pars magnocellularis (Imc) in owls, we discovered that Imc neurons encode visual space with receptive fields that have multiple excitatory hot spots (“lobes“). Such multilobed encoding is necessitated by scarcity of Imc neurons. Although distributed seemingly randomly, the locations of these lobes are optimized across the high-firing Imc neurons, allowing them to combinatorially solve selection across space. This strategy minimizes metabolic and wiring costs, a principle that also accounts for observed asymmetries between azimuthal and elevational coding. Combinatorially optimized inhibition may be a general neural principle for efficient stimulus selection.

Mahajan et al. show that a sparse set of midbrain inhibitory neurons encodes visual space with unusual multilobed receptive fields. This results in a combinatorially optimized solution for selection at all pairs of stimulus locations, which minimizes metabolic and neural wiring costs.

Path integration maintains spatial periodicity of grid cell firing in a 1D circular track

Entorhinal grid cells are thought to provide a 2D spatial metric of the environment. In this study we demonstrate that in a familiar 1D circular track (i.e., a continuous space) grid cells display a novel 1D equidistant firing pattern based on integrated distance rather than travelled distance or time. In addition, field spacing is increased compared to a 2D open field, probably due to a reduced access to the visual cue in the track. This metrical modification is accompanied by a change in LFP theta oscillations, but no change in intrinsic grid cell rhythmicity, or firing activity of entorhinal speed and head-direction cells. These results suggest that in a 1D circular space grid cell spatial selectivity is shaped by path integration processes, while grid scale relies on external information.

In an open field, the preferential firing of grid cells on a hexagonal lattice is formed by integrating external as well as self-motion cues. Here, the authors show that on a 1D circular track, path integration cues shape the spatial selectivity of grid cells while external cues determine the scale of the grid.

Efficient neural decoding of self-location with a deep recurrent network

Place cells in the mammalian hippocampus signal self-location with sparse spatially stable firing fields. Based on observation of place cell activity it is possible to accurately decode an animal’s location. The precision of this decoding sets a lower bound for the amount of information that the hippocampal population conveys about the location of the animal. In this work we use a novel recurrent neural network (RNN) decoder to infer the location of freely moving rats from single unit hippocampal recordings. RNNs are biologically plausible models of neural circuits that learn to incorporate relevant temporal context without the need to make complicated assumptions about the use of prior information to predict the current state. When decoding animal position from spike counts in 1D and 2D-environments, we show that the RNN consistently outperforms a standard Bayesian approach with either flat priors or with memory. In addition, we also conducted a set of sensitivity analysis on the RNN decoder to determine which neurons and sections of firing fields were the most influential. We found that the application of RNNs to neural data allowed flexible integration of temporal context, yielding improved accuracy relative to the more commonly used Bayesian approaches and opens new avenues for exploration of the neural code.

Being able to accurately self-localize is critical for most motile organisms. In mammals, place cells in the hippocampus appear to be a central component of the brain network responsible for this ability. In this work we recorded the activity of a population of hippocampal neurons from freely moving rodents and carried out neural decoding to determine the animals’ locations. We found that a machine learning approach using recurrent neural networks (RNNs) allowed us to predict the rodents’ true positions more accurately than a standard Bayesian method with flat priors (i.e. maximum likelihood estimator, MLE) as well as a Bayesian approach with memory (i.e. with priors informed by past activity). The RNNs are able to take into account past neural activity without making assumptions about the statistics of neuronal firing. Further, by analyzing the representations learned by the network we were able to determine which neurons, and which aspects of their activity, contributed most strongly to the accurate decoding.

Spatial cell firing during virtual navigation of open arenas by head-restrained mice

We present a mouse virtual reality (VR) system which restrains head-movements to horizontal rotations, compatible with multi-photon imaging. This system allows expression of the spatial navigation and neuronal firing patterns characteristic of real open arenas (R). Comparing VR to R: place and grid, but not head-direction, cell firing had broader spatial tuning; place, but not grid, cell firing was more directional; theta frequency increased less with running speed, whereas increases in firing rates with running speed and place and grid cells' theta phase precession were similar. These results suggest that the omni-directional place cell firing in R may require local-cues unavailable in VR, and that the scale of grid and place cell firing patterns, and theta frequency, reflect translational motion inferred from both virtual (visual and proprioceptive) and real (vestibular translation and extra-maze) cues. By contrast, firing rates and theta phase precession appear to reflect visual and proprioceptive cues alone.

Robust and efficient coding with grid cells

The neuronal code arising from the coordinated activity of grid cells in the rodent entorhinal cortex can uniquely represent space across a large range of distances, but the precise conditions for optimal coding capacity are known only for environments with finite size. Here we consider a coding scheme that is suitable for unbounded environments, and present a novel, number theoretic approach to derive the grid parameters that maximise the coding range in the presence of noise. We derive an analytic upper bound on the coding range and provide examples for grid scales that achieve this bound and hence are optimal for encoding in unbounded environments. We show that in the absence of neuronal noise, the capacity of the system is extremely sensitive to the choice of the grid periods. However, when the accuracy of the representation is limited by neuronal noise, the capacity quickly becomes more robust against the choice of grid scales as the number of modules increases. Importantly, we found that the capacity of the system is near optimal even for random scale choices already for a realistic number of grid modules. Our study demonstrates that robust and efficient coding can be achieved without parameter tuning in the case of grid cell representation and provides a solid theoretical explanation for the large diversity of the grid scales observed in experimental studies. Moreover, we suggest that having multiple grid modules in the entorhinal cortex is not only required for the exponentially large coding capacity, but is also a prerequisite for the robustness of the system.

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

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

On Decoding Grid Cell Population Codes Using Approximate Belief Propagation

Neural systems are inherently noisy. One well-studied example of a noise reduction mechanism in the brain is the population code, where representing a variable with multiple neurons allows the encoded variable to be recovered with fewer errors. Studies have assumed ideal observer models for decoding population codes, and the manner in which information in the neural population can be retrieved remains elusive. This letter addresses a mechanism by which realistic neural circuits can recover encoded variables. Specifically, the decoding problem of recovering a spatial location from populations of grid cells is studied using belief propagation. We extend the belief propagation decoding algorithm in two aspects. First, beliefs are approximated rather than being calculated exactly. Second, decoding noises are introduced into the decoding circuits. Numerical simulations demonstrate that beliefs can be effectively approximated by combining polynomial nonlinearities with divisive normalization. This approximate belief propagation algorithm is tolerant to decoding noises. Thus, this letter presents a realistic model for decoding neural population codes and investigates fault-tolerant information retrieval mechanisms in the brain.

Modelling effects on grid cells of sensory input during self-motion

The neural coding of spatial location for memory function may involve grid cells in the medial entorhinal cortex, but the mechanism of generating the spatial responses of grid cells remains unclear. This review describes some current theories and experimental data concerning the role of sensory input in generating the regular spatial firing patterns of grid cells, and changes in grid cell firing fields with movement of environmental barriers. As described here, the influence of visual features on spatial firing could involve either computations of self-motion based on optic flow, or computations of absolute position based on the angle and distance of static visual cues. Due to anatomical selectivity of retinotopic processing, the sensory features on the walls of an environment may have a stronger effect on ventral grid cells that have wider spaced firing fields, whereas the sensory features on the ground plane may influence the firing of dorsal grid cells with narrower spacing between firing fields. These sensory influences could contribute to the potential functional role of grid cells in guiding goal-directed navigation.

Neural mechanisms of self-location

The ability to self-localise and to navigate to remembered goals in complex and changeable environments is crucial to the survival of many mobile species. Electrophysiological investigations of the mammalian hippocampus and associated brain structures have identified several classes of neurons which represent information about an organism's position and orientation. These include place cells, grid cells, head direction cells, and boundary vector cells, as well as cells representing aspects of self-motion. Understanding how these neural representations are formed and updated from environmental sensory information and from information relating to self-motion is an important topic attracting considerable current interest. Here we review the computational mechanisms thought to underlie the formation of these different spatial representations, the interactions between them, and their use in guiding behaviour. These include some of the clearest examples of computational mechanisms of general interest to neuroscience, such as attractor dynamics, temporal coding and multi-modal integration. We also discuss the close relationships between computational modelling and experimental research which are driving progress in this area.

A principle of economy predicts the functional architecture of grid cells

Grid cells in the brain respond when an animal occupies a periodic lattice of ‘grid fields’ during navigation. Grids are organized in modules with different periodicity. We propose that the grid system implements a hierarchical code for space that economizes the number of neurons required to encode location with a given resolution across a range equal to the largest period. This theory predicts that (i) grid fields should lie on a triangular lattice, (ii) grid scales should follow a geometric progression, (iii) the ratio between adjacent grid scales should be √e for idealized neurons, and lie between 1.4 and 1.7 for realistic neurons, (iv) the scale ratio should vary modestly within and between animals. These results explain the measured grid structure in rodents. We also predict optimal organization in one and three dimensions, the number of modules, and, with added assumptions, the ratio between grid periods and field widths.

DOI:http://dx.doi.org/10.7554/eLife.08362.001

In the 1930s, neuroscientists studying how rodents find their way through a maze proposed that the animals could construct an internal map of the maze inside their heads. The map was thought to enable the animals to navigate between familiar locations and also to identify shortcuts and alternative routes whenever familiar ones were blocked.

In the 1960s, recordings of electrical activity in the rat brain provided the first clues as to which nerve cells form this spatial map. In a region of the brain called the hippocampus, nerve cells called ‘place cells’ are active whenever the rat finds itself in a specific location. However, place cells alone are not able to support all types of navigation. Some spatial tasks also require cells in a region of the brain called the medial entorhinal cortex (MEC), which supplies most of the information that the hippocampus receives.

Cells in the MEC called ‘grid cells’ represent two-dimensional space as a repeating grid of triangles. A given grid cell is activated if the animal is located at a particular distance and angle away from the center of any of these triangles. The size of the triangles in these grids varies systematically throughout the MEC. Individual grid cells at one end of the structure encode space in finer detail than grid cells at the opposite end.

Wei et al. have now used mathematical modeling to explore how grid cells are organized. The model assumes that the brain seeks to encode space at whatever resolution an animal requires using as few nerve cells as possible. The model successfully reproduces several known features of grid cells, including the triangular shape of the grid, and the fact that the size of the triangles increases in steps of a specific size across the MEC.

In addition to providing a mathematical basis for the way that grid cells are organized in the brain, the model makes a number of testable predictions. These include predictions of the number of grid cells in the rat brain, as well as the pattern that grid cells adopt in three-dimensions: a question that is currently being studied in bats. Wei et al.'s findings suggest that the code used by the grid to represent space is an analog of a decimal number system—except that space is not subdivided by factors of 10 to form decimal ‘digits’, but by a quantity related to a famous constant in the field of mathematics called Euler's number.

DOI:http://dx.doi.org/10.7554/eLife.08362.002

Controlling phase noise in oscillatory interference models of grid cell firing

Oscillatory interference models account for the spatial firing properties of grid cells in terms of neuronal oscillators with frequencies modulated by the animal's movement velocity. The phase of such a "velocity-controlled oscillator" (VCO) relative to a baseline (theta-band) oscillation tracks displacement along a preferred direction. Input from multiple VCOs with appropriate preferred directions causes a grid cell's grid-like firing pattern. However, accumulating phase noise causes the firing pattern to drift and become corrupted. Here we show how multiple redundant VCOs can automatically compensate for phase noise. By entraining the baseline frequency to the mean VCO frequency, VCO phases remain consistent, ensuring a coherent grid pattern and reducing its spatial drift. We show how the spatial stability of grid firing depends on the variability in VCO phases, e.g., a phase SD of 3 ms per 125 ms cycle results in stable grids for 1 min. Finally, coupling N VCOs with similar preferred directions as a ring attractor, so that their relative phases remain constant, produces grid cells with consistently offset grids, and reduces VCO phase variability of the order square root of N. The results suggest a viable functional organization of the grid cell network, and highlight the benefit of integrating displacement along multiple redundant directions for the purpose of path integration.

A hybrid oscillatory interference/continuous attractor network model of grid cell firing

Grid cells in the rodent medial entorhinal cortex exhibit remarkably regular spatial firing patterns that tessellate all environments visited by the animal. Two theoretical mechanisms that could generate this spatially periodic activity pattern have been proposed: oscillatory interference and continuous attractor dynamics. Although a variety of evidence has been cited in support of each, some aspects of the two mechanisms are complementary, suggesting that a combined model may best account for experimental data. The oscillatory interference model proposes that the grid pattern is formed from linear interference patterns or "periodic bands" in which velocity-controlled oscillators integrate self-motion to code displacement along preferred directions. However, it also allows the use of symmetric recurrent connectivity between grid cells to provide relative stability and continuous attractor dynamics. Here, we present simulations of this type of hybrid model, demonstrate that it generates intracellular membrane potential profiles that closely match those observed in vivo, addresses several criticisms aimed at pure oscillatory interference and continuous attractor models, and provides testable predictions for future empirical studies.

Space in the brain: how the hippocampal formation supports spatial cognition

Over the past four decades, research has revealed that cells in the hippocampal formation provide an exquisitely detailed representation of an animal's current location and heading. These findings have provided the foundations for a growing understanding of the mechanisms of spatial cognition in mammals, including humans. We describe the key properties of the major categories of spatial cells: place cells, head direction cells, grid cells and boundary cells, each of which has a characteristic firing pattern that encodes spatial parameters relating to the animal's current position and orientation. These properties also include the theta oscillation, which appears to play a functional role in the representation and processing of spatial information. Reviewing recent work, we identify some themes of current research and introduce approaches to computational modelling that have helped to bridge the different levels of description at which these mechanisms have been investigated. These range from the level of molecular biology and genetics to the behaviour and brain activity of entire organisms. We argue that the neuroscience of spatial cognition is emerging as an exceptionally integrative field which provides an ideal test-bed for theories linking neural coding, learning, memory and cognition.