Multiscale codes in the nervous system: the problem of noise correlations and the ambiguity of periodic scales

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.

The Structure of Hippocampal CA1 Interactions Optimizes Spatial Coding across Experience

Although much is known about how single neurons in the hippocampus represent an animal's position, how circuit interactions contribute to spatial coding is less well understood. Using a novel statistical estimator and theoretical modeling, both developed in the framework of maximum entropy models, we reveal highly structured CA1 cell-cell interactions in male rats during open field exploration. The statistics of these interactions depend on whether the animal is in a familiar or novel environment. In both conditions the circuit interactions optimize the encoding of spatial information, but for regimes that differ in the informativeness of their spatial inputs. This structure facilitates linear decodability, making the information easy to read out by downstream circuits. Overall, our findings suggest that the efficient coding hypothesis is not only applicable to individual neuron properties in the sensory periphery, but also to neural interactions in the central brain.SIGNIFICANCE STATEMENT Local circuit interactions play a key role in neural computation and are dynamically shaped by experience. However, measuring and assessing their effects during behavior remains a challenge. Here, we combine techniques from statistical physics and machine learning to develop new tools for determining the effects of local network interactions on neural population activity. This approach reveals highly structured local interactions between hippocampal neurons, which make the neural code more precise and easier to read out by downstream circuits, across different levels of experience. More generally, the novel combination of theory and data analysis in the framework of maximum entropy models enables traditional neural coding questions to be asked in naturalistic settings.

The chicken and egg problem of grid cells and place cells

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

Efficient and flexible representation of higher-dimensional cognitive variables with grid cells

We shed light on the potential of entorhinal grid cells to efficiently encode variables of dimension greater than two, while remaining faithful to empirical data on their low-dimensional structure. Our model constructs representations of high-dimensional inputs through a combination of low-dimensional random projections and "classical" low-dimensional hexagonal grid cell responses. Without reconfiguration of the recurrent circuit, the same system can flexibly encode multiple variables of different dimensions while maximizing the coding range (per dimension) by automatically trading-off dimension with an exponentially large coding range. It achieves high efficiency and flexibility by combining two powerful concepts, modularity and mixed selectivity, in what we call "mixed modular coding". In contrast to previously proposed schemes, the model does not require the formation of higher-dimensional grid responses, a cell-inefficient and rigid mechanism. The firing fields observed in flying bats or climbing rats can be generated by neurons that combine activity from multiple grid modules, each representing higher-dimensional spaces according to our model. The idea expands our understanding of grid cells, suggesting that they could implement a general circuit that generates on-demand coding and memory states for variables in high-dimensional vector spaces.

Untethered firing fields and intermittent silences: Why grid-cell discharge is so variable

Grid cells in medial entorhinal cortex are notoriously variable in their responses, despite the striking hexagonal arrangement of their spatial firing fields. Indeed, when the animal moves through a firing field, grid cells often fire much more vigorously than predicted or do not fire at all. The source of this trial-to-trial variability is not completely understood. By analyzing grid-cell spike trains from mice running in open arenas and on linear tracks, we characterize the phenomenon of "missed" firing fields using the statistical theory of zero inflation. We find that one major cause of grid-cell variability lies in the spatial representation itself: firing fields are not as strongly anchored to spatial location as the averaged grid suggests. In addition, grid fields from different cells drift together from trial to trial, regardless of whether the environment is real or virtual, or whether the animal moves in light or darkness. Spatial realignment across trials sharpens the grid representation, yielding firing fields that are more pronounced and significantly narrower. These findings indicate that ensembles of grid cells encode relative position more reliably than absolute position.

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.

Information-Theoretical Analysis of the Neural Code in the Rodent Temporal Lobe

In the study of the neural code, information-theoretical methods have the advantage of making no assumptions about the probabilistic mapping between stimuli and responses. In the sensory domain, several methods have been developed to quantify the amount of information encoded in neural activity, without necessarily identifying the specific stimulus or response features that instantiate the code. As a proof of concept, here we extend those methods to the encoding of kinematic information in a navigating rodent. We estimate the information encoded in two well-characterized codes, mediated by the firing rate of neurons, and by the phase-of-firing with respect to the theta-filtered local field potential. In addition, we also consider a novel code, mediated by the delta-filtered local field potential. We find that all three codes transmit significant amounts of kinematic information, and informative neurons tend to employ a combination of codes. Cells tend to encode conjunctions of kinematic features, so that most of the informative neurons fall outside the traditional cell types employed to classify spatially-selective units. We conclude that a broad perspective on the candidate stimulus and response features expands the repertoire of strategies with which kinematic information is encoded.

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.

Ten Years of Grid Cells

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

Representation of memories in the cortical-hippocampal system: Results from the application of population similarity analyses

Here we consider the value of neural population analysis as an approach to understanding how information is represented in the hippocampus and cortical areas and how these areas might interact as a brain system to support memory. We argue that models based on sparse coding of different individual features by single neurons in these areas (e.g., place cells, grid cells) are inadequate to capture the complexity of experience represented within this system. By contrast, population analyses of neurons with denser coding and mixed selectivity reveal new and important insights into the organization of memories. Furthermore, comparisons of the organization of information in interconnected areas suggest a model of hippocampal-cortical interactions that mediates the fundamental features of memory.

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

Mammalian grid cells fire when an animal crosses the points of an imaginary hexagonal grid tessellating the environment. We show how animals can navigate by reading out a simple population vector of grid cell activity across multiple spatial scales, even though neural activity is intrinsically stochastic. This theory of dead reckoning explains why grid cells are organized into discrete modules within which all cells have the same lattice scale and orientation. The lattice scale changes from module to module and should form a geometric progression with a scale ratio of around 3/2 to minimize the risk of making large-scale errors in spatial localization. Such errors should also occur if intermediate-scale modules are silenced, whereas knocking out the module at the smallest scale will only affect spatial precision. For goal-directed navigation, the allocentric grid cell representation can be readily transformed into the egocentric goal coordinates needed for planning movements. The goal location is set by nonlinear gain fields that act on goal vector cells. This theory predicts neural and behavioral correlates of grid cell readout that transcend the known link between grid cells of the medial entorhinal cortex and place cells of the hippocampus.

Using Grid Cells for Navigation

Mammals are able to navigate to hidden goal locations by direct routes that may traverse previously unvisited terrain. Empirical evidence suggests that this “vector navigation” relies on an internal representation of space provided by the hippocampal formation. The periodic spatial firing patterns of grid cells in the hippocampal formation offer a compact combinatorial code for location within large-scale space. Here, we consider the computational problem of how to determine the vector between start and goal locations encoded by the firing of grid cells when this vector may be much longer than the largest grid scale. First, we present an algorithmic solution to the problem, inspired by the Fourier shift theorem. Second, we describe several potential neural network implementations of this solution that combine efficiency of search and biological plausibility. Finally, we discuss the empirical predictions of these implementations and their relationship to the anatomy and electrophysiology of the hippocampal formation.

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Grid cells (GCs) are believed to provide a path integration input to place cells

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However, GCs also provide a powerful context-independent metric for large-scale space

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Hence, we show how GCs can be used for vector navigation between arbitrary locations

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We simulate various neural implementations and make testable experimental predictions

Visual Decisions in the Presence of Measurement and Stimulus Correlations

Humans and other animals base their decisions on noisy sensory input. Much work has been devoted to understanding the computations that underlie such decisions. The problem has been studied in a variety of tasks and with stimuli of differing complexity. However, how the statistical structure of stimuli, along with perceptual measurement noise, affects perceptual judgments is not well understood. Here we examine how correlations between the components of a stimulus-stimulus correlations-together with correlations in sensory noise, affect decision making. As an example, we consider the task of detecting the presence of a single or multiple targets among distractors. We assume that both the distractors and the observer's measurements of the stimuli are correlated. The computations of an optimal observer in this task are nontrivial yet can be analyzed and understood intuitively. We find that when distractors are strongly correlated, measurement correlations can have a strong impact on performance. When distractor correlations are weak, measurement correlations have little impact unless the number of stimuli is large. Correlations in neural responses to structured stimuli can therefore have a strong impact on perceptual judgments.

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

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

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

Correlations and functional connections in a population of grid cells

We study the statistics of spike trains of simultaneously recorded grid cells in freely behaving rats. We evaluate pairwise correlations between these cells and, using a maximum entropy kinetic pairwise model (kinetic Ising model), study their functional connectivity. Even when we account for the covariations in firing rates due to overlapping fields, both the pairwise correlations and functional connections decay as a function of the shortest distance between the vertices of the spatial firing pattern of pairs of grid cells, i.e. their phase difference. They take positive values between cells with nearby phases and approach zero or negative values for larger phase differences. We find similar results also when, in addition to correlations due to overlapping fields, we account for correlations due to theta oscillations and head directional inputs. The inferred connections between neurons in the same module and those from different modules can be both negative and positive, with a mean close to zero, but with the strongest inferred connections found between cells of the same module. Taken together, our results suggest that grid cells in the same module do indeed form a local network of interconnected neurons with a functional connectivity that supports a role for attractor dynamics in the generation of grid pattern.

The way mammals navigate in space is hypothesized to depend on neural structures in the temporal lobe including the hippocampus and medial entorhinal cortex (MEC). In particular, grid cells, neurons whose firing is mostly restricted to regions of space that form a hexagonal pattern, are believed to be an important part of this circuitry. Despite several years of work, not much is known about the correlated activity of neurons in the MEC and how grid cells are functionally coupled to each other. Here, we have taken a statistical approach to these questions and studied pairwise correlations and functional connections between simultaneously recorded grid cells. Through careful statistical analysis, we demonstrate that grid cells with nearby firing vertices tend to have positive effects on eliciting responses in each other, while those further apart tend to have inhibitory or no effects. Cells that respond similarly to manipulations of the environment are considered to belong to the same module. Cells belonging to a module have stronger interactions with each other than those in different modules. These results are consistent with and shed light on the population-based mechanisms suggested by models for the generation of grid cell firing.

Hippocampal remapping is constrained by sparseness rather than capacity

Grid cells in the medial entorhinal cortex encode space with firing fields that are arranged on the nodes of spatial hexagonal lattices. Potential candidates to read out the space information of this grid code and to combine it with other sensory cues are hippocampal place cells. In this paper, we investigate a population of grid cells providing feed-forward input to place cells. The capacity of the underlying synaptic transformation is determined by both spatial acuity and the number of different spatial environments that can be represented. The codes for different environments arise from phase shifts of the periodical entorhinal cortex patterns that induce a global remapping of hippocampal place fields, i.e., a new random assignment of place fields for each environment. If only a single environment is encoded, the grid code can be read out at high acuity with only few place cells. A surplus in place cells can be used to store a space code for more environments via remapping. The number of stored environments can be increased even more efficiently by stronger recurrent inhibition and by partitioning the place cell population such that learning affects only a small fraction of them in each environment. We find that the spatial decoding acuity is much more resilient to multiple remappings than the sparseness of the place code. Since the hippocampal place code is sparse, we thus conclude that the projection from grid cells to the place cells is not using its full capacity to transfer space information. Both populations may encode different aspects of space.

The mammalian brain represents space in the population of hippocampal place cells as well as in the population of medial entorhinal cortex grid cells. Since both populations are active at the same time, space information has to be synchronized between the two. Both brain areas are reciprocally connected, and it is unclear how the two codes influence each other. In this paper, we analyze a theoretical model of how a place code processes inputs from the grid cell population. The model shows that the sparseness of the place code poses a much stronger constraint than maximal information transfer. We thus conclude that the potentially high spatial acuity of the grid code cannot be efficiently conveyed to a sparse place cell population and thus propose that sparseness and spatial acuity are two independent objectives of the neuronal place representation.

Grid cells and cortical representation

One of the grand challenges in neuroscience is to comprehend neural computation in the association cortices, the parts of the cortex that have shown the largest expansion and differentiation during mammalian evolution and that are thought to contribute profoundly to the emergence of advanced cognition in humans. In this Review, we use grid cells in the medial entorhinal cortex as a gateway to understand network computation at a stage of cortical processing in which firing patterns are shaped not primarily by incoming sensory signals but to a large extent by the intrinsic properties of the local circuit.

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

We examined the accuracy with which the location of an agent moving within an environment could be decoded from the simulated firing of systems of grid cells. Grid cells were modelled with Poisson spiking dynamics and organized into multiple ‘modules’ of cells, with firing patterns of similar spatial scale within modules and a wide range of spatial scales across modules. The number of grid cells per module, the spatial scaling factor between modules and the size of the environment were varied. Errors in decoded location can take two forms: small errors of precision and larger errors resulting from ambiguity in decoding periodic firing patterns. With enough cells per module (e.g. eight modules of 100 cells each) grid systems are highly robust to ambiguity errors, even over ranges much larger than the largest grid scale (e.g. over a 500 m range when the maximum grid scale is 264 cm). Results did not depend strongly on the precise organization of scales across modules (geometric, co-prime or random). However, independent spatial noise across modules, which would occur if modules receive independent spatial inputs and might increase with spatial uncertainty, dramatically degrades the performance of the grid system. This effect of spatial uncertainty can be mitigated by uniform expansion of grid scales. Thus, in the realistic regimes simulated here, the optimal overall scale for a grid system represents a trade-off between minimizing spatial uncertainty (requiring large scales) and maximizing precision (requiring small scales). Within this view, the temporary expansion of grid scales observed in novel environments may be an optimal response to increased spatial uncertainty induced by the unfamiliarity of the available spatial cues.