Can Grid Cell Ensembles Represent Multiple Spaces?

The way grid cells represent space in the rodent brain has been a striking discovery, with theoretical implications still unclear. Unlike hippocampal place cells, which are known to encode multiple, environment-dependent spatial maps, grid cells have been widely believed to encode space through a single low-dimensional manifold, in which coactivity relations between different neurons are preserved when the environment is changed. Does it have to be so? Here, we compute, using two alternative mathematical models, the storage capacity of a population of grid-like units, embedded in a continuous attractor neural network, for multiple spatial maps. We show that distinct representations of multiple environments can coexist, as existing models for grid cells have the potential to express several sets of hexagonal grid patterns, challenging the view of a universal grid map. This suggests that a population of grid cells can encode multiple noncongruent metric relationships, a feature that could in principle allow a grid-like code to represent environments with a variety of different geometries and possibly conceptual and cognitive spaces, which may be expected to entail such context-dependent metric relationships.

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

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

Transitions between Spatial Attractors in Place-Cell Models

The spontaneous transitions between D-dimensional spatial maps in an attractor neural network are studied. Two scenarios for the transition from one map to another are found, depending on the level of noise: (i) through a mixed state, partly localized in both maps around positions where the maps are most similar, and (ii) through a weakly localized state in one of the two maps, followed by a condensation in the arrival map. Our predictions are confirmed by numerical simulations and qualitatively compared to recent recordings of hippocampal place cells during quick-environment-changing experiments in rats.

Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: collective motion of the activity

The dynamics of a neural model for hippocampal place cells storing spatial maps is studied. In the absence of external input, depending on the number of cells and on the values of control parameters (number of environments stored, level of neural noise, average level of activity, connectivity of place cells), a "clump" of spatially localized activity can diffuse or remains pinned due to crosstalk between the environments. In the single-environment case, the macroscopic coefficient of diffusion of the clump and its effective mobility are calculated analytically from first principles and corroborated by numerical simulations. In the multienvironment case the heights and the widths of the pinning barriers are analytically characterized with the replica method; diffusion within one map is then in competition with transitions between different maps. Possible mechanisms enhancing mobility are proposed and tested.

Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: phase diagram

We study the stable phases of an attractor neural network model, with binary units, for hippocampal place cells encoding one-dimensional (1D) or 2D spatial maps or environments. Different maps correspond to random allocations (permutations) of the place fields. Based on replica calculations we show that, below critical levels for the noise in the neural response and for the number of environments, the network activity is spatially localized in one environment. For high noise and loads the network activity extends over space, either uniformly or with spatial heterogeneities due to the crosstalk between the maps, and memory of environments is lost. Remarkably the spatially localized regime is very robust against the neural noise until it reaches its critical level. Numerical simulations are in excellent quantitative agreement with our theoretical predictions.