Can Grid Cell Ensembles Represent Multiple Spaces?

Abstract

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.

Utility of Environmental Complexity as a Predictor of Alzheimer’s Disease Diagnosis: A Big-Data Machine Learning Approach

“…Multiple cognitive spatial maps and sub-maps provide reference frames for subdivisions of our environments, each of which includes a collection of landmarks, geometric relationships, and additional non-geometric properties (13). The fragment fitting theory posits that the brain rotates and assembles selected sub-maps to form a local map during navigation (14,15). As such, a complex urban environment may offer daily opportunities for building and operating mental maps to stimulate these spatial cells.…”

Grid cells, border cells, and discrete complex analysis

“…An independently developed notion of discrete conformal transformations, based on rearrangements of regular circle packings in planar domains (Köbe, 1936 ; Thurston, 1985 ; Rodin and Sullivan, 1987 ; Bücking, 2008 ) may therefore offer a complementary venue for establishing correspondences between network activity and discrete-complexity. Several recent experimental (Savelli et al, 2008 ; Zhang et al, 2014 ; Krupic et al, 2016 , 2018 ; Savelli et al, 2017 ; Wernle et al, 2018 ; Bellmund et al, 2020 ) and theoretical (Urdapilleta et al, 2015 ; Santos-Pata et al, 2017 ; Spalla et al, 2019 ; Monsalve-Mercado and Leibold, 2020 ; Zhang et al, 2023 ) studies suggest that conformal transformations of the navigated spaces may induce compensatory discrete-conformal transformations of the grid field maps, similar to how the hippocampal place cells tend to preserve coactivity patterns in morphing environments (Gothard et al, 1996 ; Dabaghian et al, 2014 ; Rueckemann et al, 2021 ). If the latter is verified experimentally, it can be argued that the grid cell inputs constrain the hippocampal topological map (Dabaghian et al, 2014 ; Rueckemann et al, 2021 ), to a net conformal map of the navigated space.…”

Partial coherence and frustration in self-organizing spherical grids

“…Partial coherence, together with the realization that a network of grid cells may be endowed with a significant storage capacity (Spalla et al, 2019), in a sense brings back grid cells to the fold of memory systems, next to the place cells, with their multiple charts (Samsonovich and McNaughton, 1997;Battaglia and Treves, 1998). For years, it has been thought that grid cells may afford long-distance path integration (Fuhs and Touretzky, 2006;McNaughton et al, 2006).…”