Toroidal topology of population activity in grid cells

Abstract

The medial entorhinal cortex is part of a neural system for mapping the position of an individual within a physical environment1. Grid cells, a key component of this system, fire in a characteristic hexagonal pattern of locations2, and are organized in modules3 that collectively form a population code for the animal’s allocentric position1. The invariance of the correlation structure of this population code across environments4,5 and behavioural states6,7, independent of specific sensory inputs, has pointed to intrinsic, recurrently connected continuous attractor networks (CANs) as a possible substrate of the grid pattern1,8–11. However, whether grid cell networks show continuous attractor dynamics, and how they interface with inputs from the environment, has remained unclear owing to the small samples of cells obtained so far. Here, using simultaneous recordings from many hundreds of grid cells and subsequent topological data analysis, we show that the joint activity of grid cells from an individual module resides on a toroidal manifold, as expected in a two-dimensional CAN. Positions on the torus correspond to positions of the moving animal in the environment. Individual cells are preferentially active at singular positions on the torus. Their positions are maintained between environments and from wakefulness to sleep, as predicted by CAN models for grid cells but not by alternative feedforward models12. This demonstration of network dynamics on a toroidal manifold provides a population-level visualization of CAN dynamics in grid cells.

Excitatory-Inhibitory Recurrent Dynamics Produce Robust Visual Grids and Stable Attractors

“…The optical flow can be implemented in the early stage of visual system (Wang et al 1989;Wurtz 1998), and the heading direction can be estimated from optical flow in the visual cortex (Lappe and Rauschecker 1992) or sensorimotor circuit (Leinweber et al 2017). Furthermore, our E/I-RNN models reveal torus-like manifolds and attractors in a low-dimensional space, consistent with the other experimental and computational findings (Gardner et al 2022;Sorscher et al 2020). Interestingly, multistable torus-like manifolds may emerge from the E/I-RNN with increasing inter-or intra-subnetwork connectivity, suggesting that sparsely connected biological networks may use multiple ring attractors to store information.…”

“…In both cases, the population activity was confined to lie close to a 2D manifold regardless the initial condition. Alternatively, we constructed the 3D embedding of N -dimensional population activity using a two-step hybrid dimensionality reduction procedure (Gardner et al 2022): linear PCA (with the first 6 PCs) followed by a nonlinear dimensionality reduction method known as UMAP (Uniform Manifold Approximation and Projection). This visualization step revealed a 3D torus-like structure (Figure 4C for Setup #2; see Figure S5 for more results).…”

Functional network topography of the medial entorhinal cortex

“…The segregation of OV and grid cells might thereby reflect the different affordances of underlying network computations. These involve, in the case of grid cells, the formation of internally generated spatial firing patterns anchored to the overall layout of the environment the animal is in (20,54) and, in the case of OV cells, the formation of firing fields that anchor to and move flexibly with objects within that same environment (24). Although we were not able to resolve the exact number and location of grid or OV territories in this study, future studies using technological advances, such as 2P miniscopes with even larger FOVs (46), will extend the anatomically observable areas and thereby help to elucidate these large-scale organizational principles further.…”

Learnable latent embeddings for joint behavioral and neural analysis

“…Although there are a wealth of excellent tools aimed at inferring spike trains from calcium data, currently the pseudo-R 2 of algorithms on paired spiking and calcium data tops out at around 0.6 (31). Nonetheless, it is clear that recording with either modality has lead to similar global conclusions-for example, grid cells can be uncovered in spiking or calcium signals (32,33), reward prediction errors can be found in dopamine neurons across species and recording modalities (34)(35)(36), and visual cortex shows orientation tuning across species and modalities (37)(38)(39).…”

“…Maximum distance considered for filtration was set to infinity. To decide the number of co-cycles in each co-homology dimension with a significant lifespan, we trained 500 CEBRA embeddings with shuffled labels, similar to the approach in (33). We took the maximum lifespan of each dimension across these 500 runs as a threshold to determine robust Betti numbers.…”