Path integration is essential for spatial navigation. Experimental studies have identified neural correlates for path integration, but exactly how the neural system accomplishes this computation remains unresolved. Here, we adopt recurrent neural networks (RNNs) trained to perform a path integration task to explore this issue. After training, we borrow neuroscience prior knowledge and methods to unfold the black box of the trained model, including: clarifying neuron types based on their receptive fields, dissecting information flows between neuron groups by pruning their connections, and analyzing internal dynamics of neuron groups using the attractor framework. Intriguingly, we uncover a hierarchical information processing pathway embedded in the RNN model, along which velocity information of an agent is first forwarded to band cells, band and grid cells then coordinate to carry out path integration, and finally grid cells output the agent location. Inspired by the RNN-based study, we construct a neural circuit model, in which band cells form one-dimensional (1D) continuous attractor neural networks (CANNs) and serve as upstream neurons to support downstream grid cells to carry out path integration in the 2D space. Our study challenges the conventional view of considering grid cells as the principal velocity integrator, and supports a neural circuit model with the hierarchy of band and grid cells.

Grid cells fire in triangular grid patterns, while place cells fire at specific locations and respond to contextual cues. How do these interacting systems support not only spatial encoding but also internally driven path planning, such as navigating to locations recalled from cues? Here, we propose a system-level theory of MEC-HC wiring that explains how grid and place cell patterns could be connected to enable cue-triggered goal retrieval, path planning, and reconstruction of sensory experience along planned routes. We suggest that place cells autoassociate sensory inputs with grid cell patterns, allowing sensory cues to trigger recall of goal-location grid patterns. We show analytically that grid-based planning permits shortcuts through unvisited locations and generalizes local transitions to long-range paths. During planning, intermediate grid states trigger place cell pattern completion, reconstructing sensory experiences along the route. Using a single-layer RNN modeling the HC-MEC loop with a planning subnetwork, we demonstrate these effects in both biologically grounded navigation simulations using RatatouGym and visually realistic navigation tasks using Habitat Sim. Codes for experiments, simulations, and vision encoder are available at 1,2,3.

Path integration is essential for spatial navigation. Experimental studies have identified neural correlates for path integration, but exactly how the neural system accomplishes this computation remains unresolved. Here, we adopt recurrent neural networks (RNNs) trained to perform a path integration task to explore this issue. After training, we borrow neuroscience prior knowledge and methods to unfold the black box of the trained model, including: clarifying neuron types based on their receptive fields, dissecting information flows between neuron groups by pruning their connections, and analyzing internal dynamics of neuron groups using the attractor framework. Intriguingly, we uncover a hierarchical information processing pathway embedded in the RNN model, along which velocity information of an agent is first forwarded to band cells, band and grid cells then coordinate to carry out path integration, and finally grid cells output the agent location. Inspired by the RNN-based study, we construct a neural circuit model, in which band cells form one-dimensional (1D) continuous attractor neural networks (CANNs) and serve as upstream neurons to support downstream grid cells to carry out path integration in the 2D space. Our study challenges the conventional view of considering grid cells as the principal velocity integrator, and supports a neural circuit model with the hierarchy of band and grid cells.

Squared Wasserstein distance is a frequently used tool to measure discrepancy between probability distributions. This distance is typically computed between empirical measures of size $n$ from two underlying random samples. Unfortunately, even in lower dimensional Euclidean space problems $\left( d \in \{2,3\} \right)$, algorithms for Wasserstein distance computation with approximate or exact precision guarantees scale poorly in the runtime as a function of $n$ and the desired precision. In response, we consider the computational-statistical runtime, where the goal is to estimate from samples the Wasserstein distance between potentially smooth measures up to $ε$-additive error in expectation with respect to the sampling; we allow $O(1)$ computational cost for collecting a sample. Towards this, we develop a Sample-Sketch-Solve paradigm where we introduce a regular cartesian grid sketch of the samples. We show that (especially under $α$-Hölder smooth distributions) this can compress the data without increasing asymptotic error, and also regularizes the structure which enables faster exact algorithms. Ultimately, we approximate $W_2^2(P,Q)$ within $ε$ error in $ε^{-\max(2,\frac{d+1+o(1)}{1+α})}$ time for $0 < α< 1$ Hölder smooth distributions $P,Q$ on $(0,1)^{d}$; an optimal $Θ(ε^{-2})$ for $α> 1/2$ when $d=2$ and nearly optimal as $α\to 1$ when $d = 3$.

Causal graph discovery for space-time systems is challenging in high-dimensional gridded data, which often has many more grid cells than temporal observations per cell. The Causal Space-Time Stencil Learning (CaStLe) meta-algorithm was developed to address that niche under space-time locality and stationarity assumptions, but it is currently limited to univariate analyses. In this work, we present M-CaStLe. M-CaStLe generalizes the local embedding and parent-identification phases of CaStLe to jointly model local within-variable and cross-variable space-time causal structures in gridded data. Like CaStLe, by constraining candidate parents to a constant-size space-time neighborhood and pooling spatial replicates, M-CaStLe increases effective sample size to make discovery tractable in high-dimensional settings. We further decompose the resulting multivariate stencil graph into reaction and spatial graphs to aid interpretation in complex settings. We study M-CaStLe in four settings: a multivariate space-time vector autoregression benchmark with known ground truth, an advective-diffusive-reaction partial differential equation verification problem with derived physical reference structure, an atmospheric chemistry case study in a low-temporal-sample regime, and an El Niño Southern Oscillation study on reanalysis data, identifying phase-dependent ocean--atmosphere coupling. Across these settings, M-CaStLe more accurately recovers multivariate causal structure in controlled settings and identifies important physical dynamics in real-world case studies. Overall, M-CaStLe advances causal discovery for multivariate space-time systems while retaining interpretability at the grid level.

Understanding how grid cells perform path integration calculations remains a fundamental problem. In this paper, we conduct theoretical analysis of a general representation model of path integration by grid cells, where the 2D self-position is encoded as a higher dimensional vector, and the 2D self-motion is represented by a general transformation of the vector. We identify two conditions on the transformation. One is a group representation condition that is necessary for path integration. The other is an isotropic scaling condition that ensures locally conformal embedding, so that the error in the vector representation translates conformally to the error in the 2D self-position. Then we investigate the simplest transformation, i.e., the linear transformation, uncover its explicit algebraic and geometric structure as matrix Lie group of rotation, and explore the connection between the isotropic scaling condition and a special class of hexagon grid patterns. Finally, with our optimization-based approach, we manage to learn hexagon grid patterns that share similar properties of the grid cells in the rodent brain. The learned model is capable of accurate long distance path integration.

Grid cells in the mammalian brain are fundamental to spatial navigation, and therefore crucial to how animals perceive and interact with their environment. Traditionally, grid cells are thought support path integration through highly symmetric hexagonal lattice firing patterns. However, recent findings show that their firing patterns become distorted in the presence of significant spatial landmarks such as rewarded locations. This introduces a novel perspective of dynamic, subjective, and action-relevant interactions between spatial representations and environmental cues. Here, we propose a practical and theoretical framework to quantify and explain these interactions.

Grid cells in the medial entorhinal cortex create remarkable periodic maps of explored space during navigation. Recent studies show that they form similar maps of abstract cognitive spaces. Examples of such abstract environments include auditory tone sequences in which the pitch is continuously varied or images in which abstract features are continuously deformed (e.g., a cartoon bird whose legs stretch and shrink).

Together, grid cells' activity within a grid cell module forms a 2D torus [9, 20].