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. To this end, we train path-integrating recurrent neural networks (piRNNs) on a spatial navigation task, whose goal is to predict the agent's position with a special focus on rewarded locations. Grid-like neurons naturally emerge from the training of piRNNs, which allows us to investigate how the two aspects of the task, space and reward, are integrated in their firing patterns. We find that geometry, but not topology, of the grid cell population code becomes distorted. Surprisingly, these distortions are global in the firing patterns of the grid cells despite local changes in the reward. Our results indicate that after training with location-specific reward information, the preserved representational topology supports successful path integration, whereas the emergent heterogeneity in individual responses due to global distortions may encode dynamically changing environmental cues. By bridging the gap between computational models and the biological reality of spatial navigation under reward information, we offer new insights into how neural systems prioritize environmental landmarks in their spatial navigation code.

Success in collaborative and competitive environments, where agents must work with or against each other, requires individuals to encode the position and trajectory of themselves and others. Decades of neurophysiological experiments have shed light on how brain regions [e.g., medial entorhinal cortex (MEC), hippocampus] encode the self's position and trajectory. However, it has only recently been discovered that MEC and hippocampus are modulated by the positions and trajectories of others. To understand how encoding spatial information of multiple agents shapes neural representations, we train a recurrent neural network (RNN) model that captures properties of MEC to path integrate trajectories of two agents simultaneously navigating the same environment. We find significant differences between these RNNs and those trained to path integrate only a single agent. At the individual unit level, RNNs trained to path integrate more than one agent develop weaker grid responses, stronger border responses, and tuning for the relative position of the two agents. At the population level, they develop more distributed and robust representations, with changes in network dynamics and manifold topology. Our results provide testable predictions and open new directions with which to study the neural computations supporting spatial navigation.