We present a minimalistic representation model for the head direction (HD) system, aiming to learn a high-dimensional representation of head direction that captures essential properties of HD cells. Our model is a representation of rotation group $U(1)$, and we study both the fully connected version and convolutional version. We demonstrate the emergence of Gaussian-like tuning profiles and a 2D circle geometry in both versions of the model. We also demonstrate that the learned model is capable of accurate path integration.

This paper investigates the conformal isometry hypothesis as a potential explanation for the emergence of hexagonal periodic patterns in the response maps of grid cells. The hypothesis posits that the activities of the population of grid cells form a high-dimensional vector in the neural space, representing the agent's self-position in 2D physical space. As the agent moves in the 2D physical space, the vector rotates in a 2D manifold in the neural space, driven by a recurrent neural network. The conformal isometry hypothesis proposes that this 2D manifold in the neural space is a conformally isometric embedding of the 2D physical space, in the sense that local displacements of the vector in neural space are proportional to local displacements of the agent in the physical space. Thus the 2D manifold forms an internal map of the 2D physical space, equipped with an internal metric. In this paper, we conduct numerical experiments to show that this hypothesis underlies the hexagon periodic patterns of grid cells. We also conduct theoretical analysis to further support this hypothesis. In addition, we propose a conformal modulation of the input velocity of the agent so that the recurrent neural network of grid cells satisfies the conformal isometry hypothesis automatically. To summarize, our work provides numerical and theoretical evidences for the conformal isometry hypothesis for grid cells and may serve as a foundation for further development of normative models of grid cells and beyond.

The responses of the population of grid cells collectively form a vector in a high-dimensional neural activity space, and this vector represents the self-position of the agent in the 2D physical space. As the agent moves, the vector is transformed by a recurrent neural network that takes the velocity of the agent as input. In this paper, we propose a simple and general conformal normalization of the input velocity for the recurrent neural network, so that the local displacement of the position vector in the high-dimensional neural space is proportional to the local displacement of the agent in the 2D physical space, regardless of the direction of the input velocity. Our numerical experiments on the minimally simple linear and non-linear recurrent networks show that conformal normalization leads to the emergence of the hexagon grid patterns. Furthermore, we derive a new theoretical understanding that connects conformal normalization to the emergence of hexagon grid patterns in navigation tasks.

The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of the recurrent transformation as a representation of self-motion. Geometrically, we study the conformal isometry of the Lie group representation where the local displacement of the activity vector in the neural space is proportional to the local displacement of the agent in the 2D physical space. Topologically, the compact abelian Lie group representation automatically leads to the torus topology commonly assumed and observed in neuroscience. We then focus on a simple non-linear recurrent model that underlies the continuous attractor neural networks of grid cells. Our numerical experiments show that conformal isometry leads to hexagon periodic patterns in the grid cell responses and our model is capable of accurate path integration.

Experiments reveal that in the dorsal medial superior temporal (MSTd) and the ventral intraparietal (VIP) areas, where visual and vestibular cues are integrated to infer heading direction, there are two types of neurons with roughly the same number. One is “congruent” cells, whose preferred heading directions are similar in response to visual and vestibular cues; and the other is “opposite” cells, whose preferred heading directions are nearly “opposite” (with an offset of 180 degree) in response to visual vs. vestibular cues. Congruent neurons are known to be responsible for cue integration, but the computational role of opposite neurons remains largely unknown. Here, we propose that opposite neurons may serve to encode the disparity information between cues necessary for multisensory segregation. We build a computational model composed of two reciprocally coupled modules, MSTd and VIP, and each module consists of groups of congruent and opposite neurons. In the model, congruent neurons in two modules are reciprocally connected with each other in the congruent manner, whereas opposite neurons are reciprocally connected in the opposite manner. Mimicking the experimental protocol, our model reproduces the characteristics of congruent and opposite neurons, and demonstrates that in each module, the sisters of congruent and opposite neurons can jointly achieve optimal multisensory information integration and segregation. This study sheds light on our understanding of how the brain implements optimal multisensory integration and segregation concurrently in a distributed manner.

Psychophysical experiments have demonstrated that the brain integrates information from multiple sensory cues in a near Bayesian optimal manner. The present study proposes a novel mechanism to achieve this. We consider two reciprocally connected networks, mimicking the integration of heading direction information between the dorsal medial superior temporal (MSTd) and the ventral intraparietal (VIP) areas. Each network serves as a local estimator and receives an independent cue, either the visual or the vestibular, as direct input for the external stimulus. We find that positive reciprocal interactions can improve the decoding accuracy of each individual network as if it implements Bayesian inference from two cues. Our model successfully explains the experimental finding that both MSTd and VIP achieve Bayesian multisensory integration, though each of them only receives a single cue as direct external input. Our result suggests that the brain may implement optimal information integration distributively at each local estimator through the reciprocal connections between cortical regions.