The Shape of Experience and the Hyperbolic Geometry in the Brain’s Spatial Maps

Tatyana O. Sharpee’s work in the realm of geometric cognition—a field dedicated to understanding how the brain interprets and represents spatial information—has yielded groundbreaking insights into how our minds map the world around us. Continuing her previous argument for hyperbolic geometry in neural circuits, the recent paper, co-authored with Huanqiu Zhang, P. Dylan Rich, and Albert K. Lee, sheds light on the hyperbolic geometry of hippocampal spatial representations.

The hippocampus, a crucial part of the brain involved in memory and spatial navigation, houses ‘place cells’—neurons that fire when an animal is in a particular location. The geometry of these spatial representations, however, has remained largely unknown. Breaking new ground, Sharpee and her team reveal that the hippocampus does not represent space according to a linear geometry, as might be expected. Instead, they discovered a hyperbolic representation.

We investigated whether hyperbolic geometry underlies neural networks by analyzing responses of sets of neurons from the dorsal CA1 region of the hippocampus. This region is considered essential for spatial representation and trajectory planning

Imagine holding a map of your city. A linear representation would be akin to the map’s scale, where an inch on paper corresponds to a fixed number of miles in reality. A hyperbolic representation, in contrast, changes the scale depending on where you are on the map. The implications of this discovery are profound. A hyperbolic representation provides more positional information than a linear one, potentially aiding in complex navigation tasks.

This hyperbolic representation isn’t static—it dynamically expands with experience. As an animal spends more time exploring its environment, the spatial representations in its brain expand. The expansion is proportional to the logarithm of time spent exploring, suggesting our brains continually refine our spatial maps based on our experiences. As we spend more time in an area, our mental map of that area becomes more detailed and expansive. This dynamic updating could be crucial for efficiently navigating familiar environments.

Fig. 1 | Construction of hierarchical organization of place cell responses that reflects underlying hyperbolic geometry. a, A Poincaré disk model of 2D hyperbolic geometry is shown for visualization of its similarity to a tree structure. Each curve represents the geodesic between the two connected points, and all triangles have the same size. b, Illustration of the construction of hierarchical representation from neuronal response properties. The tree structure does not have to be perfect to allow for mapping onto a hyperbolic geometry. Some loops can be present (dashed lines) due to partial overlap between disks of neurons from different orders in the hierarchy. c, Place field size versus location of 264 place fields from 63 putative pyramidal cells from dorsal CA1 of a rat running on a 48-m-long linear track (Fig. 3b and Supplementary Fig. 1)20. d, Histogram of place field sizes shown in c.

Fig. 7 | Schematic illustration of how increased hyperbolic radius affects neural representation of space. As radius grows over time, exponential distribution of
place field sizes shifts toward smaller place fields (Extended Data Fig. 7d) or larger depth in a discrete tree structure (Eq. 1), to improve spatial localization.

The study also shows that the size of the hippocampal representation matches the optimal predictions for the number of CA1 neurons. This suggests that the brain might optimize its spatial representations to maximize information while minimizing resource usage—a striking example of the brain’s efficiency.

In addition to illuminating the complex geometry of our spatial understanding, these findings shed light on the role of experience in shaping our perceptions. It seems our experiences don’t just create memories—they also shape our spatial understanding, helping us navigate the world with greater accuracy and efficiency.

This work also opens up new avenues for exploring how different areas of the brain contribute to our spatial understanding. While this study focused on the CA1 region of the hippocampus, future research could explore how other regions, like the entorhinal cortex or the prefrontal cortex, contribute to our spatial maps. It’s a vast, intricate puzzle that Sharpee and her colleagues are helping us solve, piece by piece.

Importantly, these findings could also have significant implications for our understanding of certain neurological conditions. Conditions like Alzheimer’s disease and epilepsy often involve disruptions in the hippocampus, which can lead to disorientation and difficulties with spatial navigation. Understanding how the hippocampus represents space could therefore be a crucial step towards developing better treatments for these conditions.

In the broader context, Sharpee’s work underscores the beauty and complexity of the brain. It’s a testament to the brain’s remarkable adaptability and the intricate, dynamic ways it represents the world around us. Far from being a passive recipient of information, the brain actively shapes our perceptions, molding our understanding of space in response to our experiences.

The study also raises fascinating questions for future research. For instance, how quickly do these hyperbolic representations form? Are there individual differences in the formation and expansion of these representations? What role do other sensory inputs, like sight or smell, play in shaping these maps?

While the paper answers some questions, it opens up many more—signifying a true scientific advancement. It moves us a step closer to understanding the complexities of the brain and the enigmatic process of cognition.

Sharpee’s research also has potential implications for fields like artificial intelligence and robotics. Just as understanding bird flight has inspired aeronautical engineering, understanding the brain’s spatial representations could inspire new algorithms for machine learning and autonomous navigation. By learning how the brain represents space, we might be able to design more intelligent and adaptable machines.

Paper abstract

Daily experience suggests that we perceive distances near us linearly. However, the actual geometry of spatial representation in the brain is unknown. Here we report that neurons in the CA1 region of rat hippocampus that mediate spatial perception represent space according to a non-linear hyperbolic geometry. This geometry uses an exponential scale and yields greater positional information than a linear scale. We found that the size of the representation matches the optimal predictions for the number of CA1 neurons. The representations also dynamically expanded proportionally to the logarithm of time that the animal spent exploring the environment, in correspondence with the maximal mutual information that can be received. The dynamic changes tracked even small variations due to changes in the running speed of the animal. These results demonstrate how neural circuits
achieve efficient representations using dynamic hyperbolic geometry.

Hippocampal spatial representations exhibit a hyperbolic geometry that expands with experience, Huanqiu Zhang, P. Dylan Rich, Albert K. Lee & Tatyana O. Sharpee
Published: 29 December 2022
DOI: https://doi.org/10.1038/s41593-022-01212-4

Via: Hippocampal spatial representations exhibit a hyperbolic geometry that expands with experience