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An argument for hyperbolic geometry in neural circuits

An argument for hyperbolic geometry in neural circuits

In her 2019 review, Tatyana O Sharpee connects several lines of research to argue that hyperbolic geometry should be broadly applicable to neural circuits as well as other biological circuits. Networks with hyperbolic geometry are most sensitive to both internal and external disturbances, which explains why. Moreover, these networks enable effective communication whether nodes are added or withdrawn.

The well-known Zipf’s rule, which is also commonly referred to as the Pareto distribution and asserts that the likelihood of observing a certain pattern is inversely linked to its rank, is another characteristic of hyperbolic geometry, according to the author. Several biological systems, including protein sequences, brain networks, and economics, exhibit Zipf’s law. These discoveries give more evidence for the universality of networks having an underlying hyperbolic metric structure.

Hyperbolic spaces approximate tree-like hierarchical networks. Poincare half space model (A) and Poincare disk (B) provide two different (but equivalent) ways for representing hyperbolic spaces in the Euclidean plane. Both representation capture exponential expansion of states produced by tree-like networks near the space boundary. This boundary is planar in (A) and circular in (B). In both cases, the lines of shortest distance (red) are curved, unlike in the Euclidean space where straight lines yield the shortest paths between points. Note that these lines approximately traces the “phylogenetic” distances between nodes. In (A) colored dots describe nodes of a network. Background squares provide discrete version of a Poincare half-space model of a hyperbolic space. (B) Points are sampled randomly from their expected density. Node position reflects their expected degree. Hub-like nodes are assigned coordinates closer to the disk center. The number of nodes increases exponentially with radius whereas their expected degree decreases exponentially. This combination leads to power-law distribution of network degrees

A three-dimensional hyperbolic space is relevant for neuronal signaling, according to recent discoveries in neuroscience. Compared to other dimensions, the three-dimensional hyperbolic space could offer more resilience. The paper provides an example of how the olfactory system’s new topographic arrangement was discovered using hyperbolic coordinates. The adoption of such coordinates could make it easier for pertinent signals to be represented elsewhere in the brain.

The link between hyperbolic geometry, Zipf’ law and maximally informative representations clarifies why hyperbolic geometry is simultaneously relevant for both the olfactory system and word distribution. In the case of olfaction, plants and animals have to produce chemical signals that will be discernible by other animals.

Tatyana O. Sharpee

Figure 2, which depicts the embedding of monomolecular odorants into a three-dimensional hyperbolic space (Poincare ball) where distances between scents represent how correlated they are across various fruit samples, serves as an illustration of this issue. This picture was created by embedding the tomato and strawberry samples’ aromas into a three-dimensional hyperbolic space using non-metric multi-dimensional scaling (nMDS). A topological study that ruled out Euclidean and spherical geometries and suggested that points placed near the surface of a hyperbolic Poincare ball were compatible with the observed distances was the first to discover the dimensionality of the space and its metric. A nMDS with a hyperbolic metric was used to find the precise distribution of points once the space parameters had been determined.

Across different systems, emergent evidence suggests that both the neural activity and behavioral outputs are described by low-dimensional smooth manifolds.

Maximally informative representations lead to hyperbolic geometry. In neural coding and information processing, the goal is to find efficient ways to represent and process information. This is particularly challenging when dealing with high-dimensional and hierarchical data, such as images or language. The paper argues that hyperbolic geometry provides a natural framework for efficient representation and processing of such data.

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One key advantage of hyperbolic geometry is that it allows for more compact representations of high-dimensional data. In Euclidean space, the volume of a ball increases exponentially with its radius, making it difficult to represent high-dimensional data in a compact way. In hyperbolic space, however, the volume of a ball increases only linearly with its radius, allowing for more efficient representation of high-dimensional data.

Moreover, hyperbolic geometry naturally supports hierarchical structures, which are common in many complex systems. For example, in natural language, words can be organized into a hierarchy based on their semantic relationships.

An argument for hyperbolic geometry in neural circuits, Tatyana O. Sharpee

Published: October 2019, Current Opinion in Neurobiology
Volume 58, Pages 101-104
DOI: 10.1016/j.conb.2019.07.008

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