Algebraic topology applied to Neuroscience

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Eugene Wigner once famously talked about the "unreasonable effectiveness of mathematics" in describing the natural world. Today again we are seeing this in action in particular with regard to the description of the biological brain from the perspective of neuroscience. Researchers from the Blue Brain Project have used applied algebraic topology to better understand the connection between brain network connections relating to its function which has so far defied all other conventional mathematical techniques.

Reimann et al. 2017, Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously unseen topology of synaptic connectivity. The synaptic network contains an abundance of cliques of neurons bound into cavities that guide the emergence of correlated activity. In response to stimuli, correlated activity binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence toward peak complexity. We propose that the brain processes stimuli by forming increasingly complex functional cliques and cavities.

This link to a recent Nature review on network neuroscience explains how applied algebraic topology can be of use in describing neural networks from the perspective of simplicial complexes.

This thread is to serve as a collection of articles regarding the use of algebraic topology to understand the brain and other artificial neural networks. Discussion and input regarding the underlying mathematics (or related concepts/techniques) is indeed also welcome. For those who prefer video, Infinite Series also has an excellent three-part introduction to this topic:

Spoiler

 

[atyy]

Intriguing, thanks!

 

[WWGD]

Is this in anyway related to Persistent Homology? Informally: We model a data set as a Topological space and compute its Homology group. The traits that remain for several consecutive groups are considered to be signal and otherwise are considered noise?