## Involve: A Journal of Mathematics

• Involve
• Volume 12, Number 5 (2019), 737-754.

### Sparse neural codes and convexity

#### Abstract

Determining how the brain stores information is one of the most pressing problems in neuroscience. In many instances, the collection of stimuli for a given neuron can be modeled by a convex set in $ℝd$. Combinatorial objects known as neural codes can then be used to extract features of the space covered by these convex regions. We apply results from convex geometry to determine which neural codes can be realized by arrangements of open convex sets. We restrict our attention primarily to sparse codes in low dimensions. We find that intersection-completeness characterizes realizable 2-sparse codes, and show that any realizable 2-sparse code has embedding dimension at most $3$. Furthermore, we prove that in $ℝ2$ and $ℝ3$, realizations of 2-sparse codes using closed sets are equivalent to those with open sets, and this allows us to provide some preliminary results on distinguishing which 2-sparse codes have embedding dimension at most $2$.

#### Article information

Source
Involve, Volume 12, Number 5 (2019), 737-754.

Dates
Revised: 24 October 2018
Accepted: 5 December 2018
First available in Project Euclid: 29 May 2019

https://projecteuclid.org/euclid.involve/1559095402

Digital Object Identifier
doi:10.2140/involve.2019.12.737

Mathematical Reviews number (MathSciNet)
MR3954293

Zentralblatt MATH identifier
07072551

Keywords
neural code sparse convexity

#### Citation

Jeffs, R. Amzi; Omar, Mohamed; Suaysom, Natchanon; Wachtel, Aleina; Youngs, Nora. Sparse neural codes and convexity. Involve 12 (2019), no. 5, 737--754. doi:10.2140/involve.2019.12.737. https://projecteuclid.org/euclid.involve/1559095402

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