## Involve: A Journal of Mathematics

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

### Sparse neural codes and convexity

R. Amzi Jeffs, Mohamed Omar, Natchanon Suaysom, Aleina Wachtel, and Nora Youngs

#### 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 ${\mathbb{R}}^{d}$. 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 ${\mathbb{R}}^{2}$ and ${\mathbb{R}}^{3}$, 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**

Received: 26 October 2017

Revised: 24 October 2018

Accepted: 5 December 2018

First available in Project Euclid: 29 May 2019

**Permanent link to this document**

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

**Subjects**

Primary: 05C62: Graph representations (geometric and intersection representations, etc.) For graph drawing, see also 68R10 52A10: Convex sets in 2 dimensions (including convex curves) [See also 53A04] 92B20: Neural networks, artificial life and related topics [See also 68T05, 82C32, 94Cxx]

**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