Sparse neural codes and convexity
| Autor: | R. Amzi Jeffs, Nora Youngs, Natchanon Suaysom, Aleina Wachtel, Mohamed Omar |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Closed set
sparse General Mathematics Dimension (graph theory) Open set Convex set 52A10 0102 computer and information sciences 01 natural sciences Convexity FOS: Mathematics Mathematics - Combinatorics 05C62 0101 mathematics Mathematics Discrete mathematics Convex geometry convexity 010102 general mathematics Regular polygon 010201 computation theory & mathematics 92C20 52A35 05C62 92B20 Embedding Combinatorics (math.CO) neural code |
| Zdroj: | Involve 12, no. 5 (2019), 737-754 |
| ISSN: | 1944-4184 1944-4176 |
| DOI: | 10.2140/involve.2019.12.737 |
| Popis: | 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 \emph{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 $2$. 13 pages, 10 figures |
| Databáze: | OpenAIRE |
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