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Algebraic signatures of convex and non-convex codes.

ABSTRACT: A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley-Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.

SUBMITTER: Curto C

PROVIDER: S-EPMC6750060 | biostudies-literature | 2019 Sep

REPOSITORIES: biostudies-literature

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Algebraic signatures of convex and non-convex codes.

Curto Carina C Gross Elizabeth E Jeffries Jack J Morrison Katherine K Rosen Zvi Z Shiu Anne A Youngs Nora N

Journal of pure and applied algebra 20181214 9

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley-Reisner ideal. Using the neural ideal together with its stand ...[more]

PMID: 31534273

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Project description:ALUs contribute to genetic diversity by altering DNA's linear sequence through retrotransposition, recombination and repair. ALUs also have the potential to form alternative non-B-DNA conformations such as Z-DNA, triplexes and quadruplexes that alter the read-out of information from the genome. I suggest here these structures enable the rapid reprogramming of cellular pathways to offset DNA damage and regulate inflammation. The experimental data supporting this form of genetic encoding is presented. ALU sequence motifs that form non-B-DNA conformations under physiological conditions are called flipons. Flipons are binary switches. They are dissipative structures that trade energy for information. By efficiently targeting cellular machines to active genes, flipons expand the repertoire of RNAs compiled from a gene. Their action greatly increases the informational capacity of linearly encoded genomes. Flipons are programmable by epigenetic modification, synchronizing cellular events by altering both chromatin state and nucleosome phasing. Different classes of flipon exist. Z-flipons are based on Z-DNA and modify the transcripts compiled from a gene. T-flipons are based on triplexes and localize non-coding RNAs that direct the assembly of cellular machines. G-flipons are based on G-quadruplexes and sense DNA damage, then trigger the appropriate protective responses. Flipon conformation is dynamic, changing with context. When frozen in one state, flipons often cause disease. The propagation of flipons throughout the genome by ALU elements represents a novel evolutionary innovation that allows for rapid change. Each ALU insertion creates variability by extracting a different set of information from the neighbourhood in which it lands. By elaborating on already successful adaptations, the newly compiled transcripts work with the old to enhance survival. Systems that optimize flipon settings through learning can adapt faster than with other forms of evolution. They avoid the risk of relying on random and irreversible codon rewrites.

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Algebraic signatures of convex and non-convex codes.

Publications

Algebraic signatures of convex and non-convex codes.

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