Neural ideals and stimulus space visualization

@article{Gross2018NeuralIA,
  title={Neural ideals and stimulus space visualization},
  author={Elizabeth Gross and Nida Kazi Obatake and Nora Youngs},
  journal={Adv. Appl. Math.},
  year={2018},
  volume={95},
  pages={65-95}
}
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18 Citations
Highly Influential Citations
3
Background Citations
11
Methods Citations
7
Results Citations
1

18 Citations

Toric Ideals, Polytopes, and Convex Neural Codes

TLDR
This work works with a special class of convex codes, known as inductively pierced codes, and seeks to identify these codes through the Gröbner bases of their toric ideals.

On the identification of $k$-inductively pierced codes using toric ideals

TLDR
The toric ideal of a code is used to show sufficient conditions for a code to be 1- or 2-inductively pierced, so that the existing algorithm to draw realizations of such codes can be used to draw Euler diagrams.

Geometry, combinatorics, and algebra of inductively pierced codes

TLDR
All inductively pierced codes are nondegenerate convex codes and nondEGenerate hyperplane codes, and it is proved that a $k-inductively pierced code on $n$ neurons has a convex realization with balls in $\mathbb R^{k+1}$ and with half spaces in $n}$.

Universal Gröbner bases of toric ideals of combinatorial neural codes

TLDR
This article examines a variety of classes of combinatorial neural codes by identifying universal Grobner bases of the toric ideal for these codes.

Neural codes, decidability, and a new local obstruction to convexity

TLDR
Giusti and Itskov prove that convex neural codes have no "local obstructions," which are defined via the topology of a code's simplicial complex, and reveal a stronger type of local obstruction that prevents a code from being convex, and prove that the corresponding decision problem is NP-hard.

State polytopes related to two classes of combinatorial neural codes

  • Robert Davis
  • Computer Science, Mathematics
    Advances in Applied Mathematics
  • 2018

Inductively pierced codes and neural toric ideals

TLDR
All inductively pierced codes are nondegenerate convex codes and nondEGenerate hyperplane codes, and it is proved that a $k-inductively pierced code on $n$ neurons has a convex realization with balls in $\mathbb R^{k+1}$ and with half spaces in $n}$.

Gröbner bases of neural ideals

TLDR
It is proved that if the canonical form of a neural ideal is a Gr\"obner basis, then it is the universal Gr\"OBner basis (that is, the union of all reduced Gr \"obner bases).

Analysis of Combinatorial Neural Codes: An Algebraic Approach

Morphisms of Neural Codes

  • R. Jeffs
  • Computer Science
    SIAM J. Appl. Algebra Geom.
  • 2020
TLDR
It is shown that morphisms can be used to remove redundant information from a code, and that Morphisms preserve convexity, and this fact leads to define "minimally non-convex" codes.

References

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TLDR
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TLDR
A selection of computer algebra packages are used to show the considerable ease with which both a simple basis and a Graver basis can be found within the use of toric ideal theory from algebraic geometry.