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}
}

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

SHOWING 1-10 OF 41 REFERENCES

What Makes a Neural Code Convex?

TLDR
This work provides a complete characterization of local obstructions to convexity and defines max intersection-complete codes, a family guaranteed to have noLocal obstructions, a significant advance in understanding the intrinsic combinatorial properties of convex codes.

The Neural Ring: An Algebraic Tool for Analyzing the Intrinsic Structure of Neural Codes

TLDR
The main finding is that the neural ring and a related neural ideal can be expressed in a “canonical form” that directly translates to a minimal description of the receptive field structure intrinsic to the code, providing the groundwork for inferring stimulus space features from neural activity alone.

Neural Ring Homomorphisms and Maps Between Neural Codes

TLDR
This work considers maps between neural codes and the associated homomorphisms of their neural rings and characterize all code maps corresponding to neural ring homomorphicisms as compositions of five elementary code maps.

Combinatorial Neural Codes from a Mathematical Coding Theory Perspective

TLDR
It is suggested that a compromise in error-correcting capability may be a necessary price to pay for a neural code whose structure serves not only error correction, but must also reflect relationships between stimuli.

A No-Go Theorem for One-Layer Feedforward Networks

TLDR
It is not possible to infer a computational role for recurrent connections from the combinatorics of neural response patterns alone, suggesting that recurrent or many-layer feedforward architectures are not necessary for shaping the (coarse) combinatorial features of neural codes.

Cell Groups Reveal Structure of Stimulus Space

TLDR
It is found that simply knowing which groups of cells fire together reveals a surprising amount of structure in the underlying stimulus space; this may enable the brain to construct its own internal representations.

Generating and drawing area-proportional Euler and Venn diagrams

  • S. Chow
  • Mathematics, Computer Science
  • 2007
TLDR
A graph-theoretic model of an Euler diagram's structure is described and this model is used to develop necessary-and-sufficient existence conditions and to prove that the Euler Diagram Generation Problem (EDGP) is NP-complete.

General Euler Diagram Generation

TLDR
A method for Euler diagram generation is described, demonstrated by implemented software, and the advances in methodology via the production of diagrams which were difficult or impossible to draw using previous approaches are illustrated.

Dimensional Analysis Using Toric Ideals: Primitive Invariants

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.