Gröbner Bases of Neural Ideals

Garcia, Rebecca; Puente, Luis David García; Kruse, Ryan C.; Liu, Jessica; Miyata, Dane; Petersen, Ethan; Phillipson, Kaitlyn; Shiu, Anne

doi:10.48550/arxiv.1612.05660

Cited by 2 publications

(5 citation statements)

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“…In this work we proved that not only are the converses of the Type 4-6 relations in [2] false as stated, but the converses of the Type 5 and 6 relations are also false even when the neural ideal J C is replaced by the larger ideal I(C). However, our modified versions of the Type 4-6 relations are if-and-only-if statements at the level of both J C and I(C).…”

Section: Discussionmentioning

confidence: 66%

“…Therefore, it remains to prove the forward implications. We do this by essentially repeating the proofs in [2].…”

Section: Modified Typementioning

confidence: 99%

“…Later, Garcia et al discovered three more such relations, known as the Type 4-6 relations, but proved only one direction of the relations [2]. It is therefore natural to ask whether the converses of any of the Type 4-6 relations might also hold.…”

Section: Introductionmentioning

confidence: 99%

“…In this section we introduce neural codes, pseudo-monomials, and neural ideals, as well as the prior results on which our work is based. Our notation matches that in [1,2]. We begin with neural codes.…”

Section: Introductionmentioning

confidence: 99%

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Strengthening Relationships between Neural Ideals and Receptive Fields

Morvant

2018

Preprint

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Neural codes are collections of binary vectors that represent the firing patterns of neurons.The information given by a neural code C can be represented by its neural ideal J C . In turn, the polynomials in J C can be used to determine the relationships among the receptive fields of the neurons. In a paper by Curto et al., three such relationships, known as the Type 1-3 relations, were linked to the neural ideal by three if-and-only-if statements. Later, Garcia et al.discovered the Type 4-6 relations. These new relations differed from the first three in that they were related to J C by one-way implications. In this paper, we first show that the converses of these new implications are false at the level of both the neural ideal J C and the larger ideal I(C) of a code. We then present modified statements of these relations that, like the first three, can be related by if-and-only-if statements to both J C and I(C). Using the modified relations, we uncover a new relationship involving J C , I(C), and the Type 1-6 relations.

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Section: Discussionmentioning

confidence: 66%

“…Therefore, it remains to prove the forward implications. We do this by essentially repeating the proofs in [2].…”

Section: Modified Typementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Strengthening Relationships between Neural Ideals and Receptive Fields

Morvant

2018

Preprint

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“…There has been much recent work on determining which neural codes can arise in this way, and figuring out how to reconstruct data about the ambient space from this information; see [3,4,5,2,10,8]. To this end, the following tool was introduced.…”

Section: Neural Rings Setmentioning

confidence: 99%

Polarization of neural rings

2019

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The "neural code" is the way the brain characterizes, stores, and processes information. Unraveling the neural code is a key goal of mathematical neuroscience. Topology, coding theory, and, recently, commutative algebra are some the mathematical areas that are involved in analyzing these codes. Neural rings and ideals are algebraic objects that create a bridge between mathematical neuroscience and commutative algebra. A neural ideal is an ideal in a polynomial ring that encodes the combinatorial firing data of a neural code. Using some algebraic techniques one hopes to understand more about the structure of a neural code via neural rings and ideals. In this paper, we introduce an operation, called "polarization," that allows us to relate neural ideals with squarefree monomial ideals, which are very well studied and known for their nice behavior in commutative algebra.2010 Mathematics Subject Classification. 13F20, 13P25, 13L99, 92B20.

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Gröbner Bases of Neural Ideals

Cited by 2 publications

References 0 publications

Strengthening Relationships between Neural Ideals and Receptive Fields

Strengthening Relationships between Neural Ideals and Receptive Fields

Polarization of neural rings

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