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Max Intersection-Complete Codes Molly Hoch Wellesley College July 17, 2017 Molly Hoch (Wellesley College) Max -Complete Codes July 17, 2017 1 / 24 Motivation The 2014 Nobel Prize in Physiology or Medicine was awarded for the discovery

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Max Intersection-Complete Codes

Molly Hoch

Wellesley College

July 17, 2017

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 1 / 24

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Motivation

The 2014 Nobel Prize in Physiology or Medicine was awarded for the discovery of place cells and grid cells

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 2 / 24

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Motivation

The 2014 Nobel Prize in Physiology or Medicine was awarded for the discovery of place cells and grid cells Place cells represent an animal’s location Multiple place cells can fire at once

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 2 / 24

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Notation

Definition

A neural code C on n neurons is a set of subsets of [n]. Given n neurons, we build neural codes from their respective receptive fields, living in Rd. The receptive field of a neuron i is denoted Ui.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 3 / 24

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Notation

Definition

A neural code C on n neurons is a set of subsets of [n]. Given n neurons, we build neural codes from their respective receptive fields, living in Rd. The receptive field of a neuron i is denoted Ui. On 5 neurons, one codeword could be {2,4}; this is where the receptive fields U2 and U4 overlap; we write this as 24.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 3 / 24

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Neural Codes and Convexity

We call a code convex if all the receptive fields from which it is built are convex.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 4 / 24

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Neural Codes and Convexity

We call a code convex if all the receptive fields from which it is built are convex. Certain types of codes are known to be convex, notably max intersection-complete codes.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 4 / 24

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Types of Codes

Definition

A code C is intersection-complete if all intersections of its codewords are present in the code.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 5 / 24

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Types of Codes

Definition

A code C is intersection-complete if all intersections of its codewords are present in the code.

Definition

A code C is max intersection-complete if all intersections of its facets (codewords maximal up to inclusion) are present in the code.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 5 / 24

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Types of Codes

Definition

A code C is intersection-complete if all intersections of its codewords are present in the code.

Definition

A code C is max intersection-complete if all intersections of its facets (codewords maximal up to inclusion) are present in the code.

Definition

The maximal code on n neurons is Cmax(n) = {σ : σ ⊆ [n]}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 5 / 24

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Max Intersection-Complete Codes

U1 U2 U3 U4

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 6 / 24

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Max Intersection-Complete Codes

1

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Max Intersection-Complete Codes

1 12 1 1

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Max Intersection-Complete Codes

1 12 1 1 13 123

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Max Intersection-Complete Codes

1 12 1 1 13 123 124 14

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 7 / 24

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13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅}

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

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13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅} Intersection-complete?

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

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13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅} Intersection-complete? No!

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

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13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅} Intersection-complete? No! Max intersection-complete?

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

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13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅} Intersection-complete? No! Max intersection-complete? Yes!

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

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Neural Ideals

From a neural code C, we obtain its neural ideal JC, defined to be JC :=

  • i∈σ

xi

  • j∈τ

(1 − xj) : σ / ∈ C, τ = [n] − σ.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 9 / 24

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Neural Ideals

From a neural code C, we obtain its neural ideal JC, defined to be JC :=

  • i∈σ

xi

  • j∈τ

(1 − xj) : σ / ∈ C, τ = [n] − σ. In our example, 24 is not a codeword of C, so x2x4(1 + x1)(1 + x3) ∈ JC

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 9 / 24

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The Canonical Form

The canonical form CF(JC) of a neural ideal consists of the minimal pseudomonomials with respect to divisibility present in the neural ideal.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 10 / 24

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The Canonical Form

The canonical form CF(JC) of a neural ideal consists of the minimal pseudomonomials with respect to divisibility present in the neural ideal. The canonical form has three types of elements, but we focus on only two: Type 1 relations:

i xi

Type 2 relations:

i xi

  • j (1 − xj)

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 10 / 24

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The Canonical Form

The canonical form CF(JC) of a neural ideal consists of the minimal pseudomonomials with respect to divisibility present in the neural ideal. The canonical form has three types of elements, but we focus on only two: Type 1 relations:

i xi

Type 2 relations:

i xi

  • j (1 − xj)

If a Type 1 relation xa1 . . . xan is in the CF of JC, then the codeword c = a1 . . . an is not in C, nor is any codeword containing c. If a Type 2 relation xa1 . . . xan(1 − xb1) . . . (1 − xbm) is in the CF, then

  • i∈{a1,...,an}

Ui ⊆

  • j∈{b1,...,bm}

Uj

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 10 / 24

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Canonical Form Example

Recall our code C = {123, 124, 12, 14, 13, ∅}. Here, CF(JC) = {x2(1−x1), x3(1−x1), x4(1−x1), x3x4, x1(1−x2)(1−x3)(1−x4)}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 11 / 24

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Canonical Form Example

Recall our code C = {123, 124, 12, 14, 13, ∅}. Here, CF(JC) = {x2(1−x1), x3(1−x1), x4(1−x1), x3x4, x1(1−x2)(1−x3)(1−x4)}. Because x3x4 ∈ CF(JC), we can’t have 34 ∈ C, nor can we have 134, 234,

  • r 1234 ∈ C.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 11 / 24

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Canonical Form Example

Recall our code C = {123, 124, 12, 14, 13, ∅}. Here, CF(JC) = {x2(1−x1), x3(1−x1), x4(1−x1), x3x4, x1(1−x2)(1−x3)(1−x4)}. Because x3x4 ∈ CF(JC), we can’t have 34 ∈ C, nor can we have 134, 234,

  • r 1234 ∈ C.

Further, an element like x2(1 − x1) tells us that U2 ⊆ U1.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 11 / 24

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Canonical Form Example

Similarly, because x1(1 − x2)(1 − x3)(1 − x4) ∈ CF(JC), we have that U1 ⊆ U2 ∪ U3 ∪ U4.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 12 / 24

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Canonical Form Example

Similarly, because x1(1 − x2)(1 − x3)(1 − x4) ∈ CF(JC), we have that U1 ⊆ U2 ∪ U3 ∪ U4. 123 124 12 13 14

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 12 / 24

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An Existing Signature for Intersection-Complete Codes

The following theorem gives a signature in the canonical form for intersection-complete codes:

Theorem (Curto, Gross, et al. 2015)

A code C is intersection-complete if and only if CF(JC) contains only monomials and pseudomonomials of the form (1 − xj)

i xi.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 13 / 24

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Question

Research Question

Does there exist a signature in the canonical form for maximum intersection-complete codes?

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 14 / 24

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Finding the Facets

We have been able to develop an algorithm for finding the facets of a code C from CF(JC). We use the fact that if a monomial appears in CF(JC) then no codeword containing the indices of that monomial appears in C.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 15 / 24

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Example of Facet Algorithm

Recall our earlier example: C = {123, 124, 12, 13, 14, ∅}. The only monomial in CF(JC) is x3x4. On 4 neurons, Cmax = {∅, 1, 2, 3, 4, 12, 13, 14, 23, 24, 34, 123, 124, 134, 234, 1234}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 16 / 24

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Example of Facet Algorithm

Recall our earlier example: C = {123, 124, 12, 13, 14, ∅}. The only monomial in CF(JC) is x3x4. Removing all codewords eliminated by this monomial gives us C′

max = {∅, 1, 2, 3, 4, 12, 13, 14, 23, 24, 34, 123, 124, 134, 234, 1234}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 17 / 24

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Example of Facet Algorithm

Recall our earlier example: C = {123, 124, 12, 13, 14, ∅}. The only monomial in CF(JC) is x3x4. This leaves us with C′

max = {∅, 1, 2, 3, 4, 12, 13, 14, 23, 24, 123, 124}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 18 / 24

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Example of Facet Algorithm

Recall our earlier example: C = {123, 124, 12, 13, 14, ∅}. The only monomial in CF(JC) is x3x4. This leaves us with C′

max = {∅, 1, 2, 3, 4, 12, 13, 14, 23, 24, 123, 124}.

We see that the facets of C and our reduced C′

max are the same.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 18 / 24

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Sufficient Condition for Non-maximality

Proposition (Franke-H)

Let C be a neural code, JC be its neural ideal, and CF(JC) be the corresponding canonical form. If there exist τ ⊂ [n] and σ ⊆ [n] − τ such that

i∈τ xi ∈ CF(JC) and j∈σ xj

  • i∈τ (1 − xi) ∈ CF(JC), then C is not

convex.

Corollary

For a code to be max intersection-complete, it cannot have the above condition.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 19 / 24

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Example

Let C = {4, 5, 1234, 1235, ∅} be a code on five neurons. The canonical form contains both x4x5 and x1(1 − x4)(1 − x5). This tells us that U4 ∩ U5 = ∅

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 20 / 24

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Example

Let C = {4, 5, 1234, 1235, ∅} be a code on five neurons. The canonical form contains both x4x5 and x1(1 − x4)(1 − x5). This tells us that U4 ∩ U5 = ∅ U4 U5

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 20 / 24

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Example

Let C = {4, 5, 1234, 1235, ∅} be a code on five neurons. The canonical form contains both x4x5 and x1(1 − x4)(1 − x5). This tells us that U4 ∩ U5 = ∅, and that U1 ⊆ U4 ∪ U5.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 21 / 24

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Example

Let C = {4, 5, 1234, 1235, ∅} be a code on five neurons. The canonical form contains both x4x5 and x1(1 − x4)(1 − x5). This tells us that U4 ∩ U5 = ∅, and that U1 ⊆ U4 ∪ U5. U4 U5 U1

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 21 / 24

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An Algorithm for Determining Missing Codewords

  • 1. Pick a complex pseudomonomial

This is a pseudomonomial with multiple (1 − xj) factors, e.g., xi(1 − xj1) . . . (1 − xjm). Write ∩k∈[m]ijk = i.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 22 / 24

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An Algorithm for Determining Missing Codewords

  • 1. Pick a complex pseudomonomial

This is a pseudomonomial with multiple (1 − xj) factors, e.g., xi(1 − xj1) . . . (1 − xjm). Write ∩k∈[m]ijk = i.

  • 2. Add “equivalent” neurons

Neurons which always fire together are equivalent, e.g. if xi(1 − xj) and xj(1 − xi) ∈ CF(JC), then neurons i and j are equivalent.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 22 / 24

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An Algorithm for Determining Missing Codewords

  • 1. Pick a complex pseudomonomial

This is a pseudomonomial with multiple (1 − xj) factors, e.g., xi(1 − xj1) . . . (1 − xjm). Write ∩k∈[m]ijk = i.

  • 2. Add “equivalent” neurons

Neurons which always fire together are equivalent, e.g. if xi(1 − xj) and xj(1 − xi) ∈ CF(JC), then neurons i and j are equivalent.

  • 3. Add all other possible neurons not prevented by monomials.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 22 / 24

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Example

Let C = {12345, 1236, 2345, ∅}

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 23 / 24

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Example

Let C = {12345, 1236, 2345, ∅}

  • 1. Pick x2(1 − x4)(1 − x6) ∈ CF(JC). This gives us U2 ⊆ U4 ∪ U6.

Potentially, we have 24 ∩ 26 = 2.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 23 / 24

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Example

Let C = {12345, 1236, 2345, ∅}

  • 1. Pick x2(1 − x4)(1 − x6) ∈ CF(JC). This gives us U2 ⊆ U4 ∪ U6.

Potentially, we have 24 ∩ 26 = 2.

  • 2. x4(1 − x5), x5(1 − x4), x2(1 − x3), x3(1 − x2) ∈ CF(JC), so

U4 = U5 and U2 = U3 Potentially, we then have 2345 ∩ 236

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 23 / 24

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Example

Let C = {12345, 1236, 2345, ∅}

  • 1. Pick x2(1 − x4)(1 − x6) ∈ CF(JC). This gives us U2 ⊆ U4 ∪ U6.

Potentially, we have 24 ∩ 26 = 2.

  • 2. x4(1 − x5), x5(1 − x4), x2(1 − x3), x3(1 − x2) ∈ CF(JC), so

U4 = U5 and U2 = U3 Potentially, we then have 2345 ∩ 236

  • 3. x4x6, x5x6 are the only monomials in CF(JC), so we can add 1 to 2345

and 236 to get 1236 ∩ 12345 = 123

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 23 / 24

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Acknowledgments

Thank you to:

  • Dr. Anne Shiu

Megan Franke Nida Obatake Texas A&M University Department of Mathematics National Science Foundation

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