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 of place cells and grid cells Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 2 / 24
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
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 R d . The receptive field of a neuron i is denoted U i . Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 3 / 24
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 R d . The receptive field of a neuron i is denoted U i . On 5 neurons, one codeword could be { 2,4 } ; this is where the receptive fields U 2 and U 4 overlap; we write this as 24. Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 3 / 24
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
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
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
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
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 C max ( n ) = { σ : σ ⊆ [ n ] } . Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 5 / 24
Max Intersection-Complete Codes U 1 U 2 U 3 U 4 Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 6 / 24
Max Intersection-Complete Codes 1
Max Intersection-Complete Codes 1 12 1 1
Max Intersection-Complete Codes 13 1 123 12 1 1
Max Intersection-Complete Codes 13 1 123 12 1 124 14 1 Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 7 / 24
13 123 12 124 14 C = { 123 , 124 , 12 , 13 , 14 , ∅} Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 8 / 24
13 123 12 124 14 C = { 123 , 124 , 12 , 13 , 14 , ∅} Intersection-complete? Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 8 / 24
13 123 12 124 14 C = { 123 , 124 , 12 , 13 , 14 , ∅} Intersection-complete? No! Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 8 / 24
13 123 12 124 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
13 123 12 124 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
Neural Ideals From a neural code C , we obtain its neural ideal J C , defined to be � � J C := � (1 − x j ) : σ / ∈ C , τ = [ n ] − σ � . x i i ∈ σ j ∈ τ Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 9 / 24
Neural Ideals From a neural code C , we obtain its neural ideal J C , defined to be � � J C := � (1 − x j ) : σ / ∈ C , τ = [ n ] − σ � . x i i ∈ σ j ∈ τ In our example, 24 is not a codeword of C , so x 2 x 4 (1 + x 1 )(1 + x 3 ) ∈ J C Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 9 / 24
The Canonical Form The canonical form CF ( J C ) 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
The Canonical Form The canonical form CF ( J C ) 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 x i Type 2 relations: � j (1 − x j ) i x i � Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 10 / 24
The Canonical Form The canonical form CF ( J C ) 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 x i Type 2 relations: � j (1 − x j ) i x i � If a Type 1 relation x a 1 . . . x a n is in the CF of J C , then the codeword c = a 1 . . . a n is not in C , nor is any codeword containing c . If a Type 2 relation x a 1 . . . x a n (1 − x b 1 ) . . . (1 − x b m ) is in the CF, then � � U i ⊆ U j i ∈{ a 1 ,..., a n } j ∈{ b 1 ,..., b m } Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 10 / 24
Canonical Form Example Recall our code C = { 123 , 124 , 12 , 14 , 13 , ∅} . Here, CF ( J C ) = { x 2 (1 − x 1 ) , x 3 (1 − x 1 ) , x 4 (1 − x 1 ) , x 3 x 4 , x 1 (1 − x 2 )(1 − x 3 )(1 − x 4 ) } . Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 11 / 24
Canonical Form Example Recall our code C = { 123 , 124 , 12 , 14 , 13 , ∅} . Here, CF ( J C ) = { x 2 (1 − x 1 ) , x 3 (1 − x 1 ) , x 4 (1 − x 1 ) , x 3 x 4 , x 1 (1 − x 2 )(1 − x 3 )(1 − x 4 ) } . Because x 3 x 4 ∈ CF ( J C ), we can’t have 34 ∈ C , nor can we have 134 , 234 , or 1234 ∈ C . Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 11 / 24
Canonical Form Example Recall our code C = { 123 , 124 , 12 , 14 , 13 , ∅} . Here, CF ( J C ) = { x 2 (1 − x 1 ) , x 3 (1 − x 1 ) , x 4 (1 − x 1 ) , x 3 x 4 , x 1 (1 − x 2 )(1 − x 3 )(1 − x 4 ) } . Because x 3 x 4 ∈ CF ( J C ), we can’t have 34 ∈ C , nor can we have 134 , 234 , or 1234 ∈ C . Further, an element like x 2 (1 − x 1 ) tells us that U 2 ⊆ U 1 . Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 11 / 24
Canonical Form Example Similarly, because x 1 (1 − x 2 )(1 − x 3 )(1 − x 4 ) ∈ CF ( J C ), we have that U 1 ⊆ U 2 ∪ U 3 ∪ U 4 . Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 12 / 24
Canonical Form Example Similarly, because x 1 (1 − x 2 )(1 − x 3 )(1 − x 4 ) ∈ CF ( J C ), we have that U 1 ⊆ U 2 ∪ U 3 ∪ U 4 . 13 123 12 124 14 Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 12 / 24
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 ( J C ) contains only monomials and pseudomonomials of the form (1 − x j ) � i x i . Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 13 / 24
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
Finding the Facets We have been able to develop an algorithm for finding the facets of a code C from CF ( J C ). We use the fact that if a monomial appears in CF ( J C ) then no codeword containing the indices of that monomial appears in C . Molly Hoch (Wellesley College) Max ∩ -Complete Codes July 17, 2017 15 / 24
Example of Facet Algorithm Recall our earlier example: C = { 123 , 124 , 12 , 13 , 14 , ∅} . The only monomial in CF ( J C ) is x 3 x 4 . On 4 neurons, 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 16 / 24
Example of Facet Algorithm Recall our earlier example: C = { 123 , 124 , 12 , 13 , 14 , ∅} . The only monomial in CF ( J C ) is x 3 x 4 . 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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