Inductively Pierced Codes and Toric Ideals Samuel Muthiah Westmont - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . Inductively Pierced Codes and Toric Ideals Samuel Muthiah Westmont College July 17, 2017 Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 20

. . . . . . . . . . . . . . Place Cells In 2014 John O’Keefe received the Nobel Prize for his discovery of place cells Place cells are part of the way certain mammals’ brains identify where there are spatially Place cells fjre in approximately convex regions Figure: Place Cells Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 20

. . . . . . . . . . . . . . . . Neural Codes Defjnition Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . 3 / 20 A neural code on n neurons is a set of binary strings C ⊆ { 0 , 1 } n . The elements of C are called codewords . C = { 000 , 100 , 001 , 101 , 011 , 111 }

. . . . . . . . . . . . . . . . Neural Codes Defjnition Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . 4 / 20 . . . . . A realization of a code C on n neurons is a collection of sets U = { U 1 , . . . , U n } such that C ( U ) = C . C = { 000 , 100 , 001 , 101 , 011 , 111 } 2 1 3

. . . . . . . . . . . . . . . . Neural Codes Defjnition Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . 5 / 20 A code C is convex if there exists a realization of C by convex sets. C = { 000 , 100 , 001 , 101 , 011 , 111 } 3 2 1

. . . . . . . . . . . . . . . Neural Codes Defjnition Curves intersect at a fjnite number of points At any given point, at most two curves intersect Each zone is connected Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 20 The realization of a code C is well-formed if C = { 000 , 100 , 001 , 101 , 011 , 111 } 3 2 1

. . . . . . . . . . . . . . . Need for Algorithms {000000, 100000, 010000, 001000, 000100, 000010, 110000, 100010, 011000, 010100, 010010, 001100, 000110, 000011, 110010, 011100, 010110} Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 20

. . . . . . . . . . . . . . . Need for Algorithms {000000, 100000, 010000, 001000, 000100, 000010, 110000, 100010, 011000, 010100, 010010, 001100, 000110, 000011, 110010, 011100, 010110} Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 20

. . . . . . . . . . . . . . . k -Piercings Defjnition A k-piercing is a curve that pierces (intersects) k other curves and that Figure: Example of a 1-Piercing Figure: Example of a 2-Piercing Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . 8 / 20 adds 2 k zones when added to an existing diagram.

. . . . . . . . . . . . . . . k -Piercings Defjnition A k-piercing is a curve that pierces (intersects) k other curves and that Figure: Example of a 1-Piercing Figure: Example of a 2-Piercing Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . 8 / 20 adds 2 k zones when added to an existing diagram.

. . . . . . . . . . . . . . . . k-Inductively Pierced Defjnition Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . 9 / 20 A neural code C is k-inductively pierced if C has a 0 − , 1 − , ..., or k − piercing λ and C − λ is k-inductively pierced.

. . . . . . . . . . . . . . . . k-Inductively Pierced Defjnition Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . 9 / 20 A neural code C is k-inductively pierced if C has a 0 − , 1 − , ..., or k − piercing λ and C − λ is k-inductively pierced.

. is I . . . . . . . . Toric Ideals x i Defjnition The toric ideal of the neural code ker . c p p p p x x x x x x x x Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 / 20 Let C = { c 1 , ..., c m } be a neural code on n neurons Let φ c : K [ p c | c ∈ C\ (0 , .., 0)] → K [ x i | i ∈ [ n ]] ∏ p c �→ i ∈ supp ( c )

. . . . . . . . . . . . . . . . Toric Ideals x i Defjnition Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . 10 / 20 . . . Let C = { c 1 , ..., c m } be a neural code on n neurons Let φ c : K [ p c | c ∈ C\ (0 , .., 0)] → K [ x i | i ∈ [ n ]] ∏ p c �→ i ∈ supp ( c ) The toric ideal of the neural code C is I C := ker φ c p 101 p 110 − p 111 p 100 �→ x 1 x 3 · x 1 x 2 − x 1 x 2 x 3 · x 1 = 0

. . . . . . . . . . . . . . . . Identifying k-Piercings Theorem (Gross-Obatake-Youngs) generated by quadratics. Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . 11 / 20 Let C be well formed. The neural code C is 0-inductively pierced if and only if I C = ⟨ 0 ⟩ . If the neural code C is 0- or 1- inductively pierced then I C = ⟨ 0 ⟩ or If the neural code C has a 2-piercing then I C contains a binomial of degree 3 of particular form, in particular p 111 w p 2 000 v − p 100 v p 010 v p 001 w or p 111 w − p 100 ... 0 p 010 ... 0 p 001 w where v , w are zones in C ( U ) .

. . . . . . . . . . . . . . . . Identifying k-Piercings One set of generators of its toric ideal is: Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . 12 / 20 . . . . Take the code C = { 0001 , 1001 , 0101 , 0011 , 1101 , 1011 , 0111 , 1111 } . ⟨− p 1011 p 0111 + p 0011 p 1111 , − p 1101 p 0111 + p 0101 p 1111 , − p 0101 p 1011 + p 1001 p 0111 , − p 0011 p 1101 + p 1001 p 0111 , − p 1011 p 1011 + p 1001 p 1111 , − p 1001 p 0101 + p 0001 p 1101 , − p 1001 p 0011 + p 0001 p 1011 , − p 1001 p 0111 + p 0001 p 1111 , − p 0101 p 0011 + p 0001 p 0111 ⟩ .

. . . . . . . . . . . . . . . . . Identifying k-Piercings Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . 12 / 20 . . . . . Take the code C = { 0001 , 1001 , 0101 , 0011 , 1101 , 1011 , 0111 , 1111 } . 4 3 1 2 Figure: Realization of C

. . . . . . . . . . . . . . . . . Identifying k-Piercings Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . 12 / 20 ⟨− p 1011 p 0111 + p 0011 p 1111 , − p 1101 p 0111 + p 0101 p 1111 , − p 0101 p 1011 + p 1001 p 0111 , − p 0011 p 1101 + p 1001 p 0111 , − p 1011 p 1011 + p 1001 p 1111 , − p 1001 p 0101 + p 0001 p 1101 , − p 1001 p 0011 + p 0001 p 1011 , − p 1001 p 0111 + p 0001 p 1111 , − p 0101 p 0011 + p 0001 p 0111 ⟩

. p . . . . . . . Constructing Cubics and p p p p . p p p p p p p p p p Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 / 20 p 0001 p 1111 − p 1001 p 0111 , p 0001 p 0111 − p 0101 p 0011 ∈ I C ⇓ p 0001 ( p 1111 p 0001 − p 1001 p 0111 ) + p 1001 ( p 0111 p 0001 − p 0101 p 0011 ) ∈ I C

. . . . . . . . . . . . . . . . Constructing Cubics and Samuel Muthiah (Westmont College) Inductively Pierced Codes and Toric Ideals July 17, 2017 . . . . . . . . . . . . . . . . . . . . . . . . 13 / 20 p 0001 p 1111 − p 1001 p 0111 , p 0001 p 0111 − p 0101 p 0011 ∈ I C ⇓ p 0001 ( p 1111 p 0001 − p 1001 p 0111 ) + p 1001 ( p 0111 p 0001 − p 0101 p 0011 ) ∈ I C p 0001 ( p 1111 p 0001 − p 1001 p 0111 ) + p 1001 ( p 0111 p 0001 − p 0101 p 0011 ) = p 1111 p 2 0001 − p 1001 p 0101 p 0011