Convex Codes and Minimal Embedding Dimensions Megan Franke UC Santa - PowerPoint PPT Presentation

Convex Codes and Minimal Embedding Dimensions Megan Franke UC Santa Barbara July 17, 2017 Franke (UCSB) Just Convex Realization July 17, 2017 1 / 21

Neural Codes 2014 Nobel Prize in Physiology or Medicine: Place Cells Franke (UCSB) Just Convex Realization July 17, 2017 2 / 21

Neural Codes 2014 Nobel Prize in Physiology or Medicine: Place Cells Each place cell corresponds to a receptive field Franke (UCSB) Just Convex Realization July 17, 2017 2 / 21

Neural Codes 2014 Nobel Prize in Physiology or Medicine: Place Cells Each place cell corresponds to a receptive field The receptive fields from a set of neurons give us a neural code Figure: Place Cells Franke (UCSB) Just Convex Realization July 17, 2017 2 / 21

Neural Code Example Convex Code: {∅ , 1 , 2 , 12 } U 1 U 2 Franke (UCSB) Just Convex Realization July 17, 2017 3 / 21

Convex Neural Codes Definition We say that a code C is a convex code on n neurons if there exists a collection of sets U = { U 1 , U 2 , . . . , U n } such that for each i ∈ [ n ], U i is a convex subset of R d and C ( U ) = C . A code C = C ( U ) is open convex or closed convex if the U i ∈ U are all open or all closed. Franke (UCSB) Just Convex Realization July 17, 2017 4 / 21

When are codes just convex? Goal Classify which codes are convex open, convex closed, just convex, or not convex at all. Franke (UCSB) Just Convex Realization July 17, 2017 5 / 21

When are codes just convex? Goal Classify which codes are convex open, convex closed, just convex, or not convex at all. Theorem (F., Muthiah) Every neural code is just convex. Franke (UCSB) Just Convex Realization July 17, 2017 5 / 21

Definitions Definition Let X 1 , X 2 , . . . , X n be subsets of R d . Define the convex hull of X 1 , X 2 , . . . , X n to be the smallest convex set in R d containing X 1 , X 2 , . . . , X n , denoted by conv( X 1 , X 2 , . . . , X n ). Franke (UCSB) Just Convex Realization July 17, 2017 6 / 21

Definitions Definition Let X 1 , X 2 , . . . , X n be subsets of R d . Define the convex hull of X 1 , X 2 , . . . , X n to be the smallest convex set in R d containing X 1 , X 2 , . . . , X n , denoted by conv( X 1 , X 2 , . . . , X n ). Let X 1 = (0 , 0 , 0), X 2 = (1 , 0 , 0), X 3 = (0 , 1 , 0), and X 4 = (0 , 0 , 1). Franke (UCSB) Just Convex Realization July 17, 2017 6 / 21

Definitions Definition Let X 1 , X 2 , . . . , X n be subsets of R d . Define the convex hull of X 1 , X 2 , . . . , X n to be the smallest convex set in R d containing X 1 , X 2 , . . . , X n , denoted by conv( X 1 , X 2 , . . . , X n ). Let X 1 = (0 , 0 , 0), X 2 = (1 , 0 , 0), X 3 = (0 , 1 , 0), and X 4 = (0 , 0 , 1). Then the convex hull of { X 1 , X 2 , X 3 , X 4 } is X 3 X 1 X 2 X 4 Franke (UCSB) Just Convex Realization July 17, 2017 6 / 21

Just Convex Construction Let C be a code on n neurons where C \ {∅} = { σ 1 , σ 2 , . . . , σ k } and let { e 1 , ..., e k − 1 } be the standard basis for R k − 1 . Franke (UCSB) Just Convex Realization July 17, 2017 7 / 21

Just Convex Construction Let C be a code on n neurons where C \ {∅} = { σ 1 , σ 2 , . . . , σ k } and let { e 1 , ..., e k − 1 } be the standard basis for R k − 1 . Take σ 1 . Then for every j ∈ [ n ], if j ∈ σ 1 define V 1 j to be the closed point at the origin. V 1 j Otherwise, define V 1 j = ∅ . Franke (UCSB) Just Convex Realization July 17, 2017 7 / 21

Just Convex Construction Take σ 2 . Then for every j ∈ [ n ], if j ∈ σ 2 define V 2 j to be conv { 0 , e 1 } − { 0 } . V 2 j e 1 Otherwise, define V 2 j = ∅ . Franke (UCSB) Just Convex Realization July 17, 2017 8 / 21

Just Convex Construction Next take σ 3 . Then for every j ∈ [ n ], if j ∈ σ 3 define V 3 j to be conv { 0 , e 1 , e 2 } , but open along its intersection with conv { 0 , e 1 } . e 2 V 3 j e 1 Otherwise, define V 3 j = ∅ . Franke (UCSB) Just Convex Realization July 17, 2017 9 / 21

Just Convex Construction Continuing in this way, for all j ∈ [ n ], if j ∈ σ m , define V m to be j conv { 0 , e 1 , e 2 , . . . , e m − 1 } , but open along its intersection with conv { 0 , e 1 , e 2 , . . . , e m − 2 } . Otherwise, define V m = ∅ . j Franke (UCSB) Just Convex Realization July 17, 2017 10 / 21

Just Convex Construction When this has been completed for all σ j ∈ C , define � j = V 1 j ∪ V 2 V i j ∪ . . . ∪ V k U j = j i ∈ [ k ] for all j ∈ [ n ]. Franke (UCSB) Just Convex Realization July 17, 2017 11 / 21

Example Let C = {∅ , 12 , 13 , 23 } . Franke (UCSB) Just Convex Realization July 17, 2017 12 / 21

Example Let C = {∅ , 12 , 13 , 23 } . Then σ 1 = 12, σ 2 = 13, σ 3 = 23. Franke (UCSB) Just Convex Realization July 17, 2017 12 / 21

Example Let C = {∅ , 12 , 13 , 23 } . Then σ 1 = 12, σ 2 = 13, σ 3 = 23. V codeword# neuron Franke (UCSB) Just Convex Realization July 17, 2017 12 / 21

Example Let C = {∅ , 12 , 13 , 23 } . Then σ 1 = 12, σ 2 = 13, σ 3 = 23. V codeword# neuron V 1 3 = ∅ V 1 V 1 1 2 Franke (UCSB) Just Convex Realization July 17, 2017 12 / 21

Example Let C = {∅ , 12 , 13 , 23 } . Then σ 1 = 12, σ 2 = 13, σ 3 = 23. V codeword# neuron V 1 3 = ∅ V 1 V 1 1 2 V 2 V 2 V 2 2 = ∅ 1 3 Franke (UCSB) Just Convex Realization July 17, 2017 12 / 21

Example Let C = {∅ , 12 , 13 , 23 } . Then σ 1 = 12, σ 2 = 13, σ 3 = 23. V codeword# neuron V 1 3 = ∅ V 1 V 1 1 2 V 2 V 2 V 2 2 = ∅ 1 3 V 3 1 = ∅ V 3 V 3 2 3 Franke (UCSB) Just Convex Realization July 17, 2017 12 / 21

Example C = {∅ , 12 , 13 , 23 } U 1 U 2 U 3 Franke (UCSB) Just Convex Realization July 17, 2017 13 / 21

Minimal Embedding Dimension {∅ , 1, 2, 3, 4, 5, 12, 15, 23, 24, 25, 34, 45, 56, 125, 234, 245 } Franke (UCSB) Just Convex Realization July 17, 2017 14 / 21

Minimal Embedding Dimension {∅ , 1, 2, 3, 4, 5, 12, 15, 23, 24, 25, 34, 45, 56, 125, 234, 245 } Franke (UCSB) Just Convex Realization July 17, 2017 14 / 21

Minimal Embedding Dimension Definition Let C be a convex code on n neurons. Suppose C is realized by U = { U 1 , U 2 , . . . , U n } where each U i ⊂ R d is convex. The minimal such d is the minimal embedding dimension of C . If we require all U i ∈ U to be open, the minimal such d is the minimal open embedding dimension of C . If we require all U i ∈ U to be closed, the minimal such d is the minimal closed embedding dimension of C . Franke (UCSB) Just Convex Realization July 17, 2017 15 / 21

Minimal Embedding Dimension Definition Define C n to be the code on n neurons containing all codewords of length n − 1, C n = { σ | σ ⊆ [ n ] , | σ | = n − 1 } . � n � Note that |C n | = = n . n − 1 Franke (UCSB) Just Convex Realization July 17, 2017 16 / 21

Minimal Embedding Dimension Definition Define C n to be the code on n neurons containing all codewords of length n − 1, C n = { σ | σ ⊆ [ n ] , | σ | = n − 1 } . � n � Note that |C n | = = n . n − 1 Theorem (F., Muthiah) For every n, C n has minimal embedding dimension n − 1 . Franke (UCSB) Just Convex Realization July 17, 2017 16 / 21

Example Let C 3 = {∅ , 12 , 13 , 23 } and U = { U 1 , U 2 , U 3 } be a realization of C 3 . Franke (UCSB) Just Convex Realization July 17, 2017 17 / 21

Example Let C 3 = {∅ , 12 , 13 , 23 } and U = { U 1 , U 2 , U 3 } be a realization of C 3 . Then there exists points a 12 , a 13 , and a 23 such that a 12 ∈ U 1 ∩ U 2 , a 13 ∈ U 1 ∩ U 3 , a 23 ∈ U 2 ∩ U 3 . Franke (UCSB) Just Convex Realization July 17, 2017 17 / 21

Example Let C 3 = {∅ , 12 , 13 , 23 } and U = { U 1 , U 2 , U 3 } be a realization of C 3 . Then there exists points a 12 , a 13 , and a 23 such that a 12 ∈ U 1 ∩ U 2 , a 13 ∈ U 1 ∩ U 3 , a 23 ∈ U 2 ∩ U 3 . Suppose toward contradiction that C 3 has a realization in 1 dimension. Then, a 12 , a 13 , and a 23 must be collinear. Franke (UCSB) Just Convex Realization July 17, 2017 17 / 21

Example C 3 = {∅ , 12 , 13 , 23 } a 12 ∈ U 1 ∩ U 2 a 13 ∈ U 1 ∩ U 3 a 23 ∈ U 2 ∩ U 3

Example C 3 = {∅ , 12 , 13 , 23 } a 12 ∈ U 1 ∩ U 2 a 13 ∈ U 1 ∩ U 3 a 23 ∈ U 2 ∩ U 3 a 12

Example C 3 = {∅ , 12 , 13 , 23 } a 12 ∈ U 1 ∩ U 2 a 13 ∈ U 1 ∩ U 3 a 23 ∈ U 2 ∩ U 3 a 12 a 13

Example C 3 = {∅ , 12 , 13 , 23 } a 12 ∈ U 1 ∩ U 2 a 13 ∈ U 1 ∩ U 3 a 23 ∈ U 2 ∩ U 3 a 12 a 13

Example C 3 = {∅ , 12 , 13 , 23 } a 12 ∈ U 1 ∩ U 2 a 13 ∈ U 1 ∩ U 3 a 23 ∈ U 2 ∩ U 3 a 12 a 13 a 23

Example C 3 = {∅ , 12 , 13 , 23 } a 12 ∈ U 1 ∩ U 2 a 13 ∈ U 1 ∩ U 3 a 23 ∈ U 2 ∩ U 3 ⊆ U 2 a 12 a 13 a 23

Example C 3 = {∅ , 12 , 13 , 23 } a 12 ∈ U 1 ∩ U 2 a 13 ∈ U 1 ∩ U 3 a 23 ∈ U 2 ∩ U 3 ⊆ U 2 a 23 a 12 a 13 a 23

Example C 3 = {∅ , 12 , 13 , 23 } a 12 ∈ U 1 ∩ U 2 a 13 ∈ U 1 ∩ U 3 a 23 ∈ U 2 ∩ U 3 ⊆ U 2 a 23 a 12 a 13 a 23 ⊆ U 3 Franke (UCSB) Just Convex Realization July 17, 2017 18 / 21