SLIDE 1 Convex Codes and Minimal Embedding Dimensions
Megan Franke
UC Santa Barbara
July 17, 2017
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SLIDE 2 Neural Codes
2014 Nobel Prize in Physiology or Medicine: Place Cells
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SLIDE 3 Neural Codes
2014 Nobel Prize in Physiology or Medicine: Place Cells Each place cell corresponds to a receptive field
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SLIDE 4 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
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SLIDE 5 Neural Code Example
Convex Code: {∅, 1, 2, 12} U1 U2
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SLIDE 6 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 = {U1, U2, . . . , Un} such that for each i ∈ [n], Ui is a convex subset of Rd and C(U) = C. A code C = C(U) is open convex or closed convex if the Ui ∈ U are all open or all closed.
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SLIDE 7 When are codes just convex?
Goal
Classify which codes are convex open, convex closed, just convex, or not convex at all.
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SLIDE 8 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.
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SLIDE 9 Definitions
Definition
Let X1, X2, . . . , Xn be subsets of Rd. Define the convex hull of X1, X2, . . . , Xn to be the smallest convex set in Rd containing X1, X2, . . . , Xn, denoted by conv(X1, X2, . . . , Xn).
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SLIDE 10 Definitions
Definition
Let X1, X2, . . . , Xn be subsets of Rd. Define the convex hull of X1, X2, . . . , Xn to be the smallest convex set in Rd containing X1, X2, . . . , Xn, denoted by conv(X1, X2, . . . , Xn). Let X1 = (0, 0, 0), X2 = (1, 0, 0), X3 = (0, 1, 0), and X4 = (0, 0, 1).
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SLIDE 11 Definitions
Definition
Let X1, X2, . . . , Xn be subsets of Rd. Define the convex hull of X1, X2, . . . , Xn to be the smallest convex set in Rd containing X1, X2, . . . , Xn, denoted by conv(X1, X2, . . . , Xn). Let X1 = (0, 0, 0), X2 = (1, 0, 0), X3 = (0, 1, 0), and X4 = (0, 0, 1). Then the convex hull of {X1, X2, X3, X4} is X2 X3 X4 X1
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SLIDE 12 Just Convex Construction
Let C be a code on n neurons where C \ {∅} = {σ1, σ2, . . . , σk} and let {e1, ..., ek−1} be the standard basis for Rk−1.
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SLIDE 13 Just Convex Construction
Let C be a code on n neurons where C \ {∅} = {σ1, σ2, . . . , σk} and let {e1, ..., ek−1} be the standard basis for Rk−1. Take σ1. Then for every j ∈ [n], if j ∈ σ1 define V 1
j to be the closed point
at the origin. V 1
j
Otherwise, define V 1
j = ∅.
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SLIDE 14 Just Convex Construction
Take σ2. Then for every j ∈ [n], if j ∈ σ2 define V 2
j to be
conv{0, e1} − {0}. V 2
j
e1 Otherwise, define V 2
j = ∅.
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SLIDE 15 Just Convex Construction
Next take σ3. Then for every j ∈ [n], if j ∈ σ3 define V 3
j to be
conv{0, e1, e2}, but open along its intersection with conv{0, e1}. V 3
j
e1 e2 Otherwise, define V 3
j = ∅.
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SLIDE 16 Just Convex Construction
Continuing in this way, for all j ∈ [n], if j ∈ σm, define V m
j
to be conv{0, e1, e2, . . . , em−1}, but open along its intersection with conv{0, e1, e2, . . . , em−2}. Otherwise, define V m
j
= ∅.
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SLIDE 17 Just Convex Construction
When this has been completed for all σj ∈ C, define Uj =
V i
j = V 1 j ∪ V 2 j ∪ . . . ∪ V k j
for all j ∈ [n].
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SLIDE 18 Example
Let C = {∅, 12, 13, 23}.
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SLIDE 19 Example
Let C = {∅, 12, 13, 23}. Then σ1 = 12, σ2 = 13, σ3 = 23.
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SLIDE 20 Example
Let C = {∅, 12, 13, 23}. Then σ1 = 12, σ2 = 13, σ3 = 23. V codeword#
neuron
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SLIDE 21 Example
Let C = {∅, 12, 13, 23}. Then σ1 = 12, σ2 = 13, σ3 = 23. V codeword#
neuron
V 1
1
V 1
2
V 1
3 = ∅
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SLIDE 22 Example
Let C = {∅, 12, 13, 23}. Then σ1 = 12, σ2 = 13, σ3 = 23. V codeword#
neuron
V 1
1
V 1
2
V 1
3 = ∅
V 2
1
V 2
2 = ∅
V 2
3
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SLIDE 23 Example
Let C = {∅, 12, 13, 23}. Then σ1 = 12, σ2 = 13, σ3 = 23. V codeword#
neuron
V 1
1
V 1
2
V 1
3 = ∅
V 2
1
V 2
2 = ∅
V 2
3
V 3
1 = ∅
V 3
2
V 3
3
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SLIDE 24 Example
C = {∅, 12, 13, 23} U1 U2 U3
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SLIDE 25 Minimal Embedding Dimension
{∅, 1, 2, 3, 4, 5, 12, 15, 23, 24, 25, 34, 45, 56, 125, 234, 245}
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SLIDE 26 Minimal Embedding Dimension
{∅, 1, 2, 3, 4, 5, 12, 15, 23, 24, 25, 34, 45, 56, 125, 234, 245}
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SLIDE 27 Minimal Embedding Dimension
Definition
Let C be a convex code on n neurons. Suppose C is realized by U = {U1, U2, . . . , Un} where each Ui ⊂ Rd is convex. The minimal such d is the minimal embedding dimension of C. If we require all Ui ∈ U to be open, the minimal such d is the minimal open embedding dimension of C. If we require all Ui ∈ U to be closed, the minimal such d is the minimal closed embedding dimension of C.
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SLIDE 28 Minimal Embedding Dimension
Definition
Define Cn to be the code on n neurons containing all codewords of length n − 1, Cn = {σ | σ ⊆ [n], |σ| = n − 1}. Note that |Cn| = n
n−1
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SLIDE 29 Minimal Embedding Dimension
Definition
Define Cn to be the code on n neurons containing all codewords of length n − 1, Cn = {σ | σ ⊆ [n], |σ| = n − 1}. Note that |Cn| = n
n−1
Theorem (F., Muthiah)
For every n, Cn has minimal embedding dimension n − 1.
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SLIDE 30 Example
Let C3 = {∅, 12, 13, 23} and U = {U1, U2, U3} be a realization of C3.
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SLIDE 31 Example
Let C3 = {∅, 12, 13, 23} and U = {U1, U2, U3} be a realization of C3. Then there exists points a12, a13, and a23 such that a12 ∈ U1 ∩ U2, a13 ∈ U1 ∩ U3, a23 ∈ U2 ∩ U3.
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SLIDE 32 Example
Let C3 = {∅, 12, 13, 23} and U = {U1, U2, U3} be a realization of C3. Then there exists points a12, a13, and a23 such that a12 ∈ U1 ∩ U2, a13 ∈ U1 ∩ U3, a23 ∈ U2 ∩ U3. Suppose toward contradiction that C3 has a realization in 1 dimension. Then, a12, a13, and a23 must be collinear.
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SLIDE 33
Example
C3 = {∅, 12, 13, 23} a12 ∈ U1 ∩ U2 a13 ∈ U1 ∩ U3 a23 ∈ U2 ∩ U3
SLIDE 34
Example
C3 = {∅, 12, 13, 23} a12 ∈ U1 ∩ U2 a13 ∈ U1 ∩ U3 a23 ∈ U2 ∩ U3 a12
SLIDE 35
Example
C3 = {∅, 12, 13, 23} a12 ∈ U1 ∩ U2 a13 ∈ U1 ∩ U3 a23 ∈ U2 ∩ U3 a12 a13
SLIDE 36
Example
C3 = {∅, 12, 13, 23} a12 ∈ U1 ∩ U2 a13 ∈ U1 ∩ U3 a23 ∈ U2 ∩ U3 a12 a13
SLIDE 37
Example
C3 = {∅, 12, 13, 23} a12 ∈ U1 ∩ U2 a13 ∈ U1 ∩ U3 a23 ∈ U2 ∩ U3 a12 a13 a23
SLIDE 38
Example
C3 = {∅, 12, 13, 23} a12 ∈ U1 ∩ U2 a13 ∈ U1 ∩ U3 a23 ∈ U2 ∩ U3 a12 a13 a23 ⊆ U2
SLIDE 39
Example
C3 = {∅, 12, 13, 23} a12 ∈ U1 ∩ U2 a13 ∈ U1 ∩ U3 a23 ∈ U2 ∩ U3 a12 a13 a23 ⊆ U2 a23
SLIDE 40 Example
C3 = {∅, 12, 13, 23} a12 ∈ U1 ∩ U2 a13 ∈ U1 ∩ U3 a23 ∈ U2 ∩ U3 a12 a13 a23 ⊆ U2 a23 ⊆ U3
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SLIDE 41 Discussion
New Questions: Since every code is convex, what is the minimal embedding dimension
When is the minimal open/closed embedding dimension strictly greater than the minimal embedding dimension of a code? When is the minimal open/closed embedding dimension equal to the minimal embedding dimension of a code?
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SLIDE 42 Acknowledgements
I would like to thank: Advisor: Dr. Anne Shiu Graduate Student Mentor: Ola Sobieska Project Partner: Samuel Muthiah Funding: National Science Foundation Host: Texas A&M University
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SLIDE 43 Thank you!
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