Toric ideals of neural codes Elizabeth Gross San Jos´ e State University Joint work with Nida Obatake (SJSU) Nora Youngs (Harvey Mudd) Elizabeth Gross, SJSU Toric ideals of neural codes
Neural activity and place fields The freely moving rat (O’Keefe, 1979) Elizabeth Gross, SJSU Toric ideals of neural codes
Receptive field code 2) C C ⊂ {0,1} 4 1 codewords:)) 3) 1000)0100)0010)0001) 1100)0110)1001)0011) 0101)1101)0111) 4) ac-vity)pa3ern) codeword) 0)))))1))))))0))))))1) The code C is the collection of the activity patterns of neurons Let U = { U 1 , ..., U n } with U i ⊂ R 2 . The neural code associated to U is C ( U ) = { c ∈ { 0 , 1 } n | � � � � � � U i \ U j � = ∅} . i ∈ supp ( c ) j / ∈ supp ( c ) Note: The code C is the data that is returned from an experiment. There is no knowledge of the sets in U . Elizabeth Gross, SJSU Toric ideals of neural codes
Drawing receptive fields 1' Neural'code' C :' 3' ' 100000'000010'000001' 2' 6' 110000'101000'100100' 100010'100001'000011' ? 111000'110100'100011''' 4' 5' Drawing realizations: Given a neural code, is there an algorithm to draw a realization (a Euler diagram) with convex fields? Convexity: Not all codes are realizable with convex place fields. When is a neural code convexly realizable? (Curto–Gross–Jeffries–Morrison–Omar–Rosen–Shiu–Youngs) Dimension: What is the minimum dimension for which a convex neural code is realizable? (Rosen–Zhang, full characterization for dimension 1) Elizabeth Gross, SJSU Toric ideals of neural codes
Drawing Euler diagrams Automatically drawing Euler diagrams using “nice” shapes is quite tricky. It is a topic of current interest in the field of Information Visualization. C = { 0000 , 1000 , 0100 , 1100 , 1110 , 1011 , 1111 } Images from Stapleton–Zhang–Howse–Rodgers 2013 Elizabeth Gross, SJSU Toric ideals of neural codes
Drawing Euler diagrams Automatically drawing Euler diagrams using “nice” shapes is quite tricky. It is a topic of current interest in the field of Information Visualization. C = { 1000 , 0100 , There exists an algorithm for drawing Euler diagrams with circles 1100 , 1110 , 1011 , 1111 } (Stapleton–Flower–Rodgers–Howse, 2013) for inductively pierced codes. Use toric ideals to iden- Goal: tify inductively pierced codes. Elizabeth Gross, SJSU Toric ideals of neural codes
0-piercings D = 0-piercings of D : Elizabeth Gross, SJSU Toric ideals of neural codes
1-piercings D = 1-piercings of D : Elizabeth Gross, SJSU Toric ideals of neural codes
2-piercings D = 2-piercing of D : Elizabeth Gross, SJSU Toric ideals of neural codes
1-inductively pierced diagrams Elizabeth Gross, SJSU Toric ideals of neural codes
k -inductively pierced Definition Let C be a neural code on n neurons. Let Λ = { λ 1 , . . . , λ k } ⊆ { 1 , 2 , . . . , n } = [ n ]. Then λ k +1 ∈ [ n ] is a k -piercing of Λ in C if there exists c ∗ ∈ C such that λ i / ∈ supp( c ∗ ) for i ≤ k + 1 1 { supp( c ) : λ k +1 ∈ supp( c ) } = { supp( c ∗ ) ∪ { λ k +1 } ∪ Λ i : Λ i ⊆ Λ } 2 { supp( c ∗ ) ∪ Λ i : Λ i ∈ Λ } ⊆ { supp( c ) : c ∈ C} . 3 Definition A neural code C is k -inductively pierced if C has a 0 , 1 , . . . , or k piercing λ and C − λ = { ( c 1 , . . . , c λ − 1 , ˆ c λ , c λ +1 , . . . , c n ) : ( c 1 , . . . , c n ) ∈ C} is k -inductively pierced. We can explore k -inductively pierced codes using neural ideals and their canonical form (Curto–Ikskov–Veliz-Cuba–Youngs 2013) or toric ideals of neural codes . Elizabeth Gross, SJSU Toric ideals of neural codes
Toric ideals of neural codes Let C = { c 1 , . . . , c m } be a neural code on n neurons. Let φ C : K [ p c | c ∈ C ] → K [ x i | i ∈ { 1 , . . . , n } ] � p c �→ x i . i ∈ supp ( c ) The toric ideal of the neural code C is I C := ker φ C . Example C = { 1000 , 0100 , 1100 , 1110 , 1011 , 1111 } I C = � p 0110 p 1011 − p 1111 , p 1000 p 0100 − p 1100 � Elizabeth Gross, SJSU Toric ideals of neural codes
Detecting 0-inductively pierced codes Theorem (Gross-Obatake-Youngs) Let C be well-formed (think: convexly realizable in dimension 2). Then I C = � 0 � if and only if the neural code C is 0 -inductively pierced. Proof. ( ⇒ ) I C = � 0 � ⇒ realizations of C have no crossings ⇒ 0-inductively pierced. ( ⇐ ) Prove by induction by using theory on toric ideals of hypergraphs (Petrovi´ c–Stasi 2014, Petrovi´ c–Gross 2013). Elizabeth Gross, SJSU Toric ideals of neural codes
Necessary condition for 1-inductively pierced codes Theorem (Gross-Obatake-Youngs) Let C be well-formed. If the neural code C is 1 -inductively pierced, then the toric ideal I C is generated by quadratics or I C = � 0 � . Fact Converse is not true! • C = { 100 , 010 , 001 , 110 , 101 , 011 , 111 } 1 2 • C is not 1-inductively pierced • I C = � p 111 − p 110 p 001 , p 110 − p 100 p 010 , p 101 − p 100 p 001 , p 011 − 3 p 010 p 001 � Elizabeth Gross, SJSU Toric ideals of neural codes
Necessary condition for 1-inductively pierced codes Theorem (Gross-Obatake-Youngs) Let C be well-formed. If the neural code C is 1 -inductively pierced, then the toric ideal I C is generated by quadratics or I C = � 0 � . Fact Converse is not true! • C = { 100 , 010 , 001 , 110 , 101 , 011 , 111 } 1 2 • C is not 1-inductively pierced • I C = � p 111 − p 100 p 010 p 001 , p 110 − p 100 p 010 , p 101 − p 100 p 001 , p 011 − 3 p 010 p 001 � Elizabeth Gross, SJSU Toric ideals of neural codes
Cubic signatures of 2-piercings 1 2 • C = { 100 , 010 , 001 , 110 , 101 , 011 , 111 } • Notice: p 111 − p 100 p 010 p 001 ∈ I C . 3 Theorem (Gross-Obatake-Youngs) Let C be well-formed. If C has a realization that contain three circles that intersect as in the figure above, then I C contains a cubic of the form p 111 z p 000 z p 000 z − p 100 z p 010 z p 001 z or p 1110 ... 0 − p 1000 ... 0 p 0100 ... 0 p 0010 ... 0 Elizabeth Gross, SJSU Toric ideals of neural codes
Using Gr¨ obner bases Proposition (Gross-Obatake-Youngs) A neural code C on 3 neurons is 1-inductively pierced if and only if the Gr¨ obner basis of I C with respect to the term order determined by the weight vector [0 , 0 , 0 , 1 , 1 , 1 , 0] and GRevLex contains only binomials of degree 2 or less. Conjecture For all n, there exists a term order such that a code is 1-inductively pierced if and only if the Gr¨ obner basis contains only binomials of degree 2 or less. Elizabeth Gross, SJSU Toric ideals of neural codes
Thank you! Elizabeth Gross, SJSU Toric ideals of neural codes
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