Toric ideals of neural codes
Elizabeth Gross
San José 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
Receptiverealizations.
field code
Receptive fields of neural codes
2)
C ⊂ {0, 1}4
1
codewords:))
1000)0100)0010)0001)
1100)0110)1001)0011)
0101)1101)0111)
3)
4)
ac-vity)pa3ern)
codeword)
0)))))1))))))0))))))1)
The code C is the collection of the activity patterns of neurons that
we get from an experiment.
2
Let U = {U1 , ..., Un } with Ui ⊂ R . The neural code associated to U is
\
[
C(U)
= {c ∈ {0,receptive
1}n |
Ufields
Uj 6= ∅}.
i \
Drawing
i∈supp(c)
j ∈supp(c)
/
Note: The code C is the data that is returned from an experiment. There
1'
Neural'code'C:'
is no knowledge of the 3'sets in U.
'
Elizabeth Gross, SJSU
Toric ideals of neural
codes
100000'000010'000001'
To
Drawing receptive fields
Drawing receptive fields
1'
Neural'code'C:'
3'
'
2'
6'
?
4'
100000'000010'000001'
110000'101000'100100'
100010'100001'000011'
111000'110100'100011'''
Toric ideals
• Let C = {c1, . . . , cm
• Let
fC : K
5'
Drawing realizations: Given a neural code, is there an algorithm
• The toric ideal of the
to draw a realization
(an a
Euler
diagram)
withisconvex
Drawing realizations:
Given
neural
code,
therefields?
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
of the layout improvements applied to the lefthand
diagram in Fig. 5 can be seen on the right.
Automatically drawing Euler diagrams using “nice” shapes is quite
tricky.
It isofaFlower
topic
of current
interest
inofthe
field
of Information
and Howse
[5] were
ex- number
times
conditions are broken
of Flower The
and techniques
Howse [5] were
ex- number
of times
wellformedness
conditions
arewellformedness
broken
tended
to
enhance
the
layout
[20].
First,
some
modificaand
guarantees
the
absence
of certain conditions (such
e the layout
[20].
First,
some
modificaand
guarantees
the
absence
of
certain
conditions
(such
Visualization.
tions were made
the implementation
of the generation
non-simple zones).
curve and no ‘disconnected’ zones).
o the implementation
of the to
generation
as no non-simple
curve and as
nono
‘disconnected’
OF LTEX CLASS
SS FILES, VOL.JOURNAL
6, NO. 1, JANUARY
2007 FILES, VOL. 6, NO. 1, JANUARY 2007
A
3
3
Fig. 5. Using the layout improvement methods of [20].
Fi
extensions toa the
generation
of
method;
our the
running
example
gives the
lefthand
An alternative
dual
graphmethods
is
unning example
thisingives
lefthand
An this
alternative
method
for choosing
a dual method
graph Further
isfor choosing
Flower and Howse allow the drawing of abstract de5, althoughdiagram
the labels
are not
shown, the
developed
by Simonetto
and Auber
[22], which
has been
in Fig.
5, although
labels are
not shown,
developed
by Simonetto
and Auber
which
has been
scriptions
that need[22],
not have
a completely
wellformed
embedding.
was
done by Rodgers et al.
[21], where
he diagramasinopposed
Fig. 4. Also
Flower
et in
implemented
[6].Flower
Outputetfromimplemented
that implementation
can fromThis
to the
diagram
Fig. 4. Also
[6]. Output
that
implementation
can
techniques to allow the generation of any abstract deut metrics al.
and[20]
hillused
climbing
algorithms
behill
seen
in Fig. 7,
where the labels
have
been7,manually
layout
metrics and
climbing
algorithms
be seen
in Fig.
where
the labels have been manually
scription were developed; output from the software of
1
1
iagrams’ aesthetic
qualities;
the resultaesthetic
added qualities;
post drawing
.
to improve
the diagrams’
the result
added post drawing .Rodgers at al. can be seen in Fig. 6. All of the methods
described so far use a dual graph based approach and are
mprovements
applied
to the
lefthand
of the
layout
improvements
applied to the lefthand
computationally complex, having an NP-complete step.
can be seen
on
the
right.
diagram in Fig. 5 can be seen on the right.
This means that some diagrams take a significant time
C = {0000, 1000, 0100, 1100, 1110, 1011, 1111}
to draw.
Fi
ayout improvement methods of [20].
Fig. 6. Generation using the methods of [21].
Fig. 7. Generation
Fig. 5. Using the layout improvement
methods of using
[20]. the methods of [6].
Fig. 7. Generation using the methods of [6].
ions to the generation methods of
methods
of
se allow theFurther
drawingextensions
of abstracttode-the generation
Images
from
Flower
and Howse
allow the drawing of abstract deed not have
a completely
wellformed
scriptions
that
need
not
have
a
completely
wellformed
was done by Rodgers et al. [21], where
embedding.
Thisabstract
was done
ow the generation
of any
de- by Rodgers et al. [21], where
techniques
the generation
of any abstract developed; output
fromtotheallow
software
of
scription
were
from the
software
Elizabeth
Gross,
SJSU of
n be seen in
Fig. 6. All
of developed;
the methodsoutput
Indeed, the dual graph method requires one to choose
a dual graph from the infinitely many that are caStapleton–Zhang–Howse–Rodgers
2013
pable of generating the required Euler diagram. The
chosen graph directly impacts the aesthetic quality of
the drawn diagrams and finding a suitable dual can
be difficult. A substantial part of Rodgers et al. [21]
focusses on the task of finding a dual that minimizes the
Toric ideals of neural codes
th
ot
th
ge
ar
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ht
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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,
1100, 1110, 1011, 1111}
There exists an algorithm for
drawing Euler diagrams with circles
(Stapleton–Flower–Rodgers–Howse,
2013) for inductively pierced codes.
Goal: Use toric ideals to identify 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 = {λ1 , . . . , λk } ⊆ {1, 2, . . . , n} = [n]. Then λk+1 ∈ [n] is a k-piercing
of Λ in C if there exists c∗ ∈ C such that
1
2
3
λi ∈
/ supp(c∗ ) for i ≤ k + 1
{supp(c) : λk+1 ∈ supp(c)} = {supp(c∗ ) ∪ {λk+1 } ∪ Λi : Λi ⊆ Λ}
{supp(c∗ ) ∪ Λi : Λi ∈ Λ} ⊆ {supp(c) : c ∈ C}.
Definition
A neural code C is k-inductively pierced if C has a 0, 1, . . . , or k piercing
λ and C − λ = {(c1 , . . . , cλ−1 , ĉλ , cλ+1 , . . . , cn ) : (c1 , . . . , cn ) ∈ 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 = {c1 , . . . , cm } be a neural code on n neurons.
Let
φC : K[pc | c ∈ C] → K[xi | i ∈ {1, . . . , n}]
Y
pc 7→
xi .
i∈supp(c)
The toric ideal of the neural code C is IC := ker φC .
Example
C = {1000, 0100,
1100, 1110, 1011, 1111}
IC = hp0110 p1011 − p1111 ,
p1000 p0100 − p1100 i
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 IC = h0i if and only if the neural code C is 0-inductively
pierced.
Proof.
(⇒) IC = h0i ⇒ realizations of C have no crossings ⇒
0-inductively pierced.
(⇐) Prove by induction by using theory on toric ideals of
hypergraphs (Petrović–Stasi 2014, Petrović–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 IC is generated by quadratics or IC = h0i.
Fact
Converse is not true!
1
2
• C = {100, 010, 001, 110, 101, 011, 111}
• C is not 1-inductively pierced
3
• IC = hp111 − p110 p001 , p110 −
p100 p010 , p101 − p100 p001 , p011 −
p010 p001 i
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 IC is generated by quadratics or IC = h0i.
Fact
Converse is not true!
1
2
• C = {100, 010, 001, 110, 101, 011, 111}
• C is not 1-inductively pierced
3
• IC = hp111 − p100 p010 p001 , p110 −
p100 p010 , p101 − p100 p001 , p011 −
p010 p001 i
Elizabeth Gross, SJSU
Toric ideals of neural codes
Cubic signatures of 2-piercings
1
2
• C = {100, 010, 001, 110, 101, 011, 111}
• Notice: p111 − p100 p010 p001 ∈ IC .
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 IC contains a
cubic of the form p111z p000z p000z − p100z p010z p001z or
p1110...0 − p1000...0 p0100...0 p0010...0
Elizabeth Gross, SJSU
Toric ideals of neural codes
Using Gröbner bases
Proposition (Gross-Obatake-Youngs)
A neural code C on 3 neurons is 1-inductively pierced if and only if
the Gröbner basis of IC 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 Gröbner 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