Math 197: Senior Thesis
Realizing Convex Neural
Codes with Polytopes
Caitlin Lienkaemper
Place Cells
How does your brain know where you are? The 2014 Nobel Prize was
awarded to John O’Keefe, May Britt Moser, and Edvard Moser for a
partial answer to this question. Hippocampal neurons called place cells
become associated to convex regions of space, known as their place
fields. When an animal is in the place field of a given place cell, that
place cell will fire.
Mathematics
@
Harvey Mudd College
neural codes in general position.
Definition 4 (General Position) A realization U = {U1, . . . , Un } is in
general position if there exists e > 0 such that for all covers
V = {V1, . . . , Vn } such that Ui and Vi are within Hausdorff distance e,
the neural codes C (V ) and C (U ) are equal.
Realizing Codes With Convex
Polytopes
Definition 5 (Convex Polytope) The convex hull of a set of points is the
smallest convex set containing these points. A convex polytope is the
convex hull of a fine number of points in Rn.
Figure 1: Place fields were discovered when researchers observed the activity of single neurons in rodents and found that some neurons fired only
when the animal was in a specific region of space.
Convex Neural Codes
We describe the activity of a population of n neurons with a neural
code.
Definition 1 (Neural Code) A codeword is a binary vector c ∈ F2n. A
neural code is a collection of codewords, and therefore is a subset
C ⊂ F2n.
Example Suppose we have monitored the activity of three place cells
over some interval of time. At various times in our interval, we find
that, no neurons fire, neuron 1 fires alone, neuron 2 fires alone, neurons
1 and 2 fire together, neurons 1 and 3 fire together, neurons 1, 2, and 3
all fire.
We can summarize this behavior with the neural code
C = {(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (1, 0, 1), (1, 1, 1)},
which we abbreviate as
C = {000, 100, 010, 110, 101, 111}.
Figure 3: Convex poytopes in 2 and 3 dimensions.
Theorem: Any convex neural code in general position can be realized
using convex polytopes.
Proof Sketch:
If a neural code C can be realized in general position with sets
U1, . . . , Un, it is also realized with the sets U10 , . . . , Un0 , where Ui0 is the set
Ui “inflated” by some e > 0. We can find a convex polytope Pi such that
Ui ⊂ Pi ⊂ Ui0.
Figure 4: Finding a polytope between two copies of a convex set.
Since the code is in general position and Pi is in hausforff distance e of
Ui, P1, . . . , Pn is a convex realization of C .
Future Directions
Figure
2:
A
convex
{000, 100, 010, 110, 101, 111}.
realization
of
the
neural
code
The neural code of a collection of subsets: Suppose
U = {U1, . . . , Un } ⊂ Rd. For each x ∈ Rd, let c x be a vector that has a 1
at the ith index if x ∈ Ui and a 0 otherwise. Define the neural code
C U = { c x | x ∈ Rd } .
Definition 2 (Convex Neural Code): A neural code C is convex if there
exists a collection of convex sets U ⊂ Rd such that C = CU .
Neural Codes in General Position
Realistic neural codes are likely to be robust to small deformations in
the shapes of the place fields. We follow Cruz et al. (2016) in defininig
Advisor: Mohamed Omar
Reader: Dagan Karp
There are several examples of neural codes which cannot be realized in
general position, but can be realized with convex polygons. This
motivates us to conjecture that any convex neural code can be realized
with convex polytopes.
References
Cruz, Joshua, Chad Giusti, Vladimir Itskov, and Bill Kronholm. 2016.
On open and closed convex codes. arXiv preprint arXiv:160903502 .
Acknowledgments
First and foremost, I’d like to thank my advisor Mohamed Omar for his
suggestions of research directions and his consistently helpful advice.
https://www.math.hmc.edu/ clienkaemper/thesis/