Caitlin Lienkaemper

I am a fifth year graduate student in mathematics at Penn State. I am a student of Carina Curto. I am primarily interested in geometry, combinatorics, and dynamical systems, especially as applied to problems in mathematical biology and mathematical neuroscience.

Office: 402 McAllister Building

Papers and Preprints

  1. "Order-forcing in Neural Codes." With Amzi Jeffs and Nora Youngs (2020) arXiv.
  2. "Oriented Matroids and Combinatorial Neural Codes." With Alex Kunin and Zvi Rosen (2020) arXiv.
  3. "The geometry of partial fitness orders and an efficient method for detecting genetic interactions." With Lisa Lamberti, Dawn Drain, Niko Beerenwinkel, and Alex Gavryushkin. Journal of Mathematical Biology (2018). bioRxiv.
  4. "Obstructions to convexity in neural codes." With Anne Shiu and Zev Woodstock. Advances in Applied Mathematics (2017). arXiv.


Convex Neural Codes

Combinatorial neural codes describe the joint activity of a collection of neurons in terms of which neurons fire together and which do not. Convex neural codes model the activity of neurons with convex receptive fields, such as place cells in the hippocamps. Characterizing convex neural codes is mathematically difficult, and work in this area is ongoing.

In [4], Anne Shiu, Zev Woodstock, and I provided the first example of a non-convex code which did not have topological local obstrutions.

In [2], Alex Kunin, Zvi Rosen, and I connect the theory of convex neural codes to the theory of oriented matroids. Using this connection, we show that it is computationally difficult to check whether a code is convex. A recording of a talk I gave based on this paper is available here .

In [1], Amzi Jeffs, Nora Youngs, and I introduce the idea of order-forcing, and use it to construct some new, very simple examples of non-convex codes.

Underlying Rank

Scientists frequently aim to measure the intrinsic dimensionality of a dataset. In many cases, this is trivial: we just compute the rank of the matrix containing our data. However, in many cases, we do not have access to accurate measurements of the quantity we are really interested in, and instead, have access to some proxy measurement which has a monotone, nonlinear relationship with it.

In upcoming work, Carina Curto, Juliana Londono Alvarez, Hannah Rocio Santa Cruz and I define the underlying rank of a matrix A, which is the minimum rank r such that there is a rank r matrix B whose entries are in the same order as A. By associating matrices to point configurations, we are able to use results about random polytopes, oriented matroids, and allowable sequences to estimate underlying rank. A recording of a talk I've given on this work is available here.

Threshold Linear Networks

Threshold linear networks (TLNs) are a model for networks of neurons which are simple enough to be mathematically tractable, but complex enough to display the key features of nonlinear dynamics needed to replicate real neural networks, such as multistability, limit cycles, and chaos. Combinatorial threshold linear networks (CTLNs) are a subclass of TLN whose behavior is determined by a directed graph.

In upcoming work with Carina Curto and Katie Morrison, I relate the structure of the network to its dynamics. For instance, we show that if the underlying graph of a CTLN is a directed acyclic graph, then all trajectories of the dynamical system approach a stable fixed point.