DYNAMICS OF MINIMAL PERMITTED MOTIFS IN COMBINATORIAL THRESHOLD-LINEAR NETWORKS

Open Access
Author:
Falk, David Benjamin
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
  • Carina Pamela Curto, Thesis Supervisor
  • Mark Levi, Honors Advisor
Keywords:
  • CTLN
  • neural network
  • threshold linear
  • mathematical modeling
  • mathematical neuroscience
  • mathematical biology
  • minimal permitted motif
  • nonlinear dynamics
  • differential equations
  • graph theory
  • combinatorics
  • discrete mathematics
  • symmetry
  • neurons
  • clique union
  • cyclic union
Abstract:
At the intersection of mathematics and neuroscience, neural network models allow us to study how the connections between neurons shape neural circuit activity. In this work, we examine Combinatorial Threshold-Linear Networks (CTLNs), which are neural network models defined from directed graphs, and whose nonlinear dynamics are determined by the structure of the graph. We focus our attention on the dynamics of minimal permitted motifs. These are subnetworks whose dynamics are in some sense irreducible. Motivated by a recent theorem which tells us how to construct new minimal permitted motifs from smaller "building blocks," we analyze two new families of minimal permitted motifs: clique unions and cyclic unions. Each component of a clique union or cyclic union is a smaller minimal permitted motif whose dynamics are more or less understood. Our main goal is to understand how the dynamics of these families depend on their components as well as the overall network structure. Through an exhaustive analysis of constructed networks of sizes n=5, 6, and 7 that are built from components of size less than or equal to 4, we uncover several patterns that predict qualitative aspects of the dynamics. We find that cyclic unions typically produce cyclic-looking attractors, with neurons firing in a regular order following the cyclic architecture of the graph. On the other hand, clique unions produce what we call fusion attractors, that appear to combine attractors coming from the individual components. Interestingly, a pair of distinct building blocks can produce nearly identical behavior inside a larger network in some constructions, but completely different behavior in the context of a different construction. Finally, we collect observations about the emergence of synchrony among various subsets of neurons, and explore the role of symmetry in shaping the dynamics. Our findings provide a set of preliminary data that will allow us to test and refine conjectures about CTLNs in future work.