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Core motifs predict dynamic attractors in combinatorial threshold-linear networks.

Caitlyn Parmelee1, Samantha Moore2, Katherine Morrison3

  • 1Mathematics Department, Keene State College, Keene, NH, United States of America.

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|March 4, 2022
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Summary
This summary is machine-generated.

Combinatorial threshold-linear networks (CTLNs) exhibit complex dynamics. Core motifs within these networks predict attractors, revealing how graph structure influences nonlinear system behavior.

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Area of Science:

  • Computational neuroscience
  • Network theory
  • Dynamical systems

Background:

  • Combinatorial threshold-linear networks (CTLNs) are a subset of threshold-linear networks (TLNs) defined by directed graphs.
  • CTLNs exhibit diverse nonlinear dynamics, including multistability, limit cycles, quasiperiodic attractors, and chaos.
  • Previous research established a mathematical framework linking fixed points of CTLNs to graph properties.

Purpose of the Study:

  • To investigate the relationship between specific fixed points (core motifs) and attractors in CTLNs.
  • To test the hypothesis that dynamic attractors correspond to unstable fixed points on core motifs.
  • To explore the predictability of nonlinear dynamics based on graph structure.

Main Methods:

  • Analysis of fixed points in CTLNs, focusing on core motifs.
  • Testing the hypothesis on directed graphs of size n=5.
  • Classifying attractors based on graph families derived from similar core motif embeddings.

Main Results:

  • Core motifs were found to be predictive of both static and dynamic attractors.
  • Small perturbations of fixed points corresponding to core motifs yielded network attractors.
  • A strong agreement was observed between the hypothesis and results for n=5 graphs.
  • Similar core motif embeddings led to nearly identical attractors, enabling classification.

Conclusions:

  • Unstable fixed points on core motifs are strong predictors of dynamic attractors in CTLNs.
  • Network attractors can be effectively predicted and classified using graph-theoretic properties.
  • The study highlights the potential of using graphical structure to understand complex nonlinear dynamics in biological and computational networks.