Fixed points of competitive threshold-linear networks

Carina Curto, Jesse Geneson, Katherine Morrison

Research output: Contribution to journalLetterpeer-review

18 Scopus citations


Threshold-linear networks (TLNs) are models of neural networks that consist of simple, perceptron-like neurons and exhibit nonlinear dynamics determined by the network's connectivity. The fixed points of a TLN, including both stable and unstable equilibria, play a critical role in shaping its emergent dynamics. In this work,we provide two novel characterizations for the set of fixed points of a competitive TLN: the first is in terms of a simple sign condition, while the second relies on the concept of domination. We apply these results to a special family of TLNs, called combinatorial threshold-linear networks (CTLNs),whose connectivity matrices are defined from directed graphs. This leads us to prove a series of graph rules that enable one to determine fixed points of a CTLN by analyzing the underlying graph. In addition, we study larger networks composed of smaller building block subnetworks and prove several theorems relating the fixed points of the full network to those of its components. Our results provide the foundation for a kind of graphical calculus to infer features of the dynamics from a network's connectivity.

Original languageEnglish (US)
Pages (from-to)94-155
Number of pages62
JournalNeural computation
Issue number1
StatePublished - Jan 1 2019

All Science Journal Classification (ASJC) codes

  • Arts and Humanities (miscellaneous)
  • Cognitive Neuroscience


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