Core motifs predict dynamic attractors in combinatorial threshold-linear networks

TL;DR

This paper shows that core motifs in combinatorial threshold-linear networks can predict both static and dynamic attractors, linking network structure to complex nonlinear behaviors.

Contribution

It introduces the concept that core motifs are predictive of attractors and demonstrates this through extensive testing on directed graphs, advancing understanding of network dynamics.

Findings

Core motifs predict static and dynamic attractors.

Attractors can be located by perturbing fixed points.

Similar core motifs lead to similar attractors.

Abstract

Combinatorial threshold-linear networks (CTLNs) are a special class of inhibition-dominated TLNs defined from directed graphs. Like more general TLNs, they display a wide variety of nonlinear dynamics including multistability, limit cycles, quasiperiodic attractors, and chaos. In prior work, we have developed a detailed mathematical theory relating stable and unstable fixed points of CTLNs to graph-theoretic properties of the underlying network. Here we find that a special type of fixed points, corresponding to core motifs, are predictive of both static and dynamic attractors. Moreover, the attractors can be found by choosing initial conditions that are small perturbations of these fixed points. This motivates us to hypothesize that dynamic attractors of a network correspond to unstable fixed points supported on core motifs. We tested this hypothesis on a large family of directed graphs of size $n=5$, and found remarkable agreement. Furthermore, we discovered that core motifs with similar embeddings give rise to nearly identical attractors. This allowed us to classify attractors based on structurally-defined graph families. Our results suggest that graphical properties of the connectivity can be used to predict a network's complex repertoire of nonlinear dynamics.