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.
Highlights
The vast majority of the literature on attractor neural networks has focused on fixed point attractors
We have shown that in the case of Combinatorial threshold-linear networks (CTLNs), the problem is surprisingly tractable
We hypothesized that core fixed points can be used to predict attractors in CTLNs
Summary
The vast majority of the literature on attractor neural networks has focused on fixed point attractors. The typical scenario is that of a network that contains either a discrete set of stable fixed points, as in the Hopfield model, or a continuum of marginally stable fixed points, as in continuous attractor networks. In response to external inputs, the activity of the network converges to one of these fixed points. These are sometimes referred to as static attractors, because the fixed point is in an equilibrium or steady state. Even very simple ones like threshold-linear networks (TLNs), can exhibit dynamic attractors with periodic, quasiperiodic, or even chaotic orbits.
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