Core motifs predict dynamic attractors in combinatorial threshold-linear networks

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.

Heterogeneous and higher-order cortical connectivity undergirds efficient, robust and reliable neural codes

“…Experimental research has demonstrated several ways in which neuronal connectivity is nonrandom with significant higher-order structure 18,21,28,29,31 . Moreover, small models with simple dynamics have shown the relevance of higher-order motifs in the observed activity [60][61][62] while in silico research has showed that the complexity is largely driven by neuronal morphologies 63,64,20 .…”

“…Second, counting n-simplices can be done systematically for all dimensions, and the maximum dimension attained (at a given graph density) is an indicator of network complexity. Third, counting n-simplices can be efficiently implemented as opposed to counting the "core motifs" or "robust motifs" of Parmelee et al 61 and Curto et al 62 , for which even listing them concretely above dimension 5/6 remains an open problem. Moreover, whilst counting n-simplices and counting directed cycles on n nodes have the same worst-case complexity, increasing exponentially with the number of nodes 66 ; in practice, counting n-simplices can be computed more efficiently, particularly in large sparse graphs.…”

“…Experimental research has demonstrated several ways in which neuronal connectivity is nonrandom with significant higher-order structure 30,33,42,43,45 . Moreover, small models with simple dynamics have shown the relevance of higher-order motifs in the observed activity [69][70][71] while in silico research has shown that the complexity is largely driven by neuronal morphologies 72,73,32 . At the same time, in the field of computational neuroscience, the bulk of research is conducted in models with simple connectivity based mostly on pairwise statistics that do not capture the entire complexity of the structure.…”

“…Song et al 28 , and Brunel 15 count all motifs on 3 and up to 5 nodes respectively. Parmelee et al 70 and Curto et al 71 , define other types of functionally meaningful motifs which they call "core motifs" and "robust motifs" respectively. In contrast to these, counting simplices can be done more systematically and has been implemented efficiently 75 .…”

Data-driven motif discovery in biological neural networks

“…The question of the functional and dynamical implications of structural motif must be a major consideration for future work. This prospect, however, may bring with it some considerable advantages: Recent work has shown that attractor dynamics may correlate closely with certain structural motifs [38,39], simultaneously narrowing the search space for structural motifs and broadening our ability to empirically validate results in simulation. These challenges will likely not be surmounted by individual research teams.…”