Diversity of emergent dynamics in competitive threshold-linear networks

TL;DR

This paper explores the diverse nonlinear dynamics possible in competitive threshold-linear networks, revealing conditions that prevent steady states and enable complex behaviors like chaos and limit cycles, with implications for understanding neural network connectivity.

Contribution

It introduces conditions for the absence of steady states in competitive threshold-linear networks and demonstrates their capacity for rich nonlinear dynamics, including coexisting behaviors.

Findings

Networks can exhibit chaos, limit cycles, and quasiperiodic attractors.

Multiple dynamic patterns can coexist within the same network.

A new MATLAB package facilitates simulation of these dynamics.

Abstract

Threshold-linear networks consist of simple units interacting in the presence of a threshold nonlinearity. Competitive threshold-linear networks have long been known to exhibit multistability, where the activity of the network settles into one of potentially many steady states. In this work, we find conditions that guarantee the absence of steady states, while maintaining bounded activity. These conditions lead us to define a combinatorial family of competitive threshold-linear networks, parametrized by a simple directed graph. By exploring this family, we discover that threshold-linear networks are capable of displaying a surprisingly rich variety of nonlinear dynamics, including limit cycles, quasiperiodic attractors, and chaos. In particular, several types of nonlinear behaviors can co-exist in the same network. Our mathematical results also enable us to engineer networks with multiple dynamic patterns. Taken together, these theoretical and computational findings suggest that threshold-linear networks may be a valuable tool for understanding the relationship between network connectivity and emergent dynamics. The new Matlab package CTLN Basic 2.0 can be used to reproduce the simulations in Figures 1-13. The package is available at https://github.com/nebneuron/CTLN-Basic-2.0