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.Comment: 12 pages, 9 figures. Preliminary repor