  The asymptotic attractors of a nonlinear dynamical system play a key role in
the long-term physically observable behaviors of the system. The study of
attractors and the search for distinct types of attractor have been a central
task in nonlinear dynamics. In smooth dynamical systems, an attractor is often
enclosed completely in its basin of attraction with a finite distance from the
basin boundary. Recent works have uncovered that, in neuronal networks,
unstable attractors with a remote basin can arise, where almost every point on
the attractor is locally transversely repelling. Herewith we report our
discovery of a class of attractors: partially unstable attractors, in
pulse-coupled integrate-and-fire networks subject to a periodic forcing. The
defining feature of such an attractor is that it can simultaneously possess
locally stable and unstable sets, both of positive measure. Exploiting the
structure of the key dynamical events in the network, we develop a symbolic
analysis that can fully explain the emergence of the partially unstable
attractors. To our knowledge, such exotic attractors have not been reported
previously, and we expect them to arise commonly in biological networks whose
dynamics are governed by pulse (or spike) generation.
