# Partially unstable attractors in networks of forced integrate-and-fire oscillators

## TL;DR

This paper introduces a new class of attractors called partially unstable attractors in forced integrate-and-fire neural networks, characterized by coexisting stable and unstable regions, expanding understanding of complex neural dynamics.

## Contribution

The study discovers and characterizes partially unstable attractors in pulse-coupled neural networks, providing a symbolic analysis of their emergence, a novel finding in nonlinear dynamics.

## Key findings

- Identified partially unstable attractors with mixed stability regions.
- Developed a symbolic analysis explaining their emergence.
- Suggests these attractors are common in biological neural networks.

## Abstract

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.