Stable Synchronous Propagation of Signals by Feedforward Networks

TL;DR

This paper analyzes the stability of signal propagation in feedforward networks driven by a central pattern generator, with applications in biology and robotics, introducing new stability conditions for synchronized and traveling waves.

Contribution

It introduces a comprehensive stability analysis framework for synchronized signal propagation in feedforward networks, including transverse stability conditions and their relation to Floquet stability.

Findings

Stable standing and traveling waves are characterized by new Lyapunov, asymptotic, and Floquet conditions.

Transverse stability conditions imply stability for long chains of nodes.

Comparison shows transverse stability of the synchrony subspace is equivalent to Floquet stability for 1D nodes.

Abstract

We analyse the dynamics of networks in which a central pattern generator (CPG) transmits signals along one or more feedforward chains in a synchronous or phase-synchronous manner. Such propagating signals are common in biology, especially in locomotion and peristalsis, and are of interest for continuum robots. We construct such networks as feedforward lifts of the CPG. If the CPG dynamics is periodic, so is the lifted dynamics. Synchrony with the CPG manifests as a standing wave, and a regular phase pattern creates a travelling wave. We discuss Liapunov, asymptotic, and Floquet stability of the lifted periodic orbit and introduce transverse versions of these conditions that imply stability for signals propagating along arbitrarily long chains. We compare these notions to a simpler condition, transverse stability of the synchrony subspace, which is equivalent to Floquet stability when nodes are 1-dimensional.