Rigorous formulation of transverse stability on average

Develop a rigorous mathematical formulation of "transverse stability on average" for periodic orbits in network dynamical systems, specifically for feedforward lifts and central pattern generator-driven chains, so that the notion can be precisely stated and analyzed within the standard framework of coupled ODEs and Floquet theory.

Background

The paper studies stability of periodic signals propagating through feedforward lifts of central pattern generators in networks of model neurons. Beyond classical Floquet stability and transverse stability of synchrony subspaces, the authors discuss a heuristic notion they call "transverse stability on average," which informally describes cases where brief repelling segments along a periodic orbit are compensated by longer attracting segments, yielding overall stability of the lifted orbit. While numerical evidence suggests this phenomenon occurs in several neuron models, especially when node spaces have dimension greater than one, the concept lacks a formal definition within the network-dynamics and Floquet frameworks.

The authors explicitly note that this average notion of transverse stability has not been made rigorous and, elsewhere in the paper, emphasize that it is unclear how to formalize it for higher-dimensional node spaces. A precise definition and accompanying analytical criteria would enable systematic use of this stability concept and clarify its relationship to standard Floquet stability.