Phase response approach to networked neural activity models with delays

Networks of delay-coupled neural mass models alternate between synchrony and anti-phase locking as the delay between brain regions increases. The origin of these delay-dependent bands has remained unclear. Here, we show that a single function, the phase interaction function $H$ of an isolated node, predicts where the bands occur and explains why they repeat. For a Wilson--Cowan model with intra-nodal delay $τ$ and inter-nodal delay $ρ$, we compute the limit cycle and phase response, and hence $H$, by harmonic balance. Each band boundary is a zero of $H'$ shifted by the phase lag $Ωρ$, so the bands repeat in $ρ$ with the oscillation period. Simulations of the full delay-differential networks confirm these predictions, from two nodes, where predicted frequencies agree to second order in the coupling strength, to rings, where the theory also identifies coexisting stable twisted states that random initial conditions rarely reach. At weak coupling, the delays at which synchrony gains or loses stability are set by the node dynamics alone, whereas stable anti-phase states require a bipartite network. For a macaque connectome, metastability peaks at the edges of the synchrony bands, where the stability margin of phase-locking vanishes. The reduction is accurate for coupling strengths well below the slowest Floquet exponent of the node, beyond which a phase--amplitude description is needed. More broadly, vanishing phase-locking stability provides a simple route to metastability in delay-coupled oscillator networks and a candidate mechanism for the flexible coordination of brain dynamics.