Phase dynamics of delay-coupled quasi-cycles with application to brain rhythms

We consider the phase locking of two delay-coupled quasi-cycles. A coupled envelope-phase system obtained via stochastic averaging enables a stability analysis. While for deterministic limit-cycle oscillators the coupling can produce in-phase, antiphase, and the intermediate “out-of-phase” locking (OPL) behavior via spontaneous symmetry breaking, such outcomes for the quasi-cycle case are shown to require instead both noise and coupling delay. The theory, which applies the stochastic averaging method to delayed dynamics, generates stochastic stability functions that predict the numerically observed OPL behavior as a function of all the system parameters. OPL for coupled quasi-cycles occurs for additive or multiplicative noise, and for coupled networks of excitatory and inhibitory neurons as well as networks of inhibitory neurons coupled to one another. Our theory also predicts that the bifurcation at which the in-phase state becomes unstable lies at smaller delays for stronger noise. The noise produces the realistic quasi-cycle rhythms and out-of-phase behavior, all the while causing random reversals of the phase leader. Asymmetry in the coupling between networks, as well as heterogeneity within each network, also allows for quasi-cycle OPL, although it produces asymmetric bifurcations that bias the leadership towards one of the networks. These results are relevant to communication between brain areas and other networks that rely on noise-induced rather than noise-perturbed rhythms.