Amplitude-phase description of stochastic neural oscillators across the Hopf bifurcation

Abstract

We derive a unified amplitude-phase decomposition for both noisy limit cycles and quasicycles; in the latter case, the oscillatory motion has no deterministic counterpart. We extend a previous amplitude-phase decomposition approach using the stochastic averaging method (SAM) for quasicycles by taking into account nonlinear terms up to order 3. We further take into account the case of coupled networks where each isolated network can be in a quasi- or noisy limit-cycle regime. The method is illustrated on two models which exhibit a deterministic supercritical Hopf bifurcation: the Stochastic Wilson-Cowan model of neural rhythms, and the Stochastic Stuart-Landau model in physics. At the level of a single oscillatory module, the amplitude process of each of these models decouples from the phase process to the lowest order, allowing a Fokker-Planck estimate of the amplitude probability density. The peak of this density captures well the transition between the two regimes. The model describes accurately the effect of Gaussian white noise as well as of correlated noise. Bursting epochs in the limit-cycle regime are in fact favored by noise with shorter correlation time or stronger intensity. Quasicycle and noisy limit-cycle dynamics are associated with, respectively, Rayleigh-type and Gaussian-like amplitude densities. This provides an additional tool to distinguish quasicycle from limit-cycle origins of bursty rhythms. The case of multiple oscillatory modules with excitatory all-to-all delayed coupling results in a system of stochastic coupled amplitude-phase equations that keeps all the biophysical parameters of the initial networks and again works across the Hopf bifurcation. The theory is illustrated for small heterogeneous networks of oscillatory modules. Numerical simulations of the amplitude-phase dynamics obtained through the SAM are in good agreement with those of the original oscillatory networks. In the deterministic and nearly identical oscillators limits, the stochastic Stuart-Landau model leads to the stochastic Kuramoto model of interacting phases. The approach can be tailored to networks with different frequency, topology, and stochastic inputs, thus providing a general and flexible framework to analyze noisy oscillations continuously across the underlying deterministic bifurcation.