Breaking the bonds of weak coupling: the dynamic causal modelling of oscillator amplitudes

Models of coupled oscillators are useful in describing a wide variety of phenomena in physics, biology and economics. These models typically rest on the premise that the oscillators are weakly coupled, meaning that amplitudes can be assumed to be constant and dynamics can therefore be described purely in terms of phase differences. Whilst mathematically convenient, the restrictive nature of the weak coupling assumption can limit the explanatory power of these phase-coupled oscillator models. We therefore propose an extension to the weakly-coupled oscillator model that incorporates both amplitude and phase as dependent variables. We use the bilinear neuronal state equations of dynamic causal modelling as a foundation in deriving coupled differential equations that describe the activity of both weakly and strongly coupled oscillators. We show that weakly-coupled oscillator models are inadequate in describing the processes underlying the temporally variable signals observed in a variety of systems. We demonstrate that phase-coupled models perform well on simulations of weakly coupled systems but fail when connectivity is no longer weak. On the other hand, using Bayesian model selection, we show that our phase-amplitude coupling model can describe non-weakly coupled systems more effectively despite the added complexity associated with using amplitude as an extra dependent variable. We demonstrate the advantage of our phase-amplitude model in the context of model-generated data, as well as of a simulation of inter-connected pendula, neural local field potential recordings in rodents under anaesthesia and international economic gross domestic product data.