  Models of coupled oscillators are used to describe a wide variety of
phenomena in neuroimaging. These models typically rest on the premise that
oscillator dynamics do not evolve beyond their respective limit cycles, and
hence that interactions can be described purely in terms of phase differences.
Whilst mathematically convenient, the restrictive nature of phase-only models
can limit their explanatory power. We therefore propose a generalisation of
dynamic causal modelling that incorporates both phase and amplitude. This
allows for the separate quantifications of phase and amplitude contributions to
the connectivity between neural regions. We establish, using model-generated
data and simulations of coupled pendula, that phase-only models perform well
only under weak coupling conditions. We also show that, despite their higher
complexity, phase-amplitude models can describe strongly coupled systems more
effectively than their phase-only counterparts. We relate our findings to four
metrics commonly used in neuroimaging: the Kuramoto order parameter,
cross-correlation, phase-lag index, and spectral entropy. We find that, with
the exception of spectral entropy, the phase-amplitude model is able to capture
all metrics more effectively than the phase-only model. We then demonstrate,
using local field potential recordings in rodents and functional magnetic
resonance imaging in macaque monkeys, that amplitudes in oscillator models play
an important role in describing neural dynamics in anaesthetised brain states.
