  We present a didactic introduction to spectral Dynamic Causal Modelling
(DCM), a Bayesian state-space modelling approach used to infer effective
connectivity from non-invasive neuroimaging data. Spectral DCM is currently the
most widely applied DCM variant for resting-state functional MRI analysis. Our
aim is to explain its technical foundations to an audience with limited
expertise in state-space modelling and spectral data analysis. Particular
attention will be paid to cross-spectral density, which is the most distinctive
feature of spectral DCM and is closely related to functional connectivity, as
measured by (zero-lag) Pearson correlations. In fact, the model parameters
estimated by spectral DCM are those that best reproduce the cross-correlations
between all measurements--at all time lags--including the zero-lag correlations
that are usually interpreted as functional connectivity. We derive the
functional connectivity matrix from the model equations and show how changing a
single effective connectivity parameter can affect all pairwise correlations.
To complicate matters, the pairs of brain regions showing the largest changes
in functional connectivity do not necessarily coincide with those presenting
the largest changes in effective connectivity. We discuss the implications and
conclude with a comprehensive summary of the assumptions and limitations of
spectral DCM.
