  This technical note introduces parametric dynamic causal modelling, a method
for inferring slow changes in biophysical parameters that control fluctuations
of fast neuronal states. The application domain we have in mind is inferring
slow changes in variables (e.g., extracellular ion concentrations or synaptic
efficacy) that underlie phase transitions in brain activity (e.g., paroxysmal
seizure activity). The scheme is efficient and yet retains a biophysical
interpretation, in virtue of being based on established neural mass models that
are equipped with a slow dynamic on the parameters (such as synaptic rate
constants or effective connectivity). In brief, we use an adiabatic
approximation to summarise fast fluctuations in hidden neuronal states (and
their expression in sensors) in terms of their second order statistics; namely,
their complex cross spectra. This allows one to specify and compare models of
slowly changing parameters (using Bayesian model reduction) that generate a
sequence of empirical cross spectra of electrophysiological recordings.
Crucially, we use the slow fluctuations in the spectral power of neuronal
activity as empirical priors on changes in synaptic parameters. This introduces
a circular causality, in which synaptic parameters underwrite fast neuronal
activity that, in turn, induces activity-dependent plasticity in synaptic
parameters. In this foundational paper, we describe the underlying model,
establish its face validity using simulations and provide an illustrative
application to a chemoconvulsant animal model of seizure activity.
