  We consider the theoretical constraints on interactions between coupled
cortical columns. Each column comprises a set of neural populations, where each
population is modelled as a neural mass. The existence of semi-stable states
within a cortical column has been shown to be dependent on the type of
interaction between the constituent neuronal subpopulations, i.e., the form of
the implicit synaptic convolution kernels. Current-to-current coupling has been
shown, in contrast to potential-to-current coupling, to create semi-stable
states within a cortical column. In this analytic and numerical study, the
interaction between semi-stable states is characterized by equations of motion
for ensemble activity. We show that for small excitations, the dynamics follow
the Kuramoto model. However, in contrast to previous work, we derive coupled
equations between phase and amplitude dynamics. This affords the possibility of
defining connectivity as a dynamic variable. The turbulent flow of phase
dynamics found in networks of Kuramoto oscillators indicate turbulent changes
in dynamic connectivity for coupled cortical columns. Crucially, this is
something that has been recorded in epileptic seizures. We used the results we
derived to estimate a seizure propagation model, which allowed for relatively
straightforward inversions using variational Laplace (a.k.a., Dynamic Causal
Modelling). The face validity of the ensuing seizure propagation model was
established using simulated data as a prelude to future work; which will
investigate dynamic connectivity from empirical data. This model also allows
predictions of seizure evolution, induced by virtual lesions to synaptic
connectivity: something that could be of clinical use, when applied to epilepsy
surgical cases.
