  Electro-cortical activity in patients with epilepsy may show abnormal
rhythmic transients in response to stimulation. Even when using the same
stimulation parameters in the same patient, wide variability in the duration of
transient response has been reported. These transients have long been
considered important for the mapping of the excitability levels in the
epileptic brain but their dynamic mechanism is still not well understood.
  To understand the occurrence of abnormal transients dynamically, we use a
thalamo-cortical neural population model of epileptic spike-wave activity and
study the interaction between slow and fast subsystems.
  In a reduced version of the thalamo-cortical model, slow wave oscillations
arise from a fold of cycles (FoC) bifurcation. This marks the onset of a region
of bistability between a high amplitude oscillatory rhythm and the background
state. In vicinity of the bistability in parameter space, the model has
excitable dynamics, showing prolonged rhythmic transients in response to
suprathreshold pulse stimulation. We analyse the state space geometry of the
bistable and excitable states, and find that the rhythmic transient arises when
the impending FoC bifurcation deforms the state space and creates an area of
locally reduced attraction to the fixed point. This area essentially allows
trajectories to dwell there before escaping to the stable steady state, thus
creating rhythmic transients. In the full thalamo-cortical model, we find a
similar FoC bifurcation structure.
  Based on the analysis, we propose an explanation of why stimulation induced
epileptiform activity may vary between trials, and predict how the variability
could be related to ongoing oscillatory background activity.
