  We consider a heterogeneous, globally coupled population of excitatory
quadratic integrate-and-fire neurons with excitability adaptation due to a
metabolic feedback associated with ketogenic diet, a form of therapy for
epilepsy. Bifurcation analysis of a three-dimensional mean-field system derived
in the framework of next-generation neural mass models allows us to explain the
scenarios and suggest control strategies for the transitions between the
neurophysiologically desired asynchronous states and the synchronous,
seizure-like states featuring collective oscillations. We reveal two
qualitatively different scenarios for the onset of synchrony. For weaker
couplings, a bistability region between the lower- and the higher-activity
asynchronous states unfolds from the cusp point, and the collective
oscillations emerge via a supercritical Hopf bifurcation. For stronger
couplings, one finds seven co-dimension two bifurcation points, including pairs
of Bogdanov-Takens and generalized Hopf points, such that both lower- and
higher-activity asynchronous states undergo transitions to collective
oscillations, with hysteresis and jump-like behavior observed in vicinity of
subcritical Hopf bifurcations. We demonstrate three control mechanisms for
switching between asynchronous and synchronous states, involving parametric
perturbation of the adenosine triphosphate (ATP) production rate, external
stimulation currents, or pulse-like ATP shocks, and indicate a potential
therapeutic advantage of hysteretic scenarios.
