  Low-dimensional yet rich dynamics often emerge in the brain. Examples include
oscillations and chaotic dynamics during sleep, epilepsy, and voluntary
movement. However, a general mechanism for the emergence of low dimensional
dynamics remains elusive. Here, we consider Wilson-Cowan networks and
demonstrate through numerical and analytical work that a type of homeostatic
regulation of the network firing rates can paradoxically lead to a rich
dynamical repertoire. The dynamics include mixed-mode oscillations, mixed-mode
chaos, and chaotic synchronization. This is true for single recurrently coupled
node, pairs of reciprocally coupled nodes without self-coupling, and networks
coupled through experimentally determined weights derived from functional
magnetic resonance imaging data. In all cases, the stability of the homeostatic
set point is analytically determined or approximated. The dynamics at the
network level are directly determined by the behavior of a single node system
through synchronization in both oscillatory and non-oscillatory states. Our
results demonstrate that rich dynamics can be preserved under homeostatic
regulation or even be caused by homeostatic regulation.
