  Brain activities are featured by spatially distributed neural clusters of
coherent firings and a spontaneous switching of the clusters between the
synchrony and asynchrony states. Evidences from {\it in vivo} experiments
suggest that astrocytes, a type of glial cell regarded previously as providing
only structural and metabolic supports to neurons, participate actively in
brain functions and play a crucial role in regulating the neural firing
activities, yet the mechanism remains unknown. Introducing astrocyte as a
reservoir of the glutamate released from neuron synapses, here we propose the
model of complex neuron-astrocyte network and employ it to explore the roles of
astrocyte in regulating the synchronization behaviors of networked neurons. It
is found that a fraction of neurons on the network can be synchronized as a
cluster, while the remaining neurons are kept as desynchronized. Moreover,
during the course of network evolution, the cluster is switching between the
synchrony and asynchrony states intermittently, henceforth the phenomenon of
``breathing cluster". By the method of symmetry-based analysis, we conduct a
theoretical investigation on the stability of the cluster and the mechanism
generating the breathing activities. It is revealed that the contents of the
cluster are determined by the network symmetry and the breathing activities are
due to the interplay between the neural network and the astrocyte. The
breathing phenomenon is demonstrated in network models of different structures
and neural dynamics. The studies give insights into the cellular mechanism of
astrocytes in regulating neural activities, and shed lights onto the
spontaneous state switching of the neocortex.
