  Macroscopic oscillations of different brain regions show multiple phase
relationships that are persistent across time and have been implicated routing
information. Various cellular level mechanisms influence the network dynamics
and structure the macroscopic firing patterns. Key question is to identify the
biophysical neuronal and synaptic properties that permit such motifs to arise
and how the different coherence states determine the communication between the
neural circuits. We analyse the emergence of phase locking within
bidirectionally delayed-coupled spiking circuits showing global gamma band
oscillations. We consider both the interneuronal (ING) and the
pyramidal-interneuronal (PING) gamma rhythms and the inter coupling targeting
the pyramidal or the inhibitory interneurons. Using a mean-field approach
together with an exact reduction method, we break down each spiking network
into a low dimensional nonlinear system and derive the macroscopic phase
resetting-curves (mPRCs) that determine how the phase of the global oscillation
responds to incoming perturbations. Depending on the type of gamma oscillation,
we show that incoming excitatory inputs can either only speed up the
oscillation (phase advance; type I PRC) or induce both an advance and a delay
the macroscopic oscillation (phase delay; type II PRC). From there we determine
the structure of macroscopic coherence states (phase locking) of two weakly
synaptically-coupled networks. To do so we derive a phase equation for the
coupled system which links the synaptic mechanisms to the coherence state of
the system. We show that the transmission delay is a necessary condition for
symmetry breaking, i.e. a non-symmetric phase lag between the macroscopic
oscillations, potentially giving an explanation to the experimentally observed
variety of gamma phase-locking modes.
