  We analyze the collective dynamics of hierarchically structured networks of
densely connected spiking neurons. These networks of sub-networks may represent
interactions between cell assemblies or different nuclei in the brain. The
dynamical activity pattern that results from these interactions depends on the
strength of synaptic coupling between them. Importantly, the overall dynamics
of a brain region in the absence of external input, so called ongoing brain
activity, has been attributed to the dynamics of such interactions. In our
study, two different network scenarios are considered: a system with one
inhibitory and two excitatory sub-networks, and a network representation with
three inhibitory sub-networks. To study the effect of synaptic strength on the
global dynamics of the network, two parameters for relative couplings between
these sub-networks are considered. For each case, a co-dimension two
bifurcation analysis is performed and the results have been compared to
large-scale network simulations. Our analysis shows that Generalized
Lotka-Volterra (GLV) equations, well-known in predator-prey studies, yield a
meaningful population-level description for the collective behavior of spiking
neuronal interaction, which have a hierarchical structure. In particular, we
observed a striking equivalence between the bifurcation diagrams of spiking
neuronal networks and their corresponding GLV equations. This study gives new
insight on the behavior of neuronal assemblies, and can potentially suggest new
mechanisms for altering the dynamical patterns of spiking networks based on
changing the synaptic strength between some groups of neurons.
