  This report is concerned with the relevance of the microscopic rules, that
implement individual neuronal activation, in determining the collective
dynamics, under variations of the network topology. To fix ideas we study the
dynamics of two cellular automaton models, commonly used, rather
in-distinctively, as the building blocks of large scale neuronal networks. One
model, due to Greenberg \& Hastings, (GH) can be described by evolution
equations mimicking an integrate-and-fire process, while the other model, due
to Kinouchi \& Copelli, (KC) represents an abstract branching process, where a
single active neuron activates a given number of postsynaptic neurons according
to a prescribed "activity" branching ratio. Despite the apparent similarity
between the local neuronal dynamics of the two models, it is shown that they
exhibit very different collective dynamics as a function of the network
topology. The GH model shows qualitatively different dynamical regimes as the
network topology is varied, including transients to a ground (inactive) state,
continuous and discontinuous dynamical phase transitions. In contrast, the KC
model only exhibits a continuous phase transition, independently of the network
topology. These results highlight the importance of paying attention to the
microscopic rules chosen to model the inter-neuronal interactions in large
scale numerical simulations, in particular when the network topology is far
from a mean field description. One such case is the extensive work being done
in the context of the Human Connectome, where a wide variety of types of models
are being used to understand the brain collective dynamics.
