  In neuronal systems, inhibition contributes to stabilizing dynamics and
regulating pattern formation. Through developing mean field theories of
neuronal models, using complete graph networks, inhibition is commonly viewed
as one ``control parameter'' of the system, promoting an absorbing phase
transition. Here, we show that for low connectivity sparse networks, inhibition
weight is not a control parameter of the transition. We present analytical and
simulation results using generic stochastic integrate-and-fire neurons that,
under specific restrictions, become other simpler stochastic neuron models
common in literature, which allow us to show that our results are valid for
those models as well. We also give a simple explanation about why the
inhibition role depends on topology, even when the topology has a
dimensionality greater than the critical one. The absorbing transition
independence of the inhibitory weight may be an important feature of a sparse
network, as it will allow the network to maintain a near-critical regime,
self-tuning average excitation, but at the same time, have the freedom to
adjust inhibitory weights for computation, learning, and memory, exploiting the
benefits of criticality.
