  In self-organized criticality (SOC) models, as well as in standard phase
transitions, criticality is only present for vanishing external fields $h \to
0$. Considering that this is rarely the case for natural systems, such a
restriction poses a challenge to the explanatory power of these models. Besides
that, in models of dissipative systems like earthquakes, forest fires, and
neuronal networks, there is no true critical behavior, as expressed in clean
power laws obeying finite-size scaling, but a scenario called "dirty"
criticality or self-organized quasi-criticality (SOqC). Here, we propose simple
homeostatic mechanisms which promote self-organization of coupling strengths,
gains, and firing thresholds in neuronal networks. We show that with an
adequate separation of the timescales for the coupling strength and firing
threshold dynamics, near criticality (SOqC) can be reached and sustained even
in the presence of significant external input. The firing thresholds adapt to
and cancel the inputs ($h$ decreases towards zero). Similar mechanisms can be
proposed for the couplings and local thresholds in spin systems and cellular
automata, which could lead to applications in earthquake, forest fire, stellar
flare, voting, and epidemic modeling.
