In self-organized criticality (SOC) models, as well as in standard phase transitions, criticality is only present for vanishing driving 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. Here, we propose simple homeostatic mechanisms which promote self-organization of coupling strengths, gains, and firing thresholds (which control firing rate adaptation) in neuronal networks. We show that with adequate timescale separation between coupling strength and firing threshold dynamics, near criticality can be reached and sustained even with external inputs I. The firing thresholds adapt to and cancel the inputs (the effective field $h = I – \theta$ decreases towards zero), a phenomenon similar to perfect adaptation in sensory systems.

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