We establish a general framework that explains how leaky, dissipative systems, such as neuronal networks (NN), can exhibit robust self-organized criticality (SOC). Consistent with recent experiments, we propose that persistent membrane potential fluctuations allow NNs to transform from a sub-critical to a critical state. Our results also account for the tendency in small networks to tip towards an epileptiform state (the case of largely synchronized neurons) when background activity is strong.