  Criticality is hypothesized as a physical mechanism underlying efficient
transitions between cortical states and remarkable information processing
capacities in the brain. While considerable evidence generally supports this
hypothesis, non-negligible controversies persist regarding the ubiquity of
criticality in neural dynamics and its role in information processing. Validity
issues frequently arise during identifying potential brain criticality from
empirical data. Moreover, the functional benefits implied by brain criticality
are frequently misconceived or unduly generalized. These problems stem from the
non-triviality and immaturity of the physical theories that analytically derive
brain criticality and the statistic techniques that estimate brain criticality
from empirical data. To help solve these problems, we present a systematic
review and reformulate the foundations of studying brain criticality, i.e.,
ordinary criticality (OC), quasi-criticality (qC), self-organized criticality
(SOC), and self-organized quasi-criticality (SOqC), using the terminology of
neuroscience. We offer accessible explanations of the physical theories and
statistic techniques of brain criticality, providing step-by-step derivations
to characterize neural dynamics as a physical system with avalanches. We
summarize error-prone details and existing limitations in brain criticality
analysis and suggest possible solutions. Moreover, we present a forward-looking
perspective on how optimizing the foundations of studying brain criticality can
deepen our understanding of various neuroscience questions.
