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, nonnegligible 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 nontriviality 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, that is, 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 statistical 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. Author Summary The brain criticality hypothesis is one of the most focused and controversial topics in neuroscience and biophysics. This research develops a unified framework to reformulate the physics theories of four basic types of brain criticality, ordinary criticality (OC), quasi-criticality (qC), self-organized criticality (SOC), and self-organized quasi-criticality (SOqC), into more accessible and neuroscience-related forms. For the statistic techniques used to validate the brain criticality hypothesis, we also present comprehensive explanations of them, summarize their error-prone details, and suggest possible solutions. This framework may help resolve potential controversies in studying the brain criticality hypothesis, especially those arising from the misconceptions about the theoretical foundations of brain criticality.

The brain consists of billions of neurons connected by ultra-dense synapses, showing remarkable efficiency, robust flexibility, and economy in information processing. It is generally believed that these advantageous properties are rooted in brain connectivity; however, direct evidence remains absent owing to technical limitations or theoretical vacancy. This research explores the origins of these properties in the largest yet brain connectome of the fruit fly. We reveal that functional connectivity formation in the brain can be explained by a percolation process controlled by synaptic excitation-inhibition (E/I) balance. By increasing the E/I balance gradually, we discover the emergence of these properties as byproducts of percolation transition when the E/I balance arrives at 3:7. As the E/I balance keeps increase, an optimal E/I balance 1:1 is unveiled to ensure these three properties simultaneously, consistent with previous in vitro experimental predictions. Once the E/I balance reaches over 3:2, an intrinsic limitation of these properties determined by static (anatomical) brain connectivity can be observed. Our work demonstrates that percolation, a universal characterization of critical phenomena and phase transitions, may serve as a window toward understanding the emergence of various brain properties. Author Summary This research presents a novel framework to study functional connectivity on the largest yet brain connectome of the fruit fly, revealing that synaptic excitation-inhibition (E/I) balance characterizes the formation of dynamic brain connectivity as a percolation process. Various remarkable properties of brain functions, such as information transmission efficiency, robust flexibility, and economy, emerge as byproducts of percolation transition. These advantages can be simultaneously ensured at an optimal E/I balance 1:1, consistent with previous in vitro experimental predictions. Our work demonstrates percolation as a potential way to understand the emergence of brain function characteristics through connectivity.