Human brain organoids replicate much of the cellular diversity and developmental anatomy of the human brain. However, the physiology of neuronal circuits within organoids remains under-explored. With high-density CMOS microelectrode arrays and shank electrodes, we captured spontaneous extracellular activity from brain organoids derived from human induced pluripotent stem cells. We inferred functional connectivity from spike timing, revealing a large number of weak connections within a skeleton of significantly fewer strong connections. A benzodiazepine increased the uniformity of firing patterns and decreased the relative fraction of weakly connected edges. Our analysis of the local field potential demonstrate that brain organoids contain neuronal assemblies of sufficient size and functional connectivity to co-activate and generate field potentials from their collective transmembrane currents that phase-lock to spiking activity. These results point to the potential of brain organoids for the study of neuropsychiatric diseases, drug action, and the effects of external stimuli upon neuronal networks.