Human brain organoids replicate much of the cellular diversity and developmental anatomy of the human brain. However, the physiological behavior of neuronal circuits within organoids remains relatively under-explored. With high-density CMOS microelectrode arrays and shank electrodes, we probed broadband and three-dimensional spontaneous activity of human brain organoids. These recordings simultaneously captured local field potentials (LFPs) and single unit activity. From spiking activity, we estimated a directed functional connectivity graph of synchronous neural network activity which showed a large number of weak functional connections enmeshed within a network skeleton of significantly fewer strong connections. Increasing the intrinsic inhibitory tone with a benzodiazepine altered the functional network graph of the organoid by suppressing the network skeleton. Simultaneously examining the spontaneous LFPs and their phase alignment to spiking showed that spike bursts were coherent with theta oscillations in the LFPs. An ensemble of spikes phase-locked to theta frequency oscillations were strongly interconnected as a sub-network within the larger network in which they were embedded. Our results demonstrate that human brain organoids have self-organized neuronal assemblies of sufficient size, cellular orientation, 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 mechanisms, and the effects of external stimuli upon neuronal networks.