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Cortical oscillations play a fundamental role in organizing large-scale functional brain networks. Noninvasive brain stimulation with temporally patterned waveforms such as repetitive transcranial magnetic stimulation (rTMS) and transcranial alternating current stimulation (tACS) have been proposed to modulate these oscillations. Thus, these stimulation modalities represent promising new approaches for the treatment of psychiatric illnesses in which these oscillations are impaired. However, the mechanism by which periodic brain stimulation alters endogenous oscillation dynamics is debated and appears to depend on brain state. Here, we demonstrate with a static model and a neural oscillator model that recurrent excitation in the thalamo-cortical circuit, together with recruitment of cortico-cortical connections, can explain the enhancement of oscillations by brain stimulation as a function of brain state. We then performed concurrent invasive recording and stimulation of the human cortical surface to elucidate the response of cortical oscillations to periodic stimulation and support the findings from the computational models. We found that (1) stimulation enhanced the targeted oscillation power, (2) this enhancement outlasted stimulation, and (3) the effect of stimulation depended on behavioral state. Together, our results show successful target engagement of oscillations by periodic brain stimulation and highlight the role of nonlinear interaction between endogenous network oscillations and stimulation. These mechanistic insights will contribute to the design of adaptive, more targeted stimulation paradigms., This study presents mathematical models that explain the effect of temporally patterned electrical stimulation on cortical oscillations and provides supporting evidence using data recorded directly from human cortex during transcranial electrical stimulation., Author Summary Rhythms in the brain are believed to play an important role in cognition. Disruptions in these oscillations are associated with a number of neurological and psychiatric disorders. Therefore, noninvasive brain stimulation techniques that target these oscillations offer promise as therapeutic tools. In particular, transcranial alternating current stimulation (tACS) applies a periodic stimulation waveform to engage specific oscillations in the cortex. Although recent studies provide evidence for the modulation of cortical oscillations by tACS, the exact mechanism by which the effects are produced is poorly understood. We propose two mathematical models of interaction between periodic electrical stimulation and ongoing brain activity that may explain the effects of tACS. In addition, we present a unique dataset in which we stimulated the patients’ cortical surface with subdural electrodes and observed the responses to stimulation in neighboring electrodes. We found that stimulation enhanced ongoing oscillations both during and immediately after stimulation. This enhancement depended on the brain state, thereby supporting our proposed models. Our results demonstrate the effect of electrical stimulation on cortical oscillations and highlight the importance of considering the state of the brain when designing electrical stimulation therapies for disorders of the central nervous system.