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Synchronous neural activity in the brain in the "gamma" and "beta" frequency bands (15-70 Hz) is thought to be important for sensory processing and is altered in schizophrenia. In a previous study, gamma/beta click-train auditory stimuli were used to probe cortical oscillatory activity in control and schizophrenic subjects. We found that control subjects exhibited preferential 40 Hz responses to both 20 and 40 Hz stimulations, while schizophrenic subjects had enhanced 20 Hz responses to the same stimuli (D. Vierling-Claassen, P. Siekmeier, S. Stufflebeam, and N. Kopell, J. Neurophysiol., 99 (2008), pp. 2656-2671). High-dimensional computational network models constructed previously, which were based on evidence of altered inhibition in schizophrenia, numerically generated the entrainment behaviors observed experimentally. However, questions regarding the dynamic origin of model behaviors remained. It was not clear that the 20 Hz response to 40 Hz drive in the schizophrenic network was robust to parameter changes, which would be necessary for the predicted mechanism to explain data from a heterogeneous subject population. In the schizophrenic network we observed 30 Hz drive responses with a frequency component below 30 Hz, for which no analogue appeared in experimental data, and wondered if these were dynamically distinct from the modeled 20 Hz response to 40 Hz drive. We also wished to explore the role of background noise in model behavior. To address these questions, we consider a system of two mutually coupled oscillators representative of neural cells, driven periodically in the gamma/beta frequency band. We show that there is a one-parameter family of discontinuous discrete maps, whose dynamics clarifies issues of robustness, classifies entrainment patterns, and provides insight into the role of excitatory noise.

The Dynamics of a Periodically Forced Cortical Microcircuit, With an Application to Schizophrenia