Theta-gamma coupling: nonlinearity as a universal cross-frequency coupling mechanism

The Cross Frequency Coupling (CFC) phenomenon is defined as a statistical correlation between characteristic parameters neural oscillations. This study demonstrates and analyzes the nonlinear mechanism of the CFC, with a focus on the coupling between slow and fast oscillations, as a model for theta-gamma coupling. We first discuss the usage of the spectrum/bispectrum CFC measure using experimental data. As a physical paradigm, we propose the concept of a Class II neural population at low activity: neurons fire intermittently, and the time spent in the subthreshold regime is much larger that the duration of an action potential. We verify the emergence of fast oscillations (gamma) using a direct numerical simulations (DNS) of a population of Hodgkin-Huxley neurons forced by a slow theta oscillation. To deconstruct the mechanism, we derive a mean field approximation based on a reduction of the Hodgkin-Huxley model to a two-equation leaky-integrate-and-fire (LIF) model. Under theta forcing, mean field model generates gamma oscillations; the solutions exhibit spectrum/bispectrum CFC patterns that agree qualitatively with both the DNS model and experimental data. For the theta-gamma coupling problem, the mean field model may be linearized using an asymptotic expansion. The analytical solution of the linear system describes theta-gamma interaction as a gamma stabilization/destabilization cycle and provides explicit expressions of the gamma amplitude and frequency modulation by theta. The results demonstrate that nonlinearity as a universal/unifying mechanism of all CFC types.