On the complex interplay between spectral harmonicity and different types of cross frequency couplings in non linear oscillators and biologically plausible neural network models

Background: Cross-frequency coupling (CFC) refers to the non linear interaction between oscillations in different frequency bands, and it is a rather ubiquitous phenomenon that has been observed in a variety of physical and biophysical systems. In particular, the coupling between the phase of slow oscillations and the amplitude of fast oscillations, referred as phase-amplitude coupling (PAC), has been intensively explored in the brain activity recorded from animals and humans. However, the interpretation of these CFC patterns remains challenging since harmonic spectral correlations characterizing non sinusoidal oscillatory dynamics can act as a confounding factor. Methods: Specialized signal processing techniques are proposed to address the complex interplay between spectral harmonicity and different types of CFC, not restricted only to PAC. For this, we provide an in-depth characterization of the Time Locked Index (TLI) as a novel tool aimed to efficiently quantify the harmonic content of noisy time series. It is shown that the proposed TLI measure is more robust and outperform traditional phase coherence metrics (e.g. Phase Locking Value) in several aspects. Results: We found that a non linear oscillator under the effect of additive noise can produce spurious CFC with low spectral harmonic content. On the other hand, two coupled oscillatory dynamics with independent fundamental frequencies can produce true CFC with high spectral harmonic content via a rectification mechanism or other post-interaction nonlinear processing mechanisms. These results reveal a complex interplay between CFC and harmonicity emerging from biologically plausible neural network models and more generic non linear and parametric oscillators, which in turn suggests that the harmonicity-CFC interplay is more complex than previously thought. Conclusions: We show that, contrary to what is usually assumed in the literature, the high harmonic content observed in non sinusoidal oscillatory dynamics, is neither sufficient nor necessary condition to interpret the associated CFC patterns as epiphenomenal. There is mounting evidence suggesting that the combination of multimodal recordings, specialized signal processing techniques and theoretical modeling is becoming a required step to completely understand CFC patterns observed in oscillatory rich dynamics of physical and biophysical systems.