Untangling cross-regional cross-frequency coupling in dynamic neural oscillations

<i>Objective.</i>Brain networks communicate through long range phase coupling of low frequency oscillations (LFOs, less than 35 Hz) between brain regions. At the same time, phase-amplitude cross-frequency coupling (CFC), in which the phase of the same LFO have been shown to modulate the power of high frequency activity has also been reported across brain regions as a critical regulator of neural activity and excitability. While cross-regional CFC has been reported as a potential mechanism of long-distance modulation of neural excitability, the mechanism underlying this phenomenon has yet to be understood and methods to dissociate the effect of local vs remote LFO have not been developed. Cross-regional CFC can be a result of either LFOs in one region directly modulating high frequency oscillations in another region or due to a chain effect, in which apparent cross-regional CFC results from coupling of LFO across sites.<i>Approach.</i>A novel method of partial modulation index (PMI) is proposed as a derivation of modulation index (MI) and based on Pearl's do-calculus to remove the mathematical bias of simultaneous phase coupling and CFC measurements. Here, we first test the PMI on a simulated dataset, showing it can differentiate between biased and unbiased CFC. We then evaluate the method on intracranially collected local field potentials recorded simultaneously from thalamus and cortex in a patient undergoing deep brain stimulator implantation for essential tremor, demonstrating that the observed thalamocortical CFC was partially biased.<i>Main results.</i>For both simulated and human datasets, the PMI was compared to the conventional MI. In simulated data, the PMI was able to disentangle cross-regional phase coupling and focal CFC which is not possible using conventional MI. While there is no ground truth for comparison in human data, the results from the simulated data demonstrate the value of the proposed method in removing mathematical bias.<i>Significance.</i>This novel method facilitates a mathematically rigorous characterization of residual CFC, enabling investigations of differential contributions and roles of brain-wide LFO to CFC, which can lead to a more complete understanding of the pathophysiology of neurological processes and disorders.