A General Approach for Quantifying Nonlinear Connectivity in the Nervous System Based on Phase Coupling

Abstract

Interaction between distant neuronal populations is essential for communication within the nervous system and can occur as a highly nonlinear process. To better understand the functional role of neural interactions, it is important to quantify the nonlinear connectivity in the nervous system. We introduce a general approach to measure nonlinear connectivity through phase coupling: the multi-spectral phase coherence (MSPC). Using simulated data, we compare MSPC with existing phase coupling measures, namely n : m synchronization index and bi-phase locking value. MSPC provides a system description, including (i) the order of the nonlinearity, (ii) the direction of interaction, (iii) the time delay in the system, and both (iv) harmonic and (v) intermodulation coupling beyond the second order; which are only partly revealed by other methods. We apply MSPC to analyze data from a motor control experiment, where subjects performed isotonic wrist flexions while receiving movement perturbations. MSPC between the perturbation, EEG and EMG was calculated. Our results reveal directional nonlinear connectivity in the afferent and efferent pathways, as well as the time delay (43 ± 8 ms) between the perturbation and the brain response. In conclusion, MSPC is a novel approach capable to assess high-order nonlinear interaction and timing in the nervous system.

A Generalized Coherence Framework for Detecting and Characterizing Nonlinear Interactions in the Nervous System

“…Compared to previous studies using single-sine stimuli [7,11], multi-sine stimulus can evoke richer nonlinearity including the intermodulation coupling among multiple stimulation frequencies [17,21]. In line with previous studies [14,21,41], intermodulation and harmonic couplings were detected in all subjects, indicating that integer multiplication mapping is an important nonlinear mapping in the system. Nevertheless, this indication alone does not mean the proprioceptive system is static, since in this study subharmonic coupling was also found in the majority of the subjects (8 out of 11).…”

“…To the best of our knowledge, only Langdon and his colleagues has reported 2:3 subharmonic coupling between brain responses and tactile vibrations [11]. There are only a very few previous studies investigating nonlinear interactions between brain responses and proprioceptive stimuli [21,23,24,42]. None of them has reported subharmonic coupling.…”

“…Thus, the CSC framework provides a more complete description of input-output interactions in the nervous system than existing coherence measures (linear coherence and bicoherence). Different from generalized phase synchrony measures [20,21], the CSC framework reflects not only phase but also amplitude relation between the input and the output; therefore the CSC framework is more suitable for probing stimulus-response interactions regarding both phase and amplitude relation in the nervous system [21]. For researchers who are interested in phase synchrony, please refer to our recent work in [21].…”

“…Subharmonic coupling is typically assessed by phase synchrony (PS) measures such as the n:m phase synchronization index [20]. However, PS measures assess nonlinear interactions (as well as linear interactions) between two signals purely based on their relative phase, independent of signal amplitude (see [20,21] for details and [22] for a review). Therefore, they are more often used to investigate the synchronized discharge of neuronal populations (e.g.…”

Critical fluctuations and coupling of stochastic neural mass models

“…Measures based on asymmetry of the phase lag distribution have been proposed (Stam et al, 2007;Vinck et al, 2011) to disregard spurious zero-lag coherence due to volume conduction. Nonlinear cross-frequency coupling can also be determined for both phase (Jirsa and Müller, 2013;Yang et al, 2015) and amplitude (Bruns et al, 2000).…”