Directional coherence disentangles causality within the sensorimotor loop, but cannot open the loop

Abstract

More and more studies indicate that corticomuscular coherence in the beta band (15–30 Hz), which expresses the functional coupling between the cortex and the muscles, originates from the interaction within the sensorimotor loop (e.g. Witham et al. 2011). The phase of the corticomuscular coherence expresses the relative time–frequency relationship and is often explained as to result from the efferent delay between the cortex and the muscles. In a recent issue of The Journal of Physiology, Witham and co-workers (2011) demonstrated that the slope of the phase of corticomuscular coherence is less negative than would be expected of pure efferent pathways and even becomes positive in some subjects (negative slopes indicate that the muscle lags the brain). This is a clear indication of a bidirectional coupling between EEG and EMG; in other words, the signals are part of a closed-loop system. However, the authors also use the phase of the directional coherence to assess the delays in the efferent and afferent pathways, which will give erroneous results in a closed-loop system, like the sensorimotor loop. The causality of signals within a closed loop is difficult to assess. For example, in the sensorimotor loop it is not obvious whether cortical activity leads muscle activity – suggesting an efferent pathway; or cortical activity lags muscle activity – suggesting an afferent pathway. In the sensorimotor loop EEG and EMG signals will contain a combination of afferent and efferent influences. As the authors demonstrate, directional coherence provides a good measure to disentangle the causal relationships of the signals within the sensorimotor loop. With directional coherence multivariate autoregressive (MVAR) modelling is used to uncover causality. MVAR modelling is a common technique which disentangles the recorded signals at a certain time instant as a weighted sum of the signals’ previous values and (unknown) external noise sources, which enter the model just before the signals (see Fig. 1). The directed coherence is calculated using the directional transfer function Hij(f) (e.g. Witham et al. eqn (3)). The directional transfer function Hij(f) is calculated, which represents: ‘the causal influence of signal j on signal i’. Figure 1 Schematic representation of the sensorimotor loop as a closed-loop system Although this is a widely accepted expression, it is a simplified expression. The precise expression would be that the direction transfer function represents the causal influence of external noise source j on signal i (Kaminski & Blinowska, 1991). In other words the directional transfer function expresses how much signal i depends on the unknown external noise source which enters the model just before signal j. The simplification on what directional transfer function represents has a tremendous effect on the understanding of the phase of the directional coherence. In Witham et al. (2011), the authors assume that the phase of the directional coherence represents the relative delay between signal i and signal j. With this assumption the slope of the phase would represent the delays in the open-loop transfer functions, i.e. the relative delay of the efferent (EEG to EMG: Heff) and afferent (EMG to EEG: Haff) pathways. However the directional transfer function is a closed-loop transfer function between noise source j and signal i. In that sense directional transfer functions allow the disentangling of the causality within a closed loop, but the phase of the directional coherence presents the relative delay between the signal within the sensorimotor loop (EEG and EMG) and the unknown external noise sources ɛ (i.e. the cortical and afferent drive). This effect contributes to the observation of Witham and co-workers that the delays measured by using the directed coherence were often larger than would be expected the known conduction delays from the cortex to muscle assesses with stimulation and peripheral nerve stimulation. In conclusion, the technique based on multivariate AR modelling – like directional coherence – decomposes signals within a closed loop as a weighted combination of external sources and the signals’ past, and allows disentangling of the causality. The phase of the directional coherence, however, describes the relative delay between the signals and the unknown external sources, and is not a direct measure of the phase of the inferred open-loop transfer function, like the efferent and afferent pathways. New techniques which are able to assess the open-loop transfer functions are highly desirable. The application of controlled external perturbations could be a promising way (Campfens et al. 2011).