Direct modulation index: A measure of phase amplitude coupling for neurophysiology data

Abstract Neural communication across different spatial and temporal scales is a topic of great interest in clinical and basic science. Phase‐amplitude coupling (PAC) has attracted particular interest due to its functional role in a wide range of cognitive and motor functions. Here, we introduce a novel measure termed the direct modulation index (dMI). Based on the classical modulation index, dMI provides an estimate of PAC that is (1) bound to an absolute interval between 0 and +1, (2) resistant against noise, and (3) reliable even for small amounts of data. To highlight the properties of this newly‐proposed measure, we evaluated dMI by comparing it to the classical modulation index, mean vector length, and phase‐locking value using simulated data. We ascertained that dMI provides a more accurate estimate of PAC than the existing methods and that is resilient to varying noise levels and signal lengths. As such, dMI permits a reliable investigation of PAC, which may reveal insights crucial to our understanding of functional brain architecture in key contexts such as behaviour and cognition. A Python toolbox that implements dMI and other measures of PAC is freely available at https://github.com/neurophysiological-analysis/FiNN.


| INTRODUCTION
The investigation of the communication between structures across different spatial and temporal scales has been a major area of interest in the field of cognitive and motor neuroscience (Siems et al., 2016;Siems & Siegel, 2020). In particular, a growing body of research regards phase-amplitude coupling (PAC) as a phenomenon reflective of a multi-frequency communication mode across and within neural structures (Canolty & Knight, 2010;Jensen & Colgin, 2007). The level of PAC between neural structures is quantified by the degree to which the phase of a low-frequency neural oscillation reflects the shape of the amplitude of a high-frequency oscillation (Bragin et al., 1995;Lakatos et al., 2005). Several studies have reported a close relationship between PAC of high-gamma amplitude with alpha phase and behavioral performance on cognitive, motor, and sensory tasks (e.g., Schroeder & Lakatos, 2009;Voytek et al., 2010;Yanagisawa et al., 2012).
Given the large interest in PAC, many methods have been developed to provide an accurate quantification of this cross-frequency neural communication (Tort et al., 2010). A non-exhaustive overview of established methods is presented in Table 1. The methods include Abbreviations: dMI, direct modulation index; EEG, electroencephalography; MI, modulation index; MVL, mean vector length; PAC, phase-amplitude coupling; PLV, phase-locking value. the modulation index (MI; Tort et al., 2008), mean vector length (MVL; Canolty et al., 2006) and phase locking value (PLV; Mormann et al., 2005). Although these methods were successful in revealing relevant brain-behaviour relationships (e.g., Canolty & Knight, 2010;Penny et al., 2008), they share two main limitations. First, none of these methods provides a bounded output measure, which prohibits the interpretation of absolute PAC values across different studies (Hülsemann et al., 2019;Tort et al., 2010). Second, some methods may erroneously detect high PAC values at harmonic multiples of the frequency of the enveloped amplitude signal (e.g., Giehl et al., 2021;Kramer et al., 2008), as suggested by the findings of the present investigation.
This work introduces the direct modulation index (dMI) as a novel measure of PAC, which aims to circumvent the limitations of the aforementioned methods. The dMI is a bound variation of the MI as introduced by Tort et al. (2008). A dMI value of 1 indicates strong PAC, while a value of 0 indicates no PAC. Furthermore, dMI is highly sensitive to the target frequency only, and therefore avoids the pitfall of assuming significant PAC changes at harmonic frequencies to be actual findings. In the next section, we begin with a description of the proposed measure, followed by an illustration of its performance in comparison to a selection of established connectivity methods on simulated data.

| Direct modulation index
With the modulation index as its first step, the dMI shares the calculation of a phase-amplitude histogram . Following preprocessing, which includes bandpass filtering; a phase-amplitude histogram is constructed across the entire duration of the input signal by extracting the phase of the low-frequency signal and the amplitude of the high-frequency signal. In the current implementation, the Hilbert transform was used to extract the phase of the low-frequency signal, while the amplitude of the high-frequency signal was estimated using the rectified signal rather than the Hilbert-transform in order to speed up processing. The original study by Tort et al.  Afterwards, instead of scoring the PAC values from entropy, a sinusoid is fitted to the normalized histogram. We used a non-linear leastsquares algorithm, implemented in the LMFIT package in Python (Newville et al., 2015). The frequency of the sine is set to 1 cycle (per 360 ), while the phase and amplitude are preset to 0 and 1, respectively. During the fit, the phase is allowed to vary between À180 and +180 , and the amplitude is allowed to vary between 0.95 and 1.05 in order to obtain the best fit. We selected a sinusoidal function because the phase-amplitude histogram of two signals with an ideal PAC relationship was observed to default to a sinusoid shape. Fitting a sinusoid through the phase-amplitude histogram renders the measure highly sensitive to the targeted frequency only, whereas the entropy metric is also sensitive to the harmonic frequencies. Finally, an error value is calculated by taking the squared difference between the height of each individual phase bin and the amplitude of the sinusoidal fit at the corresponding bin. The errors are averaged across phase bins, capped at 1, and then subtracted from 1 to arrive at the dMI. This final step sets the lower and upper bounds of the dMI to 0 and 1, respectively. It simplifies the interpretation of the PAC estimate, as values approaching zero indicate low-coupling strength, while values approaching 1 indicate strong coupling.

| Validation data
dMI was evaluated in comparison to the following PAC methods: MI (Tort et al., , 2010, MVL (Canolty et al., 2006), and PLV (Mormann et al., 2005). The reader may refer to the corresponding articles for a description of the evaluated measures. dMI and the aforementioned PAC methods were evaluated using simulated data.
As opposed to using experimental data-where it is unclear whether any detected PAC at harmonic frequencies is reflective of a true coupling-simulated data enabled us to absolutely determine any signal properties, including the confirmable absence of PAC at the higher harmonic frequencies. We generated a high frequency signal of 200 Hz with an amplitude that was modulated by a 10 Hz oscillation To test the performance of each PAC method at different signalto-noise ratios, Gaussian noise, with amplitudes that are 0%, 25%, 50%, 100%, or 150% of the amplitude of the signal, was introduced.
To 3 | RESULTS

| DISCUSSION
The current work presents the dMI as a novel measure of PAC. Our dMI has been designed to be easily interpretable on a stationary interval between 0 and +1, and specific to the frequency of interest only.
The performance of dMI, PLV, MVL, and MI was investigated using artificial data under increasing levels of noise and with decreasing amounts of data. The results indicate that dMI is more robust towards varying levels of Gaussian noise and short signal durations than the other PAC methods investigated in the scope of our evaluations (Table 1). The dMI measure, as well as other reliable measures to estimate neurophysiological interactions for example, in the same frequency band (Scherer et al., 2022a), is freely available as a Python toolbox at https://github.com/neurophysiological-analysis/FiNN (Scherer et al., 2022b).
First, one characteristic specific to dMI is that its PAC estimates are bound to a stationary interval between 0 and 1, as opposed to the other investigated methods where the estimates can theoretically take on a wide range of values. Bounding the output to a specific, information is essential for any meaningful interpretation and for the discussion of any results (Lakens, 2013;Stankovski et al., 2017).
Second, dMI was found to be highly resilient towards high levels of Gaussian noise and performed well with decreasing amounts of data within the extent of the current investigations. By contrast, PAC estimates from MI, PLV, and MVL strongly deteriorated as the levels of noise increased. While the decreasing signal duration had no effect on dMI, for PLV it led to a high number of erroneously elevated PAC estimates at lower frequencies.
Finally, in our investigations, we observed that MI tends to sys- The current implementation of dMI assumes a sinusoidal distribution of amplitudes across the individual phase bins. This assumption is likely to hold, provided a sufficiently large sample size with independent measurements is available (Nixon et al., 2010). Our implementation of dMI also enables the user to easily visualize the phaseamplitude histogram to understand the shape of the PAC fit. In the event that the observed histogram is not Gaussian, the user may conveniently define another function for the shape for the line-fitting.

| CONCLUSIONS
Here, we presented dMI as a new measure to estimate PAC of neurophysiological data on the basis of a sinusoidal fit of the phaseamplitude histogram between two signals. The dMI has been designed to be resistant against varying levels of noise, to perform well with short signal durations, and to be easily interpretable due to the absolute boundary values of 0 and +1. Furthermore, through configurations of the parameters and/or changing of the sinusoidal scoring function, dMI is easily adaptable to the question at hand. We used simulated data to show that dMI provides a more reliable estimate of PAC than a number of other established measures. This novel measure may therefore provide a useful tool for the investigation of brain dynamics with implications for basic and clinical science. Future studies are required to test the performance of dMI in real-life signal processing scenarios in comparison to the other PAC metrics.