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Nonlinear interaction decomposition (NID): A method for separation of

cross-frequency coupled sources in human brain

Mina Jamshidi Idaji

, Klaus-Robert Müller

, Guido Nolte

, Burkhard Maess

Arno Villringer

, Vadim V. Nikulin

Department of Neurology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany

Machine Learning Group, Technical University of Berlin, Berlin, Germany

International Max Planck Research School NeuroCom, Leipzig, Germany

Department of Brain and Cognitive Engineering, Korea University, Anam-dong, Seongbuk-gu, Seoul, Republic of Korea

Max Planck Institute for Informatics, Stuhlsatzenhausweg, Saarbrücken, Germany

Department of Neurophysiology and Pathophysiology, University Medical Center Hamburg-Eppendorf, Hamburg, Germany

Department of Cognitive Neurology, University Hospital Leipzig, Leipzig, Germany

Centre for Cognition and Decision Making, Institute for Cognitive Neuroscience, National Research University Higher School of Economics, Moscow, Russia

Neurophysics Group, Department of Neurology, Charit



e-Universit

€

atsmedizin Berlin, Berlin, Germany

MEG and Cortical Networks Group, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany

ARTICLE INFO

Keywords:

Nonlinear interaction decomposition

NID

Cross-frequency coupling

MEG

EEG

Nonlinear neuronal interactions

Independent component analysis

ICA

ABSTRACT

Cross-frequency coupling (CFC) between neuronal oscillations reﬂects an integration of spatially and spectrally

distributed information in the brain. Here, we propose a novel framework for detecting such interactions in

Magneto- and Electroencephalography (MEG/EEG), which we refer to as Nonlinear Interaction Decomposition

(NID). In contrast to all previous methods for separation of cross-frequency (CF) sources in the brain, we propose

that the extraction of nonlinearly interacting oscillations can be based on the statistical properties of their linear

mixtures. The main idea of NID is that nonlinearly coupled brain oscillations can be mixed in such a way that the

resulting linear mixture has a non-Gaussian distribution. We evaluate this argument analytically for amplitude-

modulated narrow-band oscillations which are either phase-phase or amplitude-amplitude CF coupled. We

validated NID extensively with simulated EEG obtained with realistic head modelling. The method extracted

nonlinearly interacting components reliably even at SNRs as small as 15 dB. Additionally, we applied NID to the

resting-state EEG of 81 subjects to characterize CF phase-phase coupling between alpha and beta oscillations. The

extracted sources were located in temporal, parietal and frontal areas, demonstrating the existence of diverse local

and distant nonlinear interactions in resting-state EEG data. All codes are available publicly via GitHub.

1. Introduction

Oscillatory neuronal activity has been associated with almost all brain

operations including sensory, motor and cognitive processes (Buzs



aki

and Draguhn, 2004). In humans, these oscillations can be measured with

magneto- and electroencephalography (MEG/EEG), where the frequency

content is classically divided into speciﬁc frequency bands, namely δ

(0.5–4 Hz), θ (4–8 Hz),

(8–12 Hz), β (12–25 Hz), γ (25–70 Hz).

Each

frequency band has been associated with speciﬁc functional roles. For

example, alpha oscillations are known to be relevant for attention/sen-

sory processing (Groppe et al., 2013; Klimesch, 2012), while beta-band

activity is primarily associated with sensorimotor processing (Bayr-

aktaroglu et al., 2011; Kilavik et al., 2013; Klimesch, 2012; Salmelin and

Hari, 1994). While speciﬁc neuronal operations can be carried out by

oscillations in distinct frequency bands, there should be neuronal

mechanisms integrating such spatially and spectrally distributed pro-

cessing (Palva et al., 2005). In this way, neuronal communications can be

considerably enriched via coupling of neuronal oscillations within one

frequency band (Engel and Fries, 2010; Fries, 2015) as well as between

different frequency bands. Various types of cross-frequency (CF) in-

teractions among neural oscillations, namely phase-phase, amplitu-

de-amplitude, phase-amplitude coupling have been observed in human

* Corresponding author. Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany.

E-mail address: [email protected] (V.V. Nikulin).

The range of frequencies in each frequency band slightly differs in different references.

Contents lists available at ScienceDirect

NeuroImage

journal homepage: www.elsevier.com/locate/neuroimage

https://doi.org/10.1016/j.neuroimage.2020.116599

Received 25 November 2019; Received in revised form 16 January 2020; Accepted 31 January 2020

Available online 5 February 2020

nc-nd/4.0/).

NeuroImage 211 (2020) 116599

electrophysiological recordings (e.g. MEG/EEG) (Canolty and Knight,

2010; Jensen and Colgin, 2007; Nikulin and Brismar, 2006; Palva et al.,

2005) and have been linked to diverse perceptual and cognitive processes

(Canolty and Knight, 2010; Fell and Axmacher, 2011; Hyaﬁl et al., 2015;

Palva et al., 2005; Sauseng et al., 2008; Siebenhühner et al., 2016). In this

study, we focus on the extraction of these interactions from

multi-channel MEG/EEG. While the novel approach introduced here is

applicable to different types of CFC, a special emphasis is dedicated to

phase-phase coupling for the following reasons.

The phase of neuronal oscillations is known to represent the timing of

the ﬁring of a neuronal population generating the oscillation (Fries,

2009, 2015; Palva et al., 2005; Siegel et al., 2012), while its amplitude

reﬂects the strength of local spatial synchronization (Siegel et al., 2012).

The interaction of the activities of distinct neuronal populations is

manifested in the locking of phase/amplitude of the observed oscilla-

tions. Phase-phase coupling is a type of CFC that operates with milli-

second precision for both oscillations (Fell and Axmacher, 2011; Marzetti

et al., 2019; Palva et al., 2005; Siegel et al., 2012) and investigating it

with MEG/EEG recordings can provide a unique possibility to study

synchronization of the spiking of distinct neuronal populations

non-invasively (Palva and Palva, 2018).

A number of previous studies have investigated CF phase synchro-

nization in sensor-space (Darvas et al., 2009; Nikulin and Brismar, 2006;

Palva et al., 2005; Tass et al., 1998). However, volume conduction does

not allow the disentanglement of individual components. In order to

resolve this issue, some previous studies have investigated the phase

synchrony in the source-space using inverse modelling (Siebenhühner

et al., 2016; Tass et al., 2003). Yet, source-space analysis is computa-

tionally exhausting and source reconstruction methods are ill-posed,

which may lead to inconsistent outcomes (Mahjoory et al., 2017). On

the other hand, due to a linear mapping of the neuronal source signals to

the sensors, multivariate methods can increase the signal-to-noise ratio

(SNR) and accuracy of localizing the neuronal activity (Parra et al.,

2005). At the same time, these methods alleviate the problem of multiple

testing in sensor- or source-space analysis. While most of the multivariate

source-separation methods focus on the extraction of independent sour-

ces (e.g. independent component analysis - ICA), there are only a few

studies utilizing multivariate methods to extract dependent sources from

the electrophysiological recordings of the human brain (Chella et al.,

2016;

Cohen, 2017; D

€

ahne et al., 2014; Nikulin et al., 2012; Volk et al.,

2018). These methods optimize a contrast function of the desired type of

coupling. However, we show that the coupling can be reﬂected in the

statistical properties of the signal constructed through the linear mixing

of nonlinearly coupled processes. We refer to our method as Nonlinear

Interaction Decomposition (NID).

The rest of the manuscript is organized as follows. In section 2 we

provide some preliminary background about the amplitude-modulated

narrowband oscillations and their linear mixture. Section 3 is dedi-

cated to explaining the proposed method (NID) and its algorithmic steps.

In section 4 the experimental data and the analysis/testing approaches

are described. The results of applying NID to simulated as well as resting-

state EEG data are presented in section 5. Finally, a discussion and a

conclusion are provided in the last section.

2. Preliminary background

In this section we introduce the main assumptions and the core idea of

NID.

2.1. Nomenclature

We start with deﬁning the key phrases used throughout the manu-

script. An amplitude-modulated (AM) signal is a signal with a non-

constant envelope. An amplitude-modulated narrow-band (AM narrow-

band) signal is an AM signal whose energy is concentrated in a speciﬁc

narrow bandwidth. For instance, alpha-waves in M/EEG are AM narrow-

band signals, whose energy is in the bandwidth of 8  12 Hz.

2.2. Linear mixture of cross-frequency coupled brain oscillations

In order to understand the idea of NID, it is helpful to consider non-

sinusoidal brain oscillations. Note that NID does not require oscilla-

tions to be non-sinusoidal, they are rather used here for the demonstra-

tion of the method. The frequency content of such signals is concentrated

at two (or more) narrow bandwidths, whose central frequencies are

multiples (known as harmonic frequencies (Oppenheim et al., 1983)) of

the fundamental frequency. This means that such a non-sinusoidal signal

can be decomposed to narrow-band components which are phase-phase

coupled to each other. Fig. 1 depicts an example of this observation in

real data. Interpreting a non-sinusoidal signal as a linear mixture of

narrow-band phase-coupled signals, led to the idea that the linear

mixture of nonlinearly coupled narrow-band oscillations has a

non-Gaussian distribution, regardless of the location of the oscillation.

Fig. 2 illustrates an example from real data, where two signals in alpha

and beta frequency band are phase-coupled to each other and their linear

mixture is more non-Gaussian than each of them. Supplementary code

(2) provides some simulations for further illustration of NID’s core idea.

We assume that the distribution of AM narrow-band brain oscillations

do not deviate strongly from Gaussian distribution. It is also discussed in

(Hyv

€

arinen et al., 2010) that the amplitude modulation of brain oscil-

lations is the key property that results in the observation that the dis-

tribution of AM narrow-band oscillations does not deviate strongly from

Gaussian distribution. Note that a sufﬁcient amount of data points is

needed so that the histogram of data can become a fair estimation of its

distribution. For example, if the signal is ﬁltered with a very narrow

band-pass ﬁlter, more data points are needed to capture the ﬂuctuations

of the amplitude modulation compared to ﬁltering with a broader

band-pass

ﬁlter.

Our proposed method (nonlinear interaction decomposition-NID) is

based on the idea that if two narrow-band oscillations are independent or

only linearly coupled, the distribution of their linear mixture is closer to

Gaussian distribution in comparison to the distributions of linear mix-

tures of nonlinearly coupled oscillations. We have analytically proved

that CFC phase-phase and amplitude-amplitude coupled AM narrow-

band oscillations can be linearly mixed to a non-Gaussian distributed

signal (supplementary text, section 1) Fig. 3 illustrates the principle of

NID. Note that we assess the non-Gaussianity of a random variable by

means of kurtosis, skewness, or ﬁfth order moment, all of which are zero

for Gaussian random variables.

3. Method

3.1. Notation

We use boldface lower-case letters (e.g. x) to denote vectors, while

boldface capital letters (e.g. X) are used for matrices. Regular letters, (e.g.

x), indicate scalars. Vectors are used to denote the time series of a signal

or spatial ﬁlters/activation patterns. Matrices are used to denote the

concatenation of vectors. The operators ½:; : and ½:; : stand for horizontal

and vertical concatenation of two matrices respectively.

3.2. Measuring cross-frequency coupling

Depending on the type of the coupling, there are different measures to

quantify CFC. In this paper, we worked with phase-phase and amplitude-

amplitude coupled oscillations. As described below, the phase locking

value (PLV) was used for measuring phase-phase coupling, while

amplitude-amplitude coupling was quantiﬁed with the envelope corre-

lation. Both of these measures are calculated from the instantaneous

phase and amplitude of oscillations, which are computed as the phase

and magnitude of the complex analytic signal based on the Hilbert

transform.

M.J. Idaji et al. NeuroImage 211 (2020) 116599

Phase-phase coupling. Oscillations with frequencies f

and f

; n;

m 2 ℕ are called n:m phase-coupled if jmΦ

ðtÞnΦ

ðtÞj < const , where

ðtÞ and Φ

ðtÞ deﬁne the instantaneous phases of the two oscillations at

and f

respectively. To quantify n : m phase-phase coupling, phase-

locking value (PLV) is widely used (Palva et al., 2005; Sauseng et al.,

2008; Scheffer-Teixeira and Tort, 2016; Siebenhühner et al., 2016) and it

is deﬁned as



< e

jΨ

n;m

ðtÞ



, where Ψ

n;m

ðtÞ¼ðmΦ

ðtÞnΦ

ðtÞÞ, <:>

stands for computation of the mean over time samples, j is the imaginary

number, and j:j is the absolute value operator.

Amplitude-amplitude coupling. In the case of amplitude-amplitude

coupling, the instantaneous amplitudes of oscillations are correlated.

Therefore, the correlation coefﬁcient of the oscillations’ envelopes in-

dicates the strength of the amplitude-amplitude coupling.

3.3. Detection of cross-frequency coupling: problem formulation

We assume that there are N non-linearly coupled pairs of source

signals

fð

ðnÞ

; s

ðmÞ

Þg

i¼1

at frequencies f

and f

, where f

¼ nf

and f

. f

is a base-frequency relating f

and f

to each other. In the rest of

the paper, all the criteria and equations mentioned for frequency f

holds

for frequency f

as well. s

ðnÞ

2 R

1T

is a narrow-band source signal at f

where T is the number of time samples. The electrical (or magnetic) ac-

tivity measured at the sensors can be modeled as a linear mixture of the

sources as in the following (Baillet et al., 2001; Haufe et al., 2014):

x ¼ P

ðnÞ

þ P

ðmÞ

þ ξ (1)

where X 2 R

CT

is the matrix of multi-channel measured signal with C as

the number of channels. P

ðnÞ

¼½

ðnÞ

; ⋯; p

ðnÞ

. We call p

ðnÞ

2 R

C1

the

mixing pattern of source s

ðnÞ

. Additionally, S

ðnÞ

; ⋯; s

ðnÞ



2 R

NT

the matrix of source signals at f

, which are CF coupled to sources in

matrix S

ðmÞ

¼½

ðmÞ

; ⋯; s

ðmÞ

. In equation (1), ξ denotes the noise signal,

which cannot be explained by the linear model. Note that the superscript

of the variables is an indication of their frequency, e.g the superscript ðnÞ

in s

ðnÞ

is related to the subscript n of f

. As mentioned in section 3.2, the

coupling is called n : m coupling if ðs

ðnÞ

; s

ðmÞ

Þ are phase-phase coupled.

However, we use this notation for amplitude-amplitude coupling as well

so that we can denote the frequency ratios easier.

We provide an example here. Assume that we have two coupled

source signals in

and β frequency band, i.e. N ¼ 2, n ¼ 1; m ¼ 2, and

¼ 10 Hz; f

¼ 20 Hz. Then S

ð1Þ



ð1Þ

; s

ð1Þ



and S

ð2Þ

Fig. 1. A non-sinusoidal oscillation obtained from

spatial ﬁltering of EEG of a subject of LEMON

dataset (Babayan et al., 2019) with the spatial ﬁlter

in panel D. Panel (A) shows a segment of the time

series of the oscillation with a power spectral

density (PSD) shown in panel (B). The PSD of the

oscillation has clear peaks in alpha and beta bands.

Panel (C) shows a segment of the narrow-band

components (alpha and beta) of the oscillation in

(A). The two components are phase-coupled. Panel

(D) depicts the spatial ﬁlter and mixing pattern

(Haufe et al., 2014) of the oscillation, computed

with NID.

Fig. 2. Two phase coupled sources in alpha (x) and

beta (y) band extracted with NID using real EEG

data of a subject from LEMON dataset (Babayan

et al., 2019). Panel (A) shows a segment of alpha

and beta oscillations and their spatial patterns.

Panel (B) depicts the histogram of 2Φ

 Φ

, where

Φ stands for the phase of a signal. The fact that the

phase difference is located in a small sector of the

phase diagram indicates a strong coupling between

alpha (x) and beta (y) oscillations. Panel (C) shows

the value of the ﬁfth moment (denoted as M5) of

the narrow-band signals in panel (A) and their

linear mixture, indicating more non-Gaussianity for

the mixture than for the constituent oscillations.

Note that the ﬁfth moment is used as a measure of

non-Gaussianity in NID’s algorithm.

M.J. Idaji et al. NeuroImage 211 (2020) 116599

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