07050302

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 8, AUGUST 2015 1937

Matching Pursuit-Based Time-Variant BispectralAnalysis and its Application to Biomedical Signals

Karin Schiecke∗, Member, IEEE, Matthias Wacker, Franz Benninger, Martha Feucht, Lutz Leistritz,and Herbert Witte, Member, IEEE

Abstract—Objective: Principle aim of this study is to investi-gate the performance of a matching pursuit (MP)-based bispectralanalysis in the detection and quantification of quadratic phasecouplings (QPC) in biomedical signals. Nonlinear approaches suchas time-variant bispectral analysis are able to provide informationabout phase relations between oscillatory signal components. Meth-ods: Time-variant QPC analysis is commonly performed using Ga-bor transform (GT) or Morlet wavelet transform (MWT), and is af-fected by either constant or frequency-dependent time–frequencyresolution (TFR). The matched Gabor transform (MGT), whichemerges from the incorporation of GT into MP, can overcome thisobstacle by providing a complex time–frequency plane with an in-dividually tailored TFR for each transient oscillatory component.QPC analysis was performed by MGT, and MWT was used asthe state-of-the-art method for comparison. Results: Results weredemonstrated using simulated data, which present the general caseof QPC, and biomedical benchmark data with a priori knowledgeabout specific signal components. HRV of children during temporallobe epilepsy and EEG during burst–interburst pattern of neonatesduring quiet sleep were used for the biomedical signal analysis toinvestigate the two main areas of biomedical signal analysis: Thecardiovascular–cardiorespiratory system and neurophysiologicalbrain activities, respectively. Simulations were able to show the ap-plicability and reliability of the MGT for bispectral analysis. HRVand EEG analysis demonstrate the general validity of the MGTfor QPC detection by quantifying statistically significant time pat-terns of QPC. Conclusion and Significance: Results confirm thatMGT-based bispectral analysis provides significant benefits for theanalysis of QPC in biomedical signals.

Index Terms—Electroencephalogram, epilepsy, heart rate vari-ability (HRV), quadratic phase coupling (QPC), time-variant bis-pectral analysis.

NOMENCLATURE

AAM Additive amplitude modulation.AM Amplitude modulation.BA Biamplitude.BC Bicoherence.

Manuscript received August 26, 2014; revised December 17, 2014 and Febru-ary 6, 2015; accepted February 14, 2015. Date of publication February 26, 2015;date of current version July 15, 2015. This work was supported by the DFG underGrant Wi 1166/12-1 Le 2025/6-1. Asterisk indicates corresponding author.

∗K. Schiecke is with the Institute of Medical Statistics, Computer Sciencesand Documentation, Jena University Hospital, Friedrich Schiller University,Jena 07740, Germany (e-mail: [email protected]).

M. Wacker, L. Leistritz and H. Witte are with the Institute of Medical Statis-tics, Computer Sciences and Documentation, Jena University Hospital, FriedrichSchiller University Jena, Germany.

F. Benninger is with the Department of Child and Adolescent Neuropsychi-atry, and M. Feucht is with the Department of Child and Adolescent Medicine,University Hospital Vienna, Austria.

Preliminary results were published in a conference paper at the 36th Confer-ence of the IEEE Engineering in Medicine and Biology Society, Chicago, IL,USA, August 26–30, 2014.

Digital Object Identifier 10.1109/TBME.2015.2407573

ECG Electrocardiogram.EEG Electroencephalogram.FFT Fast Fourier transform.GT Gabor transform.HF High frequency.HR Heart rate.HRV Heart rate variability.LF Low frequency.MAM Multiplicative amplitude modulation.mBA Mean biamplitude in the ROI.mBC Mean bicoherence in the ROI.MGT Matched Gabor transform.MP Matching pursuit.MWT Continuous Morlet wavelet transform.QPC Quadratic phase coupling.QRS Q-, R-, and S-ECG-waves.QS Quiet sleep.ROI Region of interest.RSA Respiratory sinus arrhythmia.STFT Short-term Fourier transform.TFD Time–frequency distribution.TFR Time–frequency resolution.TLE Temporal lobe epilepsy.tvPS Time-variant power spectrum (spectrogram).WVD Wigner–Ville distribution.

I. INTRODUCTION

QUADRATIC phase coupling (QPC) is an important prop-erty of biomedical signals indicating a specific nonlinear

coupling between two oscillatory signal components. For exam-ple, if a signal with only two sinusoidal (oscillatory) components(frequencies f1 and f2 with f2 > f1) passes through a systemwith a quadratic nonlinearity [see (15)], then the output sig-nal contains frequency components at f1 , f2 , 2f1 , 2f2 , f2 ± f1 ,and (zero) phase relations are of the same type as the fre-quency relations, i.e., Φ1 ,Φ2 , 2Φ1 , 2Φ2 ,Φ2 ± Φ1 [1]. Such acoupling configuration is called QPC. Higher order spectralanalysis is capable of detecting and characterizing QPC [2]. Afrequency triplet {f1 , f2 , f3} produces a high peak in the bis-pectrum B (fm , fn ) (bispectral plane) at the coordinates [f1 , f2 ]if the coupling conditions f3 = f2 + f1 and Φ3 = Φ2 + Φ1 arefulfilled, i.e., this coincides with the frequency and phase condi-tions defined as QPC. Additionally, bispectral peaks at the diag-onal of the bispectral plane (coupling of the first and second har-monics) and at the coordinates [f1 , f2 − f1 ] occur because forthe component f2 − f1 , the triplet-related coupling conditions

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1938 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 8, AUGUST 2015

f3 = (f2 − f1) + f1 = f2 and Φ3 = (Φ2 − Φ1) + Φ1 = Φ2are also met. For f2 � f1 , the bispectral peaks at [f1 , f2 ] and[f1 , f2 − f1 ] are in a tight neighborhood, i.e., they can be onlydistinguished by using a high-frequency (HF) resolution.

Stationary (time invariant) bispectral analysis has alreadybeen implemented for many years [3]. Biomedical signals areusually nonstationary, i.e., QPC in biomedical signals changein time and a high time resolution is desirable. This is the rea-son why time-variant implementations of bispectral analysisare standard for biomedical applications. The categorization oftime-invariant bispectral techniques (stationarity of the signal isrequired) can also be used for time-variant versions. One can dis-tinguish between nonparametric (“conventional” [1]) and para-metric approaches. Frequently used nonparametric approachesare based on the short-term Fourier transform (STFT) [4], Ga-bor transform (GT) [5], Morlet wavelet transform (MWT) [6],on third-order time–frequency distribution (TFD) (Wigner bis-pectrum [7] and its advanced versions [8], [9]). Time-variantparametric bispectral approaches require an estimation of pa-rameters for a time-variant autoregressive (AR) model, whichconsider higher order moments (e.g., [10]). For bispectral anal-ysis of interval signals (derived from point processes), the non-linear AR integrative model and other nonlinear AR approacheshave been introduced [11], [12].

A crucial point for the application of all time-variant bis-pectral techniques is their time–frequency resolution (TFR). Itwould be beneficial if for each signal component an appropri-ate (individually adapted) TFR would be available in order toseparate existing components from each other (frequency reso-lution) and to localize QPC changes in time (time resolution) [6].However, localization in time and localization in frequency arecontradictory aims (uncertainty principle), i.e., the decrease oftime resolution results in an increase of the frequency resolutionand vice versa (for an overview, see [13]), i.e., when interpret-ing results of time-variant bispectral analysis, it is mandatory tounderstand the specific effects of TFR. The TFR properties ofthe time-variant approaches mentioned above are summarizedin Section V.

Summarizing these facts, it can be assumed that the avail-ability of individually tailored TFRs, i.e., TFRs, which are in-dividually adapted to the properties of each signal component(signal adaptive), would be a big advantage for bispectral anal-ysis in order to detect and quantify the temporal organization ofQPCs. The matched Gabor transform (MGT) [14] was recentlyintroduced to obtain a linear phase analysis with these beneficialproperties. MGT emerges from the incorporation of the GT intothe matching pursuit (MP) decomposition algorithm. For theMGT, the MP with a dictionary of real-valued Gabor atoms isused (cosine multiplied with a Gauss function). In general, theMP [15] decomposes a signal into a sum of atoms from a givendictionary. The difference between the original MP (MP plusWigner–Ville distribution—WVD) and the MGT lies in gener-ation of the aggregated time–frequency plane. Every selectedatom is used to define an individual time window (time resolu-tion), and thereby, the corresponding frequency resolution forthe GT is defined, i.e., individually tailored TFR results. Theresulting complex time–frequency planes with different TFRs

Fig. 1. General design of processing scheme. Frequency-dependent MWT aswell as MGT-based strands of analysis are depicted.

are then summed up to obtain the aggregated (complex) time–frequency plane. Amplitude and information are available foreach point of the aggregated (complex) time–frequency plane.It should be mentioned here that WVD does not provide instan-taneous phase information.

The main objective of this study can be summarized as fol-lows: The application of the MGT leads to improved analysisresults for linear-phase properties [14]. An improvement canalso be expected for time-variant bispectral analysis but anyclear evidence is still pending. Consequently, we want to showthat MGT-based bispectral analysis has key advantages, whichcan be beneficially used for QPC analysis. Therefore, the firstaim of this study is to provide the evidence of this by usingsimulations, which are related to the QPC coupling conditionswhich change abruptly. These simulations demonstrate both theadvantages of the approach yet also the restrictions and pit-falls. The results of the MGT-based QPC analysis are comparedto the MWT-based results. Second, the aim is to demonstratethat this new approach yields improved results for bispectralanalysis of both heart rate variability (HRV) and EEG. By an-alyzing HRV as well as EEG data, two main application areasof biomedical signal analysis are considered: The analysis ofthe cardiovascular–cardiorespiratory system and of neurophys-iological brain activities. We have used data for which the time-variant QPC analysis on the basis of the MWT (state-of-the-art)has been already performed, and therefore, a priori knowledgeabout signal properties exists (benchmark data).

II. METHODS

The general design of the processing scheme is depictedin Fig. 1. Investigated data, applied approaches, and theirmain characteristics are given. Methodological backgroundsof MWT, MGT, and time-variant bispectral approaches, aswell as descriptions of TFR and applied statistical analyses

SCHIECKE et al.: MATCHING PURSUIT-BASED TIME-VARIANT BISPECTRAL ANALYSIS AND ITS APPLICATION TO BIOMEDICAL SIGNALS 1939

are given in Sections II-A to II-E. In general, the time-variant definition of QPC implies that a pair of two signalcomponents {x1 , x2}, with x1(t) = a1 · sin(2πf1t + Φ1) andx2(t) = a2 · sin(2πf2t + Φ2), is quadratic phase coupled whena third component x3(t) = a3 · sin(2πf3t + Φ3) with the fre-quency f3 = f1 + f2 and the zero phase Φ3 = Φ1 + Φ2 ex-ists. As the instantaneous phase of x3(t) is ϕ3(t) = 2πf3t+Φ3 = 2πf1t + 2πf2t + Φ1 + Φ2 = ϕ1(t) + ϕ2(t), the QPC’sphase condition can be generalized by using instantaneousphases instead of zero phases. The instantaneous phase triplet{ϕ1(t), ϕ2(t), ϕ3(t)} can be computed for each time point usingof time–frequency methods like GT and MWT.

A. Morlet Wavelet Transform

The frequency-dependent complex analytic signal yk (t, fn )of the data xk (t, f) is computed using MWT [13], where kdesignates the number of signals (realizations, trials). The time-variant power spectrum (tvPS) PSk (t, fn ) and phase ϕk (t, fn )can be calculated on the basis of the complex analytic signal foreach recording k by

PSk (t, fn ) =∣∣yk (t, fn )

∣∣2

(1)

and

ϕk (t, fn ) = arg yk (t, fn ) . (2)

By ensemble averaging, the representative tvPS can be esti-mated

tvPS (t, fn ) =1K

K∑

k=1

∣∣yk (t, fn )

∣∣2. (3)

The tvPS quantifies the averaged spectral characteristics ofour K realizations of QPC in a time–frequency map represen-tation.

B. Matched Gabor Transform

MGT generates in a first step (MP) for each k, a linear com-bination of real-valued Gabor atoms dk

j (t) (cosine term) from aredundant dictionary D that approximates xk (t)

xk (t) ≈M∑

j=1

akj dk

j (t) . (4)

Second, each atom is analyzed with its own analysis time

window, determined by ˜σkj , to generate a set of time–frequency

planes ykj (t, fn ), and finally, theses planes are combined to ob-

tain a time–frequency plane yk (t, fn )

yk (t, fn ) =M∑

j=1

akj yk

j (t, fn ) (5)

where

ykj (t, fn ) =

∫ ∞

−∞dk

j (τ) g(

t − τ, fn , ˜σkj , 0

)

dτ. (6)

Here, g is a complex-valued Gabor atom of the GT with thegeneral form

g (t, f, σ, ϕ)=1

σ√

2πexp (i (2πft + ϕ)) exp

(

−0.5(

t

σ

)2)

(7)

and for ˜σkj

˜σkj =

⎧

⎪⎪⎨

⎪⎪⎩

σmin , σkj ≤ σmin

σkj , σmin < σk

j < σmax

σmax , σkj ≥ σmax

(8)

with

σbounds = [σmin , σmax] (9)

applies.Regarding yk (t, fn ) in (5), the tvPS and phase φk (t, fn )

can be calculated on the basis of this complex time–frequencyplane for each recording k, and again, the representative tvPSby ensemble averaging according to (3). A detailed descriptionof MGT can be found in [14]

C. Time-Variant Bispectral Analysis

Time-variant QPC between frequency bands can be com-puted using time-variant biamplitude and bicoherence. Accord-ing to [1], for each realization/seizure (k = 1, . . . ,K), the fol-lowing triple product can be calculated for every frequency pair(fm , fn ) and at each point in time

Bk (t, fm , fn ) = yk (t, fm ) · yk (t, fn ) · yk∗(t, fm + fn ).(10)

Here, ·∗ denotes the complex conjugate. The ensemble aver-aging of B(k) (t, fm , fn ) yields an estimation of the time-variantbispectrum B̂ (t, fm , fn ).

The time-variant bispectrum depends on the amplitudes of thefrequency components {fm , fn , fn + fm}. For detecting phasecouplings, the estimation of the time-variant bicoherence is usedas an amplitude-independent measure

Γ̂ (t, fm , fn ) =∣∣∣B̂ (t, fm , fn )

∣∣∣

1K

√∑

|yk (t, fm ) · yk (t, fn )|2 ·∑

|yk (t, fm + fn )|2. (11)

To reduce the three dimensions of the time-variant bispec-tral measures resulting from (8) and (9), the mean biamplitude(mBA) and the mean bicoherence (mBC) were computed in theregions of interest (ROI) F1 × F2 according to

mB̂ (t) =

∑nu1

i=nl1

∑nu2

j=nl2B̂ (t, fi , fj )

(

nu1 − nl

1 + 1)

·(

nu2 − nl

2 + 1)

mΓ̂ (t) =

∑nu1

i=nl1

∑nu2

j=nl2Γ̂ (t, fi , fj )

(nu1 − nl

1 + 1) · (nu2 − nl

2 + 1)(12)

where nl1 , n

u1 , nl

2 , and nu2 denote the according indices of


the frequency grid of the ROI of the estimated bispec-trum (lower/upper bounds of F1 and F2). According to thetwo different frequency ranges of phase coupling in oursimulated data, ROI1 was set to F1 = [0.025Hz, 0.075 Hz]and F2 = [0.15Hz, 0.25 Hz], and ROI2 was set to F1 =[0.075Hz, 0.125 Hz] and F2 = [0.3Hz, 0.4 Hz], respectively.

Regarding the main rhythms of our data, we calculated time-variant bispectral measures in the ROI. For HRV analysis, we in-vestigated QPC between the Mayer wave-related low-frequency(LF) component and the respiration-related HF component inthe HRV, i.e., the ROI was set to F1 = [0.075Hz, 0.15 Hz]and F2 = [0.25Hz, 0.35 Hz]. For EEG analysis, we investi-gated QPC between the delta waves related component andthe theta wave related component of the burst pattern duringquiet sleep (QS), i.e., the ROI was set to F1 = [0.5Hz, 1.5 Hz]and F2 = [3Hz, 4 Hz]. A more detailed description of parame-ter extraction in case of time-variant bispectral analysis can befound in [10].

D. Time–Frequency Resolution

The most important difference between MWT and MGT con-sists in the resulting TFR: TFR of MWT is frequency dependent,and TRF of MGT is signal adaptive.

For MWT, higher frequencies lead to a better time resolu-tion σt but also a disadvantaged frequency resolution σf andvice versa. The standard deviation of the Gauss envelope in thetime domain and the standard deviation of the Gauss curve inthe frequency domain can be used as a measure for the timeand frequency resolution [13]. For practical purpose, a freelyselectable parameter ωo has to be set, σt(σf ) is bound to thefrequency and ωo (σt = ωo

2πf = 1/σf ). Parameter ωo is depen-dent on the choice of the Morlet mother wavelet. In a first step,the mother wavelet of the MWT was adapted so that the sigmaparameter of its Gaussian envelope equals one cycle for everyfrequency (ωo = 2π; denoted by “A” in Section IV). The result-ing TFR is adequate to analyze the time pattern of linear activitybut inadequate for bispectral analysis (frequency resolution istoo low; missing “side bands”; for reference, see [13]). There-fore, the mother wavelet has to be readapted to the situation. Forsimulations and HRV data, we used ωo = 2π (denoted by “A”in Figs. 4 and 6) and ωo = 10π (denoted by “B” in Figs. 4 and6). For EEG data, we used ωo = 2π (denoted by “A” in Figs. 7and 8) and ωo = 5π (denoted by “B” in Figs. 7 and 8).

For the case of MGT, the resulting time–frequency planeshave multiple intrinsic TFR and cannot be obtained by a sin-gle GT. Each signal part is analyzed with its matched analysisfunction. In case of GT, σt (σt = 1/σf ) is freely selectable.

MGT can be applied by using the so-called “sigma bounds”(see σbounds in (9)). Gabor atoms of the dictionary with sigmavalues below the lower bound (with time resolutions, which aretoo high) will be computed (GT) with a fixed (bound value),lower time resolution. Respectively, the same applies for Ga-bor atoms of the dictionary with sigma values above the upperbound (time resolutions, which are too low: computation with afixed, higher time resolution). The use of a lower sigma boundultimately leads to a smoothing in time of results, which seems

to be appropriate for our investigation. Therefore, we used onlylower sigma bounds for our investigation of the simulations andHRV data: σbounds = [5 s, Inf] (denoted by “C” in Figs. 4 and6) and σbounds = [10 s, Inf] (denoted by “D” in Figs. 4 and6), and for EEG data, respectively: σbounds = [0.1 s, Inf] andσbounds = [0.25 s, Inf] (denoted by “C” and “D” in Fig. 7.

E. Statistical Analysis

The statistical hypothesis testing for time-variant mBC anal-ysis of real data (HRV and EEG) was performed using twodifferent methods. First, a surrogate data approach was applied,and second, a bootstrapping approach was utilized.

The null hypothesis of the surrogate data approach is thatthere is no QPC between the two frequency components of theROI (“small values” of mBC). The surrogate data were obtainedby destroying the phase information of investigated signals bymeans of phase randomization [16]. This was carried out for1000 repetitions, and the 95th percentile of the surrogate mBCin the ROI was computed. The mean over time 95th percentilewas set as the “5% threshold” for statistically significant mBCvalues (see e.g., [17]).

Aim of the bootstrapping approach is to determine statisti-cal properties of the single mBC time courses estimated for Krealizations of real data according to (11) and (12). In order toestimate confidence tubes of the extracted mBC time courseswithout any particular distribution assumption 1000 bootstrapsamples of size K (each sample element contains 10-min epochsfor HRV data and 10-s epochs for EEG data, respectively) weredrawn by a case resampling with replacement. With it, 1000bootstrap replications of each extracted parameters were com-puted. Based on these replications, the lower limit (the 2.5%quantile) defined the lower bound, and the upper limit (the97.5% quantile) defined the upper bound of the 95% confidencetube of mBC. For theoretical details concerning bootstrappingapproaches, see [18].

III. SIMULATION AND DATA MATERIAL

A. QPC Simulations

The first simulation (SIMQPC ) was used to test the propertiesof the MGT-based time-variant bispectral analysis for the caseof abrupt changes in QPC simulated by a superimposition ofsinusoidal signal components. The analysis of SIMQPC servesto clarify the following three methodological issues:

1) Time dynamics of bispectral parameters can be analyzed.2) The influence of different parameter settings can be as-

sessed. The results obtained by using our new approachare compared with those from MWT (state-of-the-art).Two different parameter settings for each approach areused in order to demonstrate the influence of differentmother wavelets (for MWT) and of different time resolu-tion bounds (for MGT).

3) Criteria (rules) for the choice of ROIs, in which the pa-rameters are determined, can be identified.

SIMQPC consists of K realizations of QPC. Structure, timelength, and sampling frequency of the simulations were adapted


Fig. 2. Schematic view of frequency distributions and time courses of simula-tions. One realization of SIMQPC is depicted in (a). In the upper row, schematicamplitude spectra are given for the two different situations of phase coupling inSIMQPC . Arrows illustrate the actual phase-coupled frequency triplets. In thelower row, the time course of change between both situations of SIMQPC (fourperiods of 150-s length, each divided by dashed black lines) is depicted. (b)time course of SIMAM is given. A division between no phase coupling/phasecoupling (each period 30 s) is given by dashed black lines.

to real data (K = 18; sampling frequency: 2 Hz). Schematicrepresentation of frequency distribution and time course ofSIMQPC is given in Fig. 2(a). Each realization of simulatedsignal y(t) is made of a sequence of two alternating 150-s seg-ments, each of them containing one phase-coupled frequencytriplet and two uncoupled frequencies

y (t) =

{y1 (t) : 0−150 s and 300−450 s

y2 (t) : 150−300 s and 450−600 s(13)

with

yj (t) =∑5

i=1ai,j cos (2πfi,j t + Φi,j ) . (14)

For j = 1, phase-coupled triplet is {f1,1 = 0.05Hz, f2,1 =0.2Hz, f3,1 = 0.35Hz}, and uncoupled frequencies are setto {f4,1 = 0.1Hz, f5,1 = 0.35Hz}. For j = 2, phase-coupledtriplet is {f1,2 = 0.1Hz, f2,2 = 0.35Hz, f3,2 = 0.45Hz},and uncoupled frequencies are set to {f4,2 = 0.05Hz,f5,2 = 0.2Hz}, respectively. Phase relations are of the sametype as frequency relations (Φ3,j = Φ1,j + Φ2,j for j = 1, 2).

Phases are chosen randomly for each realization and ai,j = 1applies for i = 1, . . . , 5 and j = 1, 2.

Frequency values chosen for SIMQPC are characteristic ofthe LF and HF HRV ranges which are defined as follows (TaskForce “HRV” [19]): LF = 0.04 − 0.15 Hz; HF = 0.15 − 0.4Hz. Component 0.05 Hz lies at the lower end of the LF rangeand 0.2 Hz at the lower end of the HF range; 0.1 Hz is char-acteristic of HRV waves, which are strongly associated withthe Mayer waves of the systemic blood pressure and 0.35 Hzcan be assumed to be the frequency of the respiratory sinusarrhythmia (RSA) in children during rest. The QPCs between0.05 and 0.2 Hz as well as between 0.1 and 0.35 Hz representtwo couplings between the LF and the HF range, where the lastis specifically related to a coupling between HRV-related Mayerwaves and RSA.

Second simulations were performed to evaluate the QPCdetection features of MGT in the case of fast changes be-tween phase couplings/no phase couplings under the condi-tion that QPC is generated by nonlinear models. QPC wasgenerated by using both additive (asymmetric) and multiplica-tive (symmetric) amplitude modulation (AM). AM phenom-ena have been already described in HRV and EEG (e.g., [20]and [21]). Two sinusoidal signals x1(t) = a1 · sin(2πf1t + Φ1)and x2(t) = a2 · sin(2πf2t + Φ2) with a2 = 5 · a1 , f1 = 1Hz,f2 = 3.5Hz, and Φ1 = 0,Φ2 = π/4 were used. A frequencyratio between 1:3 and 1:4 roughly represents our previouslyvalidated bounds for the ROI for QPC detection in both appli-cations (HRV: [0.075 Hz, 0.15 Hz] × [0.25 Hz, 0.35 Hz) [22];EEG: [0.5 Hz, 1.5 Hz] × [3, 5 Hz] [23]).

For additive amplitude modulation (AAM), both signals wereadded and applied as input signal x(t) = x1(t) + x2(t) to asystem with a quadratic nonlinearity

yAAM (t) = x (t) + εx2 (t) (15)

where yAAM(t) is the resulting output signal (ε = 0.5). Thesimulated signal (hereinafter referred to as SIMAM ; Fig. 2(b))was composed by alternating segments of yAAM(t) (30-s seg-ment with QPC) and x(t) (30-s segment without QPC). AAMcorresponds exactly to the general QPC definition [1] becauseyAAM (t) contains frequency components f1 , f2 , 2f1 , 2f2 , f2 ±f1 , and the instantaneous phase relations are of the same type(ϕ1(t), ϕ2(t), 2ϕ1 (t) , 2ϕ2 (t) , ϕ2(t) ± ϕ1(t)).

In case of a multiplicative amplitude modulation (MAM), theresulting signal can be described by

yMAM (t) = [1 + c1 · x1 (t)] · x2 (t) (16)

and the condition for the product of the constants c1 · a1 < 1must be fulfilled. The signal yMAM(t) contains componentsat frequencies f2 , f2 ± f1 with the corresponding phase rela-tions ϕ2(t), ϕ2(t) ± ϕ1(t). Alternating segments of yMAM(t) +x1(t) (30-s segment with QPC) and 0-s segment without QPC)were used as the signal to be analyzed. Results from this specificsimulation study are not presented here, however are discussedin conjunction with the AAM results.


B. Real Data

1) HRV of Children During TLE: Data are derived from agroup of 18 children; each with one seizure recording of at least10 min (K = 18 seizures; median age 9 years 4 months, in-terquartile range 8 years 7 months to 12 years 1 months, range6 years 6 months to 18 years 0 months; median seizure length88 s, interquartile range 72–110 s, range 52–177 s). Presurgi-cal evaluation was performed at the Vienna pediatric epilepsycenter following a standard protocol. EEG was recorded referen-tially from gold-disc electrodes placed according to the extended10–20 system with additional temporal electrodes. One-channelECG was recorded from an electrode placed under the left clav-icle. EEG and ECG data were recorded referentially againstCPZ , filtered (1–70 Hz), converted from analog to digital (sam-pling frequency 256 Hz), and stored digitally for further dataanalysis. Video recordings of each seizure were reviewed toclassify seizure type. Complex partial seizures were included,but not auras or generalized tonic-clonic seizures. Seizure onsetand termination were determined by the EEG (independentlyby two neurologists experienced in the field of epilepsy andclinical electrophysiology). EEG and ECG recordings includ-ing 10-min epochs [5 min before (preictal state) and 5 min afterseizure onset (seizure and postictal state)] were stored for eachseizure.

QRS detection was performed after bandpass filtering (10–50 Hz) and interpolation by cubic splines (interpolated sam-pling frequency 1024 Hz, according to [24]) to detect the timepoint of the maximum amplitude of each R-wave and the re-sulting series of events was used for the HRV computation. Thelow-pass-filtered event series was computed by applying theFrench–Holden algorithm [25]. The final HRV representationwas obtained from the low-pass-filtered event series via multi-plication with the sampling rate and with 60 beats/min and downsampled to 2 Hz. An artifact rejection was performed manuallyto minimize the influence of false QRS triggering.

All results were determined by grand mean analysis over 18seizures (K = 18 children). A detailed investigation of the HRVby means of time-variant and frequency-selective linear andnonlinear methods of the same group of children was recentlypublished in this journal [22]. Original HR courses of all childrenare depicted in Fig. 3(a).

2) EEG of Neonates With Burst–Interburst Pattern DuringQS: A group of six full-term neonates (mean conceptual age39.3 weeks, range 38–41 weeks; mean birth weight 3152 g,range 2670–3420 g; mean 5 min APGAR-score 9, range 8–10)was analyzed. Recordings were performed during sleep between09.00 and 12.00 h; all neonates lay in an incubator at temper-atures adapted to maintain normal body temperature and noneshowed any EEG abnormality. Eight channel EEG (128-Hz sam-pling rate, international 10–20 system with electrodes Fp1 , Fp2 ,C3 , C4 , T3 , T4 , O1 , O2), heart rate, respiratory movements, andEOG were recorded.

Only the EEG recorded during QS was selected. The EEGwas segmented by a trained physician, the burst onset was usedas a fix point for a 10-s interval. 4 s before (interburst) and 6 safter the burst-onset were considered. For the visual detectionof the burst onset (amplitude criteria), the burst was defined as

Fig. 3. Examples of real data used for analyses. (a) HR of all children (K =18; thin gray lines; upper row) and median HR (bold black line) with interquartiletube (25% to 75% quartiles; gray filled) for all children (lower row) is depicted.Distinct time points of TLE seizures are given: dashed line designates the onsetof the preonset acceleration (240 s), full line designates the EEG seizure onset(300 s), dotted line designates the maximum of the acceleration and beginningof deceleration (340 s), and dashed-dotted line designates the median end ofseizure (390 s). (b) All investigated EEG pattern (K = 17, thin gray line) andfirst EEG pattern (bold black line) of one exemplary child (#1) are shown. Burstonset is at 4 s (bold black line).

the simultaneous appearance of a group of high amplitude (>50 μV), LF waves (0, 5–3 Hz) in more than 75% of the record-ing channels. These waves are superimposed by low amplitude(<50 μV), HF waves (4–15 Hz). From our previous study [23],10-s intervals of the Fp1 recordings were analyzed for eachneonate; the burst–interburst patterns starting with the begin-ning of the QS period were selected for analysis (K = 17 formean analysis per neonate: minimal number of available 10-sintervals in one neonate; K = 102 for grand mean analysis ofall neonates). Examples of the EEG pattern of one neonate aredepicted in Fig. 3(b).

IV. RESULTS

A. Simulations

A comparison of grand mean results of SIMQPC (K = 18trials of simulations) of frequency-related tvPS between theMWT and MGT is depicted in Fig. 4. The TFR from the MWT-based approach is dependent on frequency that is clearly visible[see Fig. 4(a-A) and (a-B)]. It is not possible to correctly detectthe QPC in (A) because of the occurrence of cross terms betweenhigher frequencies (LF resolution). The TFR from all the MGTapproaches [see Fig. 4(a-C), and (a-D)] is adequate for bispectralanalysis. Representations of BA at two defined single time points


Fig. 4. Grand mean results (K = 18) of SIMQPC . (a) tvPS, (b1) and (b2) BA at two different time points (75 and 225 s), and (c) mBA in the ROIs are shown. (A)and (B) depict results of MWT-based approaches (ω0 = 2π/10π), and (C) and (D) depict results of MGT-based approaches (σbounds = [5 s, Inf]/[10 s, Inf]).Red color designates high power, blue color: low power of tvPS or BA, respectively, in column (a), (b1), and (b2). In column (c), light gray line depictstime course of mBA in ROI1 = [0.025 Hz, 0.075 Hz] × [0.15 Hz, 0.25 Hz], dark gray line in ROI2 = [0.075 Hz, 0.125 Hz] × [0.3 Hz, 0.4 Hz]. ROIs areframed in the adequate color in (b1) and (b2). For (D), the use of a wider ROI (guideline-driven according to [19]) for mBA calculation is depicted (ROI =[0.025 Hz, 0.125] × [0.15 Hz, 0.4 Hz]; dashed red rectangle in (b1, (b2), and red line in (c)).

(t1 = 75 s and t2 = 225 s; b1 and b2; see Fig. 1(a) for simulatedQPC at these time points) as well as the time course of mBAin both investigated ROIs (c) are given in Fig. 4(b) and (c) forsimulated data and all investigated MWT [see (A) and (B)] andMGT-based approaches [see (C) and (D)]. According to oursimulations of phase couplings, there should be a peak in thebiamplitude of f1 = 0.05Hz and f2 = 0.2Hz at the time pointpresented in (b1) and a peak in the biamplitude of f1 = 0.1Hzand f2 = 0.35 Hz at the time point presented in (b2).

In general, all approaches were able to depict these peaks,but there is an increasing frequency resolution (and, therefore,decreasing time resolution) from (A) to (B) and (C) to (D).In addition, due to the occurrence of couplings at lower fre-quencies, (A) to (B) for MWT is characterized by a higherfrequency resolution (and consequently lower time resolution)at time point (b1) in comparison to (b2). Quantification of mBAin the ROIs [see Fig. 4(c)] shows a correct time course for allapproaches, but a partially false quantification of strength of theQPC [see (A) to (B)]. Note that the simulations show a defined(and equal) strength of QPC. The simulations showed better re-sults for the MWT by applying maximum and not mean BA inthe ROI. However, this procedure is not possible to use on realdata (sensitivity to outliers of BA/BC).

In Fig. 5, results of AAM-based simulation (SIMAM ) aregiven. The fast changes (30 s, 3 cycles of frequency componentf1) in the occurrence of phase coupling/no phase coupling werein the focus of interest in the analysis. To highlight both theadvantages and disadvantages of MGT, no bounds of σk

j [seeFig. 5 (a)] and adapted σbounds [see Fig. 5(b) and (c)]were used.

The results of time–frequency related PS show that the MP isable to decompose the AAM related segments into “stationary”components of an AM (f1 , f2 , f2 ± f1). This is also true for theMAM-based simulations (not shown).

The quality of the decomposition depends on the localizationof the AAM segment within the analysis interval: Quality forsegments at the beginning and at the end of the analysis intervalis somewhat low [see Fig. 5(a)–(c)]. In contrast, for segmentslocated in the central area of the interval, all components canbe clearly detected. Without smoothing [see Fig. 5(a)], the pe-ripheral segments are decomposed into short Gabor atoms thatcorrespond to the subsequent spindles of the AAM signal, i.e.,they are “artificially” fractionized into periodically repeating(f1) Gabor atoms (frequency f2).

Consequently, we used for our real-world applications anMP approach with a slight smoothing (σbounds), and we chosesegments of our data in such a way that the pattern changes


Fig. 5. Result of MGT-based frequency related tvPS of SIMAM . (a) MGTapproach without bounds of σk

j is depicted, (b) σbounds = [5 s, Inf], and (c)σbounds = [10 s, Inf]) are used, respectively.

are located in the midarea of the analysis interval, i.e., expectedQPC changes before and after the state/pattern change can bereliably quantified.

B. Real Data

1) HRV of Children During TLE: A comparison of grandmean results of all children (K = 18 recordings of the HRV)from the MWT and MGT-based tvPS of HRV is depicted inFig. 6(a). Again, the frequency-dependence of TFR from theMWT is clearly visible [see Fig. 6(a-A) and (a-B)]. Also, theTFR of (A) from the real data is not able to depict the “side bandsof frequencies of QPC” necessary for successful bispectral anal-ysis (missing “side band” at≈ 0.3 ± 0.1Hz: modulation of RSAby Mayer waves), whereas the TFR from all of the MGT-basedapproaches [see Fig. 6(a-C) and (a-D)] is adequate for bispec-tral analysis. Representation of BA at three defined time points(240, 340, and 390 s; b1 to b3; see Fig. 3(a) for relevance ofdistinct time points) as well as the time course of mBC in theROI (c) are given in columns (b1)–(b3) and (c) of Fig. 6 forHRV of children with TLE and all investigated MWT [see (A)and (B)] and MGT-based approaches [see (C) and (D)]. Notethat for real data, the estimation of the amplitude-independentmBC is necessary to quantify the occurrence of QPC in the timecourse. At our distinct time points, there should be a peak in thebiamplitude in the ROI = [0.075Hz, 0.15Hz] × [2.5Hz, 3.5Hz]at 240 s [start of preonset acceleration; Fig. 6(b1)], no peak atall at 340 s [maximum of acceleration; ictal period; Fig. 6(b2)],and again an increasing BA at 390 s [end of seizure; Fig. 6(b3)].

Red lines in Fig. 6(c) quantify statistical significance of peaksin the mBC in their time course.

In general, all approaches are able to depict peaks of BA,but there is a strongly increasing frequency resolution (and,therefore, decreasing time resolution) from A to B and not sucha strong increase from C to D. The frequency-dependence ofTFR in the MWT results in minimally interpretable BA maps[see Fig. 6(A) and (B)]. The “high” frequency at 0.3 Hz (ourfocus) has inadequat frequency resolution [see Fig. 6(A)] andas the frequency resolution increases [see Fig. 6(B)], the timeresolution is reduced (time smooting). MGT-based BA maps areeasier to interpret, and just as importantly, it is easier to find theROIs. Quantification of mBC in the ROI [see Fig. 6(c)] is able toreflect the presumed general time course showing the expectedincrease and decrease of the QPC during the preictal, ictal, andpostictal periods. The increase of time smoothing in the MWTapproach [see (A) to (B)] is clearly visible, time smoothing ofMGT through the use of different σbounds [see (C) to (D)] isless distinct.

Statistical analysis of mBC courses over time confirms re-gions with significantly increased mBC [e.g., before 100, at240, at 300, after 400 s; red line in Fig. 6(c)].

Confidence tubes of mBC achieved by the bootstrapping ap-proach supports this view and provides more insight into sta-tistical properties of mBC over time. Confidence intervals at,e.g., 240 s (preictal period), were much narrower than after theseizure (postictal period), where the narrowest confidence in-tervals appear during the seizure [ictal period; Fig. 6(c)]. Bestcombination of significantly increased mBC (red line) and dis-crimination between different mBC periods over time (confid-cence tubes) can be achieved by the second MGT approach [seeFig. 6(c) and (D)].

2) EEG of Neonates With Burst–Interburst Pattern DuringQS: Mean results of one neonate (#1; K = 17 EEG trials) ofMWT and MGT-based tvPS of burst–interburst pattern in theEEG are depicted in Fig. 7(a), time courses of mBC in theROI are given in Fig. 7(b). Grand mean results of all neonates(K = 102 EEG trials) using MWT and MGT are not shown butconfirm the results of the single neonate.

The frequency-dependence of TFR in the tvPS of MWT isclearly visible [see Fig. 7(a-a) and (a-b)]. Again, the frequencyresolution in MWT-based approach (A) is not high enough,whereas approach (B) is afflicted with strong time smoothing(low time resolution). The TFR of all MGT-based approaches[see Fig. 7(a-c) and (a-d)] is adequate for bispectral analysis.Quantification of mBC in the ROI [see Fig. 7(b)] is able to reflectthe presumed general time course of increased QPC with theonset of the burst period. Different TFR (and resulting differenttime smoothing) of the MWT approaches [see (A) to (B)] isclearly visible, as well as the time smoothing of MGT throughthe use of different σbounds [see (C) to (D)]. Statistical analysisof the mBC course over time confirms regions with significantlyincreased mBC after the burst onset [e.g., at 5, 6, 8 s; red line inFig. 7(b)].

The course of the mBC confidence tubes achieved by thebootstrapping approach shows statistical properties of mBC overtime. Best combination of significantly increased mBC (red


Fig. 6. Grand mean results (K = 18) of HRV during TLE. (a) tvPS in (b1), (b2), and (b3) BA at distinct time points (240, 340, and 390 s; see Fig. 3(a), andin (c) mBC in the ROI are shown. (A) and (B) depict results of MWT-based approaches (ω0 = 2π/10π). (C) and (D) depict results of MGT-based approaches(σbounds = [5 s, Inf]/[10 s, Inf]). Red color designates high power, blue color: low power of tvPS or BA, respectively, in columns (a), (b1), (b2), and (b3). Incolumn (c), time course of mBC in the ROI = [0.075Hz, 0.15Hz] × [0.25Hz, 0.35Hz] is shown. Choice of ROI for the mBC quantification is illustrated by agray frame in (b1) to (b3). Red lines in column (c) designate 5% threshold for statistical significance of mBC, gray filled areas indicate 95% confidence tubes ofmBC achieved by a bootstrapping approach.

Fig. 7. Mean results (K = 17) of burst–interburst pattern in the EEG ofone child (#1). (a) tvPS and in (b) time course of mBC in the ROI =[0.5 Hz, 1.5 Hz] × [3 Hz, 5 Hz] is shown. (A) and (B) depict the results ofMWT-based approaches (ω0 = 2π/5π). (C) and (D) depict results of MGT-based approaches (σbounds = [0.1 s, Inf]/[0.25 s, Inf]). Gray line in column(b) indicates the onset of burst, red lines designate 5% threshold for statisticalsignificance of mBC, gray filled areas indicate 95% confidence tubes of mBCachieved by a bootstrapping approach.

line) and discrimination between different mBC periods overtime (confidcence tubes, before and after burst onset) can beachieved by the second MGT approach [see Fig. 7(b) and (D)].

V. DISCUSSION AND SUMMARY

The methodological novelty of our study is that for the firsttime, MP decomposition has been used to achieve a time-variantbispectral analysis without having to perform a manual (exper-imental) parameter adjustment for optimizing TFR, as requiredin GT-, MWT-, and AR-based bispectral analysis. To emphasizethe advantage of our approach, the TFR properties of our methodand the more common approaches has to be compared. TheSTFT and the GT are characterized by a frequency-independentTFR, i.e., all frequency components of the signal are analyzedwith identical TFRs. MWT-based methods have a frequency-dependent TFR, where LF components are analyzed with HFand low time resolution and HF components with LF and hightime resolution. This uncertainty principle is valid for all time-variant methods. This was illustrated in an excellent manner byJamsek et al. [6]. The authors exposed the specific problemsby a comparison of STFT- and MWT-based bispectral analysis.The time and the frequency resolution of third-order TFDs andthe bispectral approaches based on them can be adjusted inde-pendently. This does not conflict with the uncertainty principlebecause the resolutions are limited by the so-called local mo-ment function. Additionally, non-QPC terms in the bispectrumoccur which can be sufficiently suppressed by a broadening ofthe time window, which causes a decrease in the time resolution,i.e., a compromise between both properties must be found. The


TFR of time-variant parametric approaches depends on the orderof the AR model (frequency resolution) and (smoothing) param-eters of the estimation algorithm (time resolution). The lowerthe model order (number of AR parameters =̂ window length ofthe AR process analyzer), the higher the time resolution is andthe more smoothed the spectrum is (LF resolution). If the modelorder is too high, then spurious peaks occur [26]. The modelorder must be estimated by estimation methods which station-ary approaches commonly use. Smoothing during time-variantAR parameter estimation improves the estimation properties butdecreases the time resolution. In practice, an acceptable compro-mise needs to be found in order to obtain the most appropriateTFR. Such an optimization procedure was described in detailby Schwab et al. [27].

To overcome such limitations characteristic to TFR, the re-cently introduced MGT is used [14], which consists of a com-bination of the MP and the GT providing a complex time–frequency plane with an optimal TFR for each signal compo-nent. The superiority of the MGT has already been demonstratedin linear-phase locking and synchronization analyses for com-parison of the GT- and MWT-approaches [14]. The results ofthis study also demonstrate that the application of the MGT hasbeneficial effects for QPC analysis.

Kus et al. [28] concluded that the limited acceptance of MPin the EEG analysis may be partly due to the lack of definedcriteria (rules) for setting the most important parameters of thealgorithm, which are the number and distribution of atoms. Ingeneral, the results of MP-based processing concepts depend to alarge extent on the dictionary used. The MGT approach is basedon a Gabor atom dictionary, which allows the representationof the signal into the complex time–frequency plane, i.e., bothtime-variant amplitude and phase information are preserved.The precondition is that the natural signal components can be ap-proximated by Gabor atoms, which is often, but not always, thecase in HRV and EEG. Three further aspects with regard to thedictionary should be discussed. First, both signals can containasymmetric components, which cannot be reliably described byGabor atoms. To detect and quantify such components, a dic-tionary which is enriched by or composed of asymmetric atomscan be used [29]. Transient HRV patterns (e.g., extrasystoles andother cardiac arrhythmias) and artifacts (incorrect QRS trigger-ing) are mostly asymmetric, and they can mask or disturb theHRV analysis. Therefore, MP with asymmetric dictionaries canbe used for the detection of “disturbing” HRV patterns. Second,if the atoms from the dictionary are shifted on a discrete timegrid to cover the whole time range, the discrete positions auto-matically introduce bias. To compensate for this, the so-calledstochastic dictionaries can be used, where the decomposition iscarried out for many different, randomly perturbed dictionar-ies [30]. Third, the size of the dictionary plays an importantrole. In general, the use of a larger dictionary of atoms “fordecomposition of the same signal should lead to more accurateparametrization” [28]. Kus et al. [28] introduced an approachfor the design of an optimal Gabor dictionary, which is availableby an open-source software package. These advancements canbe considered for further methodological improvement of ourapproach.

For MGT-based QPC analysis, the use of a lower sigma boundfor the GT is advantageous. Our AM simulation study (AAMand MAM) shows that AM-related QPC can be successfully de-tected by using this thresholding operation for time resolution,i.e., Gabor atoms of the dictionary with sigma values belowthis bound (with time resolution, which is too high) will becomputed (GT) with a fixed (bound value) lower time resolu-tion. This is more appropriate for the particular characteristicsof the AM signal, i.e., the carrier and the side-band frequen-cies occur as continuous traces in the time–frequency plane.AM phenomena have been already described in HRV and EEG(e.g., [20] and [21]) and have been investigated in “envelope-to-signal-correlation” approaches [31], which are based on AMdemodulation (envelope) [32]. In contrast, bispectral analysisis a more general approach. Bispectral analysis is sensitive tochanges in third-order statistics, i.e., any nonlinearity that al-ters the (third order) statistics of the input will be detectable byit [33].

In biomedical signal analysis, such statistical changes arecertainly an indication of a change of state within a physiologicalsystem or process. Our HRV and EEG data show abrupt changesin state, and we have shown that the MGT-based bispectralanalysis highlights the QPC changes associated with this moreclearly than the MWT approaches.

Our methodological approach is highly relevant to physiolog-ical and clinical questions. RSA or HF HRV, which can be de-fined as an HRV component of the frequency of breathing [34], isoften used as an index of cardiac vagal tone, “whereby the HRincreases during inspiration and decreases during expiration”[35]. RSA amplitude is modulated by different physiologicalparameters (depth and frequency of breathing, cardiac vagal ef-ferent effects etc.; for an overview, see [36]). The rhythmic AMof the RSA is one possible cause for QPC in HRV, e.g., a modu-lation by the LF waves of the HRV (e.g., by the 0.1-Hz waves).It can be speculated that the nonlinearity of the cardiovascular–cardiorespiratory control system contributes to the occurrenceof QPC and that a change in the physiological state is accom-panied by a change of the control behavior. In clinical practice,time–frequency features of HRV are frequently used, e.g., foran automatic detection of seizure in newborns [37]. The resultsof our study confirm our previous results that the QPC increasesbefore and, in particular, after an epileptic seizure (temporal lobeepilepsy). A neurophysiological link between the 0.1-Hz HRVwaves and the Mayer waves in systemic arterial blood pressureexists. Mayer waves are correlated with the oscillations of ef-ferent sympathetic nervous activity and the baroreflex plays amajor role in their generation. In contrast, the 0.1-Hz HRV com-ponent associated with the Mayer wave includes most probablyboth sympathetic and parasympathetic (vagal) influences [19].Our results indicate a changed coordination between LF andHF HRV waves in particular after the seizure, i.e., the auto-nomic nervous system still remains in an altered physiologicalstate.

As has been stated, “The nonlinear nature of neuronal activitycontributes to the formation of an EEG signal with very complexdynamics” [38] and as has been shown, QPC between differentfrequency components of EEG aids in discovering changes in


physiological states of the brain and helps to elucidate the com-plex dynamics of physiological systems. For example, bispec-tral analysis has been used for the investigation of EEG patternsfollowing hypoxic-asphyxic arrest [39] for the identification ofepileptic and focal ischemic cerebral EEG [40], [41] as well asto detect sleep [42], anesthesia [43], and sedation states [5]. Ourproof-of-principle application in the field of EEG analysis isrelated to QPC changes in neonatal EEG patterns derived fromthe quiet sleep state. We used the EEG of mature neonates todemonstrate the methodological improvements by using MGT.After the burst onset, the bicoherence increases, which indicatesa trigger process in which most probably the thalamus is initiallyinvolved [44]. After the burst onset, the cortical LF oscillationmodulates the amplitude of HF oscillatory activities. This canbe physiologically explained by the model of Steriade [45], inwhich a depolarization phase of a cortical LF oscillation trav-els through the corticothalamic pathway and triggers a spindlesequence in the reticular thalamic nucleus that will be deliv-ered to the cortex via the dorsal thalamus. We have previouslyshown that changes in the QPC’s degree after the burst onset areassociated with brain maturation [23].

For bispectral analysis, i.e., independent from the computa-tion method, defining the ROI for the appropriate selection ofparameters from the bispectral plane is important and is basedon a priori information. The optimal TFR of the MGT approachis helpful for this decision making because the real signal struc-ture is mapped in the time–frequency plane. This is also true forcoupling structures in the bispectral plane. Therefore, bispec-tral ROIs can be better identified, e.g., ROIs which enable theinvestigation of specific physiological signal components andtheir couplings. This has been shown by our simulation study(individually tailored versus guideline-driven ROIs; red lines inFig. 4(D)).

VI. CONCLUSION

In general, it can be concluded that the application of the newMGT approach leads to the improved QPC results because of theachieved optimal TFR, the use of sigma bounds, and thereby, theresulting advanced decision making in terms of defining specificestimation parameters. The next question arises, how can theseadvantages be optimally expanded for use in further researchof for application in the clinical setting? One concept is to usedictionary modifications or dictionary alternatives to improvemethodologies. It can be expected that the advancements inMP and application-driven MP adaptations will lead to furtherimprovement of time-variant bispectral analysis in biomedicine.

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Authors’ photographs and biographies not available at the time of publication.

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