Time-frequency methods and parametric models for brain

signal analysis

Katarzyna J. Blinowska

4th International Summer School on Emerging Technologies in BiomedicineJune 29th-July 4th, 2008

Patras, Greece

Time frequency methods•Wigner de Ville distributions

• Spectrogram – sliding window Fourier transform

•Wavelets

•Adaptive Approximations by Matching Pursuit

Multivariate autoregressive model (MVAR)

•MVAR and Granger causality

•Directed Transfer Function DTF

•Partial Directed Coherence PDC

•Short-time Directed Transfer Function SDTF

Time

Freq

uenc

y

Wigner de Ville TransformWigner de Ville TransformFourier Transform of autocorrelation function

( ) τττξ ξτdetftftW if

−⎟⎠⎞

⎜⎝⎛ −⎟

⎠⎞

⎜⎝⎛ += ∫ 22

,

Time frequency methods relying on the decomposition into basic functions:

Spectrogram - sinusoids

Wavelet transform – wavelets

Adaptive approximations by matching pursuit –broad repertoir of functions, usually functions from Gabor family including sinusoids

( ) tiI eutgtg ξ−=)(

FFT

FFTFFT

FFT

Spectrogram – windowed Fourier Transform

( ) ( ) ( ) dteutgtfuSf tiξξ −+∞

∞−

−= ∫,

Time

Freq

uenc

y

⎟⎠⎞

⎜⎝⎛ −

=s

utgs

tgI1)(

Wavelet Transform

( ) ( ) ( )ugfdts

utgs

tfsuSf s∗=⎟⎠⎞

⎜⎝⎛ −

= ∫+∞

∞−

1,

Matching PursuitMatching Pursuit

Dictionary of basic waveforms can be generated by scaling translating and modulating basic function

s>0 - scale, ξ - frequency modulation, u -translation.

Index I = (ξ, s, u) describes the set of parameters

)(tgI

tiI e

sutg

stg ξ⎟

⎠⎞

⎜⎝⎛ −

=1)(

The dictionaries of windowed Fourier transform and wavelet transform can be derived as subsets of this dictionary, defined by certain restrictions on the choice of parameters:

• In case of WT the frequency modulation is limited by the restriction on the frequency parameter ξ = ξ0/s, ξ0 = const.

• In case of the windowed Fourier transform, the scale s is constant -equal to the window length - and the parameters ξ and u are uniformly sampled.

Matching Pursuit

Introduction of algorithm,1993Mallat S. G., Zhang Z. Matching Pursuit with time-frequency dictionaries.

IEEE Transactions on Signal Processing 1993; 41:3397-3415.

First application to biological signal 1994, 1995:Blinowska, K.J and Durka, P.J. The Application of Wavelet Transform and

Match-ing Pursuit to the Time-Varying EEG signals, In: Intelligent Engineering Systems through Artificial Neural Networks. Vol.4. pp.535-540. Eds.: C.H.Dagli, 1994.

Durka P. J., Blinowska K. J. Analysis of EEG transients by means of Matching Pursuit. Annals of Biomedical Engineering 1995; 23: 608-611.

Improvement of the original algorithm removing bias of the dyadic sampling of the time frequency space:

Durka P. J., Ircha D., Blinowska K. J. Stochastic time-frequency dictionaries for Matching Pursuit. IEEE Transactions on Signal Processing, 2001; 49:507-510.

In MP repertoire of function is very broad. Best time-frequency resolution is obtained when the functions gI(t) belong to Gabor functions family.

fRggff II1

00, +=

fRggfRfR nnInI

nn 1, ++=

Finding an optimal approximation of signal by functions from such a large family is a NP-hard problem (computationally intractable). Therefore a suboptimal iterative procedure is applied . In the first step of the iterative procedure we choose the vector which gives the largest product with the signal f(t):

0Ig

Then the residual vector R1 obtained after approximating f in the direction is decomposed in a similar way. The iterative procedure is repeated on the following obtained residues:

0Ig

fRggfRf nI

m

nI

nnn

1

0, +

=+= ∑

In this way the signal f is decomposed into a sum of time-frequency waveforms, chosen to match optimally the signal’s residues:

Matching PursuitMatching Pursuit

Time [s]

0 1 2-100

-80

-60

-40

-20

0

20

40

60

80

100

[ V]μ

0 1 2-100

-80

-60

-40

-20

0

20

40

60

80

100

0 1 2-100

-80

-60

-40

-20

0

20

40

60

80

100

0 1 2-100

-80

-60

-40

-20

0

20

40

60

80

100

0 1 2-100

-80

-60

-40

-20

0

20

40

60

80

100

0 1 2-100

-80

-60

-40

-20

0

20

40

60

80

100

Statistical biasof representation was removed by introduction of stochastic dictionaries

sleep spindles noise

Durka P. J., Ircha D., Blinowska K. J. Stochastic time-frequency dictionaries for Matching Pursuit. IEEE Transactions on Signal Processing, 2001; 49:507-510.

[ ]( ) [ ]( )

[ ]( )∑ ∑

∑∞

=

∞

≠=

∞

=+=

0 ,0

0

2

,,,,

,,,,,

n nmmIII

mI

n

nIII

n

tggWgfRgfR

tggWgfRtffW

mnmn

nnn

ω

ωω

Construction of time-frequency distribution from waveforms

Choi-Williams Distribution

Time

Freq

uenc

y

TimeTime

Freq

uenc

y

Freq

uenc

yFr

eque

ncy

Freq

uenc

y

Matching Pursuit

Discrete Wavelet TransformSpectrogram

Spectrogram Composition of the Test Signal

TimeTime--Frequency distributionstions Frequency distributionstions —— comparison of methodscomparison of methods

TimeTime--Frequency distributionstions Frequency distributionstions —— comparison of methodscomparison of methods

Time-frequency (t-f) energy density distributions for: A: Matching Pursuit (MP), B: windowed Fourier transform (STFT), C: Wigner-deVille distribution (WVD), D: continuous wavlet transform (WT).

)2sin()( 230 ftett t πγγ πΓ−=

In simulations γ functions of the form given below were used.

Time [ms]

frequ

ency

[kH

z]

Time [ms]

Noise influence

Time-frequency distribution of EEG power during finger movement

50%1s

10 - 12 Hz14 - 18 Hz36 - 40 Hz

ERD/ERS

EMGi

T-F distribution of EEG power during epileptic seizure

P.J. Franaszczuk, G.K.Bergeyat al. al. Electroenceph. clin. Neurophys.106:513-521, 1998.

Resonance modes Resonance modes –– specific specific characteristics of inner ear.characteristics of inner ear.

OAE energy distribution in timeOAE energy distribution in time--frequency for tonal stimulation of frequencies frequency for tonal stimulation of frequencies shown above the pictures. The same resonance modes are excited bshown above the pictures. The same resonance modes are excited by different y different tonal stimuli.tonal stimuli.

time [ms]

frequ

ency

[kH

z]

f=12.9 Hz, t=13.6 s,

Δt= 0.9 s, A=27.5 μV

MP provides parametric description of signal structures.

Statistical evaluation of structures observed in sleep EEG

Superimposed Spindles

Matching Pursuit and Inverse Solutions.Matching Pursuit and Inverse Solutions.

As the input to inverse solutions different quantities can be used e.g.:value of electric field on the scalp in a given time moment, spectral integral, time integral representing given structure.This quantities represent cumulative activity which may have different origin - e.g. spectral integral in the 10 - 14 Hz may represent not only sleep spindles but also alpha activity.

By means of multichannel matching pursuit (MMP) particular structures can be extracted from EEG activity e.g.: sleep spindles, epileptic spikes.

Sleep spindles as input values to inverse solutions

Inverse solutions for epileptic activity

MP - examples of applications:Żygierewicz J., Kelly E. F., Blinowska K. J., Durka P. J, Folger S. E. Time-Frequency Analysis of Vibrotactile Driving Responses by Matching Pursuit. Journal of Neuroscience Methods 1998; 81:121-129.

Żygierewicz J., Blinowska K. J., Durka P. J., Szelenberger W., Niemcewicz S., Androsiuk W. High resolution study of sleep spindles. Clinical Neurophysiology 1999; 110:2136-2147.

Durka P. J., Ircha D.,Neuper C., Pfurtscheller G. Time-frequency microstructure of event-related EEG desynchronisation and synchronisation. Med. & Biol. Engin. & Comput. 2001, 39 :315-321.

K.J. Blinowska, P.J. Durka. Unbiased high resolution method of EEG analysis in time-frequency space. Acta Neurobiol. Exper. 2001, 61:157-174.

Ginter Jr., J., Blinowska K.J., Kaminski M., Durka P., J. Phase and amplitude analysis in time-frequency space – application to voluntary finger movement. J. Neurosc. Meth. 2001, 110:113-124.

P.J. Durka, W. Szelenberger, K.J. Blinowska, W. Androsiuk, M. Myszka. Adaptive time-frequency parametrisation in pharmaco EEG. J. Neurosc Meth. 2002, 117:65-71.

P.Durka, A.Matysiak,E.Martinez-Montes, P. Valdes Sosa, K.J. Blinowska. Multichannel matching pursuit and EEG inverse solutions. J.Neurosc. Meth. 2005, 148:49-59.

U.Malinowska, P.J. Durka, K.J. Blinowska, W. Szelenberger, A.Wakarow. Micro- and Macrostructure of sleep. EEG.IEEE BME Mag. 2006, 25:26-31.

Adaptive Approximations by MP

•Representation in one framework rhythmic and transient components of the signal

•Parametric description of signal in terms of parameters with clear physiological meaning

•High resolution time-frequency representation

•Detection of microstructure of signals

•Identification of components close in frequency and/or in time

•Selective input to inverse problem solutions

• MVAR model

• DTF (assumption of ergodicity)

- simulations- egzamples of applications

• SDTF (ensemble averaging)- egzamples of applications

Multivariate methods of signal analysis

Nkpp /2))log(det(2)(AIC += V

( ) ( ) ( ) ( )tEitXiAtXp

i+−=∑

=1

Multivariate autoregressive model (MVAR) fitted to thek-channel EEG process is expressed as:

Model order p is found from criteria developped on the grounds of information theory, the most popular is Akaike (AIC) criterion:

Where: X - vector of k signals recorded in time: X (t )=(X1(t ), X2(t ), ...,Xk(t )), E (t ) is the vector of white noise values, A (i ) are the model coefficients and p is the model order.

MVAR

V- matrix of noise variance

The elements of correlation matrix R(s) are found as:

psstXtXN

sRSN

tji

Sij , ,0for ,)()(1)(

1

…=+= ∑=

Depending on the version of the estimators used, we may have NS = n – |s| for unbiased correlation estimator and NS = n for biased estimator (n is the record length). The later ensures a positive definite form of R(s). The AR model parameters may be obtained by solving the set of linear equations (called Yule-Walker equations)

∑=

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−−

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−

−−−

p

jjj

pppp

pp

0)()(

)(

)2()1(

)(

)2()1(

)0()2()1(

)2()0()1()1()1()0(

RAV

R

RR

A

AA

RRR

RRRRRR

AR model coefficients are found by minimalisation of the variance matrixThis leads to the calculation of the correlation matrix

V – variance matrix of a noise

Assuming A(0) = -I (I - identity matrix), we can rewrite the autoregressive dependence in another form:

)()()()()( 1 fffff EHEAX == −

The H( f ) is called the transfer matrix of the systemf denotes frequency ,

∑=

−=p

jjtjt

0)()()( XAE

Transforming the above equation to the frequency domain by application of Z transform we get :

timfez Δ−= π2

Transfer matrix H(f ) defined in the frequency domain contains spectral information and phase dependencies

between channels.

From H(f) spectra and coherences can be found

MVAR model in the frequency domain

S(f) = H(f) V H*(f)

( ) ( )( ) ( )

χ i ji j

j j i i

fM f

M f M f2

2

=

Mij is a minor of spectral matrix

Partial coherence:

Multiple coherence: μj(f)=(1-det|S(f)|/Sjj(f)Mjj(f))1/2

Ordinary coherence: ( ) ( )( ) ( )fSfS

fSfk

jjii

ijij

22 =

From transfer matrix H(f) spectrum and coherences may be found

SpectrumV is a variance matrix of a noise

The multivariate approach allows for easy calculation of spectra, ordinary, partial and multiple coherencies

S(f) = H(f) V H*(f)

( ) ( )( ) ( )

χ i ji j

j j i i

fM f

M f M f2

2

=

Mij is a minor of spectral matrix

Partial coherence:

Multiple coherence:μj(f)=(1-det|S(f)|/Sjj(f)Mjj(f))1/2

GRANGER CAUSALITY1 is based on the predictability

of the series:

1 W.C.J. Granger, Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 1969, 37:424-438.

Nobel Prize 2005

If we try to predict a value of X(t) using p previous values of the series X only, we get a prediction error e1:

If we try to predict a value of X(t) using p previous values of the series X and q previous values of Y we get another prediction error e2:

If the variance of e2 (after including series Y to the prediction) is lower than the variance of e1 we say that Ycauses X in the sense of Granger causality.If some series X1(t ) contains information in past terms that helps in the prediction of series X2(t ), then X1(t ) is said to cause X2(t ) - this formulation is compatible with the MVAR model

∑=

+−=p

jtejtXjAtX

1111 )()()()(

( ) ( ) ( ) ( ) ( ) ( )tejtYjAjtXjAtXp

j

p

j2

112

111' +−+−= ∑∑

==

The concept of GRANGER CAUSALITY was originally defined for two signals, however as was pointed out by Granger (1980) in his next publication test of causality is impossible unless the set of of interacting channels is complete.

What , if we have more than 2 interacting channels?

This is usually the case for EEG signals.

The concept of Granger causality was extended for arbitrary number of channels by introduction of Directed Transfer Function1 (DTF) and later Partial Directed Coherence2 .

Both measures are based on Multivariate Autoregressive Model.

DTF1M. Kamiński, K.J. Blinowska, A new method of the description of the

information flow in the brain structures. Biol.Cybern. 65:203-210,(1991)

PDC2 L.A. Baccalá, K. Sameshima, Partial directed coherence: a new concept in

neural structure determination, Biol Cybern, 84:463-74 (2001).

DTF – Directed Transfer Function1

The DTF function , describing transmission from channel j to i at frequency fnormalized in respect of inflows to channel i is defined as:

In the transfer matrix H(f )not only frequency information but also phase dependencies between channels are coded. The future of one channel can be predicted from the past of other channels.

( )( )

( )∑=

= k

mim

ijij

fH

fHf

1

2

22γ

1M. Kamiński, K.J. Blinowska, A new method of the description of the information flow in the brain structures. Biol.Cybern. 1991, 65:203-210.

DTF:

22 |)(| )(θ fHf ijij =Non normalized DTF

Coupling strength between signals

Non- normalised DTF

Non-normalised DTF for two signals is

equivalent to Granger causality,

it is proportional to the coupling

strength between signals

M. Kamiński, M. Ding, W. Truccolo,S.L. Bressler,Evaluating causal relationsin neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biol. Cybern. 2001, 85:145-157

LFPs were simulated by a model composed of N coupled cortical columns where each column was made up of an excitatory and inhibitory neuronal population (Freeman 1992).

DTF

detects correctly one way and

reciprocal

connections in point processes and continous time

series.

M. Kamiński, M. Ding, W. Truccolo, S.L. Bressler., Evaluating causal relationsin neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biol. Cybern. 2001, 85:145-157

DTF is robust in respect to random noise EEG

signal in 1st chan.

noise

+

=

EEG – experimental EEG from electrode P3, relaxedstate

Δ – delay

noise – random noise from Gaussian distribution

noise amplitude/EEG amplitude = 3

The partial directed coherence was defined by Baccala and Sameshima in the following form:

)()(

)()(

* ff

fAfP

jj

ijij

aa=

In the above equation Aij( f ) is an element of A( f ) – a Fourier transform of model coefficients A(t), aj( f ) is j-th column of A(f) and the asterisk denotes the transpose and complex conjugate operation. Although PDC is a function operating in the frequency domain, the dependence of A(f) on the frequency has not a direct correspondence to the power spectrum.

PDC is a multivariate method which detects direct connections.

Partial Directed Coherence - PDC

L.A. Baccalá, K. Sameshima, Partial directed coherence: a new concept in neural structure determination, Biol Cybern, 84, 463-74 (2001).

SimulationBivariate coherence

Result

Simulation

Result

Bivariate GrangerCausality

Simulation

Result

Non-normalized DTF

Surrogate data test

MVAR Granger Causality

Surrogate of MVAR GrangerCausality

The ”leak currents” indicating the accuracy of DTF are found from surrogate data – signals with disturbed phases. Surrogate data are obtained by transforming the data to the frquency domain, randomizingtheir phases and transforming back to the time domain.

Sampling rate =128Hz

Lenght of signal =10s

Number of pointsper channel = 1280

Simulation scheme

Granger causality (pair-wise)

result:simulation:

Directed Transfer Function (DTF)

simulation: result:

dDTFDirect Directed Transfer Function

In some cases such as e.g. recording from depth electrodes, where many coupled structures are communicating along many possible pathways, the identification of direct activity flows is important. To resolve this problem direct Directed Transfer function was proposed. It is obtained by multiplication of DTF by partial coherence:

( ) ( ) ( )fff ijijij ηχδ =

dDTF(f) = (partial coherence (f)) (DTF(f))

A. Korzeniewska, M. Manczak, M. Kaminski, K.J. Blinowska, S.Kasicki, Determination of information flow direction among brain structures by a modified directed transfer function.(dDTF) method, J.Neurosc. Meth.125:195-207 (2003).

Direct Directed Transfer Function (dDTF)

result:simulation:

Application of DTF and DDTF to local field potentials

Animal walking on a runway , stressing stimulus - bellSignals registered by electrodes chronically implanted in brain structures:

BLA- basolateral amygdalaVSB- ventral subiculumACC- nucleus accumbensSPl- subpallidal area

MVAR model was fitted to 21 channels of sleep EEG(10/20 system) simultaneously and DTF functions were determined.

EEG activity propagation during different sleep stages(averaged over 9 subjects)

Comparison of different methods of directionality estimation for EEG(awake eyes open)

DTF and PDCPDC detects only direct connections, DTF direct and indirect connections (in order to identify only direct connections DDTF has to be used).Although PDC is a function operating in the frequency domain, the dependence of Aij( f ) on the frequency has not a direct correspondence to the power spectrumOther differences are connected with normalisation.In DTF the outflow from chnnel j to i are normalised by all inflows to channel i.

In case of PDC the inflow to channel idescribed by the element Aij(f) is divided by the outflows from j to all other channels. In consequence evenweak sources emitting activity infew directions are emphasised.

PDC shows rather sinks than sources,DTF shows sources.

WARNING !

The input data for the MVAR model should not be subjected to pre-processing, which introduces correlation between signals.

No „common average”, „source derivation”, Hjorth or Laplace transform may be used, since they disturb the correlations and

phases between channels.

Directed Transfer Function is insensitive in respect to volume conduction, since DTF detects phase diffrences and volume

conduction is zero phase propagation. DTF is robust in respect of random phase noise and constant phase disturbances.

Where k – number of channels, p – model order, N – number of data points

Number of channels is limited by the length of the stationary data window.

This difficulty was overcame by introduction of SDTF.

Time varying – Short time Directed Transfer Function (SDTF) can be found when multiple realisations are available.

points data od numbertscoefficienMVARofnumber

Nkp

kNpk ==

2

DTF statistical properties depend on the ratio of:

DTF and PDC are multivariate methods based on MVAR model. When fitting models to the data the number of coefficients should be much smaller than the number of the datapoints (preferably <0.1).

Limitation of methods based on MVAR

When multiple realizations are available the model coefficients ai can be estimated by an ensemble averaging of correlation matrix:

Then MVAR model can be fitted to short data epochs and Short-time Directed Transfer Function (SDTF) can be determined

SDTF

∑ ∑ ∑= =

−

=

−−

==T TN

r

N

r

sn

t

rj

ri

T

rij

Tij stXtX

snNsR

NsR

1 1

||

1

)()()( )()(||

11)(1)(~

SDTF

Time course of the propagation of beta activity after movement

sound

0 1 3 5 6 7

cue

analyzed epoch

Experiment – finger movement paradigm

Beta activity propagation during finger movement and its imagination

Gamma activity propagation during finger movement and its imagination

Gamma band, fist clenching

Gamma band – tongue movement

DTF can be used to evaluate causality between point

processes e.g.spike trainsSpikes are low-pass filtered (Butterwort filter)

and 10% noise is added

M. Kamiński, M. Ding, W. Truccolo, S.L. Bressler,Evaluating causal relationsin neural systems: Granger causality,

directed transfer function and statistical assessment of significance. Biol. Cybern. 2001, 85:145-157

Spike trains are low-pass filtered (Butterworth filter) and 10% of noise is added

Causal relations between spike trains and local field potentials

Rat hippocampal LFP were measured together with firing of 12 SUM (Supramammiliary) neurons. Neuronal population was inhomogenous – some neurons fired on positive peak of CA1 theta waves, others on their raising phase.The directions of propagation were evaluated by means of SDTF for spontaneous activity and during stimulus – tail pinch.B. Kocsis, M. Kaminski Dynamic changes in the direction of the theta rhythmic drive between supramammillary nucleus and the septohippocampal system; Hippocampus 16(6):531-540, 2006 (available online).

During rest DTF showed predominant hippocampus-to-SUM direction of influence . In the episode of sensory stimulation the direction of drive from SUM to hippocampus was observed.

The study conducted by means of SDTF revealed dynamic relationship between SUM and septohippocampal theta oscillations in which the direction of the rhythmic drive changes depending on the origin and type of the rhythm.

Results

Directed Transfer Function - DTF

• 1991 Introduction of Directed Transfer FunctionM. Kaminski, K. Blinowska. A new method of the description of the information flow in brain structures. Biol. Cybern. 65, 203-210 (1991).

• Localization of epileptic fociP. J. Franaszczuk, G. K. Bergey, M. Kaminski. Analysis of mesial temporal seizure onset and propagation using the directed transfer function. Electroenceph. Clin. Neurophys. 91, pp. 413-427, 1994

• Propagation in brain structures during locomotionA. Korzeniewska, S. Kasicki, M. Kaminski, K. J. Blinowska. Information flow between hippocampus and related structures during various types of rat's behavior. Journal of Neuroscience Methods, 73, pp. 49-60, 1997

• Topography of EEG propagation during sleepM. Kaminski, K. J. Blinowska, W. Szelenberger. Topographic analysis of coherence and propagation of EEG activity during sleep and wakefulness. Electroenceph. Clin. Neurophys. 102, pp. 216-227, 1997.

Short-time Directed Transfer Function SDTF• M. Kamiński, M. Ding, W. Truccolo. S.L. Bressler. Evaluating Causual relations in Neural Systems: Granger

Causality, Directed Transfer Function (DTF) and Statistical Assessment of Significance . Biol Cyb.85:145-157, 2001.

• J.Ginter Jr., K.J.Blinowska, M. Kamiński, P.J. Durka. Phase and Amplitude analysis in time-frequency space - application to voluntary finger movement. J. Neurosci.Meth. 110:113-124,2001.

• R. Kus, M.Kaminski, K.J.Blinowska, Determination of EEG Activity Propagation: Pair-Wise Versus Multichannel estimate, IEEE Transactions on Biomedical Engineering 51:1501-1510, 2004.

• J.Ginter Jr., K.J. Blinowska, M. Kaminski, P.J. Durka,G. Pfurtscheller, C. Neuper, Propagation of EEG Activity in the Beta and Gamma Band during Movement Imagery in Humans. Methods In. Med.44:106-113, 2005.

• R.Kus, K.J.Blinowska, M.kaminski, A.Basinska-Starzycka, Propagation of EEG activity during continuous attention test, Bulletin of the Polish Academy of Sciences Technical Science 53:217-222, 2005.

• B.Koscis, M. Kaminski, Dynamic changes in the direction of the theta rhythmic drive, Hippocampus, 2006, 16:531-540.

For mutually dependent set of signals the correct pattern of propagations can be found only if all relevant channels are evaluated simultaneously.

DTF is an estimator robust in respect to noise, proportional to coupling between chnnels and capable to identify reciprocal connections

When multiple realizations of experiment are available SDTF can be estimated and the propagation as a function of time and frequency can be found.

alpha

beta

MP and DTF methods are complementary

• MP has higher resolution in time and frequency, makes possible identification and parametrisation of transients and rhythmical activity in the framework of the same formalism

• DTF is a multivariate method, it gives the information not only about amplitude, but also about phase dependencies between signals and hence on their propagation

• The information about both methods, references, papers and some software can be found at portal EEG.PL

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