Katarzyna J. Blinowskabiosignal.med.upatras.gr/school2008/Materials/Blinowska...Time-frequency...
Transcript of Katarzyna J. Blinowskabiosignal.med.upatras.gr/school2008/Materials/Blinowska...Time-frequency...
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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
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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
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Time
Freq
uenc
y
Wigner de Ville TransformWigner de Ville TransformFourier Transform of autocorrelation function
( ) τττξ ξτdetftftW if
−⎟⎠⎞
⎜⎝⎛ −⎟
⎠⎞
⎜⎝⎛ += ∫ 22
,
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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
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( ) tiI eutgtg ξ−=)(
FFT
FFTFFT
FFT
Spectrogram – windowed Fourier Transform
( ) ( ) ( ) dteutgtfuSf tiξξ −+∞
∞−
−= ∫,
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Time
Freq
uenc
y
⎟⎠⎞
⎜⎝⎛ −
=s
utgs
tgI1)(
Wavelet Transform
( ) ( ) ( )ugfdts
utgs
tfsuSf s∗=⎟⎠⎞
⎜⎝⎛ −
= ∫+∞
∞−
1,
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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.
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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.
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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
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Time [s]
0 1 2-100
-80
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-40
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[ V]μ
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0 1 2-100
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0
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0 1 2-100
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0
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0 1 2-100
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0
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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.
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[ ]( ) [ ]( )
[ ]( )∑ ∑
∑∞
=
∞
≠=
∞
=+=
0 ,0
0
2
,,,,
,,,,,
n nmmIII
mI
n
nIII
n
tggWgfRgfR
tggWgfRtffW
mnmn
nnn
ω
ωω
Construction of time-frequency distribution from waveforms
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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
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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]
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Noise influence
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Time-frequency distribution of EEG power during finger movement
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50%1s
10 - 12 Hz14 - 18 Hz36 - 40 Hz
ERD/ERS
EMGi
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T-F distribution of EEG power during epileptic seizure
P.J. Franaszczuk, G.K.Bergeyat al. al. Electroenceph. clin. Neurophys.106:513-521, 1998.
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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]
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f=12.9 Hz, t=13.6 s,
Δt= 0.9 s, A=27.5 μV
MP provides parametric description of signal structures.
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Statistical evaluation of structures observed in sleep EEG
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Superimposed Spindles
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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.
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Sleep spindles as input values to inverse solutions
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Inverse solutions for epileptic activity
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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.
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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
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• MVAR model
• DTF (assumption of ergodicity)
- simulations- egzamples of applications
• SDTF (ensemble averaging)- egzamples of applications
Multivariate methods of signal analysis
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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
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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
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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
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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
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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
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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
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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' +−+−= ∑∑
==
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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).
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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
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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).
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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
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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
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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).
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SimulationBivariate coherence
Result
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Simulation
Result
Bivariate GrangerCausality
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Simulation
Result
Non-normalized DTF
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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.
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Sampling rate =128Hz
Lenght of signal =10s
Number of pointsper channel = 1280
Simulation scheme
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Granger causality (pair-wise)
result:simulation:
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Directed Transfer Function (DTF)
simulation: result:
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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).
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Direct Directed Transfer Function (dDTF)
result:simulation:
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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
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MVAR model was fitted to 21 channels of sleep EEG(10/20 system) simultaneously and DTF functions were determined.
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EEG activity propagation during different sleep stages(averaged over 9 subjects)
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Comparison of different methods of directionality estimation for EEG(awake eyes open)
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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.
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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.
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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
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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)(~
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SDTF
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Time course of the propagation of beta activity after movement
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sound
0 1 3 5 6 7
cue
analyzed epoch
Experiment – finger movement paradigm
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Beta activity propagation during finger movement and its imagination
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Gamma activity propagation during finger movement and its imagination
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Gamma band, fist clenching
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Gamma band – tongue movement
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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
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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).
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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
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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.
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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
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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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THANK YOU FOR YOUR THANK YOU FOR YOUR THANK YOU FOR YOUR ATTENTION !ATTENTION !ATTENTION !