Computational Neuroscience - Lecture 3users.auth.gr/dkugiu/Teach/ComputationalNeuroscience/Lecture3.pdf · Computational Neuroscience - Lecture 3 Kugiumtzis Dimitris Department of

Computational Neuroscience - Lecture 3

Kugiumtzis Dimitris

Department of Electrical and Computer Engineering,Faculty of Engineering, Aristotle University of Thessaloniki, Greece

e-mail: [email protected] http:\\users.auth.gr\dkugiu

22 May 2018

Kugiumtzis Dimitris Computational Neuroscience - Lecture 3

Outline

1 Single-channel EEG analysis - Introduction

2 Features of signal morphology

3 Features from linear analysis

4 Features from nonlinear analysis


Outline






Outline






Outline






Outline






Single-channel EEG analysis - IntroductionRegion of Interest (ROI) of the brain

Eyeball judgement:Doctor or EEGer may diagnose abnormalities by visual inspectionof EEG signals (in ROI).

Alternatively:Quantify automatically information from EEG signal (in ROI) ⇒univariate time series analysis: study the static and dynamicproperties of the time series (signal), model the time series.

Three main perspectives:

1 Characteristics of signal morphology, e.g. identify bursts.

2 stochastic process (linear analysis), e.g.Xt = φ0 + φ1Xt−1 + . . .+ φpXt−p + εt

3 nonlinear dynamics and chaos (nonlinear analysis), e.g.St = f (St−1), St ∈ IRd ,Xt = h(St) + εt












Alternatively:Quantify automatically information from EEG signal (in ROI) ⇒

univariate time series analysis: study the static and dynamicproperties of the time series (signal), model the time series.






































Clinical applications:

predicting epileptic seizures

classifying sleep stages

measuring depth of anesthesia

detection and monitoring of brain injury

detecting abnormal brain states

others ...

Example:α-wave (8-13 Hz) is reduced in children and in the elderly,and in patients with dementia, schizophrenia, stroke, and epilepsy


Clinical applications:

predicting epileptic seizures

classifying sleep stages

measuring depth of anesthesia

detection and monitoring of brain injury

detecting abnormal brain states

others ...

Example:α-wave (8-13 Hz) is reduced in children and in the elderly,and in patients with dementia, schizophrenia, stroke, and epilepsy


Features of signal morphology

Any feature fi is computed on sliding windows i (overlapped ornon-overlapped) across the EEG signal (of length nw ).

A feature can be normalized, fi := fi/∑n

j=1 fj , where n number ofwindows.If fi is measured in several (many) channels, fi ,j , for channelsj = 1, . . . ,K : take average at each time window, e.g.fi := 1

K

∑Kj=1 fi ,j .

Features of signal morphology:Indices (statistics) that capture some characteristic of the shape ofthe signal.

Descriptive statistics:

center (mean, median)

dispersion (variance, SD, interquartile range)

higher moments (skewness, kurtosis)



Any feature fi is computed on sliding windows i (overlapped ornon-overlapped) across the EEG signal (of length nw ).A feature can be normalized, fi := fi/

∑nj=1 fj , where n number of

windows.

If fi is measured in several (many) channels, fi ,j , for channelsj = 1, . . . ,K : take average at each time window, e.g.fi := 1

K

∑Kj=1 fi ,j .










windows.If fi is measured in several (many) channels, fi ,j , for channelsj = 1, . . . ,K : take average at each time window, e.g.fi := 1

K

∑Kj=1 fi ,j .











K

∑Kj=1 fi ,j .











K

∑Kj=1 fi ,j .







Hjorth parameters:

activity = variance = total power, Var(x(t)) =∑nf

i=0 PXX (fi ).

mobility =

√Var

(dx(t)

dt

)/Var(x(t)),

estimates the SD of PXX (f ) (along the frequency axis)

complexity = mobility(

dx(t)

dt

)/mobility(x(t)),

estimates similarity of the signal to a pure sine wave(deviation of complexity from one).

[Hjorth, Electroenceph. Clin.Neuroph., 1970]


Hjorth parameters:


i=0 PXX (fi ).

mobility =

√Var

(dx(t)

dt

)/Var(x(t)),



dx(t)

dt

)/mobility(x(t)),




Hjorth parameters:


i=0 PXX (fi ).

mobility =

√Var

(dx(t)

dt

)/Var(x(t)),



dx(t)

dt

)/mobility(x(t)),




Hjorth parameters:


i=0 PXX (fi ).

mobility =

√Var

(dx(t)

dt

)/Var(x(t)),



dx(t)

dt

)/mobility(x(t)),




Line Length: L(i) =∑nw−1

t=1 |x(t + 1)− x(t)|.Normalized line length: L(i) := L(i)/

∑nj=1 L(j).

Median over all channels: L(i) := medianL(i , j), j denotes channel.

[Koolen et al, Clin. Neuroph., 2014]

Nonlinear energy: NLE(i) =∑nw−1

t=2 |x(t)2 − x(t − 1)x(t + 1)|.


Line Length: L(i) =∑nw−1

t=1 |x(t + 1)− x(t)|.Normalized line length: L(i) := L(i)/

∑nj=1 L(j).

Median over all channels: L(i) := medianL(i , j), j denotes channel.

[Koolen et al, Clin. Neuroph., 2014]

Nonlinear energy: NLE(i) =∑nw−1

t=2 |x(t)2 − x(t − 1)x(t + 1)|.Kugiumtzis Dimitris Computational Neuroscience - Lecture 3

... and many other features of signal morphology:

Fisher information (from normalized spectrum of singular values)

Katz’s fractal dimension (from line length and largest distancefrom start)

Petrosian Fractal Dimension (PFD) (from number of sign changesin the signal derivative)

Hurst exponent (the Rescaled Range statistics (R/S))

Detrended Fluctuation Analysis (DFA) (long range correlationmeasure)


Features of linear analysis - frequency domain

Features from power spectrum PXX (fk), fk = 0, . . . , fs/2,

Power in bands:δ[0, 4]Hz, θ[4, 8]Hz, α[8, 13]Hz, β[13, 30]Hz, γ > 30Hz,

e.g. for α-band power: PXX (α) =∑fk=13

fk=8 PXX (fk)

or relative α-band power: RPXX (α) = PXX (α)/∑nf

i=0 PXX (fi )

In the same way the bands are defined on wavelets.

Spectral edge frequency (SEF), e.g. SEF(90) is the frequency at

90% of total power, f 90:∑f 90

f =0 PXX (f )/∑fs/2

f =0 PXX (f ) = 0.90

median frequency: SEF(50)






fk=8 PXX (fk)


i=0 PXX (fi )




f =0 PXX (f )/∑fs/2

f =0 PXX (f ) = 0.90







fk=8 PXX (fk)


i=0 PXX (fi )




f =0 PXX (f )/∑fs/2

f =0 PXX (f ) = 0.90







fk=8 PXX (fk)


i=0 PXX (fi )




f =0 PXX (f )/∑fs/2

f =0 PXX (f ) = 0.90







fk=8 PXX (fk)


i=0 PXX (fi )




f =0 PXX (f )/∑fs/2

f =0 PXX (f ) = 0.90



Example:5 datasets of 100 single-channel EEG of N = 4096 (fs = 173.61 Hz)

A: scalp EEG, healthy eyes open B: scalp EEG, healthy eyes closed

C, D: intracranial EEG, interictal period E: intracranial EEG, ictal

period http://epileptologie-bonn.de/cms/front_content.php?idcat=193&lang=3


Example:5 datasets of 100 single-channel EEG of N = 4096 (fs = 173.61 Hz)

A: scalp EEG, healthy eyes open B: scalp EEG, healthy eyes closed

C, D: intracranial EEG, interictal period E: intracranial EEG, ictal

period http://epileptologie-bonn.de/cms/front_content.php?idcat=193&lang=3


[Bao et al, Comput Intell Neurosci, 2011]


Features of linear analysis - time domain

Autocorrelation: rX (τ) = cX (τ)s2X

=∑n−τ

t=1 (xt−x̄)(xt+τ−x̄)∑n−τt=1 (xt−x̄)2

Assuming the linear stochastic processXt = φ0 + φ1Xt−1 + . . .+ φpXt−p + εtAutoregressive model:x̂t = φ̂0 + φ̂1xt−1 + . . .+ φ̂pxt−p

Mean square error of fit: MSE= 1n−p

∑nt=p+1(xt − x̂t)

2




=∑n−τ

t=1 (xt−x̄)(xt+τ−x̄)∑n−τt=1 (xt−x̄)2




2




=∑n−τ

t=1 (xt−x̄)(xt+τ−x̄)∑n−τt=1 (xt−x̄)2




2


Features of nonlinear analysis - 1Extension of autocorrelation to linear and nonlinear correlation

Entropy: information from each sample of X (assume properdiscretization of X )

H(X ) =∑x

pX (x) log pX (x)

Mutual information: information for Y knowing X and vice versa

I (X ,Y ) = H(X )+H(Y )−H(X ,Y ) =∑x ,y

pXY (x , y) logpXY (x , y)

pX (x)pY (y)

For X → Xt and Y → Xt+τ ,Delayed mutual information:

IX (τ) = I (Xt ,Xt+τ ) =∑

xt ,xt+τ

pXtXt+τ (xt , xt+τ ) logpXtXt+τ (xt , xt+τ )

pXt (xt)pXt+τ (xt+τ )

To compute IX (τ) make a partition of {xt}nt=1, a partition of{yt}nt=1 and compute probabilities for each cell from the relativefrequency.




H(X ) =∑x

pX (x) log pX (x)


I (X ,Y ) = H(X )+H(Y )−H(X ,Y ) =∑x ,y


pX (x)pY (y)


IX (τ) = I (Xt ,Xt+τ ) =∑

xt ,xt+τ







H(X ) =∑x

pX (x) log pX (x)


I (X ,Y ) = H(X )+H(Y )−H(X ,Y ) =∑x ,y


pX (x)pY (y)


IX (τ) = I (Xt ,Xt+τ ) =∑

xt ,xt+τ







H(X ) =∑x

pX (x) log pX (x)


I (X ,Y ) = H(X )+H(Y )−H(X ,Y ) =∑x ,y


pX (x)pY (y)


IX (τ) = I (Xt ,Xt+τ ) =∑

xt ,xt+τ







H(X ) =∑x

pX (x) log pX (x)


I (X ,Y ) = H(X )+H(Y )−H(X ,Y ) =∑x ,y


pX (x)pY (y)


IX (τ) = I (Xt ,Xt+τ ) =∑

xt ,xt+τ





Features of nonlinear analysis - 2Extension of linear autoregressive models to nonlinear models

All models require the state space reconstruction, points xt ∈ IRm

from scalar xt .Delay embedding:xt = [xt , xt−τ , . . . , xt−(m−1)τ ].

Nonlinear model: xt+1 = f (xt)

1 parametric models, e.g. polynomial autoregressive models

2 semilocal (black-box) models, e.g. neural networks

3 local models, e.g. nearest neighbor models.

For the prediction of xt+1 having x1, x2, . . . , xt .

Find the k nearest neighbors of xt , {xt(1), . . . , xt(k)}Fit a linear autoregressive model on the k neighbors.xt(i)+1 = φ0 + φ1xt(i) + · · ·+ φmxt(i)−(m−1)τ + εt+1

Predict x̂t+1 = φ̂0 + φ̂1xt + · · ·+ φ̂mxt−(m−1)τ



































Features of nonlinear analysis - 3

Entropy: Estimates of the entropy ofxt = [xt , xt−τ , . . . , xt−(m−1)τ ]:

H(X) =∑

x pX(x) log pX(x)

Approximate entropy, ApEn (uses a bandwidth in theestimation of pX(x))

Sample entropy (similar to ApEn)

Permutation entropy (entropy on ranks of the components inxt)

Spectral entropy (entropy on PXX (f ))

Others, e.g. fuzzy entropy, multiscale entropy.




H(X) =∑

x pX(x) log pX(x)









H(X) =∑

x pX(x) log pX(x)









H(X) =∑

x pX(x) log pX(x)









H(X) =∑

x pX(x) log pX(x)









H(X) =∑

x pX(x) log pX(x)








Dimension and Complexity:

correlation dimension, estimates the fractal dimension ofxt = [xt , xt−τ , . . . , xt−(m−1)τ ] (assuming chaos):d(xi , xj): distance of two points in IRm

scaling of probability of distance with r , p(d(xi , xj) < r) ∝ rν ,e.g.ν: the dimension, estimated by the slope of log p(r) vs log r .

Higuchi dimension (distance is defined in a different way).

Lempel-Ziv algorithmic complexity (turning the signal to aseries of symbols by discretization and searching for newsymbol patterns in the series).

Source: [Kugiumtzis et al, Int. J of Bioelectromagnetism, 2007]

[Kugiumtzis and Tsimpiris, J. Stat. Softw., 2010], MATS module in Matlab






































Example: 5 datasets of healthy extracranial EEG (A,B) andintracranial interictal (C,D) and ictal (E) EEG

[Bao et al, Comput IntellNeurosci, 2011]


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Computational Neuroscience - Lecture 3users.auth.gr/dkugiu/Teach/ComputationalNeuroscience/Lecture3.pdf · Computational Neuroscience - Lecture 3 Kugiumtzis Dimitris Department of