Functional Brain Signal Processing: EEG & fMRI

Lesson 5

Kaushik Majumdar

Indian Statistical Institute Bangalore Center

kmajumdar@isibang.ac.in

M.Tech. (CS), Semester III, Course B50

Some Important ERPs

N100: A negative deflection peaking between 90 and 200 msec after the onset of stimulus, is observed when an unexpected stimulus is presented. It is an orienting response or a “matching process,” that is, whenever a stimulus is presented, it is matched with previously experienced stimuli. It has maximum amplitude over Cz and is therefore also called “vertex potential.”

Sur & Sinha, Ind. J. Psychiatr., 18(1): 70 – 73, 2009; http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3016705/

Important ERPs (cont.)

P300: The P3 wave was discovered by Sutton et al. in 1965 and since then has been the major component of research in the field of ERP. For auditory stimuli, the latency range is 250-400 msec for most adult subjects between 20 and 70 years of age. The latency is usually interpreted as the speed of stimulus classification resulting from discrimination of one event from another.

Sur & Sinha, Ind. J. Psychiatr., 18(1): 70 – 73, 2009; http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3016705/

Important ERPs (cont.)

Shorter latencies indicate superior mental performance relative to longer latencies. P3 amplitude seems to reflect stimulus information such that greater attention produces larger P3 waves. A wide variety of paradigms have been used to elicit the P300, of which the “oddball” paradigm is the most utilized where different stimuli are presented in a series such that one of them occurs relatively infrequently — that is the oddball.

Sur & Sinha, Ind. J. Psychiatr., 18(1): 70 – 73, 2009; http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3016705/

Important ERPs (cont.)

The subject is instructed to respond to the infrequent or target stimulus and not to the frequently presented or standard stimulus. Reduced P300 amplitude is an indicator of the broad neurobiological vulnerability that underlies disorders within the externalizing spectrum {alcohol dependence, drug dependence, nicotine dependence, conduct disorder and adult antisocial behavior} (Patrick et al., 2006).

Sur & Sinha, Ind. J. Psychiatr., 18(1): 70 – 73, 2009; http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3016705/

ERP Plot by Time Frequency Diagram

http://pubs.niaaa.nih.gov/publications/arh313/238-242.htm

Spectral Estimation

Spectral estimation can be achieved by Fourier transform or by autoregressive models (there are other methods also, see Chapter 14 of Digital Signal Processing: Principles, Algorithms & Applications, 4th ed., by J. G. Proakis and D. J. Manolakis, Pearson, 2007).

Spectral Estimation by Fourier Transform (Nonparametric)

( ) exp( 2 ) n nx t j nt dt a jb

lim ( )exp( 2 ) ( )M

n nMm M

x m j nmW m a jb

Continuous form

Discrete form

2 2n na b is the power associated with frequency n.

( ) exp( 2 ) ( ) n nx t j nt W t dt a jb

W() is a window function

Welch window

http://paulbourke.net/miscellaneous/windows/

2

2( ) 1

2

Mm

W mM

Limitations

Long data length required. Spectral leakage masks weak signals. Poor frequency resolution. Signal has to be stationary (statistical

properties do not change over time).

Gaussian (White) Noise

A Gaussian noise signal is with 0 mean. It is wide sense stationary, that is if the

digitized Gaussian white noise signal is ω(n) then E[ω(n)ω*(n + k)] = 0 for all k > 0 (k is nonnegative).

In addition it can be made to have unit variance.

Spectral Estimation by Autoreg-ressive Moving Average (ARMA) (Parametric)

Has following advantages: Suitable for short data length. Gives better frequency resolution. Avoids spectral leakage.

ARMA (cont.)

0

1

( )( )

( )1

qk

kk

pk

kk

b zB z

H zA z

a z

H(z) is the system function of ARMA(p,q), where (p,q) is the model order.

1 0

( ) ( ) ( )p q

k kk k

y n a y n k b x n k

Corresponding difference equation with y(n) output and x(n) input.

Autoregressive (AR)

Moving average (MA)

(1)

ARMA (cont.)

2( ) ( )xx xm m For a Gaussian white noise signal, is variance of the noise x and is Dirac delta function.

2x

1( ) ( ) ( ) exp( 2 )

1 2

M M

xxk M n M

m x n x n k j mnM

222 2

2

( )( ) ( )

( )xx

B ff H f

A f

For the power spectrum of a general signal x, where ω is a Gaussian white noise signal.

2( ) ( ) ( )xx f H f f

Proakis and Manolakis, 2007, p. 987

Autocorrelation & Crosscorrelation

Gray et al., Nature, 338: 334 – 337, 23 March 1989

ARMA (cont.)

In an ARMA model if A(z) = 1 then H(z) = B(z) and the model reduces to moving average (MA) process of order q.

In an ARMA model if B(z) = 1 then H(z) = 1/A(z) and the model reduces to autoregressive (AR) process of order p.

Wold’s Decomposition Theorem

Any ARMA or MA process can be represented uniquely by an AR model of possibly infinite order.

Any ARMA or AR process can be represented by an MA model of possibly infinite order.

Proakis and Manolakis, 2007, p. 987

Model Parameters of ARMA Given by Yule-Walker Equations

1

2

( ) ( 1) . . ( 1) ( 1)

( 1) ( ) . . ( 2) ( 2)

.. . . . . .

.. . . . . .

( 1) ( 2) . . ( ) ( )

xx xx xx xx

xx xx xx xx

pxx xx xx xx

aq q q p q

aq q q p q

ap q q p q q p

Here m > q. m is the frequency at which power spectrum is to be determined.

Yule-Walker Equations (cont.)

1

2

1 0

( )

( ) ( ) ( ) 0

( ) 0

p

k xxk

p q m

xx k xx k mk k

xx

a m k m q

m a m k h k b q m

m m

Yule-Walker equations give the model parameters for the AR part in the ARMA model.

Model Parameters for the MA Part

2

1

0

( ) 0

( ) 0

p

k k mk

xx

xx

b b q m

m m q

m m

2

1

(0) ( )p

xx k xxk

a k

Proakis and Manolakis, 2007, p. 990

Derivation of Yule-Walker Equations

When the power spectral density of the stationary random process is a rational function, there is a basic relationship between the autocorrelation sequence

and the model parameters and . This is given by Yule-Walker equations.

{ ( )}xx m

ka kb

Proakis and Manolakis, 2007, p. 837

Derivation (cont.)

ARMA difference equation (1) for input as white Gaussian noise and output x is

Multiplying both sides by x*(n - m) and taking expectation we get

1 0

( ) ( ) ( )p q

k kk k

x n a x n k b n k

1 0

[ ( ) ( )] [ ( ) ( )] [ ( ) ( )]p q

k kk k

E x n x n m a E x n k x n m b E n k x n m

Derivation (cont.)

1 0

( ) ( ) ( )p q

xx k xx k xk k

m a m k b m k

1 1

2

( ) [ ( ) ( )]

( ) ( ) ( ) ( ) ( )

( )

x

p p

k k

m E x n n m

E h k n k n m E h k m k

h m

While

(2)

Proakis and Manolakis, 2007, p. 837

Derivation (cont.)

Since ω is white noise, in the last step we have used

2

0 0( )

( ) 0x

mm

h m m

(3)

Combining (2) and (3) we get

Derivation (cont.)

1

2

1 0

( )

( ) ( ) ( ) 0

( ) 0

p

k xxk

p q m

xx k xx k mk k

xx

a m k m q

m a m k h k b m q

m m

Probability Density Function Estimation

Probability density function estimation of a one dimensional data set is very similar to power spectrum estimation. The only difference is that for a PDF the integral over the whole space will have to be unity, whereas power spectrum estimation does not have any such constraint.

See for detail “Model-based probability density function estimation,” S. Kay, IEEE Sig. Proc. Lett., 5(12): 318 – 320, 1998.

References

Digital Signal Processing: Principles, Algorithms & Applications, 4th ed., J. G. Proakis and D. J. Manolakis, Pearson, 2007. Chapter 14.

Modern Spectral Estimation: Theory & Application, S. M. Kay, Pearson, 1988, for a general perusal of nonparametric spectral estimation and probability distribution function estimation methods beyond Yule-Walker equations.

THANK YOU

This lecture is available at http://www.isibang.ac.in/~kaushik