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Wavelet transformation. Emrah Duzel Institute of Cognitive Neuroscience UCL. Why analyse neural oscillations?. Temporal code of information processing (versus rate code) Functional coupling Interareal synchrony Local field potentials and their correlation with fMRI

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wavelet transformation
Wavelet transformation

Emrah Duzel

Institute of Cognitive Neuroscience

UCL

slide2

Why analyse neural oscillations?

  • Temporal code of information processing (versus rate code)
  • Functional coupling
    • Interareal synchrony
  • Local field potentials and their correlation with fMRI
  • Functional specificity of oscillations
slide3

Makeig et al., 2004

Large scale neural dynamics of higher cognitive processes:

At least three types of stimulus-responses

  • Evoked response:the addition of response amplitude to the ongoing brain activity in a time-locked manner.
  • Schah et al., 2004, Cereb Cortex
  • Phase resetting response:the resetting of ongoing oscillatory brain activity without concomitant changes in response amplitude.
  • Penny, Kiebel, Kilner, Rugg, 2002, Trends in Cog Sci. / Makeig et al., 2002, Science
  • Induced response:the addition of response amplitude that is not time-locked to stimulus onset.
  • Tallon-Baudry and Bertrand, 1998, Trends in Cog Sci.
slide4

1

0

Phase resetting

8 trials

Phase-resetting of a 10 Hz oscillation

ERP power

Penny, Kiebel, Kilner, Rugg, 2002, Trends in Cog Sci. / Makeig et al., 2002, Science / Klimesh et al., 2001, Cog Brain Res. / Burgess and Gruzelier, 2000, Psychophys.

Measure of phase alignment

slide5

Single subject analyses of M400 old/new effects

Clear evidence of evoked responses in some subjects

overview
Overview
  • Basics of digital signal processing
    • Sampling theory
  • Fourier Transforms
    • Discrete Fourier Transforms
  • Wavelet Analysis
  • Applications and online demonstrations
digital signal processing
Digital signal processing
  • Decompose a signal into simple additive components
  • Process these components in a useful manner
  • Synthesize them into a final result
sampling theory
Sampling theory
  • Nyquist theorem
  • Sample rate
  • Nyquist frequency
  • Aliasing
  • With each signal there are 4 critical parameters:
    • Highest frequency in the signal (determined by low-pass filter)
    • Twice this frequency
    • Sampling rate
    • SR / 2 (nyquist frequency/rate)
slide9
Sampling theoryNyquist theorem: a signal can be properly sampeld only if it does not contain frequencies above ½ sampling frequency
  • Aliasing: if a signal contains frequencies above the Nyquist frequency.
    • Loss of information
    • Introduces wrong information (waves take on different ‚identities‘
    • Loss of phase information (phase shift)
slide10

Single-epoch wavelet transforms

Spectral

analysis

x

Wavelet

averaging

slide12

Different morlet wavelets

Better time resolution

Good compromise

Better freq. resolution

matlab demo
Matlab demo
  • Create an artificial signal composed of several frequencies of varying time/amplitude modulation
    • continuous delta [2Hz]
    • continuous alpha [10 Hz]
    • continuous beta [20Hz]
    • theta-burst [5Hz, +200 ms]
    • gamma_burst [40 Hz, -200]
    • gamma_burst [67 Hz, -100]
    • gamma_burst [67 Hz, +200]
  • Create a wavelet
  • Convolve wavelets and signal
    • highlight the issue of amplitude normalization
    • highlight limits of time/frequency resolution
  • Plot a time/frequency spectrogramm
  • Illustrate phase resetting

67hz

40hz

beta

alpha

theta

delta

-500

+500

matlab demo1
Matlab demo
  • Create an artificial signal composed of a linear combination of several sinusoids with different frequencies and time/amplitude modulations

angular frequency

A*sin(2 pi ω t)

  • where
    • ω is the angular frequency or angular speed (measured in radians per second),
    • T is the period (measured in seconds),
    • f is the frequency (measured in hertz)
    • e.g. if T = 50 ms = 0.05 sec
    • then f = 1/0.05 = 20 Hz

delta=sin(2*pi*1/500*(t))

t=-500:500

matlab demo2
Matlab demo
  • Create a wavelet

wavelet_beta=sin(2*pi*t/50).*exp(-(t/50/strecth).^2)

complex numbers
Complex numbers

In polar notation

each point zis determined by an

angle φand a distance r

In a Cartesian coordinate system

each point z is determined by two axes

central point

is ‘pole’

r

trigonometric form

exponential form

Euler’s formula

r is called the absolute value or modulus of z

slide21

Frequency resolution

Time resolution