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

• Functional specificity of oscillations

Large scale neural dynamics of higher cognitive processes: At least three types of stimulus-responses

• Evoked response: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: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:Induced response: the addition of response amplitude that is not time-locked to stimulus onset.Tallon-Baudry and Bertrand, 1998, Trends in Cog Sci.

Makeig et al., 2004

8 trials Phase-resetting of a 10 Hz oscillation

Phase resetting

ERP power

10

Measure of phase alignment

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.

Single subject analyses of M400 old/new effectsClear evidence of evoked responses in some subjects

Overview

• Basics of digital signal processing– Sampling theory

• Fourier Transforms– Discrete Fourier Transforms

• Wavelet Analysis

• Applications and online demonstrations

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• 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)

Sampling theoryNyquist theoremNyquist theorem: a signal can be properly sampeld only if it does not contain

frequencies above ½ sampling frequency

• AliasingAliasing: 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)

Single-epoch wavelet transforms

x Spectral analysis

Wavelet

averaging

+Phase

ERP

Wavelettransformation

Different morlet wavelets

Better time resolution

Good compromise

Better freq. resolution

Time-frequency resolution of a standard Morlet-wavelet

Time-frequency resolution of a standard Morlet-wavelet

Time-frequency resolution of a standard Morlet-wavelet

convolution

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

-500 +500

67hz

40hz

theta

beta

alpha

delta

Matlab demo

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

whereω 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

angular frequency

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

A*sin(2 pi ω t)

Matlab demo

• Create a wavelet

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

Complex numbers

Euler’s formula

trigonometric form

exponential form

r

In a Cartesian coordinate systemeach point z is determined by two axes

In polar notation

each point z is determined by an angle φ and a distance r

central pointis ‘pole’

r is called the absolute value or modulus of z

Frequency resolution

Time resolution