Wavelet transformation

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

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.

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

Single subject analyses of M400 old/new effects Clear 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 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)

Single-epoch wavelet transforms Spectral analysis x Wavelet averaging

Wavelettransformation ERP + Phase

Different morlet wavelets Better time resolution Good compromise Better freq. resolution

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 67hz 40hz beta alpha theta delta -500 +500

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 demo • Create a wavelet wavelet_beta=sin(2*pi*t/50).*exp(-(t/50/strecth).^2)

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

Frequency resolution Time resolution

Wavelet transformation