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 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)

Nyquist theorem Sampling theory Nyquist 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

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 hz 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 where ω is the angular frequency or angular speed (measured in radians per second),radianssecond T is the period (measured in seconds),periodseconds f is the frequency (measured in hertz)frequencyhertz 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 system each point z is determined by two axes In polar notation each point z is determined by an angle φ and a distance r central point is ‘pole’ r is called the absolute value or modulus of z

Frequency resolution Time resolution