Fractals, Wavelets, and the Brain

• Activity Number: 367
• Type: Invited
• Date/Time: Thursday, August 15, 2002 : 8:30 AM to 10:30 AM
• Sponsor: Biopharmaceutical Section*
• Author(s): Ed Bullmore*+
• Affiliation(s): University of Cambridge
• Keywords: fractal ; wavelet ; time series ; brain ; neuroimaging ; resampling

Abstract

Generally, the human brain demonstrates fractal, statistically self-similar, or 1/f-like spectral properties, both anatomically and physiologically. In functional magnetic resonance imaging (fMRI), specifically, the noise process is typically long memory with disproportionate power at low frequencies. The discrete wavelet transform (DWT) provides an appropriate and convenient basis for fMRI time series analysis, principally because the DWT is an optimal whitening filter for 1/f-like processes. Wavelet resampling exploits the decorrelating property of the DWT to justify exchangeability of the wavelet coefficients and thereby allows valid resampling of fMRI time series, which would be complicated by autocorrelation if attempted in the time domain [1]. Wavelet generalised least squares (WLS) involves transforming both data and design matrix to the wavelet domain before iterative ML estimation of regression model parameters and the Hurst exponent and variance of the errors. We show that WLS is the best linear unbiased estimator of regression models with long memory errors [2]. [1] Bullmore et al (2001) Hum Brain Mapp 12, 61-78; [2] Fadili & Bullmore (2002) NeuroImage 15, 217-232.