Signal Processing for Magnetoencephalography Rupert Benjamin Clarke Submitted for the degree of Doctor of Philosophy University of York Department of Electronics September 2010 Abstract Magnetoencephalography (MEG) is a non-invasive technology for imaging human brain function. Contemporary methods of analysing MEG data include dipole fitting, minimum norm estimation (MNE) and beamforming. These are concerned with localising brain activity, but in isolation they do not provide concrete evidence of interaction among brain regions. Since cognitive neuroscience demands answers to this type of question, a novel signal processing framework has been developed consisting of three stages. The first stage uses conventional MNE to separate a small number of underlying source signals from a large data set. The second stage is a novel time-frequency analysis consisting of a recursive filter bank. Finally, the filtered outputs from different brain regions are compared using a unique partial cross-correlation analysis that accounts for propagation time. The output from this final stage could be used to construct conditional independence graphs depicting the internal networks of the brain. In the second processing stage, a complementary pair of high- and low-pass filters is iteratively applied to a discrete time series. The low-pass output is critically sampled at each stage, which both removes redundant information and effectively scales the filter coefficients in time. The approach is similar to the Fast Wavelet Transform (FWT), but features a more sophisticated resampling step. This, in combination with the filter design procedure leads to a finer frequency resolution than the FWT. The subsequent correlation analysis is unusual in that a latency estimation procedure is included to establish the probable transmission delays between regions of interest. This test statistic does not follow the same distribution as a conventional correlation measures, so an empirical model has been developed to facilitate hypothesis testing. 3 Contents List of Figures 7 Acknowledgements 9 1 Introduction 10 1.1 Chapter Overview . 12 2 Background 13 2.1 The Human Brain . 13 2.1.1 Macroscopic View . 13 2.1.2 Microscopic View . 17 2.2 Magnetoencephalography . 21 2.2.1 History . 21 2.2.2 Instrumentation and Operation . 23 2.2.3 Alternatives . 27 2.2.4 Comparison . 32 2.3 MEG Analysis . 34 2.3.1 Classic Model . 35 2.3.2 Minimum Norm Estimation . 37 2.3.3 Beamforming . 39 2.3.4 Model Improvements . 40 2.3.5 Discussion . 42 2.4 Summary . 43 Contents 4 3 Concepts 45 3.1 Realistic Brain Models . 45 3.2 Connectivity . 46 3.3 Combined Signal Processing Framework . 48 3.3.1 Summary . 51 4 Minimum Norm Estimation 52 4.1 Definition . 52 4.2 Solution . 55 4.3 Regularisation . 56 4.3.1 Regularisation Parameter Selection . 58 4.4 MNE In Practice . 62 4.4.1 Source Space . 63 4.4.2 Depth Weighting . 64 4.5 Examples . 65 4.5.1 Introduction . 66 4.5.2 Method . 66 4.5.3 Results . 68 4.5.4 Conclusions . 68 5 Time-frequency Analysis 71 5.1 Relevance to MEG . 71 5.2 Short-time Fourier Transform . 75 5.2.1 Discrete Fourier Transform . 75 5.2.2 Definition of STFT . 79 5.2.3 Discussion . 81 5.3 Wavelet Analysis . 83 5.3.1 Continuous Wavelet Transform . 84 5.3.2 Discrete Wavelet Transform . 86 Contents 5 5.3.3 Multiresolution analysis . 88 5.3.4 Discussion . 89 5.4 A Novel Filter-bank Analysis . 90 5.4.1 Time-domain View . 91 5.4.2 Frequency-domain View . 98 5.5 Summary . 116 6 Statistical Framework 119 6.1 Theory . 119 6.1.1 Correlation . 120 6.1.2 Partial Correlation . 121 6.1.3 Cross Correlation . 122 6.2 A Test Statistic . 123 6.2.1 Latency Estimate . 125 6.2.2 Null Model . 125 6.3 Higher order statistics . 129 6.4 Application . 135 6.4.1 Introduction . 135 6.4.2 Methods . 136 6.4.3 Results . 146 6.4.4 Discussion . 146 6.5 Summary . 150 7 Summary and Further Work 152 7.1 Summary . 152 7.1.1 Aims . 152 7.1.2 MEG in the context of Neuroimaging . 152 7.1.3 MEG Analysis . 153 7.1.4 Time-Frequency Decomposition . 153 Contents 6 7.1.5 Connectivity Analysis . 154 7.2 Further Work . 155 7.2.1 Source Signal Estimation . 155 7.2.2 Time-frequency Decomposition . 155 7.2.3 Connectivity Analysis . 156 A Spherical Head Model 157 Abbreviations 162 References 165 7 List of Figures 2.1 Gross anatomy of the brain . 15 2.2 Lateral view of cerebrum . 16 2.3 General morphology of a neuron . 19 2.4 The first MEG measurement, made in 1971 . 23 2.5 MEG System at York Neuroimaging Centre . 24 2.6 Experimental current dipole . 36 3.1 Data flow diagram of MEG signal processing framework . 50 3.2 Conditional independence graph . 51 4.1 Representative diagram of an L-curve . 60 4.2 MNE solutions from simulated dipoles . 69 5.1 Time and frequency domain representations of a MEG signal . 73 5.2 Effect of windowing on DFT . 78 5.3 Spectral leakage in DFT due to rectangular windowing . 79 5.4 Sidebands in Fourier spectrum introduced by rectangular window . 80 5.5 Spectrogram of linear chirp . 82 5.6 Two common mother wavelets . 85 5.7 Multiresolution Analysis . 89 5.8 Hann window . 93 5.9 Sequence of raised cosine wavelets . 93 5.10 Zero D.C. by symmetry . 96 List of Figures 8 5.11 Fourier transform of raised cosine wavelets . 98 5.12 Transfer function of raised cosine wavelet . 99 5.13 Asymmetrical slopes of adjacent wavelet scales . 99 5.14 Impulse responses of ideal and windowed filters . 103 5.15 Illustration of filter design and action upon MEG signal . 107 5.16 Impulse response of filter bank . 112 5.17 Examples of filter bank output . 113 5.18 Reconstruction of input signal . 114 5.19 Short-time Fourier transform . 115 5.20 Change in power with respect to baselines . 116 6.1 Bootstrapped test statistic with noise input . 128 6.2 Trends in null distribution of test statistic vs. sample size . 130 6.3 Trends in null distribution of test statistic vs. sequence length . 131 6.4 Cross correlation with simulated data . 133 6.5 Removal of confounding influence using PCA . 134 6.6 Approximate locations of auditory and speech areas on inflated brain . 136 6.7 Location of chosen regions of interest . ..