Fractal analysis of resting state functional connectivity of the brain
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The document discusses fractal analysis of resting state functional connectivity in the brain, explaining how fractal behavior is observed in both time and frequency domains. It critiques traditional models of fractal behavior and proposes a fractionally integrated process model to better estimate functional connectivity of neuronal populations. The study concludes by introducing nonfractal connectivity as a novel measure of resting state functional connectivity and suggests future research directions.
Fractal analysis of resting state functional connectivity of the brain
- 1. 1 Fractal analysis of resting state functional connectivity of the brain Wonsang You Special Lab Non-Invasive Brain Imaging Leibniz Institute for Neurobiology, Magdeburg, Germany
- 2. 2Evoked-state vs. Resting-state Evoked-state brain Resting-state brain Regions whose response is correlated with stimulation Stochastic behavior High power in low frequencies How to describe the phenomenon? - The answer is Fractal. Great power in low frequencies Power Frequency Michael et al. (2007) Nature Review Neuroscience 8
- 3. 3What is Fractal? Self-similarity and Scale-invariance Ubiquitously found in nature Tree Leaves Flow
- 4. 4Fractal Behavior in Resting-state Signals Time Domain Time shift Slowly-decaying autocorrelation Self-correlation Shifted time Correlation Self-similarity Frequency Domain 1/f-type power spectrumLogPower Great power in low frequencies Log-scale Special order of scaling Log frequency Power Frequency Additive noise Fractal Properties Maxim (2005) NeuroImage
- 5. 5Traditional Models for Fractal Behavior Autoregressive process: AR(p) Bullmore et al. (1996) Complex model: requires too many parameters. Fractional Gaussian noise (FGN) Maxim et al. (2005) tested for Alzheimers disease Simple model: represented by variance and Hurst exponent 1 p t i t i t i X c X ワ 2 2 2 2 ( ) 1 2 1 2 H H H c 器 ( )c Autocorrelation Since the brain is a complicated nonlinear biological system, it is questionable to apply the simple model such as FGN.
- 6. 6Fractionally Integrated Process (FIP) Model Long memory filter Nonfractal Input Long memory Signal tU 1 1 t d d t G d t t t tX U G 2 ( ) 1 ( ) dif uS f e S f Spectral density of FIP 1 d t tU L X L : backshift operator 0.5,0 0 0,0.5 d d d 緒 Short memory White noise Long memory Memory parameter d Controlling Fractal behavior Nonfractal part
- 7. 7Fractal-driven Distortion of Connectivity Fractional Gaussian noise (FGN) Fractionally integrated process (FIP) 1 2 1 2 , 1 2 1 2 , , cos 2 X X FIP U U d d d d 駈 1 2 1 2 , 1 2 , , 1 X X FGN U U d d 1 2,X X Correlations1 2,U U Negligible distortion in correlation Abrupt change depending on the difference of memory parameters
- 8. 8Resting-state Imaging to Neuronal Populations Long memory Artifacts Correlation of spontaneous neuronal populations Distortion by nonlinearity Resting state neuroimaging signals The FIP-based model allows us to estimate functional connectivity of neuronal populations from resting state signals.
- 9. 9Nonfractal Connectivity (1/2) Definition from a multivariate FIP Limitations Nonfractal components do not always represent spontaneous neuronal populations due to nonlinearity of biological system and additive noises. 1 1 1(1 ) 0 0 (1 ) q d d q q L X t U t X t U tL 駈 駈 件 緒 件 件 件 醐 O M M 1 2, 1 2corr ,U U U U Nonfractal Connectivity Correlation of nonfractal components
- 10. 10Nonfractal Connectivity (2/2) Correlation of neuronal populations fMRI signals Fractal parameters Correlation of non-fractal components Difference caused by nonlinearity and noises Nonfractal Connectivity
- 11. 11Radio Channels and Fractal Behavior 75 MHz 107 MHz Fractal Par. 0.3 FP 0.3 FP 0.7Fractal Par. 0.7 Neural Correlation 0.7 fMRI Correlation 0.3 Nonfractal Connectivity 0.67 75 MHz 107 MHz
- 12. 12Fractal Connectivity Wavelet covariance and correlation Fractal connectivity Scale-invariance: Wavelet correlation of long memory processes converges to a constant in low frequencies. 袖 $ $ $ 1 2 1 2 1 2 , , X X X X X X j j j j $ 1 2 1 2 ( ) ( ) , , , 1 1 2 jn X X X X j k j kj kj j W W n Wavelets 1X 2X 1( )X W 2( )X W Wavelet Covariance Wavelet Correlation j 袖 1 2,X X c Convergence of wavelet correlations is called Fractal Connectivity.
- 13. 13Estimation of Nonfractal/fractal connectivity Estimation of memory parameter fMRI Signals Nonfractal connectivity Estimation of short memory covariance matrix Fractal connectivity Wavelet maximum likelihood (ML) Wavelet least-mean-squares (LMS) SDF-based method (SDF) Covariance-based method (COV) Linearity-based method (LIN) We have six estimators by combining the above methods. (i.e. ML-SDF, LMS-LIN, )
- 14. 14Comparison of nonfractal connectivity estimators Short memory correlation ML-LIN ML-COV ML-SDF
- 15. 15Application on Anesthetized Rat Brain 1. aCG 2. CPu-L 3. CPu-R 4. MEnt+MEntV-L 5. MEnt+MEntV-R 6. HIP-L 7. HIP-R 8. S1-L 9. S1-R 10. S2-L 11. S2-R 12. LSI-MS 13. TE-L 14. TE-R 15. TH
- 16. 16Functional Connectivity Matrix Nonfractal Connectivity Pearson Correlation aCG CPu-L CPu-R MEnt-L MEnt-R HIP-L HIP-R S1-L S1-R S2-L S2-R LSI-MS TE-L TE-R TH aCG CPu-L CPu-R MEnt-L MEnt-R HIP-L HIP-R S1-L S1-R S2-L S2-R LSI-MS TE-L TE-R TH 9 1 12 8 4 10 13 6 15 3 11 7 14 2 5 1 14 8 5 10 4 12 2 3 11 13 15 6 7 9
- 17. 17Conclusion Ordinary correlation of resting state fMRI Variable over time and over subjects due to fractal behavior Nonfractal connectivity It is not identical to correlation among spontaneous neuronal activities, but provides better information on it. Thus, we suggest nonfractal connectivity as a novel measure of resting state functional connectivity. Future Works Studying the effects of training on resting state functional connectivity Modeling the resting state functional network of the human brain
- 18. 18Thanks to Dr. Andr辿 Brechmann (LIN, Germany) Dr. J旦rg Stadler (LIN, Germany) Dr. Frank Angenstein (LIN, Germany) Dr. Bernd Br端ckner (LIN, Germany) Prof. Dr. Jan Beran (University of Konstanz, Germany) Dr. Sophie Achard (CNRS, France) Prof. Dr. Udo Seiffert (Fraunhofer, Germany) All audiences