Fractal analysis of fMRI data P. Ciuciu 1,2 1: CEA/NeuroSpin/LNAO 2: IFR49

2 /23 12/17/2009 Outline I. Introduction II. Analysis of scale invariance in fMRI time series III. Fractal connectivity IV. Conclusions

3 /23 12/17/2009 The BOLD signal at rest Low frequency content of the resting BOLD signal: ➢ Physiological (cardio-respiratory cycles) artifacts ➢ Direct consequence of neocortical neuronal ongoing activity ➢ Vascular processes? ➢ Power spectrum at rest exhibits a 1/f law ➢ Does a long range-coherence in this activity reflect functional connectivity? [Biswal et al, MRM, 1995; Lowe et al, NIM, 1998] [Thurner et al, Phys. A 2003; Shumizu et al, NIM 2004; Fadili et al, NIM 2002; Ciuciu et al, JSTSP 2008] [Lowe et al, NIM 1998; Xiong et al, HBM 1999; Cordes et al, AJNR 2000] [De Luca, NIM 2006; Goldman et al, NR 2002; Leopold CC, 2003; Laufs et al, NIM 2003] [Kiviniemi et al, MRM 2000; Wise, NIM 2004 ]

4 /23 12/17/2009 Functional connectivity Study of resting state networks (RSNs) ➢ Multivariate exploratory analyses: PCA, spatial ICA, or other transformation ➢ Analysis based on measurements (EEG, fMRI signals) ➢ Lack of a coherent statistical framework ➢ Does brain dynamics reflect SOC, stochastic or chaotic systems? [Beckmann&Smith, IEEE TMI, 2004; De Luca et al, NIM 2006; Shimizu et al, NIM 2004; Perlbarg et al, ISBI'08; Varoquaux, MICCAI-WS09, sub. To NIM] [Werner, J Phys Paris 2008; Bedard and Destexhe, 2008, Destexhe and Contreras, Science 2006; Bedard et al, Phys. Rev Let, 2006 ;Piękniewski&Schreiber, NN in press] Need to probe scale invariance and to work on deconvolved neuronal signatures in fMRI or on reconstructed sources in MEG/EEG

5 /23 12/17/2009 Beyond classical analyses ➢ Statistical issues ➢ GLM-based framework no longer valid: loss of independence between “signal” and “noise” Long-memory correlation structure affects estimator performance ➢ Modulation of the scaling properties with stimulation ➢ Relative deactivations of RSNs during tasks [ Marre et al, 2008; Shimizu et al, NIM 2004; Ciuciu et al, JSTSP, 2008] [ Kincses, 2008] [ Abry et al, IEEE IT 2002] Need to define a proper statistical framework

6 /23 12/17/2009 Outline I. Introduction II. Analysis of scale invariance in fMRI time series III. Fractal connectivity: a novel functional connectivity test IV. Conclusions and perspectives

7 /23 12/17/2009 Scale invariance Evidence: Covariance under dilation operation Covariance under a change of scale The subpart and the whole are statistically indistinguishable No characteristic scale of time Implications: Non stationarity Long range dependence [Abry, et al, Lois d’échelles, fractales et ondelettes. Traité IC2, Lavoisier 2002]

8 /23 12/17/2009 Models of scale invariance Self-similarity: Multifractality:

9 /23 12/17/2009 Self-similarity and wavelets

10 /23 12/17/2009 High temporal resolution fMRI BOLD impulse response ~ 20 s Current EPI acquisitions in fMRI: Time of repetition ~ 1 or 2 s. EPI 2D Acquisition, TR = 2s (s) First high temporal resolution fMRI results obtained with a new rapid imaging method: localized EVI parallel sequence (s) EVI 3D Acquisition, TR = 200 ms robust to motion artifacts [Rabrait, Ciuciu et al, JMRI 2008]

11 /23 12/17/2009 Voxelwise Multifractal analysis Log scale diagrams based on Wavelet-leaders Time scale (s) Clear scale invariance from 1.5 to 15 s. [Ciuciu et al, ISBI’07]

12 /23 12/17/2009 Voxelwise Multifractal spectrum Quantify the reproductibility of each scaling exponent Evidence for a multifractal behavior [Ciuciu et al, IEEE JSTSP 2008]

13 /23 12/17/2009 Voxelwise Multifractal spectrum Influence of the activation level Modulation of MF properties between task-related TS, residuals and resting state signal [Ciuciu et al, IEEE JSTSP 2008] Strongly activated voxel Weakly activated voxel

14 /23 12/17/2009 Ongoing vs. evoked activity Estimation of first order cumulant of [Ciuciu et al, IEEE JSTSP 2008] Rest Activ Region 1Region 2 Region 3

15 /23 12/17/2009 Ongoing vs. evoked activity Estimation of second order cumulant of RestActivRestActiv RestActiv Region 1Region 2 Region 3 [Ciuciu et al, IEEE JSTSP 2008]

16 /23 12/17/2009 Summary Multifractal analysis exhibits long memory in EVI fMRI data Multifractal analysis makes feasible the comparison of ongoing (resting state) and evoked activity Evidence of change in WL-based cumulants with activation ➢ Activation induces an increase in self-similarity ➢ Activation induces a decrease in multifractality (lower irregularity)

17 /23 12/17/2009 Outline I. Introduction II. Analysis of scale invariance in fMRI time series III. Fractal connectivity IV. Conclusions and perspectives

18 /23 12/17/2009 Fractal connectivity Fractal connectivity: Particular model where the LFs of the interspectrum of each pair of process components are determined by the autospectra of the components Adequate model for exhibiting a complex network organization (eg RSN) from the LF signatures of its components

19 /23 12/17/2009 Bivariate fractal connectivity Bivariate long memory model: [Achard et al, Phys Rev E 2008] Long memory process: Coherence function:

20 /23 12/17/2009 Bivariate fractal connectivity [Wendt et al, sub to ICASSP'09] Fractal connectivity: Intuition: ➢ A single mechanism in the system uniquely controls the independent and joint properties of the multivariate dataset Test in the wavelet domain:

21 /23 12/17/2009 Testing fractal connectivity Fisher Z statistic of : : Test of fractal connectivity Statistic for the UMPI test of Eq. of means of GRVs [Wendt et al, sub to ICASSP'09]

22 /23 12/17/2009 Outline I. Introduction II. Analysis of scale invariance in fMRI time series III. Fractal connectivity IV. Conclusions and perspectives

23 /23 12/17/2009 Conclusions & Perspectives Exploratory analysis for exhibiting RSNs ➢ Extend univariate analysis to a multivariate perspective Comparison of evoked and ongoing activity ➢ Modulation of scale-invariance properties ➢ Identify inhibition/excitation mechanisms Fractal connectivity: ➢ Analysis of functional connectivity in the original space ➢ Analysis from the reconstructed neuronal time series Functional connectivity and plasticity mechanisms

24 /23 12/17/2009 Multifractality and wavelets Scaling exponents: Estimation of MF quantities with confidence intervals [Wendt et al, IEEE SP 2007]

25 /23 12/17/2009 Introduction Low frequency content of the resting BOLD signal: ➢ Physiological (cardio-respiratory cycles) artifacts ➢ Direct consequence of neocortical neuronal ongoing activity ➢ Long range-coherence in this activity reflects functional connectivity Study of resting state networks (RSNs) ➢ Exploratory analyses: PCA, ICA, or space transformation ➢ Link with a statistical framework Consequences ➢ Modulation of the LF content with evoked activity? ➢ Statistical issues: GLM-based approaches no longer valid [Thurner et al, Phys. A 2003; Shumizu et al, NIM 2004; De Luca, NIM 2006] [Biswal et al, MRM, 1995; Lowe et al, NIM, 1998]