February 6, 2014 15:59 1450010
International Journal of Neural Systems, Vol. 24, No. 3 (2014) 1450010 (14 pages) c World Scientific Publishing Company DOI: 10.1142/S0129065714500105
SINGULAR SPECTRUM ANALYSIS AND ADAPTIVE FILTERING ENHANCE THE FUNCTIONAL CONNECTIVITY ANALYSIS OF RESTING STATE fMRI DATA
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
PAOLO PIAGGI∗,† Department of Energy and Systems Engineering University of Pisa, Largo Lucio Lazzarino Pisa, 56122, Italy [email protected] DANILO MENICUCCI† Institute of Clinical Physiology National Research Council (CNR) Via G. Moruzzi 1, Pisa, 56124, Italy [email protected] CLAUDIO GENTILI Department of Surgery Medical, Molecular and Critical Area Pathology University of Pisa, Via Bonanno Pisano 6 Pisa, 56100, Italy GIACOMO HANDJARAS Laboratory of Clinical Biochemistry and Molecular Biology University of Pisa Via Roma 55, Pisa, 56126, Italy ANGELO GEMIGNANI EXTREME Centre, Department of Surgery Medical, Molecular and Critical Area Pathology University of Pisa, Via S. Zeno 31 Pisa, 56123, Italy ALBERTO LANDI Department of Information Engineering University of Pisa, Largo Lucio Lazzarino Pisa, 56122, Italy [email protected] Accepted 8 November 2013 Published Online 10 December 2013 Sources of noise in resting-state fMRI experiments include instrumental and physiological noises, which need to be filtered before a functional connectivity analysis of brain regions is performed. These noisy components show autocorrelated and nonstationary properties that limit the efficacy of standard techniques (i.e. time filtering and general linear model). Herein we describe a novel approach based on the combination of singular spectrum analysis and adaptive filtering, which allows a greater noise reduction
∗ †
Corresponding author. These authors contributed equally to this work. 1450010-1
February 6, 2014 15:59 1450010
P. Piaggi et al.
and yields better connectivity estimates between regions at rest, providing a new feasible procedure to analyze fMRI data. Keywords: Adaptive filtering; functional magnetic resonance imaging; physiological noise; resting state; RV coefficient; singular spectrum analysis.
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
1.
Introduction
Functional Magnetic Resonance Imaging (fMRI) is commonly used to study the dynamics of brain activity.1–3 In particular, resting-state fMRI studies have demonstrated that the neural activity is indirectly reflected in amplitude changes of the Blood Oxygen Level-Dependent (BOLD) signal, which are prominent in the low-frequency band (LFB, f < 0.10 Hz)1,4 and show temporal synchronizations between spatially remote brain areas (the so-called functional connectivity 5 of the resting state networks). To uncover the underlying neural connectivity between regions, BOLD signal analysis mandatorily requires a pre-processing procedure for filtering several non-neural noisy components, whose power may exceed that of the neural activity. Indeed, non-neural sources of noise are the principal cause of flawed or false positive functional connectivity results.6 Generally, the noisy components of BOLD signal can be divided into two classes: random and physiological. The former class includes the components of instrumental noise such as those related to the instability of MRI scanner, which show a broad spectrum mainly focused in the LFB of BOLD signal. Differently, the latter class comprises the physiological processes not related to neural activity, e.g. the respiratory and cardiac processes, whose BOLD components show frequency peaks at ∼0.25 Hz7 and ∼1.00 Hz,4 respectively. In addition, the nonstationary nature of physiological processes such as the heart rate variability,8,9 slow variations of respiratory rate10 and changes of arterial carbon dioxide level,11 induces further artifactual components in a broad frequency interval that may overlap the LFB of BOLD neural correlates. In summary, both random and physiological components show frequencies peaks superimposed on an autocorrelated noise background spreading over the whole spectrum of BOLD signal. As the largest effects of non-neural noise are due to the cardiac and respiratory components, various
techniques have been extensively used for filtering the BOLD signal, depending on the frequency location of the noisy components. Accordingly, cardiac and respiratory components at higher frequencies (>0.10 Hz), along with hardware-related ultra-low frequency drifts ( 0.05). The results obtained with the novel procedure considering only active voxels by SSA may corroborate the hypothesis of lower (or even absent) coupling in WM where no neural activity is expected, as compared to GM regions that have been previously identified as part of functional resting state networks.1,7,12,40,41 These findings may also suggest that SSA identified only the voxels in GM regions that contributed to the neural coupling between homologous ROIs, resulting in higher RV coefficients than those obtained using the gold standard procedure on all available voxels. Nevertheless, the relative higher values of RV coefficients in brain ventricles indicated a residual connectivity between these non-neural regions that could be due to incomplete removal of physiological noise in the LFB. This hypothesis is supported by the fact that REF signal explained only half the variability of ventricles signals (on average, 48.6%), thus the noise could not be entirely removed by adaptive filtering. One possible solution to overcome this issue might be performing sequential stages of adaptive filtering using the first two or three PCs of ventricle signals. Accordingly, future studies should be conducted to investigate the feasibility of this method to achieve a better suppression of physiological noise in the LFB. The main limitation of this methodological study on BOLD signal processing is the acquisition of only four slices for brain volume, which limited the spatial extension of ROIs as well as the interpretation of results in terms of the underlying brain physiology. This was due to higher sampling rates and technical limitations of our MRI equipment. Yet, the choice of a lower TR was mandatory to avoid aliasing effects of cardiac components in the LFB, which in turn would have biased the results to an unknown extent. However, both the novel and the gold standard procedure were able to discriminate neural ROIs of GM from brain ventricles and WM. In fact, brain ventricles
were found to be characterized by higher levels of noise (possibly due to displacement of cerebrospinal fluid), while pair of WM regions resulted in the lowest functional connectivity as expected in non-neural regions. Despite the technical limitations that precluded a full brain coverage, our findings indicate that the gold standard procedure of BPF followed by GLM achieved consistent results and that the novel procedure of SSA followed by adaptive filtering improved the efficacy of standard techniques, overcoming their methodological limitations. 5.
Conclusion
The application of SSA to BOLD signals in the resting state led to the identification of active voxels by extracting slow-varying components, which showed significantly greater power against a null hypothesis of autocorrelated noise. In addition, the adaptive filtering of SSA-filtered signals achieved a greater reduction of noise in all brain regions. Finally, the whole procedure of SSA plus adaptive filtering led to a higher contrast between GM and WM in the functional connectivity analyses as compared to the gold standard procedure of BPF and GLM. These results suggest that the combination of SSA and adaptive filtering is a reasonable and convenient approach for removing the low-frequency fluctuations of BOLD signal due to physiological and instrumental noises of fMRI data. The usage of these advanced techniques substantially improved the results of the functional connectivity analysis, demonstrating the benefit of applying new framework of analysis for studying the brain connectivity.42–45 As a result, this novel procedure allowed to emphasize the connectivity estimates of functional networks in resting-state conditions. Further studies should be carried out to explore the feasibility of this procedure on datasets with whole brain coverage, with a greater number of subjects and on higher magnetic fields. Acknowledgments The authors wish to thank Dr. Paolo Allegrini and Dr. Andrea Piarulli for their useful comments on writing the manuscript. Appendix A The SSA algorithm consists of four steps A.1–A.4.
1450010-10
February 6, 2014 15:59 1450010
SSA and Adaptive Filtering for fMRI Data
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
A.1. Embedding step A trajectory matrix D was obtained from the meancentered BOLD signal S(t) of length N by using a time window of length W . Generally, the choice of window length — that can be set from 2 to N/246 — is a compromise between the number of signal portions and the frequency resolution. In the lack of a priori hypothesis, a rule of thumb indicates a value ranging from N/5 and N /3 in order to obtain a reliable estimate for oscillatory components.47 The ith column of D contained the samples of S(t) — from S(i) to S(i + W − 1). The number of the columns of D was equal to M = N − W + 1, hence D had a Hankel matrix structure of dimension W -by-M . From D, the lag-covariance matrix CV = (1/M ) · DDT of dimension W -by-W was derived, where T indicates the matrix transpose operation.
performed by a maximum likelihood algorithm using unbiased estimators.24 In a red noise context, the signal-to-noise separation is performed exploiting its autocorrelation properties in the SSA framework. To this aim, the analytic covariance matrix of red noise CN = c0 T was derived using the estimated noise parameters, where c0 is the noise variance and elements of Tij = γ |i−j| (i and j are time-lag indices).48 The theoretical EOFs of red noise were then obtained by diagonalizing CN as: T CN EN . ΛN = EN
(A.3)
Due to the analytic structure of the noise covariance matrix, the dominant frequencies of EN were regularly spaced in the spectrum and separated by 1/(W × Tc ), where Tc is the sampling time. The theoretical noise EOFs (columns of EN ) showed also a sinusoidal behavior in time.
A.2. Decomposition step The lag-covariance matrix CV was diagonalized as: ΛV =
EVT CV
EV ,
A.3.1. Projection onto noise EOFs
(A.1)
where ΛV was the data eigenvalue matrix with elements λV along the main diagonal in decreasing order of magnitude, while EV was the matrix of associated data eigenvectors (EOFs, columns of EV ). The dominant frequency of each EOF was estimated as that with the maximum spectral power in the frequency domain. A.3. Statistical test for EOFs We assumed that the BOLD signal S(t) was composed of oscillations at different frequencies over a background of autocorrelated noise. We considered an AR(1) model for red noise, which also allowed the case of uncorrelated white noise with a suitable choice of process parameters. The recursive equation that described the red noise model is: ut − U0 = γ(ut−1 − u0 ) + αzt ,
(A.2)
where u0 is the process mean, γ and α are the process parameters (i.e. lag-1 autocorrelation and variance of red noise, respectively) and zt is a Gaussian white noise with unit variance. In the case of γ = 0 (i.e. no autocorrelation), the model downgrades to a white noise model. The estimation of the red noise parameters (γ and α) from S(t) was
Under the assumptions of Gaussian noise distribution and sinusoidal EOFs, each diagonal element λN of ΛN had a χ2 distribution with υ = 3N/W degrees of freedom.24 These assumptions are valid for AR(1) processes and lead to the following distribution for the noise eigenvalues: χ2 (ν) . (A.4) ν From the 2.5th and 97.5th percentiles of these distributions, the 95% confidence interval was calculated for each λN . The data covariance matrix CV was then projected onto noise EOFs EN as: T CN EN ) λN ≈ (EN
T CV EN . ΛVN = EN
(A.5)
Assuming a null hypothesis of pure noise that have generated S(t), all diagonal elements λVN of ΛVN should lie within the noise confidence intervals of the related λN . Otherwise, the noise EOFs associated to the λVN being outside the corresponding confidence interval were considered not compatible with the noise model, thus indicated the presence of real oscillatory components at those frequency (Fig. A.1). A.3.2. Projection onto data EOFs Since the noise EOFs EN were not directly related to data EOFs EV , the analytic covariance matrix of
1450010-11
February 6, 2014 15:59 1450010
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
P. Piaggi et al.
Fig. A.1. Eigenspectrum based on noise EOFs in the time course of Fig. 2. The dominant frequency of each noise EOF (EN , x-axis) is plotted against the diagonal elements of ΛVN . Error bars represent noise eigenvalues with 95% confidence interval using the χ2 distribution. Significant elements of ΛVN greater than the 97.5th percentile of the corresponding noise eigenvalue λN are highlighted and identified as significant trend (blue), LFB (red) and cardiac (violet) components, according to their dominant frequency. The LFB (0.04–0.10 Hz) is highlighted in yellow; nonsignificant elements are drawn as black circles.
red noise CN was also projected onto data EOFs as: ΛNV = EVT CN EV .
(A.6)
Similarly to step A.3.1, confidence intervals of the diagonal elements λNV of ΛNV were derived using (A.4). Accordingly, all data eigenvalues λV greater than the corresponding 97.5th percentile of λNV were considered statistically significant from noise (Fig. A.2).
Fig. A.2. Eigenspectrum based on data EOFs in the time course of Fig. 2. The dominant frequency of each data EOF (EV , x-axis) is plotted against the data eigenvalues λV (y-axis). Error bars represent diagonal elements of ΛNV with 95% confidence interval using the χ2 distribution. Significant data eigenvalue λV greater than the 97.5th percentile of the corresponding diagonal elements of ΛNV are highlighted and identified as significant trend (blue), LFB (red) and cardiac (violet) components, according to their dominant frequency. The LFB (0.04–0.10 Hz) is highlighted in yellow; nonsignificant elements are drawn as black circles. From each significant data EOF whose dominant frequency was also found significant in the noise eigenspectrum (Fig. A.1), the corresponding RC was calculated. All RCs related to each band (i.e. trend, LFB and high frequencies) were summed for obtaining the extracted signals shown in Fig. 2 (blue, red and violet lines, respectively).
A.4. Reconstruction step The projection of original signal S(t) onto significant data EOFs (EV ∗ matrix) yielded to:
A.3.3. Significance of data EOFs
A∗V (t) =
We considered globally significant each data EOF of EV that satisfied both these conditions: (i) whose associated noise EOF of EN (i.e. with the same dominant frequency) had the corresponding λVN element greater than the 97.5th percentile of the distribution of the related noise eigenvalue λN (step A.3.1); (ii) whose corresponding data eigenvalue λV was greater than the 97.5th percentile of the distribution of the corresponding λNV element (step A.3.2). We named EV ∗ the matrix with only the significant data EOFs as columns.
W
S(t + j − 1) · EV∗ (j).
(A.7)
j=1
For each significant EOF, the corresponding reconstructed component (RC) matrix RV ∗ was obtained by: Rk∗ (t) =
Ut 1 A∗k (t − j + 1) · Ek∗ (j), Mt
(A.8)
j=Lt
where Mt , Lt and Ut are time-dependent parameters,46 necessary to manage the border effects of time window W . The SSA code written in MATLAB can be provided upon request to the corresponding authors.
1450010-12
February 6, 2014 15:59 1450010
SSA and Adaptive Filtering for fMRI Data
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
Appendix B The corrected RV coefficient was obtained by the following resampling procedure. Given the signal matrices of a ROIs pair with NL and NR voxels of which NaL and NaR active voxels (NaL < NL and NaR < NR ) in the left and in the right hemisphere, respectively, 10,000 pairs of surrogate signal matrices were generated to derive a distribution under the null hypothesis of random selection of voxels. In doing so, the signals in each surrogate matrix were drawn from the complete matrix of all voxels by using a sampling-without-replacement algorithm, so that each surrogate matrix contained NaL and NaR signals (instead of original NL and NR ) according to the related hemisphere. The RV coefficient was then calculated for each pair of surrogate matrices to obtain the distribution for a random selection of subsets of voxels. The corrected RV coefficient was then calculated as the mean value of the surrogate distribution.
9.
10.
11.
12.
13.
References 1. B. Biswal, F. Z. Yetkin, V. M. Haughton and J. S. Hyde, Functional connectivity in the motor cortex of resting human brain using echo-planar MRI, Magn. Reson. Med. 34(4) (1995) 537–541. 2. M. J. Lowe, B. J. Mock and J. A. Sorenson, Functional connectivity in single and multislice echoplanar imaging using resting-state fluctuations, Neuroimage 7(2) (1998) 119–132. 3. J. Xiong, L. M. Parsons, J. H. Gao and P. T. Fox, Interregional connectivity to primary motor cortex revealed using MRI resting state images, Hum. Brain Mapp. 8(2–3) (1999) 151–156. 4. D. Cordes, V. M. Haughton, K. Arfanakis, J. D. Carew, P. A. Turski, C. H. Moritz, M. A. Quigley and M. E. Meyerand, Frequencies contributing to functional connectivity in the cerebral cortex in “resting-state” data, AJNR Am. J. Neuroradiol. 22(7) (2001) 1326–1333. 5. K. J. Friston, Functional and effective connectivity in neuroimaging: A synthesis, Hum. Brain Mapp. 2(1–2) (1994) 56–78. 6. T. E. Lund, fcMRI--mapping functional connectivity or correlating cardiac-induced noise? Magn. Reson. Med. 46(3) (2001) 628–629. 7. D. Cordes, V. M. Haughton, K. Arfanakis, G. J. Wendt, P. A. Turski, C. H. Moritz, M. A. Quigley and M. E. Meyerand, Mapping functionally related regions of brain with functional connectivity MR imaging, AJNR Am. J. Neuroradiol. 21(9) (2000) 1636–1644. 8. C. Chang, J. P. Cunningham and G. H. Glover, Influence of heart rate on the BOLD signal: The
14.
15.
16.
17.
18.
19.
20.
1450010-13
cardiac response function, Neuroimage 44(3) (2009) 857–869. K. Shmueli, P. van Gelderen, J. A. de Zwart, S. G. Horovitz, M. Fukunaga, J. M. Jansma and J. H. Duyn, Low-frequency fluctuations in the cardiac rate as a source of variance in the resting-state fMRI BOLD signal, Neuroimage 38(2) (2007) 306– 320. R. M. Birn, J. B. Diamond, M. A. Smith and P. A. Bandettini, Separating respiratory-variationrelated fluctuations from neuronal-activity-related fluctuations in fMRI, Neuroimage 31(4) (2006) 1536–1548. R. G. Wise, K. Ide, M. J. Poulin and I. Tracey, Resting fluctuations in arterial carbon dioxide induce significant low frequency variations in BOLD signal, Neuroimage 21(4) (2004) 1652–1664. M. D. Fox, A. Z. Snyder, J. L. Vincent, M. Corbetta, D. C. Van Essen and M. E. Raichle, The human brain is intrinsically organized into dynamic, anticorrelated functional networks, Proc. Natl. Acad. Sci. U.S.A. 102(27) (2005) 9673–9678. M. D. Fox, D. Zhang, A. Z. Snyder and M. E. Raichle, The global signal and observed anticorrelated resting state brain networks, J. Neurophysiol. 101(6) (2009) 3270–3283. P. Fransson, Spontaneous low-frequency BOLD signal fluctuations: An fMRI investigation of the resting-state default mode of brain function hypothesis, Hum. Brain Mapp. 26(1) (2005) 15–29. M. D. Greicius, B. Krasnow, A. L. Reiss and V. Menon, Functional connectivity in the resting brain: A network analysis of the default mode hypothesis, Proc. Natl. Acad. Sci. U.S.A. 100(1) (2003) 253–258. D. Mantini, M. G. Perrucci, C. Del Gratta, G. L. Romani and M. Corbetta, Electrophysiological signatures of resting state networks in the human brain, Proc. Natl. Acad. Sci. U.S.A. 104(32) (2007) 13170– 13175. K. J. Worsley, S. Marrett, P. Neelin, A. C. Vandal, K. J. Friston and A. C. Evans, A unified statistical approach for determining significant signals in images of cerebral activation, Hum. Brain Mapp. 4(1) (1996) 58–73. T. E. Lund, K. H. Madsen, K. Sidaros, W. L. Luo and T. E. Nichols, Non-white noise in fMRI: Does modelling have an impact? Neuroimage 29(1) (2006) 54–66. K. J. Friston, A. P. Holmes, J. B. Poline, P. J. Grasby, S. C. Williams, R. S. Frackowiak and R. Turner, Analysis of fMRI time-series revisited, Neuroimage 2(1) (1995) 45–53. P. Piaggi, D. Menicucci, C. Gentili, G. Handjaras, A. Gemignani and A. Landi, Adaptive filtering and random variables coefficient for analyzing functional magnetic resonance imaging data, Int. J. Neural Syst. 23(3) (2013) 1350011.
February 6, 2014 15:59 1450010
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
P. Piaggi et al.
21. J. B. Elsner and A. A. Tsonis, Singular Spectrum Analysis: A New Tool in Time Series Analysis (Plenum Publishing Corporation, 1996). 22. N. Golyandina, V. V. Nekrutkin and A. A. Zhigliavski, Analysis of Time Series Structure: SSA and Related Techniques (CRC Press, 2001). 23. I. T. Jolliffe, Principal Component Analysis (Springer-Verlag, New York, 2002). 24. M. R. Allen and L. A. Smith, Monte Carlo SSA: Detecting irregular oscillations in the presence of coloured noise, J. Climate 9(12) (1996) 3373–3404. 25. J. P. Burg, Maximum entropy spectral analysis, in Proceedings of the 37th Annual International Meeting of the Society of Exploration Geophysicists, Oklahoma City, OK (IEEE Press, Piscataway, NJ, 1978), pp. 42–48. 26. S. S. Haykin, Adaptive Filter Theory (Prentice Hall, 2002). 27. P. Piaggi, D. Menicucci, C. Gentili, G. Handjaras, M. Laurino, A. Piarulli, M. Guazzelli, A. Gemignani and A. Landi, Adaptive filtering for removing nonstationary physiological noise from resting state fMRI BOLD signals, in Proc. Intelligent Systems Design and Applications (ISDA), Cordoba, Spain (2011), pp. 237–241. 28. P. Robert and Y. Escoufier, A unifying tool for linear multivariate statistical methods: The RV-coefficient, Appl. Stat. 25(3) (1976) 257–265. 29. H. Zhang, J. Tian, J. Li and J. Zhao, RV-coefficient and its significance test in mapping brain functional connectivity, in Proceedings of SPIE Medical Imaging, International Society for Optics and Photonics (2009), pp. 726222. 30. F. Kherif, J. B. Poline, S. Meriaux, H. Benali, G. Flandin and M. Brett, Group analysis in functional neuroimaging: Selecting subjects using similarity measures, Neuroimage 20(4) (2003) 2197–2208. 31. H. Abdi, J. P. Dunlop and L. J. Williams, How to compute reliability estimates and display confidence and tolerance intervals for pattern classifiers using the Bootstrap and 3-way multidimensional scaling (DISTATIS), Neuroimage 45(1) (2009) 89–95. 32. M. D. Fox and M. E. Raichle, Spontaneous fluctuations in brain activity observed with functional magnetic resonance imaging, Nat. Rev. Neurosci. 8(9) (2007) 700–711. 33. R. W. Cox, AFNI: Software for analysis and visualization of functional magnetic resonance neuroimages, Comput. Biomed. Res. 29(3) (1996) 162– 173. 34. J. Talairach and P. Tournoux, Co-planar Stereotaxic Atlas of the Human Brain: 3-Dimensional Proportional System: An Approach to Cerebral Imaging (G. Thieme, 1988). 35. M. Bianciardi, M. Fukunaga, P. van Gelderen, J. A. de Zwart and J. H. Duyn, Negative BOLD-fMRI signals in large cerebral veins, J. Cereb. Blood Flow Metab. 31(2) (2011) 401–412.
36. J. E. A. O’Connell, The vascular factor in intracranial pressure and the maintenance of the cerebrospinal fluid circulation, Brain 66(3) (1943) 204–228. 37. M. Piche, J. Cohen-Adad, M. K. Nejad, V. Perlbarg, G. Xie, G. Beaudoin, H. Benali and P. Rainville, Characterization of cardiac-related noise in fMRI of the cervical spinal cord, Magn. Reson. Imaging 27(3) (2009) 300–310. 38. R. H. Deckers, P. van Gelderen, M. Ries, O. Barret, J. H. Duyn, V. N. Ikonomidou, M. Fukunaga, G. H. Glover and J. A. de Zwart, An adaptive filter for suppression of cardiac and respiratory noise in MRI time series data, Neuroimage 33(4) (2006) 1072–1081. 39. V. Wu, I. M. Barbash, K. Ratnayaka, C. E. Saikus, M. Sonmez, O. Kocaturk, R. J. Lederman and A. Z. Faranesh, Adaptive noise cancellation to suppress electrocardiography artifacts during real-time interventional MRI, J. Magn. Reson. Imaging 33(5) (2011) 1184–1193. 40. J. S. Damoiseaux, S. A. Rombouts, F. Barkhof, P. Scheltens, C. J. Stam, S. M. Smith and C. F. Beckmann, Consistent resting-state networks across healthy subjects, Proc. Natl. Acad. Sci. U.S.A. 103(37) (2006) 13848–13853. 41. M. De Luca, C. F. Beckmann, N. De Stefano, P. M. Matthews and S. M. Smith, fMRI resting state networks define distinct modes of long-distance interactions in the human brain, Neuroimage 29(4) (2006) 1359–1367. 42. M. Ahmadlou and H. Adeli, Functional community analysis of brain: A new approach for EEG-based investigation of the brain pathology, NeuroImage 58(2) (2011) 401–408. 43. M. Ahmadlou, H. Adeli and A. Adeli, Graph theoretical analysis of organization of functional brain networks in ADHD, Clin. EEG Neurosci. 43(1) (2012) 5–13. 44. D. Rangaprakash, X. Hu and G. Deshpande, Phase synchronization in brain networks derived from correlation between probabilities of recurrences in functional MRI data, Int. J. Neural Syst. 23(2) (2013) 1250003-1–1250003-11. 45. T. Yamanishi, J.-Q. Liu and H. Nishimura, Modeling fluctuations in default-mode brain network using a spiking neural network, Int. J. Neural Syst. 22(4) (2012) 1250016-1–1250016-10. 46. R. Vautard, P. Yiou and M. Ghil, Singular-spectrum analysis: A toolkit for short, noisy chaotic signals, Physica D 58(1–4) (1992) 95–126. 47. M. Ghil, M. Allen, M. Dettinger, K. Ide, D. Kondrashov, M. Mann, A. Robertson, A. Saunders, Y. Tian and F. Varadi, Advanced spectral methods for climatic time series, Rev. Geophys. 40(1) (2002) 1003. 48. C. Chatfield, The Analysis of Time Series: An Introduction (CRC Press, 2004).
1450010-14
International Journal of Neural Systems, Vol. 24, No. 3 (2014) 1450010 (14 pages) c World Scientific Publishing Company DOI: 10.1142/S0129065714500105
SINGULAR SPECTRUM ANALYSIS AND ADAPTIVE FILTERING ENHANCE THE FUNCTIONAL CONNECTIVITY ANALYSIS OF RESTING STATE fMRI DATA
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
PAOLO PIAGGI∗,† Department of Energy and Systems Engineering University of Pisa, Largo Lucio Lazzarino Pisa, 56122, Italy [email protected] DANILO MENICUCCI† Institute of Clinical Physiology National Research Council (CNR) Via G. Moruzzi 1, Pisa, 56124, Italy [email protected] CLAUDIO GENTILI Department of Surgery Medical, Molecular and Critical Area Pathology University of Pisa, Via Bonanno Pisano 6 Pisa, 56100, Italy GIACOMO HANDJARAS Laboratory of Clinical Biochemistry and Molecular Biology University of Pisa Via Roma 55, Pisa, 56126, Italy ANGELO GEMIGNANI EXTREME Centre, Department of Surgery Medical, Molecular and Critical Area Pathology University of Pisa, Via S. Zeno 31 Pisa, 56123, Italy ALBERTO LANDI Department of Information Engineering University of Pisa, Largo Lucio Lazzarino Pisa, 56122, Italy [email protected] Accepted 8 November 2013 Published Online 10 December 2013 Sources of noise in resting-state fMRI experiments include instrumental and physiological noises, which need to be filtered before a functional connectivity analysis of brain regions is performed. These noisy components show autocorrelated and nonstationary properties that limit the efficacy of standard techniques (i.e. time filtering and general linear model). Herein we describe a novel approach based on the combination of singular spectrum analysis and adaptive filtering, which allows a greater noise reduction
∗ †
Corresponding author. These authors contributed equally to this work. 1450010-1
February 6, 2014 15:59 1450010
P. Piaggi et al.
and yields better connectivity estimates between regions at rest, providing a new feasible procedure to analyze fMRI data. Keywords: Adaptive filtering; functional magnetic resonance imaging; physiological noise; resting state; RV coefficient; singular spectrum analysis.
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
1.
Introduction
Functional Magnetic Resonance Imaging (fMRI) is commonly used to study the dynamics of brain activity.1–3 In particular, resting-state fMRI studies have demonstrated that the neural activity is indirectly reflected in amplitude changes of the Blood Oxygen Level-Dependent (BOLD) signal, which are prominent in the low-frequency band (LFB, f < 0.10 Hz)1,4 and show temporal synchronizations between spatially remote brain areas (the so-called functional connectivity 5 of the resting state networks). To uncover the underlying neural connectivity between regions, BOLD signal analysis mandatorily requires a pre-processing procedure for filtering several non-neural noisy components, whose power may exceed that of the neural activity. Indeed, non-neural sources of noise are the principal cause of flawed or false positive functional connectivity results.6 Generally, the noisy components of BOLD signal can be divided into two classes: random and physiological. The former class includes the components of instrumental noise such as those related to the instability of MRI scanner, which show a broad spectrum mainly focused in the LFB of BOLD signal. Differently, the latter class comprises the physiological processes not related to neural activity, e.g. the respiratory and cardiac processes, whose BOLD components show frequency peaks at ∼0.25 Hz7 and ∼1.00 Hz,4 respectively. In addition, the nonstationary nature of physiological processes such as the heart rate variability,8,9 slow variations of respiratory rate10 and changes of arterial carbon dioxide level,11 induces further artifactual components in a broad frequency interval that may overlap the LFB of BOLD neural correlates. In summary, both random and physiological components show frequencies peaks superimposed on an autocorrelated noise background spreading over the whole spectrum of BOLD signal. As the largest effects of non-neural noise are due to the cardiac and respiratory components, various
techniques have been extensively used for filtering the BOLD signal, depending on the frequency location of the noisy components. Accordingly, cardiac and respiratory components at higher frequencies (>0.10 Hz), along with hardware-related ultra-low frequency drifts ( 0.05). The results obtained with the novel procedure considering only active voxels by SSA may corroborate the hypothesis of lower (or even absent) coupling in WM where no neural activity is expected, as compared to GM regions that have been previously identified as part of functional resting state networks.1,7,12,40,41 These findings may also suggest that SSA identified only the voxels in GM regions that contributed to the neural coupling between homologous ROIs, resulting in higher RV coefficients than those obtained using the gold standard procedure on all available voxels. Nevertheless, the relative higher values of RV coefficients in brain ventricles indicated a residual connectivity between these non-neural regions that could be due to incomplete removal of physiological noise in the LFB. This hypothesis is supported by the fact that REF signal explained only half the variability of ventricles signals (on average, 48.6%), thus the noise could not be entirely removed by adaptive filtering. One possible solution to overcome this issue might be performing sequential stages of adaptive filtering using the first two or three PCs of ventricle signals. Accordingly, future studies should be conducted to investigate the feasibility of this method to achieve a better suppression of physiological noise in the LFB. The main limitation of this methodological study on BOLD signal processing is the acquisition of only four slices for brain volume, which limited the spatial extension of ROIs as well as the interpretation of results in terms of the underlying brain physiology. This was due to higher sampling rates and technical limitations of our MRI equipment. Yet, the choice of a lower TR was mandatory to avoid aliasing effects of cardiac components in the LFB, which in turn would have biased the results to an unknown extent. However, both the novel and the gold standard procedure were able to discriminate neural ROIs of GM from brain ventricles and WM. In fact, brain ventricles
were found to be characterized by higher levels of noise (possibly due to displacement of cerebrospinal fluid), while pair of WM regions resulted in the lowest functional connectivity as expected in non-neural regions. Despite the technical limitations that precluded a full brain coverage, our findings indicate that the gold standard procedure of BPF followed by GLM achieved consistent results and that the novel procedure of SSA followed by adaptive filtering improved the efficacy of standard techniques, overcoming their methodological limitations. 5.
Conclusion
The application of SSA to BOLD signals in the resting state led to the identification of active voxels by extracting slow-varying components, which showed significantly greater power against a null hypothesis of autocorrelated noise. In addition, the adaptive filtering of SSA-filtered signals achieved a greater reduction of noise in all brain regions. Finally, the whole procedure of SSA plus adaptive filtering led to a higher contrast between GM and WM in the functional connectivity analyses as compared to the gold standard procedure of BPF and GLM. These results suggest that the combination of SSA and adaptive filtering is a reasonable and convenient approach for removing the low-frequency fluctuations of BOLD signal due to physiological and instrumental noises of fMRI data. The usage of these advanced techniques substantially improved the results of the functional connectivity analysis, demonstrating the benefit of applying new framework of analysis for studying the brain connectivity.42–45 As a result, this novel procedure allowed to emphasize the connectivity estimates of functional networks in resting-state conditions. Further studies should be carried out to explore the feasibility of this procedure on datasets with whole brain coverage, with a greater number of subjects and on higher magnetic fields. Acknowledgments The authors wish to thank Dr. Paolo Allegrini and Dr. Andrea Piarulli for their useful comments on writing the manuscript. Appendix A The SSA algorithm consists of four steps A.1–A.4.
1450010-10
February 6, 2014 15:59 1450010
SSA and Adaptive Filtering for fMRI Data
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
A.1. Embedding step A trajectory matrix D was obtained from the meancentered BOLD signal S(t) of length N by using a time window of length W . Generally, the choice of window length — that can be set from 2 to N/246 — is a compromise between the number of signal portions and the frequency resolution. In the lack of a priori hypothesis, a rule of thumb indicates a value ranging from N/5 and N /3 in order to obtain a reliable estimate for oscillatory components.47 The ith column of D contained the samples of S(t) — from S(i) to S(i + W − 1). The number of the columns of D was equal to M = N − W + 1, hence D had a Hankel matrix structure of dimension W -by-M . From D, the lag-covariance matrix CV = (1/M ) · DDT of dimension W -by-W was derived, where T indicates the matrix transpose operation.
performed by a maximum likelihood algorithm using unbiased estimators.24 In a red noise context, the signal-to-noise separation is performed exploiting its autocorrelation properties in the SSA framework. To this aim, the analytic covariance matrix of red noise CN = c0 T was derived using the estimated noise parameters, where c0 is the noise variance and elements of Tij = γ |i−j| (i and j are time-lag indices).48 The theoretical EOFs of red noise were then obtained by diagonalizing CN as: T CN EN . ΛN = EN
(A.3)
Due to the analytic structure of the noise covariance matrix, the dominant frequencies of EN were regularly spaced in the spectrum and separated by 1/(W × Tc ), where Tc is the sampling time. The theoretical noise EOFs (columns of EN ) showed also a sinusoidal behavior in time.
A.2. Decomposition step The lag-covariance matrix CV was diagonalized as: ΛV =
EVT CV
EV ,
A.3.1. Projection onto noise EOFs
(A.1)
where ΛV was the data eigenvalue matrix with elements λV along the main diagonal in decreasing order of magnitude, while EV was the matrix of associated data eigenvectors (EOFs, columns of EV ). The dominant frequency of each EOF was estimated as that with the maximum spectral power in the frequency domain. A.3. Statistical test for EOFs We assumed that the BOLD signal S(t) was composed of oscillations at different frequencies over a background of autocorrelated noise. We considered an AR(1) model for red noise, which also allowed the case of uncorrelated white noise with a suitable choice of process parameters. The recursive equation that described the red noise model is: ut − U0 = γ(ut−1 − u0 ) + αzt ,
(A.2)
where u0 is the process mean, γ and α are the process parameters (i.e. lag-1 autocorrelation and variance of red noise, respectively) and zt is a Gaussian white noise with unit variance. In the case of γ = 0 (i.e. no autocorrelation), the model downgrades to a white noise model. The estimation of the red noise parameters (γ and α) from S(t) was
Under the assumptions of Gaussian noise distribution and sinusoidal EOFs, each diagonal element λN of ΛN had a χ2 distribution with υ = 3N/W degrees of freedom.24 These assumptions are valid for AR(1) processes and lead to the following distribution for the noise eigenvalues: χ2 (ν) . (A.4) ν From the 2.5th and 97.5th percentiles of these distributions, the 95% confidence interval was calculated for each λN . The data covariance matrix CV was then projected onto noise EOFs EN as: T CN EN ) λN ≈ (EN
T CV EN . ΛVN = EN
(A.5)
Assuming a null hypothesis of pure noise that have generated S(t), all diagonal elements λVN of ΛVN should lie within the noise confidence intervals of the related λN . Otherwise, the noise EOFs associated to the λVN being outside the corresponding confidence interval were considered not compatible with the noise model, thus indicated the presence of real oscillatory components at those frequency (Fig. A.1). A.3.2. Projection onto data EOFs Since the noise EOFs EN were not directly related to data EOFs EV , the analytic covariance matrix of
1450010-11
February 6, 2014 15:59 1450010
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
P. Piaggi et al.
Fig. A.1. Eigenspectrum based on noise EOFs in the time course of Fig. 2. The dominant frequency of each noise EOF (EN , x-axis) is plotted against the diagonal elements of ΛVN . Error bars represent noise eigenvalues with 95% confidence interval using the χ2 distribution. Significant elements of ΛVN greater than the 97.5th percentile of the corresponding noise eigenvalue λN are highlighted and identified as significant trend (blue), LFB (red) and cardiac (violet) components, according to their dominant frequency. The LFB (0.04–0.10 Hz) is highlighted in yellow; nonsignificant elements are drawn as black circles.
red noise CN was also projected onto data EOFs as: ΛNV = EVT CN EV .
(A.6)
Similarly to step A.3.1, confidence intervals of the diagonal elements λNV of ΛNV were derived using (A.4). Accordingly, all data eigenvalues λV greater than the corresponding 97.5th percentile of λNV were considered statistically significant from noise (Fig. A.2).
Fig. A.2. Eigenspectrum based on data EOFs in the time course of Fig. 2. The dominant frequency of each data EOF (EV , x-axis) is plotted against the data eigenvalues λV (y-axis). Error bars represent diagonal elements of ΛNV with 95% confidence interval using the χ2 distribution. Significant data eigenvalue λV greater than the 97.5th percentile of the corresponding diagonal elements of ΛNV are highlighted and identified as significant trend (blue), LFB (red) and cardiac (violet) components, according to their dominant frequency. The LFB (0.04–0.10 Hz) is highlighted in yellow; nonsignificant elements are drawn as black circles. From each significant data EOF whose dominant frequency was also found significant in the noise eigenspectrum (Fig. A.1), the corresponding RC was calculated. All RCs related to each band (i.e. trend, LFB and high frequencies) were summed for obtaining the extracted signals shown in Fig. 2 (blue, red and violet lines, respectively).
A.4. Reconstruction step The projection of original signal S(t) onto significant data EOFs (EV ∗ matrix) yielded to:
A.3.3. Significance of data EOFs
A∗V (t) =
We considered globally significant each data EOF of EV that satisfied both these conditions: (i) whose associated noise EOF of EN (i.e. with the same dominant frequency) had the corresponding λVN element greater than the 97.5th percentile of the distribution of the related noise eigenvalue λN (step A.3.1); (ii) whose corresponding data eigenvalue λV was greater than the 97.5th percentile of the distribution of the corresponding λNV element (step A.3.2). We named EV ∗ the matrix with only the significant data EOFs as columns.
W
S(t + j − 1) · EV∗ (j).
(A.7)
j=1
For each significant EOF, the corresponding reconstructed component (RC) matrix RV ∗ was obtained by: Rk∗ (t) =
Ut 1 A∗k (t − j + 1) · Ek∗ (j), Mt
(A.8)
j=Lt
where Mt , Lt and Ut are time-dependent parameters,46 necessary to manage the border effects of time window W . The SSA code written in MATLAB can be provided upon request to the corresponding authors.
1450010-12
February 6, 2014 15:59 1450010
SSA and Adaptive Filtering for fMRI Data
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
Appendix B The corrected RV coefficient was obtained by the following resampling procedure. Given the signal matrices of a ROIs pair with NL and NR voxels of which NaL and NaR active voxels (NaL < NL and NaR < NR ) in the left and in the right hemisphere, respectively, 10,000 pairs of surrogate signal matrices were generated to derive a distribution under the null hypothesis of random selection of voxels. In doing so, the signals in each surrogate matrix were drawn from the complete matrix of all voxels by using a sampling-without-replacement algorithm, so that each surrogate matrix contained NaL and NaR signals (instead of original NL and NR ) according to the related hemisphere. The RV coefficient was then calculated for each pair of surrogate matrices to obtain the distribution for a random selection of subsets of voxels. The corrected RV coefficient was then calculated as the mean value of the surrogate distribution.
9.
10.
11.
12.
13.
References 1. B. Biswal, F. Z. Yetkin, V. M. Haughton and J. S. Hyde, Functional connectivity in the motor cortex of resting human brain using echo-planar MRI, Magn. Reson. Med. 34(4) (1995) 537–541. 2. M. J. Lowe, B. J. Mock and J. A. Sorenson, Functional connectivity in single and multislice echoplanar imaging using resting-state fluctuations, Neuroimage 7(2) (1998) 119–132. 3. J. Xiong, L. M. Parsons, J. H. Gao and P. T. Fox, Interregional connectivity to primary motor cortex revealed using MRI resting state images, Hum. Brain Mapp. 8(2–3) (1999) 151–156. 4. D. Cordes, V. M. Haughton, K. Arfanakis, J. D. Carew, P. A. Turski, C. H. Moritz, M. A. Quigley and M. E. Meyerand, Frequencies contributing to functional connectivity in the cerebral cortex in “resting-state” data, AJNR Am. J. Neuroradiol. 22(7) (2001) 1326–1333. 5. K. J. Friston, Functional and effective connectivity in neuroimaging: A synthesis, Hum. Brain Mapp. 2(1–2) (1994) 56–78. 6. T. E. Lund, fcMRI--mapping functional connectivity or correlating cardiac-induced noise? Magn. Reson. Med. 46(3) (2001) 628–629. 7. D. Cordes, V. M. Haughton, K. Arfanakis, G. J. Wendt, P. A. Turski, C. H. Moritz, M. A. Quigley and M. E. Meyerand, Mapping functionally related regions of brain with functional connectivity MR imaging, AJNR Am. J. Neuroradiol. 21(9) (2000) 1636–1644. 8. C. Chang, J. P. Cunningham and G. H. Glover, Influence of heart rate on the BOLD signal: The
14.
15.
16.
17.
18.
19.
20.
1450010-13
cardiac response function, Neuroimage 44(3) (2009) 857–869. K. Shmueli, P. van Gelderen, J. A. de Zwart, S. G. Horovitz, M. Fukunaga, J. M. Jansma and J. H. Duyn, Low-frequency fluctuations in the cardiac rate as a source of variance in the resting-state fMRI BOLD signal, Neuroimage 38(2) (2007) 306– 320. R. M. Birn, J. B. Diamond, M. A. Smith and P. A. Bandettini, Separating respiratory-variationrelated fluctuations from neuronal-activity-related fluctuations in fMRI, Neuroimage 31(4) (2006) 1536–1548. R. G. Wise, K. Ide, M. J. Poulin and I. Tracey, Resting fluctuations in arterial carbon dioxide induce significant low frequency variations in BOLD signal, Neuroimage 21(4) (2004) 1652–1664. M. D. Fox, A. Z. Snyder, J. L. Vincent, M. Corbetta, D. C. Van Essen and M. E. Raichle, The human brain is intrinsically organized into dynamic, anticorrelated functional networks, Proc. Natl. Acad. Sci. U.S.A. 102(27) (2005) 9673–9678. M. D. Fox, D. Zhang, A. Z. Snyder and M. E. Raichle, The global signal and observed anticorrelated resting state brain networks, J. Neurophysiol. 101(6) (2009) 3270–3283. P. Fransson, Spontaneous low-frequency BOLD signal fluctuations: An fMRI investigation of the resting-state default mode of brain function hypothesis, Hum. Brain Mapp. 26(1) (2005) 15–29. M. D. Greicius, B. Krasnow, A. L. Reiss and V. Menon, Functional connectivity in the resting brain: A network analysis of the default mode hypothesis, Proc. Natl. Acad. Sci. U.S.A. 100(1) (2003) 253–258. D. Mantini, M. G. Perrucci, C. Del Gratta, G. L. Romani and M. Corbetta, Electrophysiological signatures of resting state networks in the human brain, Proc. Natl. Acad. Sci. U.S.A. 104(32) (2007) 13170– 13175. K. J. Worsley, S. Marrett, P. Neelin, A. C. Vandal, K. J. Friston and A. C. Evans, A unified statistical approach for determining significant signals in images of cerebral activation, Hum. Brain Mapp. 4(1) (1996) 58–73. T. E. Lund, K. H. Madsen, K. Sidaros, W. L. Luo and T. E. Nichols, Non-white noise in fMRI: Does modelling have an impact? Neuroimage 29(1) (2006) 54–66. K. J. Friston, A. P. Holmes, J. B. Poline, P. J. Grasby, S. C. Williams, R. S. Frackowiak and R. Turner, Analysis of fMRI time-series revisited, Neuroimage 2(1) (1995) 45–53. P. Piaggi, D. Menicucci, C. Gentili, G. Handjaras, A. Gemignani and A. Landi, Adaptive filtering and random variables coefficient for analyzing functional magnetic resonance imaging data, Int. J. Neural Syst. 23(3) (2013) 1350011.
February 6, 2014 15:59 1450010
Int. J. Neur. Syst. 2014.24. Downloaded from www.worldscientific.com by UNIV OF MASSACHUSETTS AMHERST on 02/02/15. For personal use only.
P. Piaggi et al.
21. J. B. Elsner and A. A. Tsonis, Singular Spectrum Analysis: A New Tool in Time Series Analysis (Plenum Publishing Corporation, 1996). 22. N. Golyandina, V. V. Nekrutkin and A. A. Zhigliavski, Analysis of Time Series Structure: SSA and Related Techniques (CRC Press, 2001). 23. I. T. Jolliffe, Principal Component Analysis (Springer-Verlag, New York, 2002). 24. M. R. Allen and L. A. Smith, Monte Carlo SSA: Detecting irregular oscillations in the presence of coloured noise, J. Climate 9(12) (1996) 3373–3404. 25. J. P. Burg, Maximum entropy spectral analysis, in Proceedings of the 37th Annual International Meeting of the Society of Exploration Geophysicists, Oklahoma City, OK (IEEE Press, Piscataway, NJ, 1978), pp. 42–48. 26. S. S. Haykin, Adaptive Filter Theory (Prentice Hall, 2002). 27. P. Piaggi, D. Menicucci, C. Gentili, G. Handjaras, M. Laurino, A. Piarulli, M. Guazzelli, A. Gemignani and A. Landi, Adaptive filtering for removing nonstationary physiological noise from resting state fMRI BOLD signals, in Proc. Intelligent Systems Design and Applications (ISDA), Cordoba, Spain (2011), pp. 237–241. 28. P. Robert and Y. Escoufier, A unifying tool for linear multivariate statistical methods: The RV-coefficient, Appl. Stat. 25(3) (1976) 257–265. 29. H. Zhang, J. Tian, J. Li and J. Zhao, RV-coefficient and its significance test in mapping brain functional connectivity, in Proceedings of SPIE Medical Imaging, International Society for Optics and Photonics (2009), pp. 726222. 30. F. Kherif, J. B. Poline, S. Meriaux, H. Benali, G. Flandin and M. Brett, Group analysis in functional neuroimaging: Selecting subjects using similarity measures, Neuroimage 20(4) (2003) 2197–2208. 31. H. Abdi, J. P. Dunlop and L. J. Williams, How to compute reliability estimates and display confidence and tolerance intervals for pattern classifiers using the Bootstrap and 3-way multidimensional scaling (DISTATIS), Neuroimage 45(1) (2009) 89–95. 32. M. D. Fox and M. E. Raichle, Spontaneous fluctuations in brain activity observed with functional magnetic resonance imaging, Nat. Rev. Neurosci. 8(9) (2007) 700–711. 33. R. W. Cox, AFNI: Software for analysis and visualization of functional magnetic resonance neuroimages, Comput. Biomed. Res. 29(3) (1996) 162– 173. 34. J. Talairach and P. Tournoux, Co-planar Stereotaxic Atlas of the Human Brain: 3-Dimensional Proportional System: An Approach to Cerebral Imaging (G. Thieme, 1988). 35. M. Bianciardi, M. Fukunaga, P. van Gelderen, J. A. de Zwart and J. H. Duyn, Negative BOLD-fMRI signals in large cerebral veins, J. Cereb. Blood Flow Metab. 31(2) (2011) 401–412.
36. J. E. A. O’Connell, The vascular factor in intracranial pressure and the maintenance of the cerebrospinal fluid circulation, Brain 66(3) (1943) 204–228. 37. M. Piche, J. Cohen-Adad, M. K. Nejad, V. Perlbarg, G. Xie, G. Beaudoin, H. Benali and P. Rainville, Characterization of cardiac-related noise in fMRI of the cervical spinal cord, Magn. Reson. Imaging 27(3) (2009) 300–310. 38. R. H. Deckers, P. van Gelderen, M. Ries, O. Barret, J. H. Duyn, V. N. Ikonomidou, M. Fukunaga, G. H. Glover and J. A. de Zwart, An adaptive filter for suppression of cardiac and respiratory noise in MRI time series data, Neuroimage 33(4) (2006) 1072–1081. 39. V. Wu, I. M. Barbash, K. Ratnayaka, C. E. Saikus, M. Sonmez, O. Kocaturk, R. J. Lederman and A. Z. Faranesh, Adaptive noise cancellation to suppress electrocardiography artifacts during real-time interventional MRI, J. Magn. Reson. Imaging 33(5) (2011) 1184–1193. 40. J. S. Damoiseaux, S. A. Rombouts, F. Barkhof, P. Scheltens, C. J. Stam, S. M. Smith and C. F. Beckmann, Consistent resting-state networks across healthy subjects, Proc. Natl. Acad. Sci. U.S.A. 103(37) (2006) 13848–13853. 41. M. De Luca, C. F. Beckmann, N. De Stefano, P. M. Matthews and S. M. Smith, fMRI resting state networks define distinct modes of long-distance interactions in the human brain, Neuroimage 29(4) (2006) 1359–1367. 42. M. Ahmadlou and H. Adeli, Functional community analysis of brain: A new approach for EEG-based investigation of the brain pathology, NeuroImage 58(2) (2011) 401–408. 43. M. Ahmadlou, H. Adeli and A. Adeli, Graph theoretical analysis of organization of functional brain networks in ADHD, Clin. EEG Neurosci. 43(1) (2012) 5–13. 44. D. Rangaprakash, X. Hu and G. Deshpande, Phase synchronization in brain networks derived from correlation between probabilities of recurrences in functional MRI data, Int. J. Neural Syst. 23(2) (2013) 1250003-1–1250003-11. 45. T. Yamanishi, J.-Q. Liu and H. Nishimura, Modeling fluctuations in default-mode brain network using a spiking neural network, Int. J. Neural Syst. 22(4) (2012) 1250016-1–1250016-10. 46. R. Vautard, P. Yiou and M. Ghil, Singular-spectrum analysis: A toolkit for short, noisy chaotic signals, Physica D 58(1–4) (1992) 95–126. 47. M. Ghil, M. Allen, M. Dettinger, K. Ide, D. Kondrashov, M. Mann, A. Robertson, A. Saunders, Y. Tian and F. Varadi, Advanced spectral methods for climatic time series, Rev. Geophys. 40(1) (2002) 1003. 48. C. Chatfield, The Analysis of Time Series: An Introduction (CRC Press, 2004).
1450010-14
