In the latest years momentous advance has been made in understanding the endogenous brain dynamics from resting state functional MRI (rs-fMRI) signals. An rs-fMRI signal tends to have long memory in time as well as the $1/f$ power spectrum at low frequencies. A few statistical models of rs-fMRI time series, such as fractional Gaussian noise (FGN), had been proposed to describe such properties called the fractal behavior. Nonetheless, the long memory properties have not been elucidated by the underlying physical mechanism. In addition, how such properties have an impact on large-scale functional networks of the brain has been unclear. This thesis develops not only a parsimonious model of long memory in rs-fMRI, which provides us hypothetical ideas on these unresolved issues, but also advanced techniques for estimating intrinsic functional connectivity among brain regions hidden beyond the long memory phenomenon of rs-fMRI signals.

1. Long Memory Model
of Resting State fMRI
Wonsang You

2. I. Long memory model of hemodynamic response
II. Nonfractal connectivity
III. Applications of fractal-based analyses

3. I. Long memory model of hemodynamic response
II. Nonfractal connectivity
III. Applications of fractal-based analyses

4. Functional MRI
Neural activity BOLD signalHemodynamics
Deoxyhemoglobin BOLD contrast depends on the content of oxygen
(deoxyhemoglobin) in blood vessels modulated by neural tissues .
Neural activity is convolved by hemodynamics in fMRI.
Blood-oxygen-level dependent
1 2
1 1 2 2( )/ ( )/
1 2
( )
r r
t s b t s bt t
g t e c e
s s
      
    
   
)(tu ( ) ( ) ( )x t g t u t 
Hemodymic Response Function (HRF)
4
HRF Impulse hemodynamic response is modeled as Hemodymic
Response Function.
𝑂2
Devlin, H. (2007) Psych Central.
∗ =

5. Evoked State vs. Resting State fMRI
5
Evoked state
Prominent correlation between stimulation and BOLD response
Resting state
1. No reference signal
• Uncertainty in neurovascular coupling
between BOLD signal and neuronal activity
2. Stochastic behavior like noise
• Less consistency in correlation between
BOLD signals
Fox and Raichle (2007) Nature Neuroscience

6. Fractal Behavior
(Long memory)
Fractality: Consistent Feature in Resting State fMRI
Time shift
Slowly-decaying
autocorrelation
Self-correlation
Shifted time
Correlation
1/f-type power spectrum
LogPower
Log frequency - Hz
Power
Frequency - Hz
Additive noise
Time domain
Frequency domain
Characterized by
Hurst exponent
Self-
similarity
Power-law
Scaling
6Maxim (2005) NeuroImage

7. Physiological Sources of 1/f Noise
Respiration
Cardiac motion
System noise
Self-organized fractal behavior
Spontaneous
Neural activity
rs-fMRI signalHemodynamics
1/f noise
Frequency
Power
Log-scale power spectrum of CBV at rest
How does hemodynamics-driven 1/f noise affect neurovascular coupling in resting state?
HYPOTHESIS
Hemodynamics is one of the
main sources for 1/f noise.
Eke et al. (2006) Cerebral Blood Flow & Metabolism 7
Hemodynamics

8. Typical HRF Model Doesn’t Incorporate 1/f Noise
Basis 1
Basis 2
Neural
activity
rs-fMRI
signal
+ +
Two basis (Gamma) functions
Spontaneous
neural activity
rs-fMRI signalHemodynamics
Typical HRF
No fractal behavior
( ) ( ) ( )x t g t u t )(tg)(tu
1 1 2 2( ) ( ) ( )g t f t f t  
LogPower
Log frequency - Hz
Power spectrum of a simulated rs-BOLD signal
8

9. Proposed Model: Resting state HRF
Basis 1
Basis 2
Neural
activity
rs-fMRI
signal
Basis 3
Basis N
+ +
Lots of basis functions
with slowly decaying coefficients
Resting State HRF
Spontaneous
neural activity
rs-fMRI signalHemodynamics
1
( ) ( )
K
k k
k
g t f t

 
Allow fractal behavior at rest
Power spectrum of a simulated rs-BOLD signal
LogPower
Log frequency - Hz
1.5H
k k 

9
   
   
0.5 0.5
0.5 1
H t H
H t
   

   
( ) ( ) ( )x t g t u t )(tg)(tu
K  

10. Heterogeneity in Hurst exponents causes the connectivity discrepancy
between fMRI signals and underlying neuronal activities.
Implication of rs-HRF on Neurovascular Coupling
Hurst 1= 0.1
Hurst 2= 0.9
Generic HRFCORX= 0.8 CORY = 0.2
H1
H2
COR
Ratio
COR
Y
X

COR CORX Y
1 2H H 
1 2H H
COR CORX Y 
Larger difference of Hurst exponents results in
greater mismatch of functional connectivity.
Ratio of BOLD correlation to Neuronal correlation
Little connectivity
distortion
10

11. FD 0.1
FD 0.5
FD 0.8
Fractal self-organized HRF
Structural
connectivity
matrix
1 Network analysis5
fMRI signalsNeural activities
Mean field
model
Fractal/Nonfractal
Transfer Entropy
Wavelet Correlation
Connectivity analysis4
Simulating neuronal populations Simulating neuroimaging signals2 3
Simulation Studies of Proposed rs-fMRI Model
The proposed rs-fMRI model has been demonstrated by simulation studies.
11

12. Effects of 1/f Noise on Functional Networks
Betweenness Centrality
in Pearson Correlation
Betweenness Centrality
in Information Flow Network
Neuronal activity
BOLDsignals
Wavelet Scales
FractionalDeviation
Difference in functional network (47 ROIs) between neuronal activities and BOLD signals
when Hurst exponents in hemodynamics are given heterogenously.
Deviation in Wavelet Correlation
Neuronal activity
• Heterogeneous fractal behaviors lead to discrepancy in local network properties.
• Information flow in non-hub regions is sensitive to hemodynamics-driven 1/f noise.
• A non-hub region may have large difference of wavelet correlation in low frequencies.
Low
Freq
High
Freq
Non-hub Hub Non-hub Hub
Non-hub
Hub
COR CORX Y 
Non-hub
12
◆ Each point corresponds to an ROI
BOLDsignals

13. Summary on Proposed rs-fMRI Model
Typical HRF
Self-organized fractal behavior
Spontaneous
neural activity
rs-fMRI signalHemodynamics
1/f noise
Proposed Resting State HRF
( ) ( ) ( )x t g t u t )(tg)(tu
1
( ) ( )
K
k k
k
g t f t

  1.5H
k k 
1 1 2 2( ) ( ) ( )g t f t f t  
▪ No implication on 1/f noise driven by
hemodynamics
▪ No implication on relationship of the
1/f noise with functional connectivity
▪ Providing the theoretical model on
hemodynamic mechanism of 1/f noise
▪ Explaining how 1/f noise affects
neurovascular coupling at rest
▪ Resting state functional connectivity may be
wrongly measured depending on difference
of Hurst exponents.
▪ Network properties of non-hub regions are
sensitive to 1/f noise in low frequencies.
Simulation studies demonstrates
Typical HRF does not explain
fractal behavior of rs-BOLD signals.
13

14. I. Long memory model of hemodynamic response
II. Nonfractal connectivity
III. Applications of fractal-based analyses

15. How to Overcome Connectivity Distortion
Power
Frequency
Band-pass Filtering (BPF)
Low frequency fluctuation is still influenced by
1/f noise from non-neuronal compounds.
Estimate of intrinsic resting state functional connectivity
Fractal Dim = 0.1
Fractal Dim = 0.9
Generic HRFCor = 0.8 Cor = 0.2
Wavelet-based
Fractal Estimation
Nonfractal
connectivity
Cor = 0.76
Nonfractal Connectivity
Proposed method
Impact of non-neuronal 1/f noise
15

16. Nonfractal Connectivity
• An rs-fMRI signal can be split into
fractal and nonfractal components. =
Fractal
Component
Nonfractal
Component
≒
Generic
HRF
Neuronal
Activity
*
*
Spontaneous
Neural activity
rs-fMRI signalHemodynamics
( ) ( ) ( )x t g t u t )(tu
   
   
0.5 0.5
( )
0.5 1
H t H
g t
H t
   

   
2 1
( ) 1 ( )
Hf
X f e U f
 
  i
Controlling
Fractal behavior
Nonfractal
partWavelet-based
Fractal
Estimation
Nonfractal
Component
( )u t
HHurst exponent
PSD
Fractionally Integrated Process Model
16
Assumption
of Similarity
rs-HRF

17. Nonfractal Connectivity
Hurst 2
Wavelet-
based
Fractal
Estimation
Cor = 0.2
Hurst 1
Nonfractal
Component 1
Nonfractal
Component 2
Nonfractal
Connectivity
• An rs-fMRI signal can be split into
fractal and nonfractal components.
• Nonfractal connectivity is defined as
correlation of nonfractal components.
Simulation studies show
Similarity between Nonfractal connectivity and
Functional connectivity of neuronal activities.
=
≒
Functional
Connectivity
of neuronal
populations
≒
Cor = 0.76Cor = 0.8
Fractal
Component
Nonfractal
Component
Generic
HRF
Neuronal
Activity
*
*
17

18. Wavelet-based Estimation of Nonfractal Connectivity
Estimation of
Hurst exponents
rs-BOLD
Signal
Estimation of
nonfractal
covariance matrix
Nonfractal
Connectivity
Hurst Exponent Nonfractal Covariance
Traditional ▪ ML: Wavelet maximum likelihood
▪ US: Univariate wavelet LMS
None
Proposed ▪ MS: Multivariate wavelet LMS ▪ SDF-based
▪ Linearity-based
▪ Covariance-based
A. Proposed MS method has better performance with additive noises.
B. Simulation demonstrates performance of nonfractal connectivity.
Covariance Linearity SDFNeuronal Correlation
SNR=1A
B
18
Bias of Hurst estimators
Nonfractal connectivity with given neuronal correlation

19. Summary on Nonfractal Connectivity
Fractal dim = 0.1
Fractal dim = 0.9
Generic HRFCor = 0.8 Cor = 0.2
Multivariate
Wavelet LMS
Nonfractal
connectivity
Cor = 0.76
1. Showing that an rs-BOLD signal can be split into fractal and nonfractal components,
2. Suggesting Nonfractal connectivity as an estimate of intrinsic functional connectivity,
3. Proposing the multivariate wavelet LMS estimator of Hurst exponents which has
better performance in the presence of additive noises,
4. Proposing the wavelet-based estimators of nonfractal connectivity.
Contributions on Nonfractal Connectivity
Nonfractal connectivity is defined as correlation of nonfractal components.
Nonfractal
Covariance
19
Wavelet-based estimator

20. I. Long memory model of hemodynamic response
II. Nonfractal connectivity
III. Applications of fractal-based analyses

21. Clinical Applications to Major Depression Disorder
21
How does depression make
change in brain networks?

22. Clinical Applications to Major Depression Disorder
Fractal/nonfractal connectivity
reveals changes in network properties
caused by depression (MDD).
Depression makes fractal behavior
more consistent over subjects.
Correlation Threshold
Q-value
MDDPatients
Healthy Controls
Subject-variance of Hurst exponents
in 95 ROIs
◆ Rs-fMRI data was acquired from a 3T Siemens MRI scanner. (Healthy 21, Patients 22)
22
Lower subject
variance
in MDD patients
FDR-corrected Q-value
of Path Length in an ROI
Significance level
SignificantUnreliable
Pearson Correlation
Nonfractal Connectivity
Fractal Connectivity

23. Impact of Stimulation on Resting State in Rat Brain
1
8
12
4
14
3
6 10
2
15
7 9
11
5
13
1
8
12
4
14
3
6 10
2
15
7 9
11
5
13
1
12
4
14
3
6 10
2
15
7
5
13
8
9
11
1
8
12
4
14
3
6 10
2
15
7 9
11
5
13
Pearson Correlation – Inconsistent/Random
Nonfractal Connectivity – Consistent modularity
A. Symmetric connectivity in S1, S2, TE
B. Overlapped with task-based connectivity
1. aCG
3. CPu-R
4. MEnt-L
5. MEnt-R
7. HIP-R
8. S1-L
10. S2-L
9. S1-R
11. S2-R
13. TE-L 14. TE-R
6. HIP-L
A B
A
Specific Patterns in
Nonfractal Connectivity
Resting state
before stimulation
Resting state
after stimulation
23
A
Electrical
Stimulation
How does stimulation affect
resting state brain networks?

24. Fractal Analysis of rs-fMRI Reveals …
Rat Brain
1
12
4
14
3
6 10
2
15
7
5
13
8
9
11
1
8
12
4
14
3
6 10
2
15
7 9
5
13
11
1. Consistent network patterns over cognitive states
2. Significant change of network properties in brain disorders
Before vs. after stimulation
Healthy vs. depressive patients
• Significant group difference
in local network properties
based on fractal/nonfractal
connectivity.
Human Brain
#ROIs with significant group difference
24
• Preserved modularity
between both hemispheres
Local network properties

25. Conclusion
1. Unlike the typical HRF, the proposed resting state HRF model describes fractal behavior
of rs-BOLD signals driven by hemodynamics.
2. The rs-HRF model suggests that heterogeneity of fractal behavior among brain regions
causes the connectivity discrepancy between fMRI signals and neuronal activities.
3. The rs-HRF model implies that an rs-BOLD signal can be split into fractal and nonfractal
components.
4. Nonfractal connectivity was suggested as an estimate of intrinsic resting state
functional connectivity between spontaneous neuronal activities.
5. Wavelet-based estimators of nonfractal connectivity were developed.
6. Fractal and nonfractal connectivity are effective to reveal not only consistent network
patterns over cognitive states but also significant change of network properties in
brain disorders.
Long memory model of hemodynamic response
Nonfractal connectivity
Biomedical Applications of fractal-based analyses
25

26. Thanks to
Supervisor
Prof. Dr. Udo Seiffert
Dr. André Brechmann (LIN)
Dr. Jörg Stadler (LIN)
Dr. Sophie Achard (Gipsa-lab, France)
Prof. Dr. Jan Beran (Universität Konstanz)
Prof. Dr. Johannes Bernarding (Otto-von-Guericke-Universität)
Prof. Dr. Frank Angenstein (DZNE)
Thanks to God and lovely wife.