A Matlab Toolbox for fMRI Data Analysis: Detection, Estimation and Brain Connectivity

A Matlab Toolbox for fMRI Data Analysis: Detection, Estimation and Brain Connectivity
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Master thesis
A Matlab Toolbox for fMRI Data
Analysis: Detection, Estimation and
Brain Connectivity
by
Kiran Kumar Budde
LiTH-ISY-EX--12/4600--SE
05-09-2012
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Master thesis
A Matlab Toolbox for fMRI Data
Analysis: Detection, Estimation and
Brain Connectivity
by
Kiran Kumar Budde
LiTH-ISY-EX--12/4600--SE
05-09-2012
Supervisor:
Dr. Sadasivan Puthusserypady, Department
of Electrical Engineering, DTU, Denmark.
Examiner:
Dr. Maria Magnusson, Department of Electrical Engineering, LiU, Sweden.
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Abstract
Functional Magnetic Resonance Imaging (fMRI) is one of the best techniques
for neuroimaging and have revolutionized the way to understand the brain
functions. It measures the changes in the blood oxygen level-dependent
(BOLD) signal which is related to the neuronal activity. Complexity of the
data, presence of different types of noises and the massive amount of data
makes the fMRI data analysis a challenging one. It demands efficient signal
processing and statistical analysis methods. The inference of the analysis
are used by the physicians, neurologists and researchers for better understanding of the brain functions.
The purpose of this study is to design a toolbox for fMRI data analysis. It
includes methods to detect the brain activity maps, estimation of the hemodynamic response (HDR) and the connectivity of the brain structures. This
toolbox provides methods for detection of activated brain regions measured
with Bayesian estimator. Results are compared with the conventional methods such as t-test, ordinary least squares (OLS) and weighted least squares
(WLS). Brain activation and HDR are estimated with linear adaptive model
and nonlinear method based on radial basis function (RBF) neural network.
Nonlinear autoregressive with exogenous inputs (NARX) neural network
is developed to model the dynamics of the fMRI data. This toolbox also
provides methods to brain connectivity such as functional connectivity and
effective connectivity. These methods are examined on simulated and real
fMRI datasets.
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Acknowledgement
I would like to express my deepest gratitude towards my supervisor Dr.
Sadasivan Puthusserypady for giving me an opportunity to work in the field
of functional magnetic resonance imaging. I would like to mention that his
support is invaluable and have patiently led me throughout my work. Without his constant support this thesis work would not be finished.
I would like to thank Dr. Luo Huaien for providing his algorithms for
fMRI Matlab toolbox design and sharing his knowledge during the project.
I thank my examiner Dr. Maria Magnsson and Dr. Göran Salerud for
the thesis proposal and allowing me to do the thesis at the Technical University of Denmark (DTU). I appreciate the help from Dr. Göran Salerud
the starting of my master’s programme.
I would like to thank all those who provided invaluable advices and the
help during my masters at Linkoping University and Technical University
of Denmark.
Last but not least, I would like to say that the support from my family
is immeasurable under all the circumstances.
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Abbreviations
AIC
AR
ARX
BOLD
DCT
DWT
EEG
EPI
fMRI
FPR
GLM
GLM
HDR
HRF
ISI
LMS
LS
MLP
MVAR
MR
MRI
NARX
OLS
PET
RBF
ROC
SNR
SPM
TPR
WLS
Akaike Information Criterion
Autoregressive
Autoregressive Model with Exogenous Inputs
Blood Oxygenation Level Dependent
Discrete Cosine Transform
Discrete Wavelet Transform
Electroencephalography
Echo Planar Imaging
Functional Magnetic Resonance Imaging
False Positive Ratio
General Linear Model
Graphical User Interface
Hemodynamic Response
Hemodynamic Response Function
Inter-Stimulus Intervals
Least Mean Square
Least Squares
Multi-Layer Perceptrons
Multivariate autoregressive
Magnetic Resonance
Magnetic Resonance Imaging
Nonlinear Autoregressive with Exogenous Inputs
Ordinary Least Squares
Positron Emission Tomography
Radial Basis Function
Receiver Operator Characteristic
Signal-to-Noise Ratio
Statistical Parametric Mapping
True Positive Ratio
Weighted Least Squares
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Contents
1 Introduction
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . .
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2 Background
2.1 Functional magnetic resonance imaging . . . . . .
2.1.1 BOLD signal generation . . . . . . . . . . .
2.1.2 Hemodynamic response (HDR) . . . . . . .
2.1.3 fMRI Experimental design . . . . . . . . . .
2.1.4 fMRI experimental data used in this thesis
2.2 fMRI data analysis . . . . . . . . . . . . . . . . . .
2.2.1 Preprocessing . . . . . . . . . . . . . . . . .
2.3 fMRI data modeling . . . . . . . . . . . . . . . . .
2.3.1 Temporal modeling . . . . . . . . . . . . . .
2.3.2 BOLD Model . . . . . . . . . . . . . . . . .
2.3.3 Noise and Drift . . . . . . . . . . . . . . . .
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3 Detection
3.1 Loading and visualization . . . . . . . . . .
3.1.1 Display . . . . . . . . . . . . . . . .
3.1.2 Loading real fMRI dataset . . . . . .
3.1.3 Loading simulated fMRI dataset . .
3.1.4 Loading stimulus . . . . . . . . . . .
3.2 Detection of activated brain regions . . . .
3.2.1 Flexible design matrix . . . . . . . .
3.2.1.1 t-test . . . . . . . . . . . .
3.2.1.2 Flexible design matrix with
method . . . . . . . . . . .
3.2.2 Nonstationary noise models . . . . .
3.2.2.1 OLS estimator . . . . . . .
3.2.2.2 WLS estimator . . . . . . .
3.2.2.3 Bayesian estimator . . . . .
3.2.3 Drift model . . . . . . . . . . . . . .
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CONTENTS
4 Estimation
4.1 Estimation of the hemodynamic response (HDR)
4.1.1 Adaptive spatiotemporal modeling . . . .
4.1.2 Neural Network . . . . . . . . . . . . . . .
4.1.3 NARX model . . . . . . . . . . . . . . . .
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5 Brain connectivity
5.1 Nonlinear cross correlation . . . . .
5.2 Granger causality . . . . . . . . . .
5.3 Results . . . . . . . . . . . . . . . .
5.3.1 Nonlinear cross correlation
5.3.2 Granger causality . . . . . .
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6 Conclusion and Future work
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6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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Chapter 1
Introduction
1.1
Introduction
Brain is the most fascinating, mysterious and least understood organ of the
human body. For the last few years, functional brain imaging techniques
have been advanced tremendously. For understanding the brain functions
and brain mappings, a powerful tool like functional magnetic resonance
(fMRI) can be used. fMRI measures the changes in blood oxygen leveldependent (BOLD) signals which are related to neural activity [1].
The purpose of this thesis is to develop a Matlab toolbox for the fMRI
data analysis. It includes methods to detect the brain activation, estimation
of hemodynamic response (HDR) and the connectivity of brain structures.
The major features of the toolbox are:
• To construct a flexible design matrix in the general linear model (GLM)
under the Bayesian framework. The Bayesian approach is extended
to nonstationary noise and drift models. These frameworks provide
accurate detection and avoid multiple comparison problems in conventional methods. This estimator detect more real activation of simulated and real fMRI datasets when compared with the traditional
methods such as t-test, ordinary least squares (OLS) and weighted
least squares (WLS).
• To provide methods for estimation of brain activation and HDR for
linear and nonlinear properties of event related designs. The linear
and non-linear properties are based on the inter-stimulus interval (ISI).
When the ISI is small, it shows non-linear properties.
• Linear adaptive spatiotemporal modelling to estimate the HDR.
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1.2. OUTLINE OF THE THESIS
• Nonlinear method based on radial basis function (RBF) neural network
to detect the spatial activation.
• Nonlinear autoregressive with exogenous inputs (NARX) neural networks to model the dynamics of fMRI data.
• It describes methods on functional connectivity analysis of brain regions using non-linear cross correlation analysis. Moreover, it measures the directional interactions between spatially separated neural
populations by using the Granger causality.
1.2
Outline of the thesis
Chapter 2: Briefly explains fMRI basics and discuss various aspects which
are related to this project.
Chapter 3: This chapter describes the design of the toolbox including visualization and loading of experimental real and simulated data. Different
brain activity detection methods are explained in the presence of noise and
drift.
Chapter 4: This chapter provides different nonlinear methods for detection
of brain activation and estimation of the HDR.
Chapter 5: This chapter describes methods to investigate brain connectivity structures: functional connectivity and effective connectivity. The
functional connectivity is investigated using the nonlinear cross correlation
and effective connectivity is investigated using the Granger causality.
Chapter 6: This chapter gives the conclusion of the thesis and the scope
for future work.
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Chapter 2
Background
2.1
Functional magnetic resonance imaging
The fMRI is one of the most advanced neuroimaging techniques which uses
the standard magnetic resonance imaging (MRI) to examine the brain functions. It is a widely used technique because of its better spatial resolution
when compared to electroencephalogram (EEG) and better temporal resolution compared to positron emission tomography (PET). Moreover, it is
non-invasive, gives non-ionizing radiation and has high sensitivity [2].
2.1.1
BOLD signal generation
As shown in Figure 2.1, when neural activity in the brain increases, neurons
consume more oxygen and demand more oxygen. This results in increased
blood flow at the activated areas. As a result, the oxygen concentration
increases and decreases the deoxyhemoglobin. The changes in oxygen concentration level alters the main magnetic field because the oxyhemoglobin
is diamagnetic and deoxyhemoglobin is paramagnetic. T2* 1 time becomes
shorter at low oxygen concentration and higher at high oxygen concentration areas. Hence, the MR signal depends on oxygen level. This effect is
referred as the Blood Oxygen Level Dependent (BOLD) effect [3].
1 MR
images contrast determined by properties of tissue being imaged, different tissues
have different relaxation times T1,T2 and T2*. T2* relaxation time is time constant that
describes the exponential decay of signal, due to spin-spin interactions, magnetic field
inhomogeneities and susceptibility effects. T2* weighted imaging is commonly used in
fMRI [4].
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2.1. FUNCTIONAL MAGNETIC RESONANCE IMAGING
Figure 2.1: Block diagram of BOLD signal generation.
2.1.2
Hemodynamic response (HDR)
The change in MR signal on T2* images triggered by the neuronal activity
is known as the HDR. It can result from the reduction in the amount of deoxygenated blood and it represents temporal properties of brain. Figure 2.2
shows the sketch of a typical HDR. Its shape varies with activation: amplitude increases with rate of neural activity and width increases with increase
in duration of the neuronal activity [5]. The HDR can be summarized in
three phases [6].
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2.1. FUNCTIONAL MAGNETIC RESONANCE IMAGING
Figure 2.2: Schematic representation of HDR.
• Intial dip: During the initial short time (1-2 sec), the MR signal decreases below baseline after beginning of neural activity. It is caused
by the transient increase in deoxyhemoglobin due to the oxygen consumption.
• Overcompensation: After the initial dip, more oxygenated blood
is supplied to the area than extracted and deoxygenated hemoglobin
decreases due to the increase of neuronal activity and results in MR
signal increase above baseline at about 2 to 5 seconds.
• Undershoot: After finishing neuronal activity, the MR signal amplitude gradually decreases below the baseline level and reaches the
baseline level due to a combination of decreased blood flow and increased blood volume.
2.1.3
fMRI Experimental design
There are two schemes of experimental designs that are generally used for
fMRI experiments: block design and event-related design [7].
In Block design, stimuli of two or more conditions are presented repeatedly for an extended period of time. It is a simple, powerful and optimum
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2.1. FUNCTIONAL MAGNETIC RESONANCE IMAGING
design for detection of brain activity. It provides large signal to noise ratio (SNR). Figure 2.3 represents a schematic diagram of the BOLD signal
(blue color) and the block design experiment (magenta color). The square
waveform represent active (stimulus ON) and rest (stimulus OFF) of brain
activation. This is not the best design for temporal activity estimation. In
event related design, discrete and short duration events are presented one at
a time and separated by random ISI that can range depending on the experiment. Figure 2.4 shows event related BOLD signal with the corresponding
timing of events.
Figure 2.3: Block design BOLD signal.
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2.1. FUNCTIONAL MAGNETIC RESONANCE IMAGING
Figure 2.4: Event-related design BOLD signal.
2.1.4
fMRI experimental data used in this thesis
In this thesis, two types of experiments are examined. One is the block
design (DATA BLOCK) and another is the event related design (DATA
EVENT). The block design dataset was obtained from the Statistical Parametric Mapping (SPM) website [8].
DATA BLOCK
The real fMRI experiment was designed for the activation of the auditory
areas. The functional data or Echo Planar imaging (EPI)images acquired
using 2T Siemens MAGNETOM Vision system. The experiment data set
and its details are available on [8]. During the experiment, bisyllabic words
are presented binaurally. The first few scans are discarded due to T1 effects
then starts from fm00223 004 image. The total number of acquisitions are
96 and it is divided into 16 blocks and each block contains 6 acquisitions.
In this experiment alternate condition of the baseline (rest) and activation
are applied for an extended period of time.
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2.2. FMRI DATA ANALYSIS
2.2
fMRI data analysis
In fMRI scanning, the whole brain or part of the brain is scanned over time
period and generates 4D data or sequence of 3D MR images.
2.2.1
Preprocessing
The purpose of preprocessing is to remove the unwanted data to prepare for
the statistical analysis and enhance the brain activity mappings. There are
several steps to be followed [9]. In this thesis the following preprocessing
steps are performed by the SPM software [10]:
• Realignment: During the data acquisition, intensity of the voxel of
resting and activation assumes that no motion of subject has occurred.
The motion is caused by the subject movement due to long period of
scanning. This problem can be reduced by the realignment procedure
with estimation and correction [11].
• Co-registration: The low resolution fMRI images are aligned with
the anatomical MRI images, which can be images of the same subject
or a standard template [12].
• Normalize: It is a procedure to wrap the functional data onto a
coordinate system or template space [13].
• Smoothing: The fMRI images are smoothed across adjacent voxels
to improve the results of brain mapping and to increase the SNR [14].
2.3
fMRI data modeling
2.3.1
Temporal modeling
From an engineering point of view, the fMRI data analysis can be considered
as the analysis of the response of the system to a given input. As shown in
Figure 2.5, the system is the human brain and measuring device, input is
the experimental design and output is the observed BOLD signal.
Figure 2.5: fMRI data acquisition system with input and output.
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2.3. FMRI DATA MODELING
To understand the complexity of fMRI data, spatial and temporal properties are required. The measured fMRI signal or voxel time series is the
combination of BOLD signal, drift and noise, i.e. the change in voxel intensity represent whether the BOLD signal is active or not.
Measured fMRI time series = BOLD signal + drift + noise.
2.3.2
(2.1)
BOLD Model
This model assumes a linear system, i.e. the BOLD response expresses the
convolution of the stimulus function and impulse response [15],
Z ∞
yb (t) = h(t) ⊗ s(t) =
h(τ )s(t − τ )dτ,
(2.2)
0
where ⊗ denotes the convolution operation, s(t) is the stimulus function
and h(t) is impulse response function and it is called hemodynamic response
function (HRF). The HRF waveform is modelled by poisson, Gaussian function and difference of gamma functions [16]. Here, the widely used difference
of gamma function model is used and it can be represented by,
t a2
t a1
−(t − d1 )
−(t − d2 )
exp
exp
−c
,
h(t) =
d1
b1
d2
b2
(2.3)
where di = ai bi represents the amplitude of the peak and c represents the undershoot. The normally used parameters are a1 = 6, a2 = 12, b1 = b2 = 0.9,
and c = 0.35 [17].
2.3.3
Noise and Drift
The BOLD signal is influenced by strong random noise which results in low
SNR. The sources of noise are scanner induced noise and subject movements
during scanning such as head and lower jaw movements. The residual of the
imperfect model is also one of the noise components [15].
Drift is another component that disturbs the fMRI time series. It shows
up as a slow varying interference or trend in the fMRI time series. This
drift often comes from physiological processes like the cardiac and inspiration process [18] as well as from instability of the magnetic field [19]. The
common way to remove the drift in fMRI time series is either by using a
high-pass filter or a drift model.
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Chapter 3
Detection
This fMRI Matlab toolbox provides different methods for detection of activated brain regions and estimation of the HDR. The proposed toolbox
consists of four functionalities. The first section describes how to display
the cross sectional brain images and to load the experimental data. The second section provides different brain detection methods. The third section
provides methods for estimation of the HDR and the fourth section describes
brain connectivity algorithms. Figure 3.1 shows the graphical user interface
(GUI) for the designed toolbox.
The first section explains the display of MRI cross sectional images and
loading of real and simulated fMRI datasets. Second section explains determination of flexible design in the GLM through the Bayesian learning
procedure and it provides accurate activation compared with conventional
t-test method. The Bayesian estimator is extended to non-stationary noise
model and it provides accurate activation results when compared to OLS
and WLS. Moreover, the Bayesian estimator can be extended to remove
drift component present in the fMRI time series. Studies on simulated and
real fMRI data show that the Bayesian estimator accurately detects the
brain activated regions to specific task. The detection and estimation algorithms developed by Dr. Luo Huaien have been adopted in the design of
this toolbox.
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Figure 3.1: This toolbox GUI comprises 3 windows, i) Main window, ii)
Subwindow, iii) Graphical window. The main window consists four sections,
i) Processing and visualization, ii) Detection of activated brain regions, iii)
Estimation of HDR and iv) Analysis of brain connectivity.
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3.1. LOADING AND VISUALIZATION
3.1
3.1.1
Loading and visualization
Display
The fMRI datasets are provided in the analyze file format. It is an image
file format for storing MRI data and consists of two types of files, an image
file and a header file, with .img and .hdr extensions, respectively. An image
file contains uncompressed pixel data or a set of cross sectional images. The
header file contains history and dimensions of the data. Before the fMRI
data analysis starts, the dataset is preprocessed by the SPM software to
apply realignment, coregistraton, normalization and smoothing. In Figure
3.2, the graphical window displays the brain cross-sectional images and the
subwindow provides information about the images.
Figure 3.2: Display of brain cross-sectional images.
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3.1. LOADING AND VISUALIZATION
3.1.2
Loading real fMRI dataset
To load real fMRI data set, the toolbox requires series of MRI scanned data
or session data1 , which are stored in the analyze data format (hdr and img).
Figure 3.3 shows the result of loaded auditory fMRI activation data. It
also provides dataset information such as the number of slices, image and
voxel dimensions. The graphical window shows only one image and other
data details. The loaded image files in Figure 3.3 are from
swrfM00223 016.img, swrfM00223 016.hdr upto swrfM00223 099.img,
swrfM00223 099.hdr.
Figure 3.3: Loading real fMRI dataset.
3.1.3
Loading simulated fMRI dataset
To load simulated data, simulate the activated regions at particular positions of one image slice, by adding the BOLD signal repeatedly over time.
As shown in Figure 3.4, the BOLD signal (block design or event related)
is added at particular positions in the image. It is also possible to simulate fMRI data with noise, such as time varying and fractional noise. In
this thesis, two types of datasets are simulated: DATA BLOCK and DATA
EVENT. DATA BLOCK is simulated with block design signal and DATA
EVENT is simulated with event related signal.
1 The session is a scanning period. During the fMRI scanning the patient brain is
scanned for extended time period in several sessions and then it acquires series of the MR
images, where each session consists of MR images sequences.
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3.1. LOADING AND VISUALIZATION
In GUI, select the Load Simulated button. It opens a subwindow. Then
choose the type of design (DATA BLOCK or DATA EVENT), type of noise
(time varying or fractional) and then simulate it. In the graphical window,
simulated data with activated positions are displayed (Figure 3.4) and noisy
images are displayed in Figure 3.5.
Figure 3.4: Loading simulated fMRI dataset with active positions.
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3.1. LOADING AND VISUALIZATION
Figure 3.5: Simulated image with time varying noise.
3.1.4
Loading stimulus
To load the stimuli or experimental paradigm, we need to have knowledge
about the experiment. For block or event-related design experiments, several
values must be entered: active and rest block length, number of blocks,
repetition time (TR), signal duration and duration of rest. Block design
experiment data and information are available in [4]. Figure 3.6 shows the
block design paradigm of auditory experiment and Figure 3.7 shows the
event-related paradigm.
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3.1. LOADING AND VISUALIZATION
Figure 3.6: Block design paradigm.
Figure 3.7: Event-related paradigm.
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3.2. DETECTION OF ACTIVATED BRAIN REGIONS
3.2
3.2.1
Detection of activated brain regions
Flexible design matrix
This section briefly explains the general linear model (GLM). Moreover, the
construction of a design matrix in GLM using Bayesian analysis is considered
and compared with the conventional methods like the t-test [20].
3.2.1.1
t-test
The t-test is a conventional test used in brain mapping. It compares the average active condition with the average rest condition. The t-test is defined
as [21]
X1 − X2
t= q 2
,
S1
S22
+
n1
n2
(3.1)
where X 1 , S12 and n1 are the mean, variance and sample number for the
active fMRI samples. For the resting fMRI time samples X 2 , S22 and n2 are
the mean, variance and sample number.
Figure 3.8: Detection results of simulated fMRI data using the t-test
method.
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3.2. DETECTION OF ACTIVATED BRAIN REGIONS
Figure 3.9: Detection results of real fMRI data using the t-test method.
Figure 3.8 and 3.9 show the detection results of the t-test method applied
to the simulated and real fMRI datasets. It can be clearly observed from
that the t-test method also detects false detection. Therefore for the real
fMRI data, analyzing the performance with this method makes it difficult.
This is because we lack a reference which could serve as the true activation
of the brain.
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3.2. DETECTION OF ACTIVATED BRAIN REGIONS
3.2.1.2
Flexible design matrix with sparse Bayesian method
General linear model (GLM)
The GLM is a fundamental method for fMRI data analysis. In this, the
measured fMRI time series is described as the combination of regressors and
their vector parameters [22].
y = Φw + ,
(3.2)
where y is an N × 1 measured fMRI time series, Φ is an N × P matrix known as the design matrix, w is a P × 1 vector of estimated unknown
parameters and is N ×1 noise vector. The design matrix contains all available knowledge about the experiment. In the design matrix each row (N )
represents one time point (one scan) of the regressor and column (P ) represents the corresponding regressor or explanatory variable like the canonical
BOLD response, constant one vector and discrete cosine transform (DCT)
basis functions [23].
Sparse Bayesian Learning Algorithm
In the classical approach, the design matrix is not flexible which may lead
to problems. More regressors leads to model over-fitting problems and few
basis functions cannot filter out the noise. The selection of regressors in the
design matrix is a critical issue. The Bayesian approach is used to construct
a flexible design matrix. During the learning procedure, the data itself estimate the regressors. It determines which regressors are useful and ignores
the ones that are irrelevant [20].
This algorithm is developed based on the assumption that the noise is
stationary. Initially, the design matrix is initialized with the BOLD signal, a
constant vector of value 1 and radial basis functions. Throughout the learning procedure, the regressors and their weight coefficients are estimated and
irrelevant regressors are discarded.
Figures 3.10 and 3.11 show that the Bayesian algorithm provides better
results for both simulated and real fMRI datasets. The performance is better
than the t-test method. Both the t-test and Bayesian estimator results
clearly show that auditory activated areas are identified. As mentioned
before, the sparse Bayesian algorithm assumes that the noise is stationary.
However, the fMRI data also contains nonstationary noise which could be
removed by nonstationary noise models.
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3.2. DETECTION OF ACTIVATED BRAIN REGIONS
Figure 3.10: Detection results of simulated fMRI data using the sparse
Bayesian learning method.
Figure 3.11: Detection results of real fMRI data using the sparse Bayesian
learning method.
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3.2. DETECTION OF ACTIVATED BRAIN REGIONS
By observing the results on real fMRI data, it is clearly seen that the auditory cortical areas has been identified. The Bayesian estimator detects the
more real activated brain regions compared to conventional t-test method.
Figure 3.12: ROC curve of simulated data of the t-test and the Bayesian
estimator.
Receiver operator characteristic (ROC) analysis is used to investigate
clear comparison of detection ability of the t-test and the Bayesian estimator
methods. The ROC curve plots the true positive ratio (TPR) against the
false positive ratio (FPR) [24]. From Figure 3.12, we can say that, the
Bayesian estimator is better compared to t-test.
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3.2.2
Nonstationary noise models
The noise sources are scanner induced noise, physical movements of the subject such as lower jaw movements, and neurophysiological processes such as
number of neurons involved in specific tasks at different time points and
the background memory process. The noise sources may cause the noise
variance to be changed [25]. The aim of fMRI data analysis is to distinguish
the activated BOLD signal from noisy data and to determine the activated
regions for particular tasks. The high amount of noise in fMRI makes it
difficult and challenging for data analysis.
This section briefly explains the Bayesian estimator for detection of activated regions in the case of time varying noise. The performance is compared
with the OLS estimator and the WLS estimator [26].
3.2.2.1
OLS estimator
As introduced in Section 3.1, the General linear model is defined as
y = Φw + ,
(3.3)
where yn and n are the observed fMRI time series and noise at the nth voxel
respectively. n is assumed to be independent and identically distributed
(i.i.d) white noise. The least square estimator of the parameter w is defined
as
ŵn(OLS) = (ΦT Φ)−1 ΦT yn .
(3.4)
The assumption of i.i.d white noise is an inappropriate assumption regarding the nonstationary nature of fMRI signals. The design matrix should
be a full rank square matrix. If the design matrix is a rectangular matrix,
the matrix inversion is not possible. Therefore in order to find the matrix
inversion, firstly we multiply the design matrix with its transpose, which
results in a square matrix [22]. The activated voxels are calculated by using
static student distribution t-test
cT ŵ
t= p
,
cT Λŵ c
(3.5)
where c is the contrast vector and Λŵ is the covariance matrix of the estimated parameter ŵ.
The Figure 3.13 shows the detection results of a simulated fMRI data
using the OLS method in presence of time varying noise, here, the noise
range is from -0.4 to 6.7 dB. The OLS method detects the activated regions
as well as false detection. From Figure 3.14, it is clearly shows the detection
of auditory cortical areas by using this method.
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Figure 3.13: Detection results of simulated fMRI data using the OLS estimator.
Figure 3.14: Detection result of auditory processing fMRI data using the
OLS estimator.
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3.2.2.2
WLS estimator
The noise variance of time varying noise model is stated to increases in a
multiplicative fashion [27]. For this reason the noise variance is modeled
with a scaling matrix and overall variance. The noise model is assumed to
be time dependent, with precision matrix Bn . The precision matrix is the
inverse of the covariance matrix and it is defined as [25],


s1 0 . . . 0
 0 s2 . . . 0 


(3.6)
Bn =  .
.. . .
.  βn = Sβn ,
 ..
. .. 
.
0
0
...
sT
where s1 , s2 , ..sT are the scaling parameters and T is the number of samples
in the fMRI time series. Here, S is the scaling diagonal matrix and Bn is
the noise precision at nth voxel.
The WLS estimator of parameter w is defined as
ŵn(WLS) = (ΦT SΦ)−1 ΦT Syn .
(3.7)
The WLS estimator requires accurate estimation of the scaling matrix. In
traditional methods, the residual of the OLS estimator is used. The residual
of the OLS estimator is given by
rn = yn − Φŵn(OLS) = yn − Φ(ΦT Φ)−1 ΦT yn .
(3.8)
The overall precision of the nth voxel time series and the inverse scaling
matrix are calculated as in [17],
β̂n =
T − rank(Φ)
,
rTn rn
(3.9)
PN
diag(βn rn rTn )
,
(3.10)
N
where, N is the total number of voxels and the operator diag(.) creates the
diagonal of a square matrix into a column vector.
ŝinv =
n=1
This method is simple to implement. The residual is equal to Sβn and
then is considered as unbiased estimator. The WLS method uses the residuals of the OLS method as a covariance matrix to estimate the scaling matrix.
It fails to estimate an accurate scaling matrix. The time varying Bayesian
estimator, however, is able to accurately find the scaling matrix.
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Figure 3.15: Detection results of simulated fMRI data using the WLS estimator.
Figure 3.16: Detection result of auditory processing fMRI data using the
WLS estimator.
Figure 3.15 shows that the detection results of a simulated fMRI data
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3.2. DETECTION OF ACTIVATED BRAIN REGIONS
in the presence of a time varying noise. It is clearly observed that the WLS
method detects the false detection too. The Figure 3.16 shows the detection
result of auditory cortical areas using the WLS estimator.
3.2.2.3
Bayesian estimator
From the Figures 3.13 and 3.15 the OLS and WLS estimators for the simulated data doesn’t give accurate result due to: i) The OLS estimator assumes
i.i.d noise and does not consider time varying noise. ii) The WLS estimator
requires accurate estimate of the noise covariance matrix. It uses the residual of a traditional method (e.g. OLS) to estimate. It is difficult to perform
an accurate estimation of the covariance matrix.
In the time varying noise model, the noise covariance matrix is a diagonal matrix. This model is spatially and temporally non-stationary. This
property is investigated by the Bayesian method. In the previous section,
the Bayesian method was used to estimate the parameters w under the
assumption of constant variance. In this section, the Bayesian method is
extended to variance of noise changing with time, which gives an estimated
noise structure closer to the true noise [26].
Figure 3.17: Detection results of simulated fMRI data using the Bayesian
estimator.
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3.2. DETECTION OF ACTIVATED BRAIN REGIONS
Figure 3.18: Detection results of activation of auditory fMRI dataset using
Bayesian estimator.
By observing Figure 3.17, it can be seen that the Bayesian estimator
detects more true activations and less false activations in the presence of a
time varying noise. By comparing Figures 3.13, 3.14, 3.15, 3.16, 3.17 and
3.18 it can be stated that the results of the Bayesian estimator is more accurate than the OLS and WLS methods.
In this section, the Bayesian estimator does not consider the drift in
fMRI data. In the following section, this method is extended to drift model.
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3.2.3
Drift model
The aim of fMRI data analysis is to detect the activated human brain regions based on inference drawn from the estimated parameter in GLM.
Drift is a low frequency signal that slowly varies across the complete period of the signal. The causes of drift are head movement during acquisition,
local changes of magnetic field and internal physiological process like cardiac and respiration movements. Drift can be removed with a preprocessing
procedure before statistical analysis, e.g. highpass filtering or the median
method. It can also be removed by the drift model [28].
The measured fMRI signal contains three types of parameters: BOLD
response, noise and drift. The GLM of measured data is
y = wb + f + ,
(3.11)
or based on description in Section 3.2.1.2,traditional way of GLM
y = Φw + ,
(3.12)
where y is the measured time series, f is the drift and is the noise with
dimensions T × 1. Here, the parameter w is the scalar which represents the
contribution of BOLD signal is to the measure the fMRI time series. The
design matrix Φ = [b, f ] with dimension T × 2 and estimated parameter
w = [w, 1].
The fMRI data exhibits fractional noise also. In the drift model, first apply the discrete wavelet transform (DWT) to the fMRI data. The resulting
wavelet coefficients at lower than fine scale (J0 ) are zero [29], because the
drift does not vary significantly below J0 . The J0 value can be estimated
by using the Confidence Interval Criterion (CIC) as model selection criterion. It is more efficient than the Akaike Information Criterion (AIC) and
the Schwartz Information Criterion (SIC). The activation parameters are accurately estimated by using the Bayesian estimator from the modified GLM.
Figure 3.20 shows that the detection results of a simulated fMRI data
by using Bayesian estimator. It can be effectively estimated and removed
drift from fMRI data and detect true activations and less false detections.
Figure 3.19 shows the result of activated detection. It is clearly seen that the
activation of auditory cortical areas of the brain are successfully detected.
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3.2. DETECTION OF ACTIVATED BRAIN REGIONS
Figure 3.19: Detection result of Drift model on the auditory activation task.
Figure 3.20: Detection result of Drift model on the simulated fMRI dataset.
In this chapter, methods are discussed on the detection of the activated
regions in the brain. In the following chapter, methods to estimate the HDR
and measure the spatial activation for nonlinear properties of fMRI data are
discussed.
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Chapter 4
Estimation
4.1
Estimation of the hemodynamic response
(HDR)
The HDR can be used to study the human brain functions. It exhibits
temporal properties of human brain activities. Generally, the HDR can
be estimated through a signal averaging procedure by assuming that the
ISI is large [30], so that consecutive stimuli do not overlap. Overlapping
of consecutive stimuli leads to bias results of the averaging methods. This
method works only under the linear assumption case, i.e. ISI is larger than 46 sec, otherwise nonlinear phenomena are observed. In this chapter, I have
described linear adaptive modelling and nonlinear method based on RBF
neural networks to estimate the HDR and detection of brain activation.
More over in this chapter describes the NARX to model the dynamics of
fMRI data.
4.1.1
Adaptive spatiotemporal modeling
The fMRI data contains spatial and temporal information. In the adaptive
spatial temporal modeling, spatial information is also included for estimating the temporal activation. Because activated brain regions extend to several voxels, surrounding voxels are more likely to be activated compared to
other voxels. This spatial information can be used to improve the SNR and
detection accuracy [31].
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Figure 4.1: Schematic diagram of the spatial smoothing and temporal modelling.
Figure 4.1 shows a sketch of the spatial smoothing and temporal modelling. The spatial adaptive filter is defined as
d(n) =
L
X
wi yi (n) = wT y(n),
(4.1)
i=0
where y(n) = [y0 (n), y1 (n), . . . , yL (n)]T , y0 (n) is the nth sample of given
voxel fMRI time series and y1 (n), y2 (n) . . . yL (n) are L number of neighboring voxels time serieses, w = [w0 (n), w1 (n), . . . , wL (n)]T is the corresponding weight vector. The smoothed signal d(n) is the desired signal for
temporal modelling. For activated voxel time series, the resulted smoothed
signal is approximated to the ideal BOLD signal r(n) to the experimental
stimuli, i.e.
d(n) = wT y(n) = r(n) + u(n),
(4.2)
where r(n) is the ideal BOLD signal which is the result of stimulus function and canonical HDR and u(n) is the white noise. The output spatial
smoothed signal d(n) is used as the desired signal for temporal modeling.
The temporal modelling is defined as
d(n) =
P
X
s(n − m)hm + (n) = hT s(n) + (n),
(4.3)
m=0
where s(n) = [s(n), s(n − 1). . . s(n − P )]T is a vector corresponding to the
stimulus function with a delay of P and h = [h0 , h1 . . . hP ]T are the filter
coefficients.
The aim of this analysis is to find an optimum filter coefficients so that
hT s(n) is the resultant BOLD signal. The mean square errors of the spatial
and temporal filters are E{e21 (n)} = E{(r(n) − d(n))2 } and E{e22 (n)} =
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4.1. ESTIMATION OF THE HEMODYNAMIC RESPONSE (HDR)
E{(d(n) − hT s(n))2 }. The cost function is calculated as
J
= E{(r(n) − d(n))2 } + E{(d(n) − hT s(n))2 }
= E{(r(n) − wT y(n))2 } + E{(wT y(n) − hT s(n))2 }.
(4.4)
For the cost function J, the optimum filter coefficients are obtained at
minimum value of J. The least mean square (LMS) algorithm is used for
finding minimum mean square error by updating the following equations,
e1 (n)
e2 (n)
= r(n) − ŵT (n)y(n),
T
T
= ŵ (n)y(n) − ĥ (n)s(n),
(4.5)
(4.6)
ŵ(n + 1)
=
ŵ(n) + 2µ1 e1 (n)y(n) − 2µ1 e2 (n)y(n),
(4.7)
ĥ(n + 1)
=
ĥ(n) + 2µ2 e2 (n)s(n).
(4.8)
Figure 4.2: Schematic diagram of the adaptive spatio-temporal model.
Figure 4.2 shows the structure of the adaptive spatio-temporal model
and the estimation errors (e1 (n) and e2 (n))are feedback to spatial filter to
adjust the estimation coefficients. The estimation error (e2 (n)) is fed back to
the temporal modelling to adjust the estimation coefficients up to algorithm
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4.1. ESTIMATION OF THE HEMODYNAMIC RESPONSE (HDR)
converges. The estimated optimum spatial filter weights ŵ approximates the
spatial smoothing and the optimum temporal filter weights ŵ approximates
HDR. The adaptive spatio-temporal model is tested on the simulated BOLD
signal.
Figure 4.3: Estimation of HDR on the simulated fMRI signal.
Figure 4.3 shows the simulated BOLD signal, simulated noisy signal and
estimation of HDR.
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4.1.2
Neural Network
As mentioned before, the fMRI signal exhibits linear and nonlinear properties. It has been observed that when the duration of ISI is less than 4
to 6 sec, the event related fMRI signals exhibit nonlinear properties. The
nonlinear properties of the BOLD signal can be expressed by the balloon
model, including variables such as, blood flow, blood volume and oxygen
concentration. It is a physiologically derived model which was introduced
by Buxton et al [32]. It is a proper model for understanding the nonlinear
mechanism underlying the neural activity and BOLD effect. The balloon
model provides an approximate estimation of the parameters and it is not
easy to estimate the parameters.
On the other hand, non-physiological models were developed for better
estimation when the signal is highly non-linear. The Volterra series model
is a dynamical input output system developed to estimate the HDR using
first and second order kernels [16]. Higher the kernel orders, better the
estimates. The number of parameters increases exponentially with higher
order kernels. The performance depends upon the selected order of Volterra
series. The lower order kernels may not capture the dynamic properties of
the system as well as higher order kernels. Estimate of Volterra kernels by
using the least squares (LS) method is difficult when the kernel order is high.
However, the RBF neural network method efficiently estimates the HDR as
well as it avoids singularity problem in the LS method [33].
Radial Basis Function (RBF) neural network
Neural networks is a powerful method for nonlinear modeling [34]. As
shown in Figure 4.4, the present and past inputs of stimulus are applied to
the neural network system and the output signal is the desired BOLD signal.
The measured BOLD signal can be determined for given input stimulus s(n)
as
ŷ(n) = F N N (s(n)),
(4.9)
the output ŷ(n) is the nonlinear function of input s(n) and the nonlinear
function is denoted by F N N . s(n) is a (P + 1) × 1 vector containing the
present and past stimulus data with delays from 1 to P , s(n) = [s(n), s(n −
1) . . . s(n − p)]T .
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Figure 4.4: Schematic diagram of the RBF neural network.
RBF network has a simple structure and is easy to implement compared
to the multilayer perceptron of neural network. The output ŷ(n) is a linear
combination of M radial basis functions defined as
ŷ(n) =
M
X
hi G(s(n), ci ),
(4.10)
i=1
where hi is the weighting coefficients and ci is the center of the radial basis
function. Commonly used basis functions are
1
G(s(n), ci ) = exp − 2 ks(n) − ci k2 .
σi
(4.11)
The center of a RBF is determined randomly from the data, and variance
is selected according to the center spread. The aim of a RBF network is to
find the weights and sum of minimized least square error. Regularization
parameter is required for the stability of the solution. The weight vector is
estimated by
h = (GT G + λI)−1 GT y.
(4.12)
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For estimating the weight vector (h) and regularization, Bayesian learning approach is used and is iteratively updated using the following equations[35],
= β 2 (GT G + λI)−1 ,
1
ĥ =
ΣGT y,
β2
λ
γ = M − 2 trace(Σ),
β
ky − Gŵk2
β2 =
,
N −γ
γβ 2
λ =
,
ŵT ŵ
Σ
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
where β 2 is the noise variance. Finally the BOLD signal is computed as
ŷb = Gĥ.
(4.18)
Figure 4.5: Simulated BOLD signal, noisy BOLD signal and reconstructed
BOLD signal by using RBF neural network.
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4.1. ESTIMATION OF THE HEMODYNAMIC RESPONSE (HDR)
Figure 4.5 shows the original simulated BOLD signal, simulated noisy
signal with Gaussian noise and reconstructed BOLD signal by using RBF
neural network method.
The detection of activated brain regions the reconstructed BOLD signal
can be used to measures the activation of each voxel. The detection of
activation index (R) is defined as
R=
kŷb k
,
ky − ŷb k
(4.19)
where ŷb is the reconstructed BOLD signal and y is the measured fMRI
time series in a voxel.
Figure 4.6: Detection results of simulated fMRI data with the RBF neural
network model.
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Figure 4.7: Detection results of auditory fMRI data using the RBF neural
network method.
Figure 4.6 and Figure 4.7 show the result of activated brain regions of real
fMRI data and simulated fMRI data using the RBF neural network method.
By observing the results of auditory activation data, it can be clearly seen
that the RBF neural network method detects the auditory cortical areas in
the brain.
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4.1.3
NARX model
The nonlinear autoregressive with exogenous inputs (NARX) neural network
is a dynamical model for mapping the input-output dynamics of the BOLD
signal. It is a powerful nonlinear model with faster convergence and better
generalization ability [36].
Figure 4.8: Block diagram of the NARX model.
See Figure 4.8. In the NARX model, the measured fMRI signal y(n)
and the input signal s(n) are applied as input to the NARX neural network.
The input signal s(n) is applied to the NARX neural networks via q − 1
delays and the fMRI signal is applied to the NARX neural networks via p
delays. The output estimated signal can be determined as
ŷb (n) = F N N (y(n), s(n)) = F N N (x(n)).
(4.20)
The estimated signal is a nonlinear transformation of the input signals s(n)
and y(n). The input signal vector x(n) = [y(n), s(n)]T has the size (p+q)×1.
The output BOLD signal is estimated by the NARX model.
This method was developed through the RBF neural network, explained
in Section 4.1.2. The BOLD signal is reconstructed after estimation of the
weights. After reconstructing the BOLD signal, the following test can be
defined for detection index(R) of activation at each voxel
R=
kŷb k
,
ky − ŷb k
39
(4.21)
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4.1. ESTIMATION OF THE HEMODYNAMIC RESPONSE (HDR)
where ŷb is the reconstructed BOLD signal and y is the measured fMRI
time series in a voxel.
Figure 4.9: Detection results of auditory fMRI data using the NARX model.
From Figure 4.9, auditory activated areas can be detected by using the
NARX model.
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Chapter 5
Brain connectivity
The brain can be described as an organized structural network. Brain connectivity refers to anatomical connectivity, functional connectivity and effective connectivity [38]. Anatomical connectivity is physical or structural
connections between neurons or anatomical brain regions. Functional connectivity is a statistical concept, which represents the statistical dependency
among different brain structures. Effective connectivity is the causal interaction between different brain structures [39].
In this chapter, in the first section the functional connectivity is investigated using the nonlinear cross correlation analysis. In the second section,
effective connectivity is investigated using the Granger causality.
5.1
Nonlinear cross correlation
The nonlinear cross-correlation coefficient describes the dependency of x on
y without any specific relation between them. Here, x and y are the two time
serieses. If the value of x is considered as function of y, then the value of y
gives x and it can be predicted [40]. The nonlinear correlation coefficient is
defined as,
h2y/x
PN
=
k=1
PN
y 2 (k) − k=1 (y(k) − f (x))2
,
PN
2
k=1 y (k)
(5.1)
where f (x) is the approximation of the nonlinear method. In this work,
f (x) is approximated with an RBF neural network. As explained in section
4.1.2, the present and past inputs of the time series x(n) is used as input for
the RBF neural network. The output signal y(n) is the nonlinear function
of input x(n) vector. The output signal can be expressed as [33]
y(n) = F N N (x(n)),
41
(5.2)
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5.2. GRANGER CAUSALITY
where y(n) = [y(n), y(n − 1) . . . y(n − p)]T is the time series y(n) vector of
dimension (p + 1) × 1 formed by the present and past inputs with delay p.
The output ŷ(n) is taken to be nonlinear function of the basis functions, it
can be defined as
M
X
ŷ(n) =
hi G(x(n), ci )
(5.3)
i=1
where hi are the weighting coefficients, ci are the centers of the radial basis
functions and M is the hidden units of the RBF network. The weights can
be estimated through the Bayesian learning procedure. When the weights
are estimated, the functional approximation function can be expressed as
ŷ = Gĥ.
(5.4)
The range of h2y/x is between 0 to 1. The value 0 means that the signal
y is independent of x. The value 1 represents that the signal y can be fully
determined by the signal x [40]. The above process is applied to all voxels
fMRI time series to the selected time series.
5.2
Granger causality
According to Norbert Wiener,“If the prediction of one time series can be
improved by inclusion of the knowledge about other time series, then second time series is said to have causal influence on the first time series” [41].
However, this idea lacks the practical implementation. Later Granger proved
practically, “If the error variance of one time series reduced by inclusion of
lagged observation of second time series, then second time series has causal
influence on first time series” [42].
Granger causality method is based on multiple linear regression for determine whether a signal useful for forecasting another signal or not. According
to Granger causality a time series x (or y) Granger causes a time series y
(or x) if past values of x helps to predict the present value of y with better
accuracy compared to considering its past values alone [43].
Jointly, two time series x and y can be represented by a bivariate regressive model
x(n) =
P
X
a2j x(n − j) +
j=1
y(n) =
P
X
j=1
P
X
b2j y(n − j) + ε1 (n),
(5.5)
d2j y(n − j) + ε2 (n),
(5.6)
j=1
c2j x(n − j) +
P
X
j=1
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5.2. GRANGER CAUSALITY
In Equation (5.5) and (5.6), P is the maximum number of lagged observations included in the model and n is the number of samples in signal.
Generally P < n. The variables a, b, c and d are the coefficients of the
model and ε1 and ε2 are the residual (prediction) errors for the each time
series. If the variance of ε1 (or ε2 ) is reduced by the inclusion of y (or x)
terms in the first (or second) equation, then it is said that y (or x) G-causes
x (or y).
The magnitude of this interaction can be measured by using the log ratio
of the prediction error variances for the restricted (R) and unrestricted (U)
models,
var(ε1R )
.
(5.7)
fy→x = ln
var(ε1U )
The restricted model can be defined as
x(n) =
P
X
a1j x(n − j) + ε1R (n),
(5.8)
d2j y(n − j) + ε2R (t),
(5.9)
j=1
y(n) =
P
X
j=1
and the unrestricted model can be defined as
x(n) =
P
X
a2j x(n − j) +
j=1
y(n) =
P
X
j=1
P
X
b2j y(n − j) + ε1U (n),
(5.10)
d2j y(n − j) + ε2U (n).
(5.11)
j=1
c2j x(n − j) +
P
X
j=1
Model order
The estimation of multivariate autoregressive (MVAR) models requires
the number of time-lags (P ) to include, i.e., the model orders. When the
lags are lesser or too many these can lead to a poor representation or problems in model estimation. A principled means to specify the model order
to minimize the criteria that balance the variance accounted against the
number of coefficients to be estimated. They can be implemented on two
criteria’s, which are Akaike Information Criterion (AIC) and the Bayesian
Information Criterion (BIC) [43]. The BIC is more useful in application to
the neural systems and it is defined as
BIC = n. ln(σe2 ) + k. ln(n),
(5.12)
where σe2 is the error variance and k is the number of parameters to be
estimated.
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5.3. RESULTS
5.3
5.3.1
Results
Nonlinear cross correlation
Simulated fMRI data
First the proposed method was tested on simulated data. A simulated
block design 3-D fMRI data set was investigated in this study. A block
design BOLD signal was added at particular positions of the image, and
then white Gaussian noise was added to the simulated signals with SNR
values from 0.49 to 8.29 dB. Figure 5.1 shows the functional connected regions (blue color) with selected time series (yellow color).
Figure 5.1: Simulated image with activated brain regions (left) and functional connectivity of brain structures (blue pixels) of selected time series
(yellow pixel) (right).
Real fMRI data
The auditory experimental data was used for investigation of functional connectivity of the brain structures. First, the activated brain regions was found
by using one of the methods for detection of activated brain region, such
as OLS, WLS or the Bayesian estimator (described in section 3.2). Then a
time series voxel of interest was selected to map the functional connectivity.
Figure 5.2 shows the reconstructed BOLD signal of functionally connected
and unconnected voxel time serieses by using RBF neural network method.
Figure 5.3 shows the functional connectivity of brain structures (dark-blue
color) of the selected time series voxel (yellow color).
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5.3. RESULTS
Figure 5.2: Time courses of functionally connected voxel time series and
functionally unconnected time series with selected time series for the real
fMRI data.
Figure 5.3: Auditory activated brain regions (left) and functional connectivity (dark-blue pixels) of selected time series (yellow pixel)(right).
By observing the results of the functional connectivity of brain regions
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5.3. RESULTS
the selected time series (yellow color voxel) is functionally connected with
blue color voxel time series for auditory areas activation. The left auditory
cortical areas is functionally connected with right auditory cortical areas
during auditory activation.
Figure 5.4: Curve of nonlinear cross correlation values for different SNR
values.
Figure 5.4 shows the nonlinear cross correlation values of two simulated
time series for different noises. One of the time series is added with white
Gaussian noise with different SNR values from −10 dB to 30 dB. The value
of nonlinear cross correlation value without noise is 1. The nonlinear cross
correlation is linear when SNR is changed from −10 dB to −2 dB, and then
it varies nonlinearly until it reaches a maximum value (1), which is value for
nonlinear cross correlation when there is no noise added.
5.3.2
Granger causality
Four simulated time series x1 -x4 signals were generated by using autoregressive (AR) modeling. The relations between these signals are that x2 is
caused by x1 (x1 causes x2 ) and x3 is caused by x4 (x4 causes x3 ), see Figure 5.5. These signals are added at particular positions of images containing
Gaussian noise.
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5.3. RESULTS
The simulated time series signals were
√
x1 (n) = 0.95 2x1 (n − 1) − 0.9025x1 (n − 2) + w1 (n),
x2 (n)
=
0.5x1 (n − 2) + w2 (n),
(5.13)
(5.14)
x3 (n)
= −0.4x4 (n − 3) + w3 (n),
(5.15)
x4 (n)
=
0.35x4 (n − 2) + w4 (n).
(5.16)
Figure 5.5: Simulated image with causal relations of brain structures.
The proposed method maps the causal relation of brain areas for selected
fMRI signal time series voxels. From the Figure 5.6, the first figure shows
that selected (yellow color) voxel time series is caused by red color voxels
time series (red color voxel time series causes to yellow color voxel time
series). In the second figure, the yellow color voxel time series (x3) is caused
by red color voxel time series (x4) (red color voxel time series causes to
yellow color voxel (x4) time series). The third figure, yellow color pixel not
caused by any time series. By observing the results the Granger causality
method gives proper result according to simulation in presence of Gaussian
noise. From the Figure 5.7, in the real dataset the red color voxel time series
or regions causes to yellow color voxel time series or region.
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5.3. RESULTS
Figure 5.6: Causal relation of brain images: i) x2 (yellow pixel) caused by x1
time series (red pixels), ii) x3 (yellow pixel) caused by x4 (red pixels) time
series iii) No causal relation with selected time series at the yellow pixel.
Figure 5.7: Real data set image with causal relation of brain structures (red
pixels) with selected time series (yellow pixel).
Figure 5.8 shows the values of Granger causality interaction of two simulated time series with different noises. One of the time series is added
with white Gaussian noise with different SNR values ranging from −10 dB
to 30 dB. For the given set of equations, the estimated Granger causality
is 0.65 (i.e, without noise). When SNR is changed from −10 dB to 25 dB
the Granger causality is nonlinear, later it reaches the maximum value 0.65,
which is value when there is no noise.
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5.3. RESULTS
Figure 5.8: Curve of Granger causality values for different SNR values.
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Chapter 6
Conclusion and Future
work
6.1
Conclusion
Functional magnetic resonance imaging (fMRI) is a powerful technique to
study the brain functions. It is MRI based technique to measure brain activity by detecting the changes in blood oxygen level. The analysis of fMRI
data are challenging due to the complexity of the data and the presence of
noise.
In the first part of thesis work, the developed Matlab toolbox provides
buttons for display and loading fMRI data for the analysis. The sparse
Bayesian learning algorithm is developed for construction of the design
matrix in the GLM. This Bayesian method provides better brain activation results and avoids multiple comparison problems in the t-test method.
Through the comparison of ROC curve, the Bayesian estimator is robust
than the t-test method. When the Bayesian approach is extended the time
varying noise, the results are better than for the OLS and WLS methods.
In the drift model, the Bayesian method was extended with the drift model.
It successfully removes the drift from the fMRI data.
In the second part of this study, the linear and nonlinear methods for
detection of brain activation and estimation of the HDR were implemented.
The linear method in spatio temporal modeling is better choice when the
Inter stimulus interval is large. The nonlinear RBF method is more flexible
and computationally efficient. The NARX method developed to capture the
dynamics of fMRI data.
The third part of study was mainly focused on brain connectivity methods. A functional connectivity method is used to map the functionally
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6.2. FUTURE WORK
connected regions of the brain structures by using nonlinear cross correlation analysis. This method could successfully map the functional connected
regions in the brain. Moreover, effective connectivity is achieved by Granger
causality approach. It determines changes in brain activation causal to the
changes in other brain regions. The proposed method maps the causal interactions between brain structures. All these methods are investigated on
both simulated and real datasets. The proposed Matlab toolbox in the thesis
allows the investigator to study the brain functions and helps to do further
research.
6.2
Future work
The developed toolbox provides results in axial view, the future work can be
providing the results in 3D view. The Bayesian methods for the detection
of brain activation were based on assumption of Gaussian noise. Normally,
in MR imaging the Rician distribution of noise also exist. In the future the
Bayesian method will be implemented in the presence of Rician distribution
of noise.
The Granger causality was only applied for the bi-variate variables and
the multivariate Granger causality was not performed. The relationship
between more than two variables analysis with the Multichannel analysis
methods such as the partial coherence, directed partial coherence need to
be implemented. In this method, the fMRI serieses are assumed to be linear
and stationary. The future work can be implemented multivariate Granger
causality and nonlinear Granger causality.
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Avdelning, Institution
Division, Department
Datum
Date
DIVISIONSHORT,
Dept. of Computer and Information Science
581 83 Linköping
Rapporttyp
Report category
ISBN
Svenska/Swedish
Licentiatavhandling
ISRN
×
Engelska/English
×
Examensarbete
Språk
Language
C-uppsats
D-uppsats
Övrig rapport
05-09-2012
LiU-Tek-Lic–2012:12/4600--SE
Serietitel och serienummer ISSN
Title of series, numbering
-
URL för elektronisk version
Linköping Studies in Science and Technology
Thesis No. -12/4600--SE
Titel
Title
A Matlab Toolbox for fMRI Data Analysis: Detection, Estimation and
Brain Connectivity
Författare
Author
Kiran Kumar Budde
Sammanfattning
Abstract
Functional Magnetic Resonance Imaging (fMRI) is one of the best techniques for neuroimaging and have revolutionized the way to understand the
brain functions. It measures the changes in the blood oxygen level-dependent
(BOLD) signal which is related to the neuronal activity. Complexity of the
data, presence of different types of noises and the massive amount of data
makes the fMRI data analysis a challenging one. It demands efficient signal
processing and statistical analysis methods. The inference of the analysis are
used by the physicians, neurologists and researchers for better understanding
of the brain functions.
The purpose of this study is to design a toolbox for fMRI data analysis. It includes methods to detect the brain activity maps, estimation of the
hemodynamic response (HDR) and the connectivity of the brain structures.
This toolbox provides methods for detection of activated brain regions measured with Bayesian estimator. Results are compared with the conventional
methods such as t-test, ordinary least squares (OLS) and weighted least
squares (WLS). Brain activation and HDR are estimated with linear adaptive
model and nonlinear method based on radial basis function (RBF) neural
network. Nonlinear autoregressive with exogenous inputs (NARX) neural
network is developed to model the dynamics of the fMRI data. This toolbox
also provides methods to brain connectivity such as functional connectivity
and effective connectivity. These methods are examined on simulated and
real fMRI datasets.
Nyckelord
Keywords
fMRI,functional Magnetic Resonance Imaging, Detection of activated
regions, Estimation of hemodynamic response, Brain connectivity,
Bayesian estimator, RBF, NARX, Granger causality
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