A Matlab Toolbox for fMRI Data Analysis: Detection ...

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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, Departmentof Electrical Engineering, DTU, Denmark.

Examiner: Dr. Maria Magnusson, Department of Electri-cal Engineering, LiU, Sweden.

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Abstract

Functional Magnetic Resonance Imaging (fMRI) is one of the best techniquesfor neuroimaging and have revolutionized the way to understand the brainfunctions. It measures the changes in the blood oxygen level-dependent(BOLD) signal which is related to the neuronal activity. Complexity of thedata, presence of different types of noises and the massive amount of datamakes the fMRI data analysis a challenging one. It demands efficient signalprocessing and statistical analysis methods. The inference of the analysisare used by the physicians, neurologists and researchers for better under-standing of the brain functions.

The purpose of this study is to design a toolbox for fMRI data analysis. Itincludes methods to detect the brain activity maps, estimation of the hemo-dynamic response (HDR) and the connectivity of the brain structures. Thistoolbox provides methods for detection of activated brain regions measuredwith Bayesian estimator. Results are compared with the conventional meth-ods such as t-test, ordinary least squares (OLS) and weighted least squares(WLS). Brain activation and HDR are estimated with linear adaptive modeland nonlinear method based on radial basis function (RBF) neural network.Nonlinear autoregressive with exogenous inputs (NARX) neural networkis developed to model the dynamics of the fMRI data. This toolbox alsoprovides methods to brain connectivity such as functional connectivity andeffective connectivity. These methods are examined on simulated and realfMRI 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 fieldof functional magnetic resonance imaging. I would like to mention that hissupport is invaluable and have patiently led me throughout my work. With-out his constant support this thesis work would not be finished.

I would like to thank Dr. Luo Huaien for providing his algorithms forfMRI Matlab toolbox design and sharing his knowledge during the project.

I thank my examiner Dr. Maria Magnsson and Dr. Goran Salerud forthe thesis proposal and allowing me to do the thesis at the Technical Uni-versity of Denmark (DTU). I appreciate the help from Dr. Goran Salerudthe starting of my master’s programme.

I would like to thank all those who provided invaluable advices and thehelp during my masters at Linkoping University and Technical Universityof Denmark.

Last but not least, I would like to say that the support from my familyis immeasurable under all the circumstances.

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Abbreviations

AIC Akaike Information Criterion

AR Autoregressive

ARX Autoregressive Model with Exogenous Inputs

BOLD Blood Oxygenation Level Dependent

DCT Discrete Cosine Transform

DWT Discrete Wavelet Transform

EEG Electroencephalography

EPI Echo Planar Imaging

fMRI Functional Magnetic Resonance Imaging

FPR False Positive Ratio

GLM General Linear Model

GLM Graphical User Interface

HDR Hemodynamic Response

HRF Hemodynamic Response Function

ISI Inter-Stimulus Intervals

LMS Least Mean Square

LS Least Squares

MLP Multi-Layer Perceptrons

MVAR Multivariate autoregressive

MR Magnetic Resonance

MRI Magnetic Resonance Imaging

NARX Nonlinear Autoregressive with Exogenous Inputs

OLS Ordinary Least Squares

PET Positron Emission Tomography

RBF Radial Basis Function

ROC Receiver Operator Characteristic

SNR Signal-to-Noise Ratio

SPM Statistical Parametric Mapping

TPR True Positive Ratio

WLS Weighted Least Squares

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Contents

1 Introduction 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 32.1 Functional magnetic resonance imaging . . . . . . . . . . . . 3

2.1.1 BOLD signal generation . . . . . . . . . . . . . . . . . 32.1.2 Hemodynamic response (HDR) . . . . . . . . . . . . . 42.1.3 fMRI Experimental design . . . . . . . . . . . . . . . . 52.1.4 fMRI experimental data used in this thesis . . . . . . 7

2.2 fMRI data analysis . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 fMRI data modeling . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 Temporal modeling . . . . . . . . . . . . . . . . . . . . 82.3.2 BOLD Model . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Noise and Drift . . . . . . . . . . . . . . . . . . . . . . 9

3 Detection 103.1 Loading and visualization . . . . . . . . . . . . . . . . . . . . 12

3.1.1 Display . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.2 Loading real fMRI dataset . . . . . . . . . . . . . . . . 133.1.3 Loading simulated fMRI dataset . . . . . . . . . . . . 133.1.4 Loading stimulus . . . . . . . . . . . . . . . . . . . . . 15

3.2 Detection of activated brain regions . . . . . . . . . . . . . . 173.2.1 Flexible design matrix . . . . . . . . . . . . . . . . . . 17

3.2.1.1 t-test . . . . . . . . . . . . . . . . . . . . . . 173.2.1.2 Flexible design matrix with sparse Bayesian

method . . . . . . . . . . . . . . . . . . . . . 193.2.2 Nonstationary noise models . . . . . . . . . . . . . . . 22

3.2.2.1 OLS estimator . . . . . . . . . . . . . . . . . 223.2.2.2 WLS estimator . . . . . . . . . . . . . . . . . 243.2.2.3 Bayesian estimator . . . . . . . . . . . . . . . 26

3.2.3 Drift model . . . . . . . . . . . . . . . . . . . . . . . . 28

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CONTENTS

4 Estimation 304.1 Estimation of the hemodynamic response (HDR) . . . . . . . 30

4.1.1 Adaptive spatiotemporal modeling . . . . . . . . . . . 304.1.2 Neural Network . . . . . . . . . . . . . . . . . . . . . . 344.1.3 NARX model . . . . . . . . . . . . . . . . . . . . . . . 39

5 Brain connectivity 415.1 Nonlinear cross correlation . . . . . . . . . . . . . . . . . . . . 415.2 Granger causality . . . . . . . . . . . . . . . . . . . . . . . . . 425.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3.1 Nonlinear cross correlation . . . . . . . . . . . . . . . 445.3.2 Granger causality . . . . . . . . . . . . . . . . . . . . . 46

6 Conclusion and Future work 506.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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Chapter 1

Introduction

1.1 Introduction

Brain is the most fascinating, mysterious and least understood organ of thehuman body. For the last few years, functional brain imaging techniqueshave been advanced tremendously. For understanding the brain functionsand brain mappings, a powerful tool like functional magnetic resonance(fMRI) can be used. fMRI measures the changes in blood oxygen level-dependent (BOLD) signals which are related to neural activity [1].

The purpose of this thesis is to develop a Matlab toolbox for the fMRIdata analysis. It includes methods to detect the brain activation, estimationof 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 extendedto nonstationary noise and drift models. These frameworks provideaccurate detection and avoid multiple comparison problems in con-ventional methods. This estimator detect more real activation of sim-ulated and real fMRI datasets when compared with the traditionalmethods such as t-test, ordinary least squares (OLS) and weightedleast squares (WLS).

• To provide methods for estimation of brain activation and HDR forlinear and nonlinear properties of event related designs. The linearand 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 networkto detect the spatial activation.

• Nonlinear autoregressive with exogenous inputs (NARX) neural net-works to model the dynamics of fMRI data.

• It describes methods on functional connectivity analysis of brain re-gions using non-linear cross correlation analysis. Moreover, it mea-sures the directional interactions between spatially separated neuralpopulations by using the Granger causality.

1.2 Outline of the thesis

Chapter 2: Briefly explains fMRI basics and discuss various aspects whichare related to this project.

Chapter 3: This chapter describes the design of the toolbox including vi-sualization and loading of experimental real and simulated data. Differentbrain activity detection methods are explained in the presence of noise anddrift.

Chapter 4: This chapter provides different nonlinear methods for detectionof brain activation and estimation of the HDR.

Chapter 5: This chapter describes methods to investigate brain connec-tivity structures: functional connectivity and effective connectivity. Thefunctional connectivity is investigated using the nonlinear cross correlationand effective connectivity is investigated using the Granger causality.

Chapter 6: This chapter gives the conclusion of the thesis and the scopefor 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 usesthe standard magnetic resonance imaging (MRI) to examine the brain func-tions. It is a widely used technique because of its better spatial resolutionwhen compared to electroencephalogram (EEG) and better temporal res-olution compared to positron emission tomography (PET). Moreover, it isnon-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, neuronsconsume more oxygen and demand more oxygen. This results in increasedblood flow at the activated areas. As a result, the oxygen concentrationincreases and decreases the deoxyhemoglobin. The changes in oxygen con-centration level alters the main magnetic field because the oxyhemoglobinis diamagnetic and deoxyhemoglobin is paramagnetic. T2* 1 time becomesshorter at low oxygen concentration and higher at high oxygen concentra-tion areas. Hence, the MR signal depends on oxygen level. This effect isreferred as the Blood Oxygen Level Dependent (BOLD) effect [3].

1MR images contrast determined by properties of tissue being imaged, different tissueshave different relaxation times T1,T2 and T2*. T2* relaxation time is time constant thatdescribes the exponential decay of signal, due to spin-spin interactions, magnetic fieldinhomogeneities and susceptibility effects. T2* weighted imaging is commonly used infMRI [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 activityis known as the HDR. It can result from the reduction in the amount of de-oxygenated blood and it represents temporal properties of brain. Figure 2.2shows the sketch of a typical HDR. Its shape varies with activation: ampli-tude increases with rate of neural activity and width increases with increasein duration of the neuronal activity [5]. The HDR can be summarized inthree phases [6].

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Figure 2.2: Schematic representation of HDR.

• Intial dip: During the initial short time (1-2 sec), the MR signal de-creases below baseline after beginning of neural activity. It is causedby the transient increase in deoxyhemoglobin due to the oxygen con-sumption.

• Overcompensation: After the initial dip, more oxygenated bloodis supplied to the area than extracted and deoxygenated hemoglobindecreases due to the increase of neuronal activity and results in MRsignal increase above baseline at about 2 to 5 seconds.

• Undershoot: After finishing neuronal activity, the MR signal am-plitude gradually decreases below the baseline level and reaches thebaseline level due to a combination of decreased blood flow and in-creased blood volume.

2.1.3 fMRI Experimental design

There are two schemes of experimental designs that are generally used forfMRI experiments: block design and event-related design [7].

In Block design, stimuli of two or more conditions are presented repeat-edly for an extended period of time. It is a simple, powerful and optimum

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design for detection of brain activity. It provides large signal to noise ra-tio (SNR). Figure 2.3 represents a schematic diagram of the BOLD signal(blue color) and the block design experiment (magenta color). The squarewaveform represent active (stimulus ON) and rest (stimulus OFF) of brainactivation. This is not the best design for temporal activity estimation. Inevent related design, discrete and short duration events are presented one ata time and separated by random ISI that can range depending on the exper-iment. Figure 2.4 shows event related BOLD signal with the correspondingtiming of events.

Figure 2.3: Block design BOLD signal.

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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 blockdesign (DATA BLOCK) and another is the event related design (DATAEVENT). The block design dataset was obtained from the Statistical Para-metric Mapping (SPM) website [8].

DATA BLOCK

The real fMRI experiment was designed for the activation of the auditoryareas. The functional data or Echo Planar imaging (EPI)images acquiredusing 2T Siemens MAGNETOM Vision system. The experiment data setand its details are available on [8]. During the experiment, bisyllabic wordsare presented binaurally. The first few scans are discarded due to T1 effectsthen starts from fm00223 004 image. The total number of acquisitions are96 and it is divided into 16 blocks and each block contains 6 acquisitions.In this experiment alternate condition of the baseline (rest) and activationare 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 timeperiod 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 forthe statistical analysis and enhance the brain activity mappings. There areseveral steps to be followed [9]. In this thesis the following preprocessingsteps are performed by the SPM software [10]:

• Realignment: During the data acquisition, intensity of the voxel ofresting and activation assumes that no motion of subject has occurred.The motion is caused by the subject movement due to long period ofscanning. This problem can be reduced by the realignment procedurewith estimation and correction [11].

• Co-registration: The low resolution fMRI images are aligned withthe anatomical MRI images, which can be images of the same subjector a standard template [12].

• Normalize: It is a procedure to wrap the functional data onto acoordinate system or template space [13].

• Smoothing: The fMRI images are smoothed across adjacent voxelsto 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 consideredas the analysis of the response of the system to a given input. As shown inFigure 2.5, the system is the human brain and measuring device, input isthe 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 prop-erties are required. The measured fMRI signal or voxel time series is thecombination of BOLD signal, drift and noise, i.e. the change in voxel inten-sity represent whether the BOLD signal is active or not.

Measured fMRI time series = BOLD signal + drift + noise. (2.1)

2.3.2 BOLD Model

This model assumes a linear system, i.e. the BOLD response expresses theconvolution of the stimulus function and impulse response [15],

yb(t) = h(t)⊗ s(t) =

∫ ∞0

h(τ)s(t− τ)dτ, (2.2)

where ⊗ denotes the convolution operation, s(t) is the stimulus functionand h(t) is impulse response function and it is called hemodynamic responsefunction (HRF). The HRF waveform is modelled by poisson, Gaussian func-tion and difference of gamma functions [16]. Here, the widely used differenceof gamma function model is used and it can be represented by,

h(t) =( t

d1

)a1

exp

(−(t− d1)

b1

)− c( t

d2

)a2

exp

(−(t− d2)

b2

), (2.3)

where di = aibi represents the amplitude of the peak and c represents the un-dershoot. 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 lowSNR. The sources of noise are scanner induced noise and subject movementsduring scanning such as head and lower jaw movements. The residual of theimperfect model is also one of the noise components [15].

Drift is another component that disturbs the fMRI time series. It showsup as a slow varying interference or trend in the fMRI time series. Thisdrift often comes from physiological processes like the cardiac and inspira-tion process [18] as well as from instability of the magnetic field [19]. Thecommon way to remove the drift in fMRI time series is either by using ahigh-pass filter or a drift model.

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Chapter 3

Detection

This fMRI Matlab toolbox provides different methods for detection of ac-tivated brain regions and estimation of the HDR. The proposed toolboxconsists of four functionalities. The first section describes how to displaythe cross sectional brain images and to load the experimental data. The sec-ond section provides different brain detection methods. The third sectionprovides methods for estimation of the HDR and the fourth section describesbrain 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 andloading of real and simulated fMRI datasets. Second section explains de-termination of flexible design in the GLM through the Bayesian learningprocedure and it provides accurate activation compared with conventionalt-test method. The Bayesian estimator is extended to non-stationary noisemodel and it provides accurate activation results when compared to OLSand WLS. Moreover, the Bayesian estimator can be extended to removedrift component present in the fMRI time series. Studies on simulated andreal fMRI data show that the Bayesian estimator accurately detects thebrain activated regions to specific task. The detection and estimation algo-rithms developed by Dr. Luo Huaien have been adopted in the design ofthis 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 Loading and visualization

3.1.1 Display

The fMRI datasets are provided in the analyze file format. It is an imagefile format for storing MRI data and consists of two types of files, an imagefile and a header file, with .img and .hdr extensions, respectively. An imagefile contains uncompressed pixel data or a set of cross sectional images. Theheader file contains history and dimensions of the data. Before the fMRIdata analysis starts, the dataset is preprocessed by the SPM software toapply realignment, coregistraton, normalization and smoothing. In Figure3.2, the graphical window displays the brain cross-sectional images and thesubwindow provides information about the images.

Figure 3.2: Display of brain cross-sectional images.

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3.1.2 Loading real fMRI dataset

To load real fMRI data set, the toolbox requires series of MRI scanned dataor 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. Italso provides dataset information such as the number of slices, image andvoxel dimensions. The graphical window shows only one image and otherdata details. The loaded image files in Figure 3.3 are fromswrfM00223 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 posi-tions 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 simu-late fMRI data with noise, such as time varying and fractional noise. Inthis thesis, two types of datasets are simulated: DATA BLOCK and DATAEVENT. DATA BLOCK is simulated with block design signal and DATAEVENT is simulated with event related signal.

1The session is a scanning period. During the fMRI scanning the patient brain isscanned for extended time period in several sessions and then it acquires series of the MRimages, where each session consists of MR images sequences.

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In GUI, select the Load Simulated button. It opens a subwindow. Thenchoose 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 noisyimages are displayed in Figure 3.5.

Figure 3.4: Loading simulated fMRI dataset with active positions.

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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 knowledgeabout the experiment. For block or event-related design experiments, severalvalues must be entered: active and rest block length, number of blocks,repetition time (TR), signal duration and duration of rest. Block designexperiment data and information are available in [4]. Figure 3.6 shows theblock design paradigm of auditory experiment and Figure 3.7 shows theevent-related paradigm.

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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 Detection of activated brain regions

3.2.1 Flexible design matrix

This section briefly explains the general linear model (GLM). Moreover, theconstruction of a design matrix in GLM using Bayesian analysis is consideredand 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 av-erage active condition with the average rest condition. The t-test is definedas [21]

t =X1 −X2√

S21

n1+

S22

n2

, (3.1)

where X1, S21 and n1 are the mean, variance and sample number for the

active fMRI samples. For the resting fMRI time samples X2, S22 and n2 are

the mean, variance and sample number.

Figure 3.8: Detection results of simulated fMRI data using the t-testmethod.

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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 appliedto the simulated and real fMRI datasets. It can be clearly observed fromthat the t-test method also detects false detection. Therefore for the realfMRI data, analyzing the performance with this method makes it difficult.This is because we lack a reference which could serve as the true activationof the brain.

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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, themeasured fMRI time series is described as the combination of regressors andtheir vector parameters [22].

y = Φw + ε, (3.2)

where y is an N × 1 measured fMRI time series, Φ is an N × P ma-trix known as the design matrix, w is a P × 1 vector of estimated unknownparameters and ε is N×1 noise vector. The design matrix contains all avail-able knowledge about the experiment. In the design matrix each row (N)represents one time point (one scan) of the regressor and column (P ) repre-sents the corresponding regressor or explanatory variable like the canonicalBOLD 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 leadto problems. More regressors leads to model over-fitting problems and fewbasis functions cannot filter out the noise. The selection of regressors in thedesign matrix is a critical issue. The Bayesian approach is used to constructa flexible design matrix. During the learning procedure, the data itself es-timate the regressors. It determines which regressors are useful and ignoresthe ones that are irrelevant [20].

This algorithm is developed based on the assumption that the noise isstationary. Initially, the design matrix is initialized with the BOLD signal, aconstant vector of value 1 and radial basis functions. Throughout the learn-ing procedure, the regressors and their weight coefficients are estimated andirrelevant regressors are discarded.

Figures 3.10 and 3.11 show that the Bayesian algorithm provides betterresults for both simulated and real fMRI datasets. The performance is betterthan the t-test method. Both the t-test and Bayesian estimator resultsclearly show that auditory activated areas are identified. As mentionedbefore, the sparse Bayesian algorithm assumes that the noise is stationary.However, the fMRI data also contains nonstationary noise which could beremoved by nonstationary noise models.

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Figure 3.10: Detection results of simulated fMRI data using the sparseBayesian learning method.

Figure 3.11: Detection results of real fMRI data using the sparse Bayesianlearning method.

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By observing the results on real fMRI data, it is clearly seen that the au-ditory cortical areas has been identified. The Bayesian estimator detects themore real activated brain regions compared to conventional t-test method.

Figure 3.12: ROC curve of simulated data of the t-test and the Bayesianestimator.

Receiver operator characteristic (ROC) analysis is used to investigateclear comparison of detection ability of the t-test and the Bayesian estimatormethods. The ROC curve plots the true positive ratio (TPR) against thefalse positive ratio (FPR) [24]. From Figure 3.12, we can say that, theBayesian 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 sub-ject such as lower jaw movements, and neurophysiological processes such asnumber of neurons involved in specific tasks at different time points andthe background memory process. The noise sources may cause the noisevariance to be changed [25]. The aim of fMRI data analysis is to distinguishthe activated BOLD signal from noisy data and to determine the activatedregions for particular tasks. The high amount of noise in fMRI makes itdifficult and challenging for data analysis.

This section briefly explains the Bayesian estimator for detection of acti-vated regions in the case of time varying noise. The performance is comparedwith 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 + ε, (3.3)

where yn and εn are the observed fMRI time series and noise at the nth voxelrespectively. εn is assumed to be independent and identically distributed(i.i.d) white noise. The least square estimator of the parameter w is definedas

wn(OLS) = (ΦTΦ)−1ΦTyn. (3.4)

The assumption of i.i.d white noise is an inappropriate assumption re-garding the nonstationary nature of fMRI signals. The design matrix shouldbe 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 matrixinversion, firstly we multiply the design matrix with its transpose, whichresults in a square matrix [22]. The activated voxels are calculated by usingstatic student distribution t-test

t =cT w√cTΛwc

, (3.5)

where c is the contrast vector and Λw is the covariance matrix of the esti-mated parameter w.

The Figure 3.13 shows the detection results of a simulated fMRI datausing the OLS method in presence of time varying noise, here, the noiserange is from -0.4 to 6.7 dB. The OLS method detects the activated regionsas well as false detection. From Figure 3.14, it is clearly shows the detectionof auditory cortical areas by using this method.

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Figure 3.13: Detection results of simulated fMRI data using the OLS esti-mator.

Figure 3.14: Detection result of auditory processing fMRI data using theOLS estimator.

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3.2.2.2 WLS estimator

The noise variance of time varying noise model is stated to increases in amultiplicative fashion [27]. For this reason the noise variance is modeledwith a scaling matrix and overall variance. The noise model is assumed tobe time dependent, with precision matrix Bn. The precision matrix is theinverse of the covariance matrix and it is defined as [25],

Bn =

s1 0 . . . 00 s2 . . . 0...

.... . .

...0 0 . . . sT

βn = Sβn, (3.6)

where s1, s2, ..sT are the scaling parameters and T is the number of samplesin the fMRI time series. Here, S is the scaling diagonal matrix and Bn isthe noise precision at nth voxel.

The WLS estimator of parameter w is defined as

wn(WLS) = (ΦTSΦ)−1ΦTSyn. (3.7)

The WLS estimator requires accurate estimation of the scaling matrix. Intraditional methods, the residual of the OLS estimator is used. The residualof the OLS estimator is given by

rn = yn −Φwn(OLS) = yn −Φ(ΦTΦ)−1ΦTyn. (3.8)

The overall precision of the nth voxel time series and the inverse scalingmatrix are calculated as in [17],

βn =T − rank(Φ)

rTnrn, (3.9)

sinv =

∑Nn=1 diag(βnrnrTn )

N, (3.10)

where, N is the total number of voxels and the operator diag(.) creates thediagonal of a square matrix into a column vector.

This method is simple to implement. The residual is equal to Sβn andthen is considered as unbiased estimator. The WLS method uses the residu-als of the OLS method as a covariance matrix to estimate the scaling matrix.It fails to estimate an accurate scaling matrix. The time varying Bayesianestimator, 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 esti-mator.

Figure 3.16: Detection result of auditory processing fMRI data using theWLS estimator.

Figure 3.15 shows that the detection results of a simulated fMRI data

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in the presence of a time varying noise. It is clearly observed that the WLSmethod detects the false detection too. The Figure 3.16 shows the detectionresult 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 simu-lated data doesn’t give accurate result due to: i) The OLS estimator assumesi.i.d noise and does not consider time varying noise. ii) The WLS estimatorrequires accurate estimate of the noise covariance matrix. It uses the resid-ual of a traditional method (e.g. OLS) to estimate. It is difficult to performan accurate estimation of the covariance matrix.

In the time varying noise model, the noise covariance matrix is a diag-onal matrix. This model is spatially and temporally non-stationary. Thisproperty is investigated by the Bayesian method. In the previous section,the Bayesian method was used to estimate the parameters w under theassumption of constant variance. In this section, the Bayesian method isextended to variance of noise changing with time, which gives an estimatednoise structure closer to the true noise [26].

Figure 3.17: Detection results of simulated fMRI data using the Bayesianestimator.

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Figure 3.18: Detection results of activation of auditory fMRI dataset usingBayesian estimator.

By observing Figure 3.17, it can be seen that the Bayesian estimatordetects more true activations and less false activations in the presence of atime varying noise. By comparing Figures 3.13, 3.14, 3.15, 3.16, 3.17 and3.18 it can be stated that the results of the Bayesian estimator is more ac-curate than the OLS and WLS methods.

In this section, the Bayesian estimator does not consider the drift infMRI 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 re-gions based on inference drawn from the estimated parameter in GLM.

Drift is a low frequency signal that slowly varies across the complete pe-riod of the signal. The causes of drift are head movement during acquisition,local changes of magnetic field and internal physiological process like car-diac and respiration movements. Drift can be removed with a preprocessingprocedure before statistical analysis, e.g. highpass filtering or the medianmethod. It can also be removed by the drift model [28].

The measured fMRI signal contains three types of parameters: BOLDresponse, noise and drift. The GLM of measured data is

y = wb + f + ε, (3.11)

or based on description in Section 3.2.1.2,traditional way of GLM

y = Φw + ε, (3.12)

where y is the measured time series, f is the drift and ε is the noise withdimensions T × 1. Here, the parameter w is the scalar which represents thecontribution of BOLD signal is to the measure the fMRI time series. Thedesign matrix Φ = [b, f ] with dimension T × 2 and estimated parameterw = [w, 1].

The fMRI data exhibits fractional noise also. In the drift model, first ap-ply the discrete wavelet transform (DWT) to the fMRI data. The resultingwavelet coefficients at lower than fine scale (J0) are zero [29], because thedrift does not vary significantly below J0. The J0 value can be estimatedby using the Confidence Interval Criterion (CIC) as model selection crite-rion. It is more efficient than the Akaike Information Criterion (AIC) andthe Schwartz Information Criterion (SIC). The activation parameters are ac-curately estimated by using the Bayesian estimator from the modified GLM.

Figure 3.20 shows that the detection results of a simulated fMRI databy using Bayesian estimator. It can be effectively estimated and removeddrift 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 theactivation of auditory cortical areas of the brain are successfully detected.

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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 activatedregions in the brain. In the following chapter, methods to estimate the HDRand measure the spatial activation for nonlinear properties of fMRI data arediscussed.

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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 exhibitstemporal properties of human brain activities. Generally, the HDR canbe estimated through a signal averaging procedure by assuming that theISI is large [30], so that consecutive stimuli do not overlap. Overlappingof consecutive stimuli leads to bias results of the averaging methods. Thismethod works only under the linear assumption case, i.e. ISI is larger than 4-6 sec, otherwise nonlinear phenomena are observed. In this chapter, I havedescribed linear adaptive modelling and nonlinear method based on RBFneural networks to estimate the HDR and detection of brain activation.More over in this chapter describes the NARX to model the dynamics offMRI data.

4.1.1 Adaptive spatiotemporal modeling

The fMRI data contains spatial and temporal information. In the adaptivespatial temporal modeling, spatial information is also included for estimat-ing the temporal activation. Because activated brain regions extend to sev-eral voxels, surrounding voxels are more likely to be activated compared toother voxels. This spatial information can be used to improve the SNR anddetection accuracy [31].

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4.1. ESTIMATION OF THE HEMODYNAMIC RESPONSE (HDR)

Figure 4.1: Schematic diagram of the spatial smoothing and temporal mod-elling.

Figure 4.1 shows a sketch of the spatial smoothing and temporal mod-elling. The spatial adaptive filter is defined as

d(n) =

L∑i=0

wiyi(n) = wTy(n), (4.1)

where y(n) = [y0(n), y1(n), . . . , yL(n)]T , y0(n) is the nth sample of givenvoxel fMRI time series and y1(n), y2(n) . . . yL(n) are L number of neigh-boring voxels time serieses, w = [w0(n), w1(n), . . . , wL(n)]T is the corre-sponding weight vector. The smoothed signal d(n) is the desired signal fortemporal modelling. For activated voxel time series, the resulted smoothedsignal is approximated to the ideal BOLD signal r(n) to the experimentalstimuli, i.e.

d(n) = wTy(n) = r(n) + u(n), (4.2)

where r(n) is the ideal BOLD signal which is the result of stimulus func-tion and canonical HDR and u(n) is the white noise. The output spatialsmoothed signal d(n) is used as the desired signal for temporal modeling.The temporal modelling is defined as

d(n) =

P∑m=0

s(n−m)hm + ε(n) = hT s(n) + ε(n), (4.3)

where s(n) = [s(n), s(n − 1). . . s(n − P )]T is a vector corresponding to thestimulus function with a delay of P and h = [h0, h1. . .hP ]T are the filtercoefficients.

The aim of this analysis is to find an optimum filter coefficients so thathT s(n) is the resultant BOLD signal. The mean square errors of the spatialand temporal filters are E{e21(n)} = E{(r(n) − d(n))2} and E{e22(n)} =

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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)−wTy(n))2}+ E{(wTy(n)− hT s(n))2}. (4.4)

For the cost function J , the optimum filter coefficients are obtained atminimum value of J . The least mean square (LMS) algorithm is used forfinding minimum mean square error by updating the following equations,

e1(n) = r(n)− wT (n)y(n), (4.5)

e2(n) = wT (n)y(n)− hT (n)s(n), (4.6)

w(n+ 1) = w(n) + 2µ1e1(n)y(n)− 2µ1e2(n)y(n), (4.7)

h(n+ 1) = h(n) + 2µ2e2(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 modeland the estimation errors (e1(n) and e2(n))are feedback to spatial filter toadjust the estimation coefficients. The estimation error (e2(n)) is fed back tothe temporal modelling to adjust the estimation coefficients up to algorithm

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converges. The estimated optimum spatial filter weights w approximates thespatial smoothing and the optimum temporal filter weights w approximatesHDR. The adaptive spatio-temporal model is tested on the simulated BOLDsignal.

Figure 4.3: Estimation of HDR on the simulated fMRI signal.

Figure 4.3 shows the simulated BOLD signal, simulated noisy signal andestimation of HDR.

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4.1.2 Neural Network

As mentioned before, the fMRI signal exhibits linear and nonlinear prop-erties. It has been observed that when the duration of ISI is less than 4to 6 sec, the event related fMRI signals exhibit nonlinear properties. Thenonlinear properties of the BOLD signal can be expressed by the balloonmodel, including variables such as, blood flow, blood volume and oxygenconcentration. It is a physiologically derived model which was introducedby Buxton et al [32]. It is a proper model for understanding the nonlinearmechanism underlying the neural activity and BOLD effect. The balloonmodel provides an approximate estimation of the parameters and it is noteasy to estimate the parameters.

On the other hand, non-physiological models were developed for betterestimation when the signal is highly non-linear. The Volterra series modelis a dynamical input output system developed to estimate the HDR usingfirst and second order kernels [16]. Higher the kernel orders, better theestimates. The number of parameters increases exponentially with higherorder kernels. The performance depends upon the selected order of Volterraseries. The lower order kernels may not capture the dynamic properties ofthe system as well as higher order kernels. Estimate of Volterra kernels byusing the least squares (LS) method is difficult when the kernel order is high.However, the RBF neural network method efficiently estimates the HDR aswell 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]. Asshown in Figure 4.4, the present and past inputs of stimulus are applied tothe 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

y(n) = FNN (s(n)), (4.9)

the output y(n) is the nonlinear function of input s(n) and the nonlinearfunction is denoted by FNN . s(n) is a (P + 1) × 1 vector containing thepresent 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 comparedto the multilayer perceptron of neural network. The output y(n) is a linearcombination of M radial basis functions defined as

y(n) =

M∑i=1

hiG(s(n), ci), (4.10)

where hi is the weighting coefficients and ci is the center of the radial basisfunction. Commonly used basis functions are

G(s(n), ci) = exp(− 1

σ2i

‖s(n)− ci‖2). (4.11)

The center of a RBF is determined randomly from the data, and varianceis selected according to the center spread. The aim of a RBF network is tofind the weights and sum of minimized least square error. Regularizationparameter is required for the stability of the solution. The weight vector isestimated by

h = (GTG + λI)−1GTy. (4.12)

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For estimating the weight vector (h) and regularization, Bayesian learn-ing approach is used and is iteratively updated using the following equa-tions[35],

Σ = β2(GTG + λI)−1, (4.13)

h =1

β2ΣGTy, (4.14)

γ = M − λ

β2trace(Σ), (4.15)

β2 =‖y −Gw‖2

N − γ, (4.16)

λ =γβ2

wT w, (4.17)

where β2 is the noise variance. Finally the BOLD signal is computed as

yb = Gh. (4.18)

Figure 4.5: Simulated BOLD signal, noisy BOLD signal and reconstructedBOLD signal by using RBF neural network.

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Figure 4.5 shows the original simulated BOLD signal, simulated noisysignal with Gaussian noise and reconstructed BOLD signal by using RBFneural network method.

The detection of activated brain regions the reconstructed BOLD signalcan be used to measures the activation of each voxel. The detection ofactivation index (R) is defined as

R =‖yb‖‖y − yb‖

, (4.19)

where yb is the reconstructed BOLD signal and y is the measured fMRItime series in a voxel.

Figure 4.6: Detection results of simulated fMRI data with the RBF neuralnetwork model.

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Figure 4.7: Detection results of auditory fMRI data using the RBF neuralnetwork method.

Figure 4.6 and Figure 4.7 show the result of activated brain regions of realfMRI data and simulated fMRI data using the RBF neural network method.By observing the results of auditory activation data, it can be clearly seenthat the RBF neural network method detects the auditory cortical areas inthe brain.

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4.1.3 NARX model

The nonlinear autoregressive with exogenous inputs (NARX) neural networkis a dynamical model for mapping the input-output dynamics of the BOLDsignal. It is a powerful nonlinear model with faster convergence and bettergeneralization 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 − 1delays and the fMRI signal is applied to the NARX neural networks via pdelays. The output estimated signal can be determined as

yb(n) = FNN (y(n), s(n)) = FNN (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, explainedin Section 4.1.2. The BOLD signal is reconstructed after estimation of theweights. After reconstructing the BOLD signal, the following test can bedefined for detection index(R) of activation at each voxel

R =‖yb‖‖y − yb‖

, (4.21)

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where yb is the reconstructed BOLD signal and y is the measured fMRItime 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 theNARX model.

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Chapter 5

Brain connectivity

The brain can be described as an organized structural network. Brain con-nectivity refers to anatomical connectivity, functional connectivity and ef-fective connectivity [38]. Anatomical connectivity is physical or structuralconnections between neurons or anatomical brain regions. Functional con-nectivity is a statistical concept, which represents the statistical dependencyamong different brain structures. Effective connectivity is the causal inter-action between different brain structures [39].

In this chapter, in the first section the functional connectivity is investi-gated 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 ony without any specific relation between them. Here, x and y are the two timeserieses. If the value of x is considered as function of y, then the value of ygives x and it can be predicted [40]. The nonlinear correlation coefficient isdefined as,

h2y/x =

∑Nk=1 y

2(k)−∑N

k=1(y(k)− f(x))2∑Nk=1 y

2(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 section4.1.2, the present and past inputs of the time series x(n) is used as input forthe RBF neural network. The output signal y(n) is the nonlinear functionof input x(n) vector. The output signal can be expressed as [33]

y(n) = FNN (x(n)), (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 ofdimension (p + 1) × 1 formed by the present and past inputs with delay p.The output y(n) is taken to be nonlinear function of the basis functions, itcan be defined as

y(n) =

M∑i=1

hiG(x(n), ci) (5.3)

where hi are the weighting coefficients, ci are the centers of the radial basisfunctions and M is the hidden units of the RBF network. The weights canbe estimated through the Bayesian learning procedure. When the weightsare estimated, the functional approximation function can be expressed as

y = Gh. (5.4)

The range of h2y/x is between 0 to 1. The value 0 means that the signaly is independent of x. The value 1 represents that the signal y can be fullydetermined by the signal x [40]. The above process is applied to all voxelsfMRI time series to the selected time series.

5.2 Granger causality

According to Norbert Wiener,“If the prediction of one time series can beimproved by inclusion of the knowledge about other time series, then sec-ond time series is said to have causal influence on the first time series” [41].However, this idea lacks the practical implementation. Later Granger provedpractically, “If the error variance of one time series reduced by inclusion oflagged observation of second time series, then second time series has causalinfluence on first time series” [42].

Granger causality method is based on multiple linear regression for deter-mine whether a signal useful for forecasting another signal or not. Accordingto 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 betteraccuracy compared to considering its past values alone [43].

Jointly, two time series x and y can be represented by a bivariate regres-sive model

x(n) =

P∑j=1

a2jx(n− j) +

P∑j=1

b2jy(n− j) + ε1(n), (5.5)

y(n) =

P∑j=1

c2jx(n− j) +

P∑j=1

d2jy(n− j) + ε2(n), (5.6)

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5.2. GRANGER CAUSALITY

In Equation (5.5) and (5.6), P is the maximum number of lagged ob-servations 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 themodel and ε1 and ε2 are the residual (prediction) errors for the each timeseries. 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-causesx (or y).

The magnitude of this interaction can be measured by using the log ratioof the prediction error variances for the restricted (R) and unrestricted (U)models,

fy→x = lnvar(ε1R)

var(ε1U ). (5.7)

The restricted model can be defined as

x(n) =P∑

j=1

a1jx(n− j) + ε1R(n), (5.8)

y(n) =

P∑j=1

d2jy(n− j) + ε2R(t), (5.9)

and the unrestricted model can be defined as

x(n) =

P∑j=1

a2jx(n− j) +

P∑j=1

b2jy(n− j) + ε1U (n), (5.10)

y(n) =

P∑j=1

c2jx(n− j) +

P∑j=1

d2jy(n− j) + ε2U (n). (5.11)

Model order

The estimation of multivariate autoregressive (MVAR) models requiresthe number of time-lags (P ) to include, i.e., the model orders. When thelags are lesser or too many these can lead to a poor representation or prob-lems in model estimation. A principled means to specify the model orderto minimize the criteria that balance the variance accounted against thenumber of coefficients to be estimated. They can be implemented on twocriteria’s, which are Akaike Information Criterion (AIC) and the BayesianInformation Criterion (BIC) [43]. The BIC is more useful in application tothe neural systems and it is defined as

BIC = n. ln(σ2e) + k. ln(n), (5.12)

where σ2e is the error variance and k is the number of parameters to be

estimated.

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5.3. RESULTS

5.3 Results

5.3.1 Nonlinear cross correlation

Simulated fMRI data

First the proposed method was tested on simulated data. A simulatedblock design 3-D fMRI data set was investigated in this study. A blockdesign BOLD signal was added at particular positions of the image, andthen white Gaussian noise was added to the simulated signals with SNRvalues from 0.49 to 8.29 dB. Figure 5.1 shows the functional connected re-gions (blue color) with selected time series (yellow color).

Figure 5.1: Simulated image with activated brain regions (left) and func-tional 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 con-nectivity of the brain structures. First, the activated brain regions was foundby using one of the methods for detection of activated brain region, suchas OLS, WLS or the Bayesian estimator (described in section 3.2). Then atime series voxel of interest was selected to map the functional connectivity.Figure 5.2 shows the reconstructed BOLD signal of functionally connectedand unconnected voxel time serieses by using RBF neural network method.Figure 5.3 shows the functional connectivity of brain structures (dark-bluecolor) 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 andfunctionally unconnected time series with selected time series for the realfMRI data.

Figure 5.3: Auditory activated brain regions (left) and functional connec-tivity (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 withblue color voxel time series for auditory areas activation. The left auditorycortical areas is functionally connected with right auditory cortical areasduring auditory activation.

Figure 5.4: Curve of nonlinear cross correlation values for different SNRvalues.

Figure 5.4 shows the nonlinear cross correlation values of two simulatedtime series for different noises. One of the time series is added with whiteGaussian noise with different SNR values from −10 dB to 30 dB. The valueof nonlinear cross correlation value without noise is 1. The nonlinear crosscorrelation is linear when SNR is changed from −10 dB to −2 dB, and thenit varies nonlinearly until it reaches a maximum value (1), which is value fornonlinear cross correlation when there is no noise added.

5.3.2 Granger causality

Four simulated time series x1-x4 signals were generated by using autore-gressive (AR) modeling. The relations between these signals are that x2 iscaused by x1 (x1 causes x2) and x3 is caused by x4 (x4 causes x3), see Fig-ure 5.5. These signals are added at particular positions of images containingGaussian 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), (5.13)

x2(n) = 0.5x1(n− 2) + w2(n), (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 selectedfMRI signal time series voxels. From the Figure 5.6, the first figure showsthat selected (yellow color) voxel time series is caused by red color voxelstime series (red color voxel time series causes to yellow color voxel timeseries). In the second figure, the yellow color voxel time series (x3) is causedby red color voxel time series (x4) (red color voxel time series causes toyellow color voxel (x4) time series). The third figure, yellow color pixel notcaused by any time series. By observing the results the Granger causalitymethod gives proper result according to simulation in presence of Gaussiannoise. From the Figure 5.7, in the real dataset the red color voxel time seriesor 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 x1time series (red pixels), ii) x3 (yellow pixel) caused by x4 (red pixels) timeseries 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 (redpixels) with selected time series (yellow pixel).

Figure 5.8 shows the values of Granger causality interaction of two sim-ulated time series with different noises. One of the time series is addedwith white Gaussian noise with different SNR values ranging from −10 dBto 30 dB. For the given set of equations, the estimated Granger causalityis 0.65 (i.e, without noise). When SNR is changed from −10 dB to 25 dBthe 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 Futurework

6.1 Conclusion

Functional magnetic resonance imaging (fMRI) is a powerful technique tostudy the brain functions. It is MRI based technique to measure brain ac-tivity by detecting the changes in blood oxygen level. The analysis of fMRIdata are challenging due to the complexity of the data and the presence ofnoise.

In the first part of thesis work, the developed Matlab toolbox providesbuttons for display and loading fMRI data for the analysis. The sparseBayesian learning algorithm is developed for construction of the designmatrix in the GLM. This Bayesian method provides better brain activa-tion results and avoids multiple comparison problems in the t-test method.Through the comparison of ROC curve, the Bayesian estimator is robustthan the t-test method. When the Bayesian approach is extended the timevarying 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 fordetection of brain activation and estimation of the HDR were implemented.The linear method in spatio temporal modeling is better choice when theInter stimulus interval is large. The nonlinear RBF method is more flexibleand computationally efficient. The NARX method developed to capture thedynamics of fMRI data.

The third part of study was mainly focused on brain connectivity meth-ods. 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 correla-tion analysis. This method could successfully map the functional connectedregions in the brain. Moreover, effective connectivity is achieved by Grangercausality approach. It determines changes in brain activation causal to thechanges in other brain regions. The proposed method maps the causal in-teractions between brain structures. All these methods are investigated onboth simulated and real datasets. The proposed Matlab toolbox in the thesisallows the investigator to study the brain functions and helps to do furtherresearch.

6.2 Future work

The developed toolbox provides results in axial view, the future work can beproviding the results in 3D view. The Bayesian methods for the detectionof brain activation were based on assumption of Gaussian noise. Normally,in MR imaging the Rician distribution of noise also exist. In the future theBayesian method will be implemented in the presence of Rician distributionof noise.

The Granger causality was only applied for the bi-variate variables andthe multivariate Granger causality was not performed. The relationshipbetween more than two variables analysis with the Multichannel analysismethods such as the partial coherence, directed partial coherence need tobe implemented. In this method, the fMRI serieses are assumed to be linearand stationary. The future work can be implemented multivariate Grangercausality and nonlinear Granger causality.

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Avdelning, InstitutionDivision, Department

DatumDate

Sprak

Language

� Svenska/Swedish

� Engelska/English

�

RapporttypReport category

� Licentiatavhandling

� Examensarbete

� C-uppsats

� D-uppsats

� Ovrig rapport

�

URL for elektronisk version

ISBN

ISRN

Serietitel och serienummerTitle of series, numbering

ISSN

Linkoping Studies in Science and Technology

Thesis No. -12/4600--SE

TitelTitle

ForfattareAuthor

SammanfattningAbstract

NyckelordKeywords

Functional Magnetic Resonance Imaging (fMRI) is one of the best tech-niques for neuroimaging and have revolutionized the way to understand thebrain functions. It measures the changes in the blood oxygen level-dependent(BOLD) signal which is related to the neuronal activity. Complexity of thedata, presence of different types of noises and the massive amount of datamakes the fMRI data analysis a challenging one. It demands efficient signalprocessing and statistical analysis methods. The inference of the analysis areused by the physicians, neurologists and researchers for better understandingof the brain functions.

The purpose of this study is to design a toolbox for fMRI data anal-ysis. It includes methods to detect the brain activity maps, estimation of thehemodynamic response (HDR) and the connectivity of the brain structures.This toolbox provides methods for detection of activated brain regions mea-sured with Bayesian estimator. Results are compared with the conventionalmethods such as t-test, ordinary least squares (OLS) and weighted leastsquares (WLS). Brain activation and HDR are estimated with linear adaptivemodel and nonlinear method based on radial basis function (RBF) neuralnetwork. Nonlinear autoregressive with exogenous inputs (NARX) neuralnetwork is developed to model the dynamics of the fMRI data. This toolboxalso provides methods to brain connectivity such as functional connectivityand effective connectivity. These methods are examined on simulated andreal fMRI datasets.

DIVISIONSHORT,Dept. of Computer and Information Science581 83 Linkoping

05-09-2012

-

LiU-Tek-Lic–2012:12/4600--SE

-

05-09-2012

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

Kiran Kumar Budde

××

fMRI,functional Magnetic Resonance Imaging, Detection of activatedregions, Estimation of hemodynamic response, Brain connectivity,Bayesian estimator, RBF, NARX, Granger causality

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