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Investigating online effects of transcranial direct current stimulation from NIRS-EEG joint-imaging using Kalman Filter based online parameter estimation of an autoregressive model

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Investigating online effects of transcranial direct current
stimulation from NIRS-EEG joint-imaging using Kalman Filter
based online parameter estimation of an autoregressive model
FINAL REPORT submitted to
INRIA-DEMAR team
for
Summer Internship
By
Mehak Sood
Graduate Student
Electronics and Communication Engineering Department
International Institute of Information Technology,
Hyderabad – 500032, INDIA
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Object: Certificate of completion of internship
It is certified that Ms. Mehak Sood, currently a student of 5th year Electronics and Communication
Engineering Department, International Institute of Information Technology, Hyderabad, India has
successfully completed her Summer Internship from May 15-Aug 15, 2015 at the INRIA DEMAR
Team of Montpellier Laboratory of Informatics, Robotics, and Microelectronics (LIRMM in French)
- a cross faculty research entity of the University of Montpellier 2 (UM2) and the National Center
for Scientific Research (CNRS) - Institut des sciences informatiques et de leurs interactions (INS2I)
Montpellier, France.
David Guiraud (PhD)
Scientist leader of INRIA DEMAR Team
University of Montpellier - LIRMM
860 Rue Saint Priest
F-34095 Montpellier Cedex 5
Tel: +33 467 418 621
Sec: +33 467 418 688
Web: http://www.lirmm.fr/demar
Perso: http://www.lirmm.fr/~guiraud
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Acknowledgement
I wish to express my sincere thanks to Dr. David Guiraud, Senior Research Scientist INRIA and
Head Team DEMAR, LIRMM INRIA Montpellier, for accepting me as an internee in his team,
supporting me throughout the course of this internship program. I wish to express my deep gratitude
to Dr. Anirban Dutta, INRIA starting research scientist, DEMAR team for giving me a chance
to pursue such an interesting project under Franco-Indian INRIA-DST funding and for constantly
guiding and supervising me in research areas which were new to me.
With deep sense of gratitude, I also wish to thank Dr. Mitsuhiro Hayashibe, INRIA Research
Scientist (tenured, CR1), DEMAR team for constantly monitoring and providing me guidance on
new concepts and developing the algorithm for my project.
I am grateful to Prof. Stephane Perrey and his group - Neuroplasticity and Rehabilitation at
the Movement to Health Laboratory, EuroMov, University of Montpellier for providing me an
opportunity to use the experiemental data collected under his supervision for my computational
modeling work.
I am very grateful to Dr. Shubhajit Roy Chowdhury, Assistant Professor at International
Institute of Information Technology, Hyderabad, India for providing me an opportunity to travel
to France and complete the project that I started in India under his guidance. The project consumed
hardwork, lot of research and dedication, and wouldn’t have been possible without the support of all
my supervisors and organizations I worked with.
My thanks and appreciations also go to my colleague Mr. Utkarsh Jindal, who willingly helped
me out with his abilities throughout the project.
I attribute the completion of this research study to the blessings, moral support, love and affection
of my parents and family members. They have always been a major source of motivation and
strength.
Finally, I would like to express my recondite thanks to all those who have rendered their much
needed services in the realization of this work.
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Abstract
Although, there has been a significant improvement in neuroimaging technologies and analysis
methods, the fundamental relationship between local changes in cerebral hemodynamics and the
underlying neural activity remains largely unknown. The focus of this computational work was to
understand the relation between electrophysiological and hemodynamic data acquired
simultaneously from the human cortex during non-invasive brain stimulation (NIBS). This
computational work was performed using preliminary data collected at Prof. Perrey's Group -
Neuroplasticity and Rehabilitation at the Movement to Health Laboratory, EuroMov, University of
Montpellier. Prof. Perrey's Group has shown the feasibility of using a combined multi-electrode
tDCS-multi channel functional NIRS setup to determine the effects of different tDCS protocols on
bilateral sensorimotor cortex activation. The respective neural activity, assessed with
electroencephalogram (EEG), is postulated to be closely related, spatially and temporally, to
cerebral blood flow (CBF) that supplies glucose via neurovascular coupling. This hemodynamic
response to neural activity can be captured by near-infrared spectroscopy (NIRS), which enables
continuous monitoring of cerebral oxygenation. In this study, the NIRS-EEG joint imaging data was
processed to investigate the relation between alterations in EEG band power (specifically, <12Hz
based on a prior work) and oxy-hemoglobin concentration (O2Hb) in the slow frequency regime
(around 0.1 Hz). Here, a computational autoregressive (ARX) model was investigated for
understanding the relationship between simultaneously acquired electroencephalographic (EEG
band-power <12Hz) and near infrared spectroscopic (O2Hb) data during anodal tDCS. The online
parameter estimation of the ARX model was performed with a Kalman filter and this online
parameter estimation technique was shown to be sensitive towards transient changes in the cross-
correlation between EEG band-power and O2Hb. The computation model for online tracking of the
relation between EEG band-power and O2Hb was developed which needs to be tested on a larger
subject pool in the future studies. This may allow quantitative assessment of the existence of a
coupling relationship between electrophysiological and hemodynamic response to NIBS in health
and disease.
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CONTENTS
CERTIFICATE
ACKNOWLEDGEMENT
ABSTRACT
CONTENTS
1
Introduction
EEG
fNIRS
Slow Oscillations in EEG and fNIRS
Anodal TDCS
2
Experimental Protocol
3
ARX Model and Structure
4
Kalman Filter
5
Results
6 Discussion
7 References
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Chapter 1: INTRODUCTION
Electrophysiological measure of brain activity was obtained via electroencephalogram (EEG) signal
recorded from bilateral sensorimotor cortex. Simultaneous changes in cortical blood oxygenation
were recorded with functional NIRS (fNIRS) co-located with EEG electrodes. High definition tDCS
(HD-tDCS) was conducted for a duration of 10 minutes simultaneously with EEG-NIRS joint
recording. In the first section, the basics of EEG analysis (frequency bands and band-power
measurement) have been explained. In the second section, basics of fNIRS (oxy-hemoglobin and
deoxy-hemoglobin) measurement are explained. In the third part, the relationship between EEG
band power and cerebral hemodynamic oscillations at low frequency has been discussed based on
prior works followed by the description of HD-tDCS in the last section of this chapter.
1.1 Electroencephalography (EEG)
EEG is the most studied non-invasive brain machine interface, mainly due to its fine temporal
resolutions, ease of use, portability and low set-up cost [1]. Electric currents from excitable
membranes of brain tissue superimpose at a given location in the extracellular medium and generate
a potential, which is referred to as the electroencephalogram (EEG) when recorded from the scalp
[2]. In clinical context, EEG refers to the continuous recording of the brain's spontaneous electrical
activity over a period of time, as recorded from multiple electrodes placed on the scalp. Diagnostic
applications generally focus on the spectral content of EEG, that is, the different spectral bands of
neural oscillations that can be observed in EEG signals [3]. Despite limited spatial resolution, EEG
continues to be a valuable tool for research and diagnosis, especially when millisecond-range
temporal resolution (not possible with CT or MRI) is required. It is the most direct correlate of on-
line brain processing that is obtainable non-invasively [3].
1.1.1 Estimating neural activity from EEG
The EEG is typically described in terms of [2][3]
Rhythmic Activity
Transients
The rhythmic activity is divided into bands by frequency based on their spectral content. To some
degree, these frequency bands are a matter of nomenclature (i.e., any rhythmic activity between 8–
12 Hz can be described as "alpha"), but these designations arose because rhythmic activity within a
certain frequency range was noted to have a certain distribution over the scalp or a certain
biological significance. Frequency bands are usually extracted using spectral methods (for instance,
Welch's power spectral density estimates; [5] as implemented for instance in freely available EEG
software such as EEGLAB [4] or the Neurophysiological Biomarker Toolbox [5], besides others.
1.1.2 EEG frequency bands
EEG power spectrum mostly falls within the range of 1–20 Hz where activity below or above this
range is likely to be artifactual under standard clinical recording techniques. Nevertheless, EEG
power spectrum is broadly divided up to 100Hz for quantitative EEG (QEEG) analysis which is
then subdivided in frequency bands called alpha, beta, theta, delta, etc. based on the major EEG
bandwidth used in clinical practice [2][3]. The EEG frequency bands used for QEEG analysis in
clinical practice follows:
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1. Alpha Band: It is the frequency range from 7 Hz to 14Hz. It is the first rythmic activity
seen by Hans Berger [6]. It is seen primarily in the posterior regions of the head on both
sides and higher in amplitude on the dominant side. The posterior basic rhythm emerges
with closing of the eyes and with relaxation, and attenuates with opening of the eye and
mental exertion. The alpha activity in the contralateral sensory and motor cortical areas is
called the mu rhythm that emerges when the hands and arms are idle. When the alpha
becomes abnormal, the person is not even responsive to external stimuli.
2. Beta Band: It is the frequency range from 15Hz to 30 Hz. It may be absent or reduced in
areas of cortical damage [6]. It is the dominant rhythm in patients who are alert or anxious
or who have their eyes open.
Fig. 1.2. EEG Alpha Band
Fig. 1.1. Electric currents from excitable membranes of brain tissue superimpose at a given
location in the extracellular medium and generate a potential, which is referred to as the
electroencephalogram (EEG) when recorded from the scalp [2].
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3. Delta Band: It is the frequency range upto 4 Hz. It tends to be the highest in amplitude and
the slowest wave. It may occur focally with subcortical lesions and in general distribution
with diffuse lesions and deep midline lesions [6]. It is found most prominent in frontal
region in adults and posterior in children.
4. Theta Band: It is the frequency range from 4Hz to 7 Hz. It can be seen as a focal
disturbance in focal sub-cortical lesions [6].
5. Gamma Band: It is the frequency range approx 30-100 Hz. Gamma rhythms represent
binding of different populations of neurons together into a network for purpose of carrying
out a certain motor or cognitive function [6].
Fig. 1.5. EEG Theta band
Fig. 1.4. EEG Delta Band
Fig. 1.3. EEG Beta Band
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1.2 Functional Near Infrared Spectroscopy (fNIRS):
Functional Near Infrared Spectroscopy (fNIRS) is a functional neuroimaging technique using Near
Infrared Spectroscopy (NIRS) technique [7]. NIRS is a cerebral monitoring method that
noninvasively and continuously measures cerebral hemoglobin oxygenation, which is widely used
for monitoring of cerebral vascular status under various clinical conditions. The photons in the near-
infrared (NIR) spectral range (650–950 nm) are able to penetrate human tissue. NIR wavelengths
can be selected such that the change in concentration of oxy-hemoglobin (oxy-Hb) and deoxy-
hemoglobin (deoxy-Hb) in the brain tissue can be detected. NIRS instrumentation works on
different measuring principles, e.g., continuous wave (CW) [8], frequency domain (FD) [9], and
time domain (TD) [10]. Absolute concentration measurements may be possible with more
expensive TD and FD techniques [8], but quantification was not a crucial factor in our application,
since we wanted to detect a relative change in oxy-Hb and deoxy-Hb in response to tDCS rather
than to quantify the hemodynamic response in absolute terms. NIRS signal is strongly contaminated
with systemic interference of superficial origin. A recent approach (see Fig. 1.7) to overcome this
problem has been the use of additional short source-detector separation optodes as regressors [11].
CW fNIRS offers a relatively non-invasive, safe, portable and low cost method of monitoring
hemodynamic correlate of brain activity and relies on the principle of neurovascular coupling (NVC)
[12]. NVC lends to changes in cerebral blood flow (or hemodynamics) that is associated with neural
activity. NIR light spectrum between 700 to 900 nm is mostly transparent to skin, tissue, and bone,
while hemoglobin (Hb) and deoxygenated-hemoglobin (deoxy-Hb) are stronger absorbers of this
Fig. 1.7. Short-channel functional near-infrared spectroscopy regressions to improve
signal to noise ratio
Fig. 1.6. EEG Gamma Band
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spectrum. Differences in the absorption spectra of oxy-Hb and deoxy-Hb enable us to measure
relative changes in hemoglobin concentration through the use of light attenuation at multiple
wavelengths [8]. Two or more wavelengths are selected, with one wavelength above and one below
the isobestic point of 810 nm at which deoxy-Hb and oxy-Hb have identical absorption coefficients.
Using the modified Beer-Lambert Law (mBLL), relative concentration can be calculated as a
function of total photon path length. Typically, the light emitter and detector are placed ipsilaterally
on the subjects skull so recorded measurements are due to back-scattered (reflected) light following
elliptical pathways.
1.3 Oscillations in EEG band power and brain hemodynamics
Prior works has shown a relationship between human electroencephalogram and hemodynamics
based on fNIRS and EEG [13]. Also, modelling studies (see Fig. 1.8) have shown how neurons can
convey “hunger” signals to the vascular network via an intervening layer of glial cells (astrocytes);
vessels dilate and release glucose which fuels neuronal firing [14]. In principal accordance, in this
study, we especially investigated slow hemodynamic oscillations around 0.1Hz that can be related
to known biological phenomena, i.e., arterial blood pressure, cerebral and skin vasomotion,
respiration and neuronal activity [15]. Indeed, hemodynamic activity recorded with NIRS shows
pronounced oscillations at 0.1 Hz which are also present in fluctuations of arterial blood pressure,
and are called Mayer waves [16]. Here, recent analysis of the various transfer functions of the rat
baroreceptor reflex suggests that Mayer waves are transient oscillatory responses to hemodynamic
perturbations rather than true feedback oscillations [16]. We specifically investigated the relation of
these slow hemodynamic oscillations around 0.1Hz with EEG band power <12Hz to elucidate its
neurovascular coupling related aspects due to tDCS perturbations [17]. Here, a computational
autoregressive (ARX) model was investigated using online parameter estimation with a Kalman
filter where prior work has shown the feasibility of a Kalman estimator- and general linear model-
based on-line brain activation mapping by near-infrared spectroscopy [18].
1.4 Anodal 4x1 high-definition transcranial direct current stimulation
Fig. 1.8. Neurons convey “hunger” signals to the vascular network via an intervening layer of
glial cells (astrocytes); vessels dilate and release glucose which fuels neuronal firing [13]
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Transcranial Direct Current Stimulation(tDCS) is a non-invasive way of modulating brain activity
[19]. It is a form of NIBS which uses low DC current delivered directly to the brain area of interest
and the modulation of cortical excitability depends on the direction of current flow with respect to
the neuronal orientation (se Fig. 1.9) [20]. Here, High-Definition transcranial Direct Current
Stimulation (HD-tDCS) using specialized small electrodes in a ring montage have been proposed as
a more focal, non-invasive neuromodulatory technique [21]. High-Definition transcranial Direct
Current Stimulation (HD-tDCS) was invented at The City University of New York with the
introduction of the 4x1 HD-tDCS montage (see Fig. 1.10). The 4x1 HD-tDCS montage
categorically transformed the possibilities with non-invasive neuromodulation but allowing precise
targeting of cortical structures [22].
Anodal HD-tDCS increases cortical excitability and is postulated to increase regional cerebral blood
flow (rCBF) with a certain response time during stimulation [23]. The hemodynamic and
Fig. 1.10. 4x1 HD-tDCS montage categorically transformed the possibilities with non-invasive
neuromodulation but allowing precise targeting of cortical structures [22]. Available:
http://soterixmedical.com/hd-tdcs
Fig. 1.9. Transcranial direct current stimulation (tDCS) is a form of NIBS which uses low DC
current delivered directly to the brain area of interest and the modulation of cortical excitability
depends on the direction of current flow with respect to the neuronal orientation [20].
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electrophysiological response was evoked with a current density of 0.526 A/sq.m which is strong
enough to elicit significant effects on cortical activity [17]. The after-effects of tDCS is an
interesting aspect that is related to neuromodulation [24] The duration of this after-effects depends
on the length of stimulation as well as the intensity of stimulation [24]. Here, one of the unsolved
issue is the role of hemodynamics in tDCS after-effects [25].
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Chapter 2: EXPERIMENTAL PROTOCOL AND DATA
PREPROCESSING
Subjects participated in the study after informed consent where they were well seated in an
armchair with adjustable height and angle in front of a table. The set-up of the HD-tDCS electrodes,
EEG, and NIRS optodes was formed on the surface of the skull according to the standard used EEG
10-20 [26] at the ipsilateral and contralateral sensorimotor (SM) areas (see Fig 2.1).
The HD-tDCS (Startim®, Neuroelectrics NE, Barcelona) cathodes were placed on FC1, FC5, CP5
and CP1 with anode in the center, C3, in a 4x1 ring configuration (Figure 2.1). Measurements of
hemodynamic changes were made from 16-channel continuous wave fNIRS system (Oxymon
MkIII, Artinis, Netherlands) at a sampling frequency of 10 Hz. Pathlength Differential Factor was
populated based on the age of the subject in order to know the variations in concentration of oxy-Hb
and deoxy-Hb [27]. The receiver-transmitter distance of 3 cm was chosen. The receivers (Rx) were
placed on FC3 and CP3 for the left hemisphere and FC4 and CP4 for the right hemisphere (Figure
2.1). Transmitters (Tx) were placed diagonally, i.e., at P1, P5, C1, C5, F5 and F1 for the left
hemisphere and at P6, P2, C6, C2, F2 and F6 for the right hemisphere.
The experiment was divided into two sessions: Pre-tDCS and HD-tDCS, for the purposes of this
computational work.
Pre-tDCS session: The experiment started with 3 minutes of rest to have a baseline acquisition.
Simultaneous acquisition of all the sensor (EEG and NIRS) signals was synchronized using
'SyncBeat.m' custom-written code in Matlab (Mathworks Inc., USA). All the synchronized sensor
data files were saved after the completion of the Pre-tDCS session.
HD-tDCS session: HD-tDCS started with an intensity of 2mA on M1 (ramp up of 30 seconds)
for 20 minutes. The first 10 minutes of HD-tDCS was without task in all the cases and the next 10
minutes of HD-tDCS was in combination with a motor task. Simultaneous acquisition of all the
Fig 2.1. NIRS-EEG/HD-tDCS montage
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sensor (EEG and NIRS) signals was synchronized using 'SyncBeat.m' custom-written code in
Matlab (Mathworks Inc., USA). All the synchronized sensor data files were saved after the
completion of the HD-tDCS session.
EEG raw signal was pre-processed with EEGLAB open-source Matlab toolbox [4]. EEG artifacts
related to tDCS are possible due to the issues with electrical (e.g., unknown electrode impedance
changes during stimulation) and mechanical compatibility (saline from sponges shunting
neighbouring electrodes) where concurrent recording is possible with an optimized device (Starstim;
Neuroelectrics, Spain) and rational experimental design [28]. The raw EEG was pre-processed
using EEGLAB functions (specifically, artefact rejection with Independent Component Analysis)
after artefactual epochs (and channels) were removed following visual inspection of the data. Here,
Independent Component Analysis (ICA) is a powerful tool for eliminating several important types
of non-brain artifacts from EEG data. Then, log-transformed mean-power of ICA pruned EEG data
within the 0.5Hz-11.25Hz range (selected based on our prior work [17]) was computed. The EEG
mean-power time-series was down sampled to 10 Hz to match oxy-Hb data. Co-located four
channels at the HD-tDCS site (see Fig 2.1), namely, T1-R2, T1-R1, T2-R2, T2-R1 NIRS channels
for oxy-Hb and 10, 11, 14, 15 EEG channels for EEG mean-power (0.5Hz-11.25Hz) time-series
were considered for the development of autoregressive model as discussed in the next chapter.
Fig 2.2. The set-up of the HD-tDCS electrodes, EEG, and NIRS optodes was formed on the
surface of the skull according to the standard used EEG 10-20 [26] at the ipsilateral and
contralateral sensorimotor (SM) areas.
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Chapter 3: ARX MODELLING
There are various methods that can be used to assess the degree of similarity or shared information
between two signals. Some of these methods depend on the type of presumptive system which
processes the one “input” signal into the other “output” signal. For linear memory less systems,
cross correlation in time domain is used. For linear systems with memory, common methods
include autoregressive models with exogenous input (ARX), autoregressive moving average
(ARMA) etc.
Complex systems such as brain are difficult to analyse because of the huge number of individual
neuronal/ synaptic paths between nuclei, the nonlinear and non-stationary nature of neuronal
connections, and the operation at multiple time scales. One approach is to use simple low order
linear models to approximate the transfer function relationship, such as autoregressive with
exogenous (ARX) models. The advantage of using linear ARX models is that there is no need to
estimate nonlinearity parameters, and less training data is required. However, the performance of
such models depends on the model order, scale and pre-filtering.
In this study, we adapt and apply the ARX model approach to evaluate the degree of correlation
between cortical EEG and oxy haemoglobin dynamics at low frequency oscillations. The ARX
model is a common method to represent output signals from an unknown system by using a linear
combination of past output signal values and past input values.
3.1 Model Structure
The linear time variant system can be described by an autoregressive model with exogenous input
(ARX), which has been shown experimentally to yield good tracking of output NIRS signal, given
EEG as the input [29].
It can be described as
)
)
)
)
)
A z y t B z u t e t
= +
(1)
with transfer function
( )
( )
( )
B z
G z
A z
= and
)
1 2
1 2
1 1 .. 1
l
l
A z a z a z a z
− −
= + + + +… + +
)
1 1
1 2
( ) ( )
1 .. 1
n n n m
m
B z b z b z b z
− + + −
= + + +… + +
(2)
where y(t) is the output and u(t) is the input at any time t. The z-1 is a back shift operator and (z-1)
y(t) is equal to y(t-1). e(t) is the zero mean and gaussian white noise affecting the system. The
model has l + m parameters/ coefficients in total (a
1
……………..a
l
, b
1
……….……….b
m
).
Substituting (2) in (1), and expanding, the output of an ARX model can be parameterized as
1 1
( , ) ( ) ( )
l m
i j
i j
y t a y t i b u t n j
θ
= =
= + + −
∑ ∑
(3)
where θ = (a
1
……………..a
l
, b
1
……….……….b
m
). The size of θ depends on complexity of the
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model. Thus, the selection of model order (l,m,n) becomes a crucial step in the estimation of
unknown parameters in θ.
The system identification toolbox in Matlab (The MathWorks Inc., USA) was used to find an
optimal model order. The pre-tDCS EEG power and Oxy-Hb NIRS data was considered for
estimating the model order. A range of ARX polynomial models were investigated (see Fig 3.1) for
all the subjects and it was found that l=4, m=5, n=5, was a simpler model structure as compared to
the best model order that we got and showed comparable performance. Finally, the model order (l =
4, m= 5 and n=5) was chosen. The elements of θ are time varying as it relates to variation in EEG
power to NIRS response. At a given time t, the model estimates are predicted using equation (3),
assuming that the system is stationary (slowly time varying), during the prediction horizon.
3.2 State Space Representation of ARX model
This is required for the Kalman filter implementation [29]. Considering an ARX (l,m,n) model as
described in equation (3), its space state form can be described as :
1. Process Equation :
1 1
k k k
x A x B u
− −
= +
(4)
2. Measurement Equation :
k k
y C x
=
(5)
where k represents the current time step . In equation (4), the current state vector x
k
=
[ x
1
……….. x
q
] where q=max(l,m) and u
k-1
is the previous model input.
(
)
A R q q
∈ ×
matrix
relates the previous state x
k-1
to the current state x
k
.
(
)
1
B R q × matrix relates the previous input
u
k-1
to the current state x
k
.
Fig 3.1. The system identification toolbox in Matlab (The MathWorks Inc., USA) was used to
find an optimal ARX Model Structure.
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A = , B =
The y
k
in equation (5) is the measurement of system output.
)
1
C R q∈ ×
matrix relates the
current state to current measurement with the following expression:
[
]
1 0 0 ..0 0 .
C= …
Matrices A, B, C might change with each time step or measurement, but in this study we assume
that they are constant for simplification.
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Chapter 4: KALMAN FILTER
The Kalman filter is an efficient filter, which consists of a set of mathematical equations that
implement a predictor-corrector type estimator that is optimal in the sense that it minimizes the
estimated error covariance—when some conditions are met. It is recursive in nature and can
efficiently estimate the internal states and parameters of a discrete time system from a series of
noisy measurements. Here, the coefficients relating to the past measured NIRS and past measured
EEG power are the model parameters that are to be identified.
The state vector x was augmented with unknown model parameters in θ. Thus, By regarding the
unknown model parameters as elements of the state vector x, the basic Kalman Filter algorithm was
followed for estimation of the state and model parameters. That is, the meta state vector
k k ; k
w = x
θ
[ ]
, where k indicating the current time step, The parameters in θ are assumed to be
locally time invariant compared to the process. Consequently, the augmented Kalman filter system
becomes:
[
]
k k-1 k-1
w = F w , u
k
k
y = Hw
where
[
]
]
k-1 k-1 k-1 k-1 k-1
F w ,u = f x , u ;
θ
[
(
)
H= C 0 0 0 … l+m terms
 
 
The recursive estimation consists of two phases:
Prediction Phase: In this phase, at step k, the a priori estimate of the state
k
w
is given by
the a posteriori state at previous step
k -1
w
.
(
)
k-1 k-1
T
k k k-1 k k-1
= F w , u
= D P D + Q
w
P
P
k
is the estimated error covariance, Q
k
is a process noise covariance diagonal matrix, and
D
k
is the process Jacobian with respect to variables involved.
Correction Phase: In this phase, K
k
is called Kalman Filter gain, R
k
is a scalar
measurement noise covariance.
(
-1
T T
k k
k k k k k
K = P H H P H + R
The updated state w
k
and updated estimate error covariance P
k
is computed as follows:
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(
)
k k k
k k k
w = w K y - H w+
)
k
k k k
P = 1- H
P
K
The limitation in the approach is the degradation of Kalman Filter’s performance in estimating time
varying parameters as it refers to the entire history of past measurements. Since the activity of
stimulated brain area may lead to transients in the NIRS measurement, therefore the tracking of this
activity becomes important with model parameters.
Also, it needs to be able to detect different tdcs-stimulated conditions. In order to track this time
varying nature, a forgetting factor lambda λ is introduced [29].
T
k k-1 k
k
D P D
=
λ
P
)
-1
T T
k k k k k k
K = P H H P H +
λ
When the forgetting factor is closer to 1 then that means the filter will forget fewer past
measurements. A trade-off between the smoothness of tracking and lag in detecting the changes in
model parameters should be considered when forgetting factor is introduced to a Kalman filter.
Usually, λ [0.9, 1] is suitable for most application.
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Chapter 5: RESULTS
The validation and simulation of the ARX model used for modelling the NIRS-EEG data has been
discussed first. Then, in the second section, the experimental results have been discussed.
5.1 Validation of Time-Varying Model Estimation with Kalman Filter
The outline of the Kalman filter has been described in the previous chapters. The time varying
parameter estimation performed by Kalman filter was evaluated in simulations. Simulation is
necessary for capturing the feasibility and robustness of the model for tracking/prediction of Oxy-
NIRS signal from EEG band power signal. In prior work, cardinal B-splines wavelets, which have
been proved to have a lot of excellent properties, are considered and employed for time- varying
parameter expansion [30]. Wavelets have been proved to show excellent approximation properties,
therefore, the time varying models established by the multi-wavelet basis function expansion
scheme can be adaptable and flexible for tracking the sharp variations of non-stationary biomedical
signals, such as EEG recordings. The main features of the multi-wavelet approach is that it enables
smooth trends to be tracked but also to capture sharp changes in the time-varying process
parameters. Simulation studies and applications to real EEG data show that the proposed algorithm
can provide important transient information on the inherent dynamics of non-stationary processes
[30]. If the parameters are time-varying, the problem of parameter estimation is under determined,
and it is difficult to find the best solution. Here, expanding the time varying parameters onto a linear
combination of a set of basis functions can solve the under determined problem. Consequently, the
parameter estimation of unknown variables can be reduced to a set of constant coefficients of the
basis functions.
For our ARX (l, m, n) model, we need to estimate the r = max (l, m) +l + m dimensional meta state.
The max (l, m) parameters relate to the internal states, the rest relate to past NIRS and EEG band
power. The elements of the meta state vector were initialized as zero. The initial output estimate
was set to zero. The estimate error covariance was initialized as P
o
= I, where I is the identity matrix.
Time-varying processes encountered in different engineering applications such as biomedical signal
processing can be characterised by parametric representations. In simulation, the invariant
parameter tracking was evaluated first with the Kalman filter to investigate the stability of the
model.
Secondly in order to investigate the filter’s robustness to the time varying phenomenon, we slowly
changed the model parameters to estimate and track time-varying properties of non stationary
signals relevant for NIRS and EEG [30]. The advantage of simulation is that the true parameters are
known and therefore can be compared with the estimated ones. The simulation model order was
Page 21 of 29
chosen as l=3, m=3 and n= 1 to reduce the model complexity, as it is difficult to know how the
model output changes when too many parameters change. Thus, six parameters a
1
, a
2
, a
3
, b
1
, b
2
and
b
3
were estimated via the Kalman filter algorithm in simulation.
Here, pseudo random binary sequence (PRBS) was chosen as model input. Model input, output and
the a Posteriori estimate of the output are shown in Fig 5.1. The corresponding parameter estimates
of the model are also shown in Fig 5.2. The solid lines and dotted lines represent true parameters
and estimated parameters from the model respectively. As the model parameters vary gradually, the
estimates track the changes well which implies that the estimation method is suitable for time
variant parameter tracking with an ARX model.
The mean absolute error (MAE), normalized root mean squared error (RMSE), and the standard
deviations (Std) of the parameter estimates (with respect to the true parameters) were estimated and
shown in Table 5.1. This scheme was based on a class of time-varying Auto Regressive with an
exogenous input (ARX) model where the associated time-varying parameters are represented by
multi-wavelet basis functions. The orthogonal least square (OLS) algorithm is then applied to refine
the model parameter estimates of the time-varying ARX model.
Table 5.1
Parameters MAE RMSE Std
a1 0.0080 0.0152 0.0130
Fig 5.1. Model input, output and the a Posteriori estimate of the output
Page 22 of 29
a2 0.0066 0.0124 0.0105
a3 0.0044 0.0057 0.0036
b1 0.0031 0.0187 0.0185
b2 0.0034 0.0177 0.0174
b3 0.0035 0.0207 0.0204
5.2 Tracking of Oxy-NIRS signal Based on EEG Data from Human
Experiments
In this section, the online tracking of time-varying parameters is performed using the
Kalman filter on the NIRS-EEG data collected from 5 subjects at the Prof. Perrey's Group -
Neuroplasticity and Rehabilitation at the Movement to Health Laboratory, EuroMov, University of
Montpellier. The model order was chosen as (4, 5, 5) using the system identification toolbox in
Matlab (The MathWorks Inc., USA) (see Fig. 3.1).
The online data during the anodal tDCS was considered for NIRS-EEG tracking where EEG band
power (0.5-11.25Hz) was the input and the Oxy-NIRS signal was the output. The relation between
EEG band power and Oxy NIRS signal around 0.1Hz for one subject during anodal tDCS and
corresponding parameter changes (captured with Kalman Filter approach presented earlier in this
report) are illustrated in Fig 5.3.
Fig 5.2. True parameters represented by bold lines, estimated parameters represented by dotted
lines
Page 23 of 29
The cross-correlation function measured the similarity between EEG band power and shifted
(lagged) copies of Oxy-NIRS signal as a function of the lag. The 250-350 sec and 400-500 sec
windows has been plotted (Fig 5.4 and 5.5) to show the transients in the cross correlation function.
Fig 5.4. Cross-correlation function measured the similarity between EEG power and shifted
(lagged) copies of Oxy NIRS as a function of the lag in the 250-350sec window.
Fig 5.3. Oxy-NIRS signal tracking performed with Kalman Filter with EEG band power as the
input
Page 24 of 29
The online tracking for the other three subjects was performed and illustrated in the figures below.
Fig 5.6. Subject 2: Oxy-NIRS signal tracking performed with Kalman Filter with EEG band
power as the input
Fig 5.5. Cross-correlation function measured the similarity between EEG power and shifted
(lagged) copies of Oxy NIRS as a function of the lag in the 400-500sec window.
Page 25 of 29
Fig 5.8. Subject 4: Oxy NIRS signal tracking performed with Kalman Filter with EEG power as
the input
Fig 5.7. Subject 3: Oxy-NIRS signal tracking performed with Kalman Filter with EEG band
power as the input
Page 26 of 29
Chapter 6 : DISCUSSION
Online effects of transcranial direct current stimulation (tDCS) from simultaneous recording of
electroencephalogram (EEG) and near infra red spectroscopy (NIRS) was presented in this report
alongwith a Kalman Filter based online parameter estimation of an autoregressive model to capture
the EEG band power to Oxy-NIRS transfer function. Here, tDCS has been shown to modulate
cortical neural activity leading to changes in the EEG power spectrum. During neural activity, the
electric currents from excitable membranes of brain tissue superimpose at a given location in the
extracellular medium and generate a potential, which is referred to as the electroencephalogram
(EEG) when recorded from the scalp. This neural activity is closely related, spatially and
temporally, to cerebral blood flow (CBF) that supplies glucose via neurovascular coupling. The
hemodynamic response to neural activity can be captured by near-infrared spectroscopy (NIRS),
which enables continuous monitoring of cerebral oxygenation and blood volume. CBF is increased
in brain regions with neural enhanced activity via metabolic coupling mechanisms. In this study, a
model was developed to capture the online effects of tDCS.
The online tracking of the ARX parameters was performed with experimental data (with transients
due to tDCS) using the Kalman filter approach. We can observe the transients in the Fig 6.1 where
input EEG band power signal becomes out of phase around 300 sec and in phase between 400-500
sec range with respect to measured Oxy-NIRS. These changes are reflected in the ARX parameters.
However, it is illustrated for one subject here, and we can observe similar changes reflected in ARX
parameters when the two signals becomes in-phase/anti-phase in other subjects as well (see Fig 5.6,
5.7 and 5.8).
The variation in ARX parameters due to change in correlation between EEG band power and Oxy-
NIRS signal is validated by performing the sliding window cross correlation analysis between the
Fig 6.1. ARX parameters change as the input EEG band power signal becomes out of phase
around 300 sec and in phase between 400-500 sec range with respect to measured Oxy-NIRS.
Page 27 of 29
two signals. In Fig 6.2, the correspondence between ARX parameter change and cross correlation
between the signals is illustrated. The window size was set at 30 seconds with overlap of 0.1
second.
In Fig. 6.2, the tDCS onset response is evident that has been presented in our prior works [31]. It
can be found in the first 10 sec of the tDCS ON periods (called "initial dip" [17]) however the
physiological and/or artefactual basis is not evident yet. Investigation on any physiological basis of
this tDCS onset response is currently undergoing.
Fig 6.2. Correspondence between ARX parameter change and the cross correlation function
between the NIRS-EEG signals.
Page 28 of 29
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