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Page 1 of 29

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

Page 2 of 29

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

Page 3 of 29

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]

Page 11 of 29

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].

Page 12 of 29

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].

Page 13 of 29

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

Page 14 of 29

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.

Page 15 of 29

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

Page 16 of 29

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.

Page 17 of 29

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.

Page 18 of 29

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:

Page 19 of 29

(

)

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.

Page 20 of 29

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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