modeling, optimization and validation for brain stimulationcj82ss01k/fulltext.pdf · (tcs). the...

Modeling, Optimization and Validation for Brain Stimulation

A Thesis Presented

by

Kimia Shayestehfard

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements

for the degree of

Master of Science

in

Electrical and Computer Engineering

Northeastern University

Boston, Massachusetts

April 2018

Contents

List of Figures iii

Abstract of the Thesis vii

1 Introduction 1

1.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Electrocorticography (ECoG) . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Transcranial current stimulation (tCS) . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Finite Element Method (FEM) . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Scope and organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Evaluation of finite element model of Electrocorticography 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Clinical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Evaluation of FEM based head model building . . . . . . . . . . . . . . . . . . . . 21

2.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Optimization of temporal interfering current injection patterns 28

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Temporal Intereference Electric Field . . . . . . . . . . . . . . . . . . . . 30

i

3.2.2 Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.4 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Conclusions and Future Work 52

Bibliography 54

ii

List of Figures

2.1 Stimulus pulse duration effect on recorded data shape. . . . . . . . . . . . . . . . . 8

2.2 Montage of the electrodes in Case 1. . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Montage of the electrodes in Case 2. . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Case 1 with source electrode ]12 and sink electrode ]20 . . . . . . . . . . . . . . 11

2.5 Idealized stimulus current for one pulse. . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Voltage recorded during one stimulating pulse. . . . . . . . . . . . . . . . . . . . 12

2.7 Estimation of noise power as a function of electrode number visualized in map3D

for the 5 different stimulation pairs for Case 1 . . . . . . . . . . . . . . . . . . . . 16

2.8 Estimation of noise power as a function of electrode number for the 4 different

stimulation pairs for Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.9 Figures in the left column show mean and standard deviation for each half pulse

while figures in the right column show the ratio of mean value to standard deviation

for each half pulse. Each panel’s caption represents the electrode and stimulation

pair as ], ]− ] respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9 Figures in the left column show mean and standard deviation for each half pulse

while figures in the right column show the ratio of the mean value to the standard

deviation for each half pulse. Each panel’s caption-caption represents electrode and

stimulation pair as ], ]− ] respectively. . . . . . . . . . . . . . . . . . . . . . . . 19

2.10 Figures in the left column indicate standard deviation and mean value for each half

pulse while figures in the right column indicate the ratio of mean value to the standard

deviation for each half pulse. Each panel’s sub-caption represents electrode and


iii

2.10 Figures in the left column indicate standard deviation and mean value for each half

pulse and figures in the right column indicate ratio of the mean value to the standard

deviation for each half pulse. Each panel’s sub-caption represents electrode and


2.11 The average electric potential on each electrode during the stimulus pulses compared

to the FEA computations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.12 Information about electrode types in Case 2. . . . . . . . . . . . . . . . . . . . . . 24


stimulation pairs in Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25


stimulation pairs in Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.15 Visualization of spatial distribution of noise power as a function of electrode locaation

using map3D, for the 5 different stimulation pairs in Case 1. . . . . . . . . . . . . 27

3.1 Low-frequency field envelopes of two higher frequencies. . . . . . . . . . . . . . . 29

3.2 A graphical representation of electric fields and −→n in 2D plane. . . . . . . . . . . . 32

3.3 Green-shaded region is showing min(|−→E f1(

−→r ).cos(β) |, |−→E f2(

−→r ).cos(α− β) |). 32

3.4 Peak of min (|−→E f1(

−→r ) |, |−→E f2(

−→r ) |). . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Peak of min (|−→E f1(

−→r ) |, |−→E f2(

−→r ) |). . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Unequal projection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.9 Green electrode is the reference electrode for lead field matrix. . . . . . . . . . . . 39

3.10 Green arrow is indicating y-axis and red region show the ROI. Electrodes and ROI

boundaries are divided w.r.t y-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.11 Electrodes have trapezoidal configuration . . . . . . . . . . . . . . . . . . . . . . 42

3.12 Electrodes have rectangular configuration . . . . . . . . . . . . . . . . . . . . . . 43

3.13 Red regions indicate the ROI inside the spherical model . . . . . . . . . . . . . . . 44

3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

iv

3.16 EAM (−→x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.20 EAM (−→x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

v

Acknowledgement

A MS thesis is truly a marathon journey which would not be possible without the support

and guidance of many individuals. I would like to express my gratitude towards all of them who

helped transform my research work into a successful MS thesis.

There is possibly no transformation capable of mapping my deep appreciation for my supervisor,

Dr. Dana H. Brooks, into few lines of acknowledgements! He has provided me with invaluable

continuous guidance and support. His insights and troubleshooting skills have strengthened this study

significantly and beyond all of these, I would like to say thank you to him for all his kindness and

making the world a better place to live for all people living around him. I would like to extend my

thanks to Dr. Stratis Ioannidis, I had the honor of being his student for one semester and also having

him as one of my committee members for my MS thesis defense. Also I appreciate Dr. Sumientra

Rampersad for all her helpful feedback, all I learned from her and also for being one of my committee

members.

Over the past two years, I have also been fortunate to share a lab with some formidably intelligent

fellow students. In particular, thanks to Biel Roig-Solvas for all his help and support during this

project. Thanks to Dr. Moritz Dannhauer for pushing me, motivating me and helping me to improve

my programming skills. Thanks to Dr. Mathew Yarossi for not only being a supportive group mate

but also being a person who goes on to make a difference, whether it is in this field or another. Thanks

to Dr. Jaume Coll Font and Dr. Seyhmus Guler. I learned a lot from them during the short time I

was their group mate. Thanks to Jesse Marsh, student service coordinator of Northeastern University

for all his help and the positive vibes he is giving to students. Finally my deepest appreciation goes

to Maman, Baba, Kiana, Reihaneh and Sahba. Thanks for being there for me whenever I needed

you. Your endless love always help me to keep going on.I feel blessed in my life because of being

surrounded with smart and kindhearted people and I am thankful of presence of all of them in my

life.

vi

Abstract of the Thesis

Modeling, Optimization and Validation for Brain Stimulation

by

Kimia Shayestehfard

Master of Science in Electrical Engineering

Northeastern University, April 2018

Dr. Dana H. Brooks, Dr. Stratis Ioannidis Advisers

For the past few years, interest in non-surgical and temporarily implanted brain stimulation

methods as tools for clinical studies and research has grown substantially. In electrocorticography

(ECoG) stimulation, an array of electrodes is placed on the cortical surface during a surgical

procedure, most typically as part of surgical planning for resection of epileptogenic tissue. These

ECoG electrodes can be used to both measure intrinsic brain activity and to stimulate superficial

cortical regions. Non-invasive electrical stimulation achieved through electrodes placed on the

scalp, generally referred to as trancranial Current Stimulation (tCS), is frequently used to stimulate

superficial brain areas. This thesis explores some questions related to modeling and optimization of

ECoG and tCS stimulation. Both topics depend on computational modeling of the distribution of

current in the head induced by the stimulation, typically carried out use the Finite Element Method

(FEM). FEM models have been validated for tCS but not for ECoG stimulation. In the first part

of this project, as part of a larger collaboration, we analyze ECoG data recorded with arrays of

electrodes during stimulation episodes and compare the results of our analysis to FEM simulations

as an initial attempt to quantify accuracy of the FEM modeling.

vii

In the second part of the thesis we focus on a recently published method that offers promise

to allow deeper and more focused tCS than has been possible to date. Specifically, in a very recent

study, Grossman and co-workers described a non-invasive method to stimulate deep brain regions

that they called Temporal interference (TI) stimulation. The intuition behind this work is that, since

neurons respond only to relatively low frequencies (typically below a few hundred Hertz), by applying

multiple different oscillating currents at nearby frequencies above that range, the superposition of

these currents can create a low frequency envelope electric field in deep areas of the brain with a large

enough magnitude to modulate neuron firing in deep brain regions while not stimulating overlying

regions. This work raises many questions regarding electrode configuration and current injection

patterns that might best stimulate a specific desired region of interest. In this thesis, we investigate

the use of a multi- electrode FEM-based optimization approach to maximize the TI effect in a region

of interest while limiting the TI effect in non-targeted regions. We study this problem applying

FEM in a spherical simulation model with multiple target regions. We analyze the relevant objective

function, which turns out to be non-convex but in a highly structured manner, and formulate and

study via numerical simulations a constrained optimization problem based on that objective. Results

indicate some ability to control delivery of TI stimulation but several aspects of this optimization are

not yet well understood.

viii

Chapter 1

Introduction

1.1 General introduction

Brain stimulation is referred to the process of stimulating the activity of specific areas in

the brain, (i.e. superficial or deep regions of the brain). Brain stimulation activates or inhibits the

brain activity. It could be either invasive or non-invasive. Electric field could be generated through

current injection (e.g transcranial current stimulation (tCS)) or it could be induced by magnetic

field (e.g transcranial magnitude stimulation (TMS)). Brain stimualtion has application in treating

certain neurological and neuropsychiatricl disorders that do not respond to other treatments and also

it has application in enhancement of cognitive function (e.g Alzeimer, parkinson’s disease, epilepsy,

depression, post stroke recovery and enhancement of motor learning function.). Two better known

brain stimulation methods are Electroconvulsive Therapy (ECT) which is a non-invasive method and

Deep Brain Stimulation (DBS) which is an invasive method. Two other brain stimulation methods

which we cover in this thesis are electrocorticography (ECoG) and transcranial current stimulation

(tCS).

The focus of this thesis is on validation of Finite Element Method (FEM)-based modeling

for ECoG stimulation and Optimization of tCS current injection pattern for deep brain stimulation

via temporally interfering electric field.

In the following we to give an in depth introduction to tCS and ECoG stimulation methods and also

we discuss the numerical solver we used to solve the problem and the mathematical equations used

for our modeling and simulation purposes. In the end of the chapter we will give an overview of the

material we are going to cover in the next chapters.

1

CHAPTER 1. INTRODUCTION

1.1.1 Electrocorticography (ECoG)

Electrocorticography (ECoG) is an invasive method in which electrodes are placed on

the surface of the cortex without penetrating the cortical tissue. This method combines both high

spatial and high temporal resolution and can record neural activity simultaneously from numerous

electrodes. There has been a great interest in applying ECoG electrodes to both record information

and to stimulate the brain. ECoG grids stimulate superficial regions of the brain. As ECoG electrodes

are on the cortical surface, we need to be careful about amplitude of the injected current in order to

protect the active brain tissue; on the other hand proximity of the electrodes to the cortical surface is

an advantage both in terms of measurement, in particular having having a better quality signal than is

possible with scalp electrodes with high signal to noise ratio and better resolution, and in terms of

stimulation, as it allows more control on delivering the current to the region of interest (ROI) than is

possible with scalp electrodes .

Studies have reported the applications of ECoG recording and stimulation as a therapeutic

tool and also to map brain function. ECoG grids are used as a brain mapping technique to localize

language cortex, sensorimotor pathways or to determine seizure foci in children with supratentorial

brain tumors [3, 4]. Also electrocorticography (ECoG) has been used as a therapeutic tool in surgical

planning for drug-resistant pediatric focal epilepsy and in post stroke rehabilitation and finally ECoG

is performed following the initial resection to assess the completeness of the resection and to identify

areas of residual epileptiform activity that may be considered for removal [34, 30, 1].

1.1.2 Transcranial current stimulation (tCS)

In contrast to invasive current brain stimulation methods, we have noninvasive methods.

Noninvasive current brain stimulation methods, namely transcranial direct current stimulation (tDCS)

and related techniques [transcranial alternating current stimulation (tACS), and transcranial random

noise stimulation (tRNS)]—all of which are referred to generically as tCS—stimulate the brain

by generating electric fields through the delivery of currents transcranially from the scalp. These

subthreshold neuromodulation techniques provide a practical, complementary alternative to invasive

stimulation [33]. During tCS stimulation, changes in neural membrane potentials are induced in a

polarity-dependent manner. For example, anodal stimulation is generally thought to enhance cortical

excitability in the motor cortex region, while cathodal stimulation inhibit it [28]—although many

reports suggest that this is not as consistent as sometimes described. The strength and duration of tCS

after-effects can be altered by changing the current intensity and duration [28]. In order to assess the

2


efficiency of stimulation, a large number of experimental and clinical studies have been performed

[36, 6, 13]. Here we describe a few reports on tCS functionality and applications, just to give the

reader a sense of the breadth of applications being investigated.tCS application in modifications of

perceptual, cognitive, and behavioral functions and treatment of various brain disorders has been

reported [18]. tACS, for example, is typically used to directly modulate the ongoing rhythmic brain

activity by applying oscillatory currents on the human scalp [36]. tACS has behavioral, excitability

effects [22] and tDCS was used in some disease treatment such as Parkinson [6], Alzheimer’s

disease [13], depression [16] and stroke [15, 2]. tACS has been reported to induce phase cancellation

of the rest tremor rhythm in Parkinson disease [17], enhanced individual Alpha activity in human

EEG [36] and alpha oscillations modulated target detection performance [20]. Diverse behavioral

effects after applying different tACS frequencies raised the possibility to causally link specific

frequencies to distinct functions [20]. Some studies reported that 20 Hz tACS (90 s; 0.14 Am2 )

could increase motor cortex excitability [14], while 15 Hz tACS (20 s; 0.80 Am2 ) could decrease

motor cortex excitability [37]. The effects of tACS applied over the primary visual cortex in a high

gamma frequency range (60 Hz) in improvement of contrast perception has been reported; however

this stimulation had no effect on spatial attention modulation in that study [25]. Also theta tACS

(with frequency in the range of 4–7 Hz) with the target electrode positioned over the left parietal

brain area significantly increased working memory (WM) storage capacity in another report [24].

Multiple factors related to neuron structure and tCS safety concerns affect the neuron response to

tACS. Some of these factors are as follow.

- There are strong directional effects in the interaction of electric fields and neurons, i.e, neurons are

influenced mostly by the component of the electric field parallel to their orientation [11].

- All neurons have some contribution from a low-pass mechanism in their frequency response [23],

however the exact functionality of the neurons which made them respond to low frequencies is not

known to us.

- Due to safety aspects of tCS concerns for both healthy people and patients there are limitations on

the maximum injected current from the individual electrode and the total injected current. Even with

these standard limitations, some research has reported minor adverse effects in healthy humans and

patients with varying neurological disorders (e.g mild tingling sensation, moderate fatigue, headache,

nausea, insomnia) [31].

Therefore using optimization techniques to find the appropriate electrode configuration and the

optimum current injection pattern for reaching to the maximum electric field in the region of interest

(ROI) is crucial. However even by applying optimization techniques, due to the thresholds on the

3


maximum current injected to each electrode and total injected current, tACS can only stimulate

superficial regions of the brain.

1.1.3 Finite Element Method (FEM)

In order to better understand and design both ECoG and tCS stimulation strategies, it is

helpful to develop tools to simulate how injected currents travel through the brain. We can write

the general equations that describe the underlying physics, but to calculate simulations we need

a numerical solver to predict the spatial distribution of unknown quanities such current density,

electric potential, and electric field. There are many numerical solvers available, including the

Boundary Element Method, the Finite Element Method, the Finite Volume Method, and the Finite

Difference Method, for this purpose. The Finite Element Method is the most common one and we

used this method in our model building in this work. Using the Finite Element Method (FEM), we

approximated the unknown values such as current density or electric field distribution by discretization

of our physiological / geometric model and numerically solving the governing equations.

In more detail, we used a quasi-static approximation and used Laplace/Poisson equations and

boundary approximation to estimate the electric field in our mode.

The relationship between electric field E and scalar potential φ is given as:

E = −∇φ (1.1)

Assuming that there is no interior current source in the head equation ?? we have:

∇.σ∇φ = 0 (1.2)

This equation stating that the sum of currents going out from any point is zero. In order to solve

the above Laplace equation, we used the FEM to approximate electrical field numerically. For this

purpose we discretized the domain into volume elements and defined boundary conditions [27].

Assuming I denotes a vector of all electrode currents and I denotes an array of electrode currents

excluding the reference electrode, one obtains

I = [I1, I2, ..., IM−1, IM ]

I = [I1, I2, ..., IM−1]

Since currents must sum to 0, there are only M-1 free current variables and it is convenient for the

4


sequel to define IM = −∑i=M−1

i=1 Ii.

The electric field difference at the electrodes can be simply calculated as E(i) − E(M) = li,M ,

where E(i) and E(M) denote electric field at electrode i and M respectively. To make the solution

to Laplace’s equation unique we need to pick a common reference. For convenience we chose the

M th electrode, the one that we left out of vector I , as the reference and li,M denotes lead vector.

Calculating the lead vectors over the elements centers in the head space, results in a lead field vector

LiM ∈ R3×N where N is the number of source space elements. Therefore by fixing the last electrode

as the reference electrode, the lead field matrix becomes L = [L2,M , L3,M , ..., LM−1,M ] where

the lead field matrix size is L ∈ R3(M−1)×N . The electric fields on the center of elements can be

calculated as

Eelements = LI (1.3)

1.2 Scope and organization of this thesis

The overall scope of this thesis is to do modeling, optimization and validation for brain

stimulation. In chapter 2 we describe our work to validate simulations of ECoG stimulation on a

FEM-based human head model.This analysis could give us a better understanding of how reliable

our FEM model is. Having an accurate model could help us to modulate the stimulation effects

in the brain. Moreover, for optimization purposes, (e.g. finding the optimum current injection

pattern for stimulating a specific region in the brain), we can calculate our optimization results on

our model. We describe two ECoG datasets, from two different subjects, that were provided to

us by our collaborators; these are high quality data from clinical implementation of ECoG grids.

We then describe our method to extract meaningful information through our clinical data that we

can compare against our FEM model to validate it. Th clinical data sets were provided to us by

Prof. Jeffery Ojemann and his group members, David J Caldwell and Jeneva Cornin at Department

of Neurological Surgery, University of Washington. For building the FEM-based head model and

visualization of the results we used open source software ( SCIRunand map3d) and the head model

was generated by Moritz Dannhauer To the best of our knowledge, there have been a few reports

on validation of finite element models of transcranial current stimulation and transcranial magnetic

stimulation [29, 10] but no reports have been published on validation of FEM based models for

ECoG grids stimulation.

In Chapter 3 we investigate a very recent non-invasive method proposed by Grossman et.

al [19] to stimulate deep regions in the brain without stimulating overlying regions. The idea is

5


to deliver multiple electric fields at high frequencies that are out of neuron firing range, such that

their intersection creates a ”low frequency” amplitude modulation envelope within the range that

they report experimentally was able to modulate neural firing. Thus the promise of this method is

that we may be able to electrically stimulate neurons if we have an amplitude larger than a specific

threshold of this envelope. They called their method stimulation by ”temporal interference” (TI).

If we have a strong TI effect in deep areas of the brain, then we could stimulate these deep regions

without stimulating overlying regions.

To the best of our knowledge [19] is the only publication to date on II stimulation.

The authors concentrated on a mouse model, reporting extensive experimental results and a few

simulations. There is also a recent paper posted on arXiv, as far as we know not yet publishd, called

”Noninvasive Dynamic Patterns of Neurostimulation using Spatio-Temporal Interference” [7] that

combines optimization with the TI idea to suggest how to optimally stimulate deep regions of the

brain. They built a Hodgkin-Huxley model for a neuron to determine firing threshold as a function of

envelope frequency and a model of current dispersion in the head and used multi-electrode-pairs for

stimulating the region of interest (ROI), while leveraging very strong symmetry assumptions. Also

they replaced electrode pairs with a “patch” of multiple electrode-pairs, with each electrode- pair in

each patch generating currents of the same frequency. By applying an optimization algorithm they

then found find the minimum injected current for stimulating a central ROI in their model.

In Chapter 3 we formulate, solve and test a significantly more general multi-constraint optimization

problem that provide stimulus patterns to maximize the amplitude of the TI electric field envelope

modulation in the targeted region. We tested our simulations on a finite element (FE) multi-electrode

spherical model. Moreover we applied safety constraints that matches with clinical experiments safety

criteria. We describe and analyze the objective function we optimize along with the constraints we

impose to encourage focality as well as ensure safety, and then present and discuss our optimization

results. Finally in Chapter 4 we talk about future directions and conclusion of works we have done in

this thesis.

6

Chapter 2

Evaluation of finite element model of

Electrocorticography

2.1 Introduction

Electrocorticography (ECoG) stimulation uses electrode arrays surgically implanted on the

cortical surface to stimulate the brain. These arrays can be used to both record brain electrical activity

from the cerebral cortex and to inject current through the same electrodes in order to modulate

brain activity They can be used for mapping of brain function for pre-surgical planning and also

for rehabilitation and brain-computer interfaces, both of which are useful in several applications.

However little is known about the spatial pattern in which current injected by these arrays flow

through the cortex or how these patterns affect brain activity.

Having an appropriate model could help us to modulate the stimulation effects in the

brain. Moreover, for optimization purposes, (e.g. finding the optimum current injection pattern for

stimulating a region of interest (ROI) in the brain), we do the analysis on our model. Therefore Model

validation could help us to understand how reliable our model is, which is useful since these models

are widely used in simulating brain activity, stimulation effects and current distribution pattern. In

this chapter, we will present analysis of human ECoG arrays recordings data and comparisons to

finite element-based human head model predictions. The data were provided to us by the University

of Washington team as mentioned in Section 1.2. We have multipoint measurements using clinical

electrocorticography (ECoG) grids, strips and depth electrodes for 2 human subjects, denoted here

7

CHAPTER 2. EVALUATION OF FINITE ELEMENT MODEL OF ELECTROCORTICOGRAPHY

Case 1 and Case 21 For each stimulus epoch, one pair of electrodes was chosen for stimulation with

biphasic, bipolar, constant-current pulses, while voltages were recorded on all the non-stimulating

electrodes.

We note that in the particular datasets chosen for analysis here, the pulses were long

enough that we consistently saw relatively “flat” signals on both positive and negative phases of the

pulses. This is important because we use linear and quasistatic assumptions in our modeling and

thus want to extract a single amplitude number per measurement channel for a particular stimulation

electrode pair If we use stimulations with short pulse widths, the voltage signals on the measurement

electrodes do not look ”flat”; evidently there are capacitive effect with time constants that are too

long to saturate during the half-pulses. Figure 2.1 clearly shows the impact of stimulation duration

on recorded voltage shape.

(a) Stimulus pulse width is 0.4 ms. (b) Stimulus pulse width is 2.4 ms.

Figure 2.1: Stimulus pulse duration effect on recorded data shape.

Figure 2.2 shows the geometry of the electrodes for Case 1 and Figure 2.3 shows the

geometry of the electrodes for Case 2. These figures are created by Nile Wilson. They were created

using clinical magnetic resonance imaging (MRI) and computed tomography (CT) scan images. The

cortical reconstructions were completed using previously described techniques ( [5], [35], [21]).1For reference, the case numbers in the Unversity of Washington system are StimulationDepths 2fd831 and

StimulationSpacing 20f8a3

8


Figure 2.2: Montage of the electrodes in Case 1.

(a) (b)

(c) (d)

Figure 2.3: Montage of the electrodes in Case 2.

9


The rest of the chapter is organized as follows. In Section 2.2, we explain the methods

we used for information extraction. Section 2.3 reports on analysis of experimental data sets. In

Section 2.4, we will present comparisons against FEM simulation results and in Section 2.5 we

include some additional data analysis plots.

2.2 Methods

In this section, we present details of how the clinical data were acquired and the data

analysis methods we used.

2.2.1 Clinical data

The clinical data were acquired with a Tucker Davis Technologies (TDT) System 3 with the

RZ5D and PZ5Neurodigitizer. The stimulation was delivered through the TDT IZ2H-16 stimulator

and LZ48-400 battery pack. The experiments were done with patients with intractable epilepsy who

underwent neurosurgery and temporary electrode placement for clinical monitoring. They remaind

in the hospital for about 1 week post implantation. Initially they were taken off anti-epileptic drugs

to enhance seizure and spike mapping. The stimulation studies was only carried out once they were

back on these medications; usually 6 or 7 days after the initial electrode implant and typcally 1 day

before the electrode removal.

It was always emphasized to patients that the ECoG grid placement was determined based on only

clinical needs and not based on research interests.

In all the data sets analyzed here, the injected current contained a train of 10 biphasic

pulses with 2.4 ms pulse width and 120 ms inter-pulse gap. This gap is the time between end of

one pulse and start of next. Peak-to-peak amplitude was 1 mA. In both cases the sampling rate was

12.207 kHz.

In Case 1, we have 5 data sets obtained by stimulating different electrode pairs. All

stimulus channels were selected from grid electrodes. In Case 2, we have 4 data sets obtained by

choosing different electrode pairs for stimulation purposes. In all the data sets for Case 2, one

stimulus channel was selected from grid electrodes and the other from depth electrodes. As our data

was noisy, we needed to do some pre-processings to improve signal conditioning. In particular there

was a large DC offset on the signals. By subtracting the mean of each signal we were able to remove

10


this effect.

V i(t) = V ioffset(t)− V

ioffset(t) (2.1)

where i denotes the electrode number, V i denotes the voltage after DC offset removal, V ioffset

denotes the voltage with DC offset and V ioffset represent the average of voltage on each channel,

and t is time. All the data had similar offests , so we illustrate the signal with DC offset and without

DC offset one specific stimulus pair from Case 1 in Figure 2.4 .

(a) Signal with DC offset. (b) Signal after DC offset removal

Figure 2.4: Case 1 with source electrode ]12 and sink electrode ]20

2.2.2 Data analysis

In Case 1 we had 5 data sets obtained by recording brain activity during current injection

with different pairs of electrodes used to deliver the stimulus. There were 128 electrodes in total,

including two electrode grids, one of which had 32 electrodes and the other 16. We only used these

48 grid electrodes in this analysis since the rest of the electrodes were disconnected. In Case 2 we

have 4 data sets. There were 128 electrodes in total, including two electrode grids. Each had 32

electrodes, 32 depth electrodes and 32 strips electrodes. Information related to electrode types in

Figure 2.3 is in Figure 2.12. In each experiment, we wanted to extract two meaningful numbers

per channel, one positive and one negative number, to be compared with our finite element analysis

(FEA) results. For this purpose we averaged all the positive phase samples together and we did the

same thing with all negative phase samples during each current injection pulse.

11


Figure 2.5: Idealized stimulus current for one pulse.

For our analysis, we selected the relatively flat parts of both positive and negative phase

half-pulses. To do this, we selected 9 samples in each half-pulse, from the 3rd sample to the 11th

sample, based on an overall visual analysis of the duration of the transient effects at the start and end

of each half-pulse. We considered only these samples and rejected the first and last two samples in

each half pulse, which typically were not close to flat. Figure 2.6 illustrates why we ignored the first

two and the last two samples of each pulse in our analysis.

Figure 2.6: Voltage recorded during one stimulating pulse.

The mathematical representation of the procedure we took for data analysis is as follows.

Tstep = Tgap + Tpulse (2.2)

12


Where Tgap = 120 ms is the inter-pulse gap, Tpulse = 2.4 ms is the pulse duration and Tstep

represents the pulse repetition period, the time between start of one pulse and start of the next.

Vifirst =

1

n

k=10∑k=1

ts+T2− 2

fs+(k−1)(Tstep)∑

ts+3fs

+(k−1)(Tstep)

V i(t) (2.3)

Visecond =

1

nVifirst =

1

n

k=10∑k=1

ts+T− 2fs

+(k−1)(Tstep)∑ts+

T2+ 3

fs+(k−1)(Tstep)

V i(t) (2.4)

In equation 2.3 and 2.4, ts denotes the moment that we start stimulation, fs is the sampling

frequency , k is the pulse order (we had a train of 10 stimulus pulses), n is the number of samples

which we averaged over per pulse, V ifirst is the average of all samples during the first half-pulses

stimulation and V isecond is the average of all sample during the second half-pulse stimulation.

Additionally we quantified the reliability of the results by computing not only the average

amplitude of each channel but also the ratio of average of samples over their standard deviation for

both positive and negative phase half-pulses. In equation 2.7 and equation 2.8, Rifirst and Risecondindicate these ratios for each electrode and first and second half-pulses.

sifirst =

√√√√√√ 1

n

k=10∑k=1

T2− 2

fs+k(Tstep)∑

ts+3fs

+k(Tstep)

(V i(t)− V ifirst)

2 (2.5)

sisecond =

√√√√√√ 1

n

k=10∑k=1

T− 2fs

+k(Tstep)∑ts+

T2+ 3

fs+k(Tstep)

(V i(t)− V isecond)

2 (2.6)

sfirst and ssecond are the standard deviation for the n samples in first half pulses and half pulses,

respectively.

Rifirst =Vifirst

sifirst(2.7)

Risecond =Visecond

sisecond(2.8)

We also looked at which electrode, for a given stimulation pair location, had the strongest amplitude.

This pattern was not obvious, so to try to approximate the observed effect, with the thought that

13


perhaps this as a purely geometric effect, we devised an algorithm to try to predict the channel with

the greatest amplitude average. Our insight was that when we averaged positive pulses and negative

pulses, the largest magnitudes typically belonged to the electrodes, which were in the neighborhood

of one stimulation channel but relatively far from the other one. Hence in our algorithm we considered

only these electrodes. Let ea represent the active stimulation channel and eg represent the ground

channel, and N represent the 4-neighborhood of both stimulation channels.

Algorithm 1: Estimating the magnitude of the difference between the distances ofeach electrode from stimulus electrode pair

for ∀j ∈ N do

d1 = ‖di − dea‖22

d2 =∥∥di − deg∥∥22

relative dj = |d1− d2|end

We define relative dj as the magnitude of the difference between the distances of each

electrode in N from the two stimulation channels. In our algorithm, the electrode that had the

highest relative dj is the candidate for having the greatest averaged amplitude. Finally we calcu-

lated the noise power on each electrode to see if amplitude strength corresponds to noise power or not.

2.3 Data Analysis

Here we present the results of implementation of the methods discussed in 2.2 and our

analysis. A summary of all results for Case 1 and Case 2 are provided in Table 2.1 and Table 2.2, re-

spectively. In both Table 2.1 and Table 2.2 we define the ”Strongest Channel” as the channel, imax=i

for max(V ifirst), and ”algorithm candidate for the strongest channel”, jmax=j for max(V j

algorithm)

, as the channel, which based on our algorithm we might expect to be the strongest channel. In

column ]6 we report the channel with the highest noise power. Column ]7 and column ]8 represent

Rimaxfirst and Rjmax

first, respectively. We considered the results obtained from first-half pulses for our

analysis since the results for second-half pulses were the essentially the same, just the sign for the

values was different.

14


Stimulus Source Sink imax jmax Channel with Rimaxfirst Rjmax

first

Pair ] electrode ] electrode ] the highest

noise power

1 12 20 4 4 17 44.28 44.28

2 20 21 22 22 17 1.99 1.99

3 18 23 22 17 17 1.98 1.99

4 19 22 23 18 17 1.99 1.99

5 4 28 29 3 17 1.99 1.98

Table 2.1: Case 1

Stimulus Source Sink imax jmax Channel with Rimaxfirst Rjmax

first

Pair ] electrode ] electrode ] the highest

noise power

1 121 63 122 122 64 1.98 2.27

2 55 121 122 122 29 1.98 1.98

3 63 121 122 122 64 1.98 1.70

4 121 55 122 122 29 1.98 1.98

Table 2.2: Case 2

We note that in Table 2.1 our algorithm corresponds to the channel with the strongest

amplitude for row ]1 and row ]2, while for the other 3 rows our algorithm picked a channel adjacent

to the stimulation electrode while the channel with the largest estimated amplitude. Across the

entire experiment channel ]17 had the highest noise power. In Case 2 in all four experiments our

algorithmic choice corresponded to the channel with the strongest amplitude.

To illustrate the spatial distribution of the noise over the array, Figure 2.7 and Figure 2.8

visualize the noise power in grid electrodes for Case 1 and Case 2, respectively using the map3D

software. Note that in these figures we replace the noise power of stimulation channels with one of

their neighborhood channels for visualization purposes since on stimulation channels the recorded

signal is purely noise .

15


(a) Active channel is e]12 and ground channel is e]20. (b) Active channel is e]20 and ground channel is e]21.

(c) Active channel is e]18 and ground channel is e]23. (d) Active channel is e]19 and ground channel is e]22.

(e) Active channel is e]4 and ground channel is e]28.

Figure 2.7: Estimation of noise power as a function of electrode number visualized in map3D for the 5different stimulation pairs for Case 1

16




Figure 2.8: Estimation of noise power as a function of electrode number for the 4 different stimulation pairsfor Case 2

Figure 2.7 and Figure 2.8 confirm that the noise was not a function of stimulus location

and Figure 2.7 shows that there was a systematic region of the array that was more noisy, perhaps

due to weaker contacts, interference, or noise in an amplifier bank.

We cannot show the the mean amplitude and the standard deviation and the ratio of these

two factors in each positive phase samples and the negative phase samples for all channels, since

we have far too many data samples and channels. Figure 2.9 and Figure 2.10 show two plots per

stimulation pair in the strongest channels which have the highest mean amplitude and relatively low

variation in Case1 and Case2, respectively. The first row shows the mean amplitude and the standard

deviation of the samples and the second row shows the ratio of mean to standard deviation during

each half-pulse.

17


(a) 4, 12− 20 (b) 4, 12− 20

(c) 22, 20− 21 (d) 22, 20− 21

(e) 23,19− 22 (f) 23, 19− 22

Figure 2.9: Figures in the left column show mean and standard deviation for each half pulse while figures inthe right column show the ratio of mean value to standard deviation for each half pulse. Each panel’s captionrepresents the electrode and stimulation pair as ], ]− ] respectively.

18


(a) 22, 18− 23 (b) 22, 18− 23

(c) 29, 4− 28 (d) 29, 4− 28

Figure 2.9: Figures in the left column show mean and standard deviation for each half pulse while figures inthe right column show the ratio of the mean value to the standard deviation for each half pulse. Each panel’scaption-caption represents electrode and stimulation pair as ], ]− ] respectively.

19


(e) 122, 121− 63 (f) 122, 121− 63

(g) 122, 63− 121 (h) 122, 63− 121

Figure 2.10: Figures in the left column indicate standard deviation and mean value for each half pulse whilefigures in the right column indicate the ratio of mean value to the standard deviation for each half pulse. Eachpanel’s sub-caption represents electrode and stimulation pair as ], ]− ] respectively.

20


(a) 122, 55− 121 (b) 122, 55− 121

(c) 122, 121− 55 (d) 122, 121− 55

Figure 2.10: Figures in the left column indicate standard deviation and mean value for each half pulse andfigures in the right column indicate ratio of the mean value to the standard deviation for each half pulse. Eachpanel’s sub-caption represents electrode and stimulation pair as ], ]− ] respectively.

2.4 Evaluation of FEM based head model building

Figure 2.11 shows the average electric potential on each electrode during the stimulus

pulses compared to the FEA computations. In the FEA model the active channel was stimulated with

positive current amplitude and the ground was stimulated with negative value of current, so to make

the comparison more accurate, we compare FEA simulation results obtained by averaging the odd

(first) half pulses which had the same polarity as the stimulation channels. In Figure 2.11(e) we can

see the clinical data and simulation results in Case 1 with different stimulus electrode pairs. We can

see that generally there is a good agreement between the experiment and the simulation, except on a

few channels. Some of these less accurately simulated electrodes were adjacent to the stimulation

channels but this is not the case for all electrodes in the neighbourhood of stimulus channels. In

21


Figure 2.7 we observed that the 8× 2 grid, comprising electrodes ]33 to ]48, had the smallest values

of noise power in Case 1. Interestingly, they all also showed good agreement when we compared the

experimental results and FEM based simulation results.

22


(e) Active channel is e]12 and ground channel is e]20. (f) Active channel is e]20 and ground channel is e]21.

(g) Active channel is e]19 and ground channel is e]22.

(h) Active channel is e]18 and ground channel is e]23. (i) Active channel is e]4 and ground channel is e]28.

Figure 2.11: The average electric potential on each electrode during the stimulus pulses compared to the FEAcomputations.

We tested a number of other factors that might have predicted which electrods were more

/ less accurately simulated in the FEM, including the degree to which the pulse tops and bottoms

23


were constant or had increasing / decreasing amplitude during the pulses as well as the residuals with

respect to both the mean and a linear fit during the half-pulses, but we did not found any behavior

that seemed to consistently correlate with the FEA agreement error.

2.5 Appendix

The figure below was provided to us to reportthe electrode types in Figure 2.3 for Case 2.

Figure 2.12: Information about electrode types in Case 2.

Here we included some additional figures of noise power and some noisy channels of each

data set, which have a relatively high variation among the mean values of the pulses. Note that y-axis

range is not similar in Figure 2.14

24





Figure 2.13: Estimation of noise power as a function of electrode number for the 5 different stimulation pairsin Case 1

25




Figure 2.14: Estimation of noise power as a function of electrode number for the 4 different stimulation pairsin Case 2

26





Figure 2.15: Visualization of spatial distribution of noise power as a function of electrode locaation usingmap3D, for the 5 different stimulation pairs in Case 1.

27

Chapter 3

Optimization of temporal interfering

current injection patterns

3.1 Introduction

As described briefly in the introduction to this thesis, transcranial current stimulation (tCS)

is a noninvasive brain stimulation technique in which weak, constant or time-varying electrical

currents are applied to the brain through the scalp. The tCS family of modaliies (tDCS, tACS,

tRNS) use scalp electrodes with electrode current intensity to area ratios of about 0.3–5 A/m2 at

low frequencies (typically < 0.5 kHz) resulting in weak induced electric fields in the brain, with

amplitudes of about 0.2–2 V/m [32]. There are many different factors influencing these cortical

current flow fields such as electrode size, location, model specifications, skull defects and lesions in

brain tissue [8, 9]. Conventional methods of using relatively large bipolar electrode montages (with

an area of 35 cm2 induce a broad cortical current flow field with the large intensities often located in

the non-target brain regions [12]). Although the effect of tCS in clinical studies is proven, due to the

poor focality in stimulating only a specific region, it is unclear whether these effects are driven by

stimulating the targeted region or non-targeted region. We face this issue especially in cases where

the region of interest (ROI) is located in deep brain regions. One approach for targeting the ROI and

improving tCS focality is to use ”dense” multi-electrode arrays instead of just two electrodes [12].

Although with this approach an enhancement in focality has been reported, still we are stimulating

only superficial regions of the brain. Grossman et. al and colleagues [19] recently presented a highly

novel solution to stimulate deep areas of the brain non-invasively. For this purpose they applied

28

CHAPTER 3. OPTIMIZATION OF TEMPORAL INTERFERING CURRENT INJECTION PATTERNS

currents with high oscillating frequencies at multiple sites on the scalp, such that these frequencies

were out of the range the neurons respond to. However these frequencies were chosen to be close to

each other, so that their sum resulted in an oscillating envelope of whose period was in the range that

neurons could be modulated by.

To analyize this idea, we assume that we have two electric fields with different high frequencies;

we call the first frequency f1 and the second frequency f2, where the relationship between f1 and

f2 is f2 = f1 + δf . The electric field generated by f1 is Ef1 = Acos(2πf1t) and the electric field

generated by f2 is Ef2 = Bcos(2πf2t). Let min(|A|, |B|) = |A|, then the superposition of these

two electric fields can be written as:

Ef1+Ef2 = Acos(2πf1t)+Bcos(2πf2t) = Acos(2πf1 + f2

2t)cos(2π

δf

2t)+(B−A)cos(2πf2t)

(3.1)

A graphical representation of this equation is presented in Figure 3.1

Figure 3.1: Low-frequency field envelopes of two higher frequencies.

If we the amplitude of the low frequency envelope modulation is large enough in deep re-

gions of the brain, we my b able to stimulate these regions without stimulating overlying regions [19].

The key observation leading to this statement is that the amplitude of this effect is governed by the

smaller of the two component fields; if the electrode arrangement is such that more superficially the

field at at least one frequency is small, but a deeper locations both are somewhat larger, then there is

the potential to stimulate deep but not superficially.

In Grossman et. al [19], the authors used a spherical phantom and computational and experimental

mouse models. For the first step in this work, we placed two simulated electrode pairs on our

FEM-based spherical head model, in a fashion similar to Grossman et. al [19] in order to see if we

29


could observe similar a temporal interference (TI) effect.

We then introduce, solve, and test an optimization problem which manages the trade-off

between focality and magnitude of the stimulation. We used a FEM-based spherical head model

with 18 electrodes, spatially divided in two groups. Our goal is to maximize the TI modulation

effect in the region of interest (ROI) while bounding this effect outside the ROI. For this purpose, we

applied a set of convex constraints on the individual electrode injection currents and the maximum

electric field on the ROI boundaries. This optimization problem imposes many challenges due to the

non-convex relationship between frequency of injected currents, size and location of the electrodes

and the resulting TI modulation effect.

In Section 3.2 we will provide the equation for calculating the interfering electric field along with a

derivation. In the rest of the chapter, we will talk about the objective function of the optimization

problem, the constraints we applied to the problem and the spherical head model characteristics we

used. In Section 3.3 we present the results of our simulation with electrode geometries similar to

Grossman et. al. Then we report on the computational experiment results for two different ROIs.

3.2 Methods

In this section, we first present the equation for estimating the magnitude of interferential

electric field envelope modulation in a particular direction and provide the proof, along with that

for the equation for maximum interfering electric field magnitude in any direction . In what follows

we give an in-depth description of the optimization approach used to determine stimulus patterns to

maximize the peak of envelop of modulated electric field in the ROI for our computational spherical

head model. We introduce the objective function and the three sets of constraints applied to the

problem.

3.2.1 Temporal Intereference Electric Field

In this section we give derivations of two key equations in

• Case 1: (−→E f1(

−→r ).−→n >−→E f2(

−→r ).−→n ) & (|−→E 1(−→r ).−→n | > |

−→E 2(−→r ).−→n |)

|−→EAM (−→n ,−→r ) |

=| αE1x+αE2x+βE1y+βE2y+γE1z+γE2z−αE1x+αE2x−βE1y+βE2y−γE1z+γE2z |=| 2αE2x + 2βE2y + 2γE2z| = 2|

−→E f2(

−→r ).−→n |= 2min(|−→E f1(

−→r ).−→n |, |−→E f2(

−→r ).−→n |)

30


• Case 2: (−→E f1(

−→r ).−→n >−→E f2(

−→r ).−→n ) & (|−→E f1(

−→r ).−→n | < |−→E f2(

−→r ).−→n |)

|−→EAM (−→n ,−→r ) |

=| −αE1x − αE2x − βE1y − βE2y − γE1z − γE2z − αE1x + αE2x − βE1y + βE2y −γE1z + γE2z |=| −2αE1x − 2βE1y − 2γE1z| = 2|

−→E f1(

−→r ).−→n |= 2min(|−→E f1(

−→r ).−→n |, |−→E f2(

−→r ).−→n |)

• Case 3: (−→E f1(

−→r ).−→n <−→E f2(

−→r ).−→n ) & (|−→E f1(

−→r ).−→n | > |−→E f2(

−→r ).−→n |)

|−→EAM (−→n ,−→r ) |

=| −αE1x − αE2x − βE1y − βE2y − γE1z − γE2z + αE1x − αE2x + βE1y − βE2y +

γE1z − γE2z |=| −2αE2x−2βE2y−2γE2z |= 2 |

−→E f2(

−→r ).−→n |= 2min(|−→E f1(

−→r ).−→n |, |−→E f2(

−→r ).−→n |)

• Case 4: (−→E f1(

−→r ).−→n <−→E f2(

−→r ).−→n ) & (|−→E f1(

−→r ).−→n | < |−→E f2(

−→r ).−→n |)

|−→EAM (−→n ,−→r ) |

=| αE1x+αE2x+βE1y+βE2y+γE1z+γE2z+αE1x−αE2x+βE1y−βE2y+γE1z−γE2z |=| 2αE1x + 2βE1y + 2γE1z| = 2|

−→E f1(

−→r ).−→n |= 2min(|−→E f1(

−→r ).−→n |, |−→E f2(

−→r ).−→n |)As shown above, in all cases the resulting interferential electric field is equal to twice the

minimum of two given electric fields in the direction of interest.

Next we address the question of in which direction of n, the minimum between |−→E f1(

−→r ).−→n | and

|−→E f2(

−→r ).−→n | is maximized.

The maximizing direction of the vector n must lie in the plane spanned by−→E f1(

−→r ) and−→E f2(

−→r ) since any component outside of that plane will not contribute to the envelope amplitude. If

we maximize |−→E f1(

−→r ).−→n | and |−→E f2(

−→r ).−→n |, then we could claim that we are maximizing the

minimum of them. We know that the dot product of two vectors will be maximum if the two vectors

are parallel. Hence n should be in−→E f1(

−→r ) and−→E f2(

−→r ) plane. Let’s call α as the angle between

two electric fields. We can assume that the angle α between vectors−→E f1(

−→r ) and−→E f2(

−→r ) is smaller

than π/2; if that is not the case, we can take−→E f1(

−→r ) as −−→E f1(

−→r ) and the assumption will hold.

We label the angle between−→E f1(

−→r ) and −→n as β. We can assume that β ≤ π, we could always

change the sign of−→E f1(

−→r ) and−→E f2(

−→r ) to make this true.

31


Figure 3.2: A graphical representation of electric fields and −→n in 2D plane.

We can reform our question to find β which maximizes minimum of |−→E f1(

−→r ).cos(β) |and |

−→E f2(

−→r ).cos(α−β) | as we can see in Figure 3.2. In Figure 3.3 we plotted |−→E f1(

−→r ).cos(β) |in blue, |

−→E f2(

−→r ).cos(α− β) | in red and their minimum shaded in green, for a particular−→E f1(

−→r ),−→E f2(

−→r ) and −→n , for 0 < β < π.

Figure 3.3: Green-shaded region is showing min(|−→E f1(

−→r ).cos(β) |, |−→E f2(

−→r ).cos(α− β) |).

Our goal is to find a way to predict the location on the x-axis where we have the maximum

value on the green-shaded region, where the minimum of the projections is maximized. Assuming

without loss of generality, |−→E f2(

−→r ) |<|−→E f1(

−→r ) |, these two electric fields can exhibit two

different behaviors:

• Case1: If |−→E f2(

−→r ) |<|−→E f1(

−→r ) | Cos(α), as we can see in Figure 3.4 the peak of the red

curve is below the blue one. In that setting the green region is maximized in that peak and we

can say that based on equation 3.3 the maximum(peak) of the minimum is:

32


|−→EmaxAM (−→r ) |= 2 |

−→E f2(

−→r ) | (3.2)

Figure 3.4: Peak of min (|−→E f1(

−→r ) |, |−→E f2(

−→r ) |).

• Case2: If |−→E f2(

−→r ) |>|−→E f1(

−→r ) | Cos(α), as we can see in Figure 3.5 the green region does

not peak at the maximum of neither curve, but somewhere in between.

Figure 3.5: Peak of min (|−→E f1(

−→r ) |, |−→E f2(

−→r ) |).

We have the peak of green shaded region for the β at which the green region is maximized, the

two curves cross each other, hence:

|−→E f1(

−→r )Cos(β) |=|−→E f2(

−→r )Cos(α− β) | (3.3)

33


Figure 3.6: Unequal projection.

(a) Unequal projection. (b) Rotation of the left figure.

Figure 3.7

We can compute the height of the triangle through the area of the triangle. We use two

formulas for the area of the triangle:

Area=0.5×base× heightArea = 0.5× side1× side2× sin(angle(side1, side2))Equating these two formulas of area above we have:

height = (side1× side2× sin(angle(side1, side2)))/base (3.4)

34


Therefor by applying equation 3.4 to Figure 3.7 we have:

|−→E f1(

−→r )Cos(β) |=|−→E f1(

−→r ) ||−→E f2(

−→r ) | sin(α)|−→E f1(

−→r )−−→E f2(

−→r ) |(3.5)

Based on the law of Sines we have:

|−→E f1(

−→r ) |sin(γ)

=|−→E f1(

−→r )−−→E f2(

−→r ) |sin(α)

(3.6)

⇒|−→E f1(

−→r ) | sin(α) =|−→E f1(

−→r )−−→E f2(

−→r ) | sin(γ)

By combining equations 3.8 and 3.6 we have:

|−→E f1(

−→r )Cos(β) |=|−→E f1(

−→r )−−→E f2(

−→r ) ||−→E f2(

−→r ) | sin(γ)|−→E f1(

−→r )−−→E f2(

−→r ) |(3.7)

|−→E f1(

−→r )Cos(β) |=|−→E f2(

−→r )× (−→E f1(

−→r )−−→E f2(

−→r )) ||−→E f1(

−→r )−−→E f2(

−→r ) |(3.8)

So in this case

|−→EmaxAM (−→r ) |= 2

|−→E f2(

−→r )× (−→E f1(

−→r )−−→E f2(

−→r )) ||−→E f1(

−→r )−−→E f2(

−→r ) |(3.9)

3.2.2 Objective function

In the problem we want to address here, we again assume that we have two groups of

electrodes injecting AC currents with frequencies of f1 and f2 in the range of KHz, which cannot

stimulate the neurons, but the envelope of the amplitude-modulated signal has a frequency (equal to

the difference between the two input frequencies) that is small enough to stimulate the brain. As we

mentioned in Section 3.1. our goal is to maximize the envelope of these two temporally interfering

electric fields in a predetermined direction for each element of a predefined ROI while keeping the

magnitude of the TI effect in non-ROI below a threshold. For this purpose, our objective function is

thus to maximize the minimum of the two frequency dependant electric fields in the same chosen

direction when summed over all elements in the ROI. We spatially divided the electrodes in two

groups. The injected current array for each frequency is represented as follow.

35


If1 ∈ Rj×1 (3.10)

If2 ∈ Rk×1 (3.11)

where j and k refer to the number of electrodes belong to frequency f1 and f2, respectively. (In

what follows we take j = k but this is not necessary.) We use n ∈ R3×1 for the desired direction of

electric field and nTAi ∈ R1×j as the transform vector that gives the injected current to electric field

on each element in direction n.(Note that we assume for simplicity that the desired direction is the

same over the ROI but again this could be relaxed as needed.)

Our objective function thus is:

max∑i∈roi

min(| nTAiIf1 |, | nTAiIf2 |) (3.12)

where i indicates the elements indices in the ROI.

We denote nTAi = Si where Si ∈ R1×j , giving us

max∑i∈roi

min(|SiIf1 |, |SiIf2 |) (3.13)

Now we want to determine if our objective function is convex or concave. A function is

convex iff:

f(αx+ (1− α)y) ≤ αf(x) + (1− α)f(y) (3.14)

for all x, y ∈ Rj×1 and α ∈ [0, 1]. And a function is concave iff:

f(αx+ (1− α)y) ≥ αf(x) + (1− α)f(y) (3.15)

for all x, y in domain andα ∈ [0, 1]. Figure 3.8 shows a plot of objective function (min(|SiIf1 |, |SiIf2 |))only for one element in the ROI, where both SiIf1 and SiIf2 are scalars. As we can clearly see in

figure 3.8 the objective function is neither convex nor concave.

36


Figure 3.8

To show this fact it is sufficient to find two points in the domain of the objective function

that break the inequality for convexity and two points that break the inequality for concavity.

• Breaking the convexity:

In Figure 3.8 we pick two points in the objective function domain as |x| = min(|x|, |y|) and

|0.5x| = min(|0.5x|, |0.5y|). Assuming x > 0, based on equation 3.14 we should show that:

min(|α×x+(1−α)×(0.5x)|, |α×(y)+(1−α)×(0.5y)|) > α×x+(1−α)×(0.5x) (3.16)

where α ∈ [0, 1].

min(|3− α2

x|, |3− α2

y|) > 1

2x (3.17)

Since α ∈ [0, 1], we know that 3−α2 > 0, hence in equation 3.18, we have:

3− α2

x >1

2x (3.18)

37


α < 2 (3.19)

which is always true. Therefor for all α ∈ [0, 1] the convexity inequality does not hold and the

objective function is not convex.

• Breaking the concavity:

In figure 3.8 we pick two points as x = min(|x|, |y|) and x = min(| − x|, | − y|), where all

these points belong to the domain of the objective function. For showing non-concavity we

should show that:

min(|α×x+(1−α)× (−x)|, |α× (y)+ (1−α)× (−y)|) < α×x+(1−α)×x (3.20)

where α ∈ [0, 1].

min(|(2α− 1)x|, |(2α− 1)y|) < x (3.21)

|(2α− 1)x| < x (3.22)

Hence:

0 < α <1

2(3.23)

Therefor we show that for any α ∈ [0, 12 ] and the two given points in domain of objectives

function, the concavity inequality does not hold and the objective function is not concave.

To summarize, we found points in the objective function domain for which there exists α’s

in the range of [0, 1] that break inequality for convexity and concavity. Hence our objective function

is neither convex nor concave. This is true for over each element and the summation of all elements

together adds more complexity to our problem. Therefore we cannot guarantee that we can find the

global maximum. Nonetheless we will explore the problem by maximizing the TI electric field in the

ROI starting from initial values of the unknown current amplitudes and applying convex constraints.

3.2.3 Simulation

The model we used in this project is a sphere with radius of 9.2 cm. This size is the typical

size used for representing the human head. The size of spherical model and human head model

are not exactly the same in coronal, sagital and radial plane ( i.e. the spherical size is wider than

human head typical size for ear to ear distance but the nose to back of the head distance and height is

smaller). However the total volume is comparable with an average human head volume [26, ?]. The

38


sphere is completely homogeneous. There are no internal compartments and the elements sizes are

roughly equal. We have 685219 elements and 125227 nodes. The sphere conductivity is σ = 1 S/m.

As our model does not have any internal compartments we used a value for conductivity which is

not as low as the skull bone conductivity (e.g 0.007 S/m) not as high as cerebrospinal fluid (CSF)

conductivity (e.g 1.7 S/m). We note that there is a lot of variability in these values as reported in the

literature..

We have 19 electrodes. Each electrode is a cylinder with a radius of 1 cm and a height of 5 mm.

Electrode ]19 is fixed as the reference electrode and the lead fields for electrodes 1 to 18 are calculated

with respect to the reference electrode as we previously discussed in 1.1.3. The reference electrode is

shown in green in Figure 3.9.

Figure 3.9: Green electrode is the reference electrode for lead field matrix.

3.2.4 Constraints

We impose three constraints on our problem. The first constraint is the safety constraint.

In order to control the injected current delivered to the brain, we limit the maximum amount through

each individual electrode current.

‖If1‖∞ ≤ thr (3.24)

‖If2‖∞ ≤ thr

where ‖‖∞ represents the infinity norm. The second constraint is to ensure that that the total current

39


going into the brain for each frequency is equal to the total current coming out of the brain:

j∑i=1

If1i = 0 (3.25)

k∑i=1

If2i = 0

where j and k denotes the number of electrodes assigned to frequency f1 and f2 respectively. The

goal of the third constraint is to stimulate the neurons on the ROI and prevent other non-ROI regions

from stimulation and keep the TI modulation effect below the threshold. The threshold is the

minimum electric field we need to make the neurons fire. For this purpose, we spatially divided

electrodes into two groups with respect to the y-plane and assigned a frequency to each group. We

divided the boundary elements of the ROI in two groups with respect to y-plane as well. The intuition

behind this constraint was to limit the electric field generated by electrodes on the opposite side

since they are smaller than electric fields of the same side over each of these two boundaries. As we

previously proved, the magnitude of the smaller electric field will be the amplitude of the envelope

of the modulation effect, therefore if we limit the minimum of the electric fields on the boundaries

of the ROI, we hope to force the TI modulation effect on non-ROI elements to be below this limit.

Assume electrodes on the left− y plane have frequency of f1 and electrodes on the right− y plane

frequency of f2. If we use e− lf1 as the electric field generated by electrodes of frequency f1, e− lf2as the electric field produced by electrodes of frequency f2, e− rf1 as the electric field generated by

electrodes of frequency f1 and e− rf2 as the electric field generated by electrodes of frequency f2

and e thr as the threshold of minimum electric field over each element of the the boundary, we have:

‖e− lf2‖∞ ≤ e thr (3.26)

‖e− rf1‖∞ ≤ e thr

In Figure 3.10 the green arrow represents y-axis and blue arrow shows z-axis. As we mentioned

before, ROI boundaries and electrodes are grouped based on their location in y-plane.

40


Figure 3.10: Green arrow is indicating y-axis and red region show the ROI. Electrodes and ROI boundariesare divided w.r.t y-axis.

3.3 Results

In the first place, in order to validate our model, we used electrode configurations and

injected current values similar to Grossman et. al [19] to compare our results with the results in

the paper. We simulated the envelope modulation of interferential electric field projected along the

x-direction and along the y-direction. These result were obtained from two electrode pairs with

rectangular and trapezoidal geometry and the magnitude of the injected current was 1 mA. Our

observations showed that there is a good agreement between our results and the results provided

in the paper in terms of shape and location of the stimulation regions. The maximum value of the

electric field in a spherical human model is smaller than reported for a mouse head model since the

size of spherical human head model is larger. The results are presented in Figure 3.11 and Figure 3.12

41


(a) TI effect projected in x direction. (b) TI effect projected in y direction.

(c) each blue-red pairs indicates the electrodes

used for stimulation.

(d) The TI modulation effect shown in Grossman’s

paper.

Figure 3.11: Electrodes have trapezoidal configuration

42


(a) TI effect projected in x direction. (b) TI effect projected in y direction.

(c) each blue-red pairs indicates the electrodes

used for stimulation.

(d) The TI modulation effect shown in Grossman’s paper.

Figure 3.12: Electrodes have rectangular configuration

In the next step, we chose two different spherical ROIs with 1 cm radius. The center for

ROI1 was the same as the center of the sphere ([0, 0, 0]) while in ROI2 the center location was

[0, 5, 0]. Figure 3.13 shows the location of these ROIs inside our spherical model.

43


(a) ROI center is (0, 0, 0). (b) ROI center is (0, 5, 0).

Figure 3.13: Red regions indicate the ROI inside the spherical model

We applied the following constraints to our problem and tried to maximize the TI modula-

tion effect:

- ‖If1‖∞ ≤ 2mA , ‖If2‖∞ ≤ 2mA

-∑j

i=1 If1i = 0 ,∑k

i=1 If2i = 0

- The third constraint is as follow.

where r is the sphere radius. We ran our optimization algorithm with different random starting values

Algorithm 2: Electric field threshold on the patch of a surface surrounding the ROI.

for y > 0 & 0.8 ≤ r ≤ 1.1cm do‖e− f2‖∞ ≤ 0.2(V/m)endfor y < 0 & 0.8 ≤ r ≤ 1.1cm do‖e− f1‖∞ ≤ 0.2(V/m)end

of the electrode currents to check the consistency of the results since we know the objective is not

convex. For ROI1 we got consistent results from our algorithm while all our constraints are active.

Figure 3.14 shows the current injection pattern and their values in (mA) unit for f1 and f2.

44


(a) Injected currents for f1. Each columns indicates

the algorithm suggested current injection for a random

input currents.

(b) Injected currents for f2. Each columns indicates the

algorithm suggested current injection for a random input

currents.

(c) Current injection pattern for f1. (d) Current injection pattern for f2.

Figure 3.14

The electric field generated by f1 and electric field caused by f2 are shown in Figure 3.15.

The black patch inside the ROI1 represents the patch for our third constraint. In this case the ratio

of∑j

i=1 |If2|∑ki=1 |If1|

is approximately 1.07.

45


(a) Ef1 . (b) Ef2 .

Figure 3.15

Figure 3.16 demonstrate the TI effect above 0.15(V/m) projected along x-direction.

Figure 3.16: EAM (−→x )

For ROI2, after executing our optimization code with five different random inputs we got

consistent results only for electrodes belong to f2 which are further from ROI2 in comparison to

electrodes connected to f1. Moreover, only the first two constraints are active in this case. Figure 3.17

shows the current injection values and also their current injection pattern for f1 and f2 for one of our

random inputs.

46


(a) If1 mA. (b) If2 mA.

(c) Current injection pattern for If1 . (d) Current injection pattern for If2 .

Figure 3.17

The electric field generated by f1 (Ef1) and f2 (Ef2) are separately shown in Figure 3.18.

The black patch inside ROI2 represents the patch for our third constraint. As we can see below in

3.18(a) and 3.18(b), the electric field on the black patch is less than 0.2(V/m) for both Ef1 and Ef2 .

Hence the TI effect inside the ROI2 is less than our predetermined threshold.

47


(a) Ef1 . (b) Ef2

(c) EAM (−→x ).

Figure 3.18

By increasing the threshold for our first, safety, constraint from 2mA to 20mA, our

algorithm results is the same for all our random input currents. Additionally, in this case all three

constraints are active. We can see the current injection pattern and their values in (mA) unit for f1

and f2 in Figure 3.19.

48


(a) If1 . Each columns indicates the algorithm suggested

current injection for a random input currents.

(b) If2 . Each columns indicates the algorithm suggested

current injection for a random input currents.

(c) If1 . (d) If2 .

Figure 3.19

The electric field generated by f1 and f2 are shown in Figure 3.20. In this case the ratio of∑ji=1 |If2|∑ki=1 |If1|

is approximately 15.32.

49


(a) Ef1 . (b) Ef2 .

(c) TI effect.

Figure 3.20: EAM (−→x )

Although by increasing the threshold for the injected current to the individual electrodes

we mathematically reach active constraints and higher TI electric field in the ROI, we cannot use such

high values in clinical setting and thus it is not practical. One solution to reach a higher TI electric

field in the ROI safely is to increase the number of electrodes and distribute this high injected currents

over multiple electrodes. We tried to look at a number of factors such as maximum, minimum,

mean, median and range of TI electric field (E AM ) in both ROI and non-ROI in Table 3.1 and

Table 3.2. For ROI1, since by applying ‖If1‖∞ ≤ 2 mA we got the injected current values such that

all constraints are active, we brought the results with respect to 2 mA threshold in Table 3.1 however

for ROI2, we needed to increase the threshold from 2 mA to 20 mA to have the active constraints

and consistent injected currents, hence we brought the results for 20 mA in Table 3.2

50


Maximum Minimum Mean Median Range

E AM(v/m) E AM(v/m) E AM(v/m) E AM(v/m) E AM(v/m)

projected in projected in projected in projected in projected in

x-direction x-direction x-direction x-direction x-direction

ROI 0.1996 0.1661 0.1863 0.1868 0.0335

non-ROI 0.3638 0 0.0795 0.0753 0.3638

Table 3.1: ROI1

Maximum Minimum Mean Median Range

E AM(v/m) E AM(v/m) E AM(v/m) E AM(v/m) E AM(v/m)

projected in projected in projected in projected in projected in

x-direction x-direction x-direction x-direction x-direction

ROI 0.4396 0.3300 0.3835 0.3822 0.1096

non-ROI 0.7362 0 0.1610 0.1257 0.7362

Table 3.2: ROI2

51

Chapter 4

Conclusions and Future Work

In Chapter 2 we introduced a method to validate the accuracy of FEM-based human head

model. For this purpose we did data analysis on clinical data provided to us by University of

Washington. In order to provide this clinical data, Electrocorticography (ECoG) grids were used to

stimulate and record brain activity at the same time. The grid electrodes were implemented on the

cortical surface of patients to determine seizure foci. Although the ECoG grids were implemented

based on clinical needs not research interests, still this data could have application in research. We

did data analysis to extract meaningful information and compare the results with data obtained from

simulation of FEM-based human head model. This is an on progress project and we are still receiving

new clinical data. We are interested in doing the model validation in the cases in which we have

grid electrodes and also depth electrodes for stimulation. In addition to model validation, one side

purpose of this project was to work with new versions of visualization tool, map3D, developed by

SCI Institute of University of Utah and gave feedbacks to them.

In Chapter 3, we formulated and solved a multi-constraint optimization problem to maximize the

TI effect in deep regions of the brain with a non-invasive method. We used a FEM-based spherical

model with no compartment for simulation and modeling current and electric field distribution. Our

results indicate that by choosing the appropriate electrode settings and injected current values we

could stimulate deep regions of the brain non-invasively. However still we have problems in terms

of enhancing focality since we are stimulating regions outside the ROI, as well. There are some

ideas we plan to explore which we suggest following to improve focality. One idea is to increase the

number of electrodes and to add one additional constraint for the total amount of current from all

sources. It may also be possible to devise more efficient constraints to shape the TI effect on the ROI

boundaries. Trying to limit the electric field in some chosen elements outside the ROI also could be

52

CHAPTER 4. CONCLUSIONS AND FUTURE WORK

helpful. Once we fix the focality issue of our problem, it would be useful to improve our approach by

adapting it for a spherical head model with multiple compartment and later on apply the optimization

method on a realistic human head model. Also we are interested in understanding neuron behavior,

therefore we may explore a new simulation software to model the neuron activity and response to the

stimulation as well.

53

Bibliography

[1] M. J. Aminoff. Aminoff’s Electrodiagnosis in Clinical Neurology E-Book. Elsevier Health

Sciences, 2012.

[2] J. M. Baker, C. Rorden, and J. Fridriksson. Using transcranial direct-current stimulation to treat

stroke patients with aphasia. Stroke, 41(6):1229–1236, 2010.

[3] M. S. Berger, J. Kincaid, G. A. Ojemann, and E. Lettich. Brain mapping techniques to

maximize resection, safety, and seizure control in children with brain tumors. Neurosurgery,

25(5):786–792, 1989.

[4] T. Blakely, K. J. Miller, R. P. Rao, M. D. Holmes, and J. G. Ojemann. Localization and

classification of phonemes using high spatial resolution electrocorticography (ecog) grids. In

Engineering in Medicine and Biology Society, 2008. EMBS 2008. 30th Annual International

Conference of the IEEE, pages 4964–4967. IEEE, 2008.

[5] T. Blakely, K. J. Miller, S. P. Zanos, R. P. Rao, and J. G. Ojemann. Robust, long-term control of

an electrocorticographic brain-computer interface with fixed parameters. Neurosurgical focus,

27(1):E13, 2009.

[6] J.-S. Brittain, P. Probert-Smith, T. Z. Aziz, and P. Brown. Tremor suppression by rhythmic

transcranial current stimulation. Current Biology, 23(5):436–440, 2013.

[7] J. Cao and P. Grover. Stimulus: Noninvasive dynamic patterns of neurostimulation using

spatio-temporal interference. bioRxiv, page 216622, 2017.

[8] A. Datta, J. M. Baker, M. Bikson, and J. Fridriksson. Individualized model predicts brain current

flow during transcranial direct-current stimulation treatment in responsive stroke patient. Brain

Stimulation: Basic, Translational, and Clinical Research in Neuromodulation, 4(3):169–174,

2011.

54

BIBLIOGRAPHY

[9] A. Datta, M. Bikson, and F. Fregni. Transcranial direct current stimulation in patients with skull

defects and skull plates: high-resolution computational fem study of factors altering cortical

current flow. Neuroimage, 52(4):1268–1278, 2010.

[10] A. Datta, X. Zhou, Y. Su, L. C. Parra, and M. Bikson. Validation of finite element model of

transcranial electrical stimulation using scalp potentials: implications for clinical dose. Journal

of neural engineering, 10(3):036018, 2013.

[11] S. Eichelbaum, M. Dannhauer, M. Hlawitschka, D. Brooks, T. R. Knosche, and G. Scheuermann.

Visualizing simulated electrical fields from electroencephalography and transcranial electric

brain stimulation: A comparative evaluation. NeuroImage, 101:513–530, 2014.

[12] P. Faria, A. Leal, and P. C. Miranda. Comparing different electrode configurations using the

10-10 international system in tdcs: a finite element model analysis. In Engineering in Medicine

and Biology Society, 2009. EMBC 2009. Annual International Conference of the IEEE, pages

1596–1599. IEEE, 2009.

[13] R. Ferrucci, F. Mameli, I. Guidi, S. Mrakic-Sposta, M. Vergari, S. Marceglia, F. Cogiamanian,

S. Barbieri, E. Scarpini, and A. Priori. Transcranial direct current stimulation improves

recognition memory in alzheimer disease. Neurology, 71(7):493–498, 2008.

[14] M. Feurra, G. Bianco, E. Santarnecchi, M. Del Testa, A. Rossi, and S. Rossi. Frequency-

dependent tuning of the human motor system induced by transcranial oscillatory potentials.

Journal of Neuroscience, 31(34):12165–12170, 2011.

[15] F. Fregni, P. S. Boggio, C. G. Mansur, T. Wagner, M. J. Ferreira, M. C. Lima, S. P. Rigonatti,

M. A. Marcolin, S. D. Freedman, M. A. Nitsche, et al. Transcranial direct current stimulation

of the unaffected hemisphere in stroke patients. Neuroreport, 16(14):1551–1555, 2005.

[16] F. Fregni, P. S. Boggio, M. A. Nitsche, M. A. Marcolin, S. P. Rigonatti, and A. Pascual-Leone.

Treatment of major depression with transcranial direct current stimulation. Bipolar disorders,

8(2):203–204, 2006.

[17] F. Fregni, P. S. Boggio, M. C. Santos, M. Lima, A. L. Vieira, S. P. Rigonatti, M. T. A. Silva,

E. R. Barbosa, M. A. Nitsche, and A. Pascual-Leone. Noninvasive cortical stimulation with

transcranial direct current stimulation in parkinson’s disease. Movement disorders, 21(10):1693–

1702, 2006.

55

BIBLIOGRAPHY

[18] F. Fregni and A. Pascual-Leone. Transcranial direct current stimulation: State of the art 2008.

Brain Stimul, 1(3):206–23, 2008.

[19] N. Grossman, D. Bono, N. Dedic, S. B. Kodandaramaiah, A. Rudenko, H.-J. Suk, A. M. Cassara,

E. Neufeld, N. Kuster, L.-H. Tsai, et al. Noninvasive deep brain stimulation via temporally

interfering electric fields. Cell, 169(6):1029–1041, 2017.

[20] R. F. Helfrich, T. R. Schneider, S. Rach, S. A. Trautmann-Lengsfeld, A. K. Engel, and C. S.

Herrmann. Entrainment of brain oscillations by transcranial alternating current stimulation.

Current Biology, 24(3):333–339, 2014.

[21] D. Hermes, K. J. Miller, H. J. Noordmans, M. J. Vansteensel, and N. F. Ramsey. Automated

electrocorticographic electrode localization on individually rendered brain surfaces. Journal of

neuroscience methods, 185(2):293–298, 2010.

[22] C. S. Herrmann, S. Rach, T. Neuling, and D. Struber. Transcranial alternating current stimula-

tion: a review of the underlying mechanisms and modulation of cognitive processes. Frontiers

in human neuroscience, 7:279, 2013.

[23] B. Hutcheon and Y. Yarom. Resonance, oscillation and the intrinsic frequency preferences of

neurons. Trends in neurosciences, 23(5):216–222, 2000.

[24] N. Jausovec and K. Jausovec. Increasing working memory capacity with theta transcranial

alternating current stimulation (tacs). Biological psychology, 96:42–47, 2014.

[25] B. Laczo, A. Antal, R. Niebergall, S. Treue, and W. Paulus. Transcranial alternating stimulation

in a high gamma frequency range applied over v1 improves contrast perception but does not

modulate spatial attention. Brain Stimulation: Basic, Translational, and Clinical Research in

Neuromodulation, 5(4):484–491, 2012.

[26] P. C. Miranda, M. Lomarev, and M. Hallett. Modeling the current distribution during transcranial

direct current stimulation. Clinical neurophysiology, 117(7):1623–1629, 2006.

[27] T. Neuling, S. Wagner, C. H. Wolters, T. Zaehle, and C. S. Herrmann. Finite-element model

predicts current density distribution for clinical applications of tdcs and tacs. Frontiers in

psychiatry, 3:83, 2012.

[28] M. A. Nitsche and W. Paulus. Excitability changes induced in the human motor cortex by weak

transcranial direct current stimulation. The Journal of physiology, 527(3):633–639, 2000.

56

BIBLIOGRAPHY

[29] A. Opitz, W. Legon, A. Rowlands, W. K. Bickel, W. Paulus, and W. J. Tyler. Physiological

observations validate finite element models for estimating subject-specific electric field distri-

butions induced by transcranial magnetic stimulation of the human motor cortex. Neuroimage,

81:253–264, 2013.

[30] E. B. Plow, J. R. Carey, R. J. Nudo, and A. Pascual-Leone. Invasive cortical stimulation to

promote recovery of function after stroke: a critical appraisal. Stroke, 40(5):1926–1931, 2009.

[31] C. Poreisz, K. Boros, A. Antal, and W. Paulus. Safety aspects of transcranial direct current

stimulation concerning healthy subjects and patients. Brain research bulletin, 72(4-6):208–214,

2007.

[32] G. Ruffini, M. D. Fox, O. Ripolles, P. C. Miranda, and A. Pascual-Leone. Optimization of

multifocal transcranial current stimulation for weighted cortical pattern targeting from realistic

modeling of electric fields. Neuroimage, 89:216–225, 2014.

[33] G. Ruffini, F. Wendling, I. Merlet, B. Molaee-Ardekani, A. Mekonnen, R. Salvador, A. Soria-

Frisch, C. Grau, S. Dunne, and P. C. Miranda. Transcranial current brain stimulation (tcs):

models and technologies. IEEE Transactions on Neural Systems and Rehabilitation Engineering,

21(3):333–345, 2013.

[34] M. Simon. 11. intraoperative electrocorticography (ecog) for seizure focus localization. Clinical

Neurophysiology, 127(9):e306, 2016.

[35] J. D. Wander, D. Sarma, L. A. Johnson, E. E. Fetz, R. P. Rao, J. G. Ojemann, and F. Darvas.

Cortico-cortical interactions during acquisition and use of a neuroprosthetic skill. PLoS

computational biology, 12(8):e1004931, 2016.

[36] T. Zaehle, S. Rach, and C. S. Herrmann. Transcranial alternating current stimulation enhances

individual alpha activity in human eeg. PloS one, 5(11):e13766, 2010.

[37] S. Zaghi, L. de Freitas Rezende, L. M. de Oliveira, R. El-Nazer, S. Menning, L. Tadini, and

F. Fregni. Inhibition of motor cortex excitability with 15 hz transcranial alternating current

stimulation (tacs). Neuroscience letters, 479(3):211–214, 2010.

57

modeling, optimization and validation for brain stimulationcj82ss01k/fulltext.pdf · (tcs). the...

Documents