Finding potential electroencephalography parameters for identifying clinical depression

UPTEC F15 025

Examensarbete 30 hp Juni 2015

Finding potential

electroencephalography parameters for identifying clinical depression

Johan Gustafsson

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Finding potential electroencephalography parameters for identifying clinical depression

Johan Gustafsson

This master thesis report describes signal processing parameters of

electroencephalography (EEG) signals with a significant difference between the signals from the animal model of clinical depression and the non-depressed animal model.

The signal from the depressed model had a weaker power in gamma (30 - 80 Hz) than the non-depressed model during awake and it had a stronger power in delta (1.5 - 4 Hz) during sleep.

The report describes the process of using visualisation to understand the shape of the signal which helps with interpreting results and helps with the development of

parameters. A generic tool for time-frequency analysis was improved to cope with the size of the weeklong EEG dataset.

A method for evaluating the quality of how well the EEG parameters are able to separate the strains with as short recordings as possible was developed. This project shows that it is possible to separate an animal model of depression from an animal model of non-depression based on its EEG and that EEG-classifiers may work as indicative classifiers for depression. Not a lot of data is needed. Further studies are needed to verify that the results are not overly sensitive to recording setup and to study to what extent the results are translational. It might be some of the EEG parameters with significant differences described here are limited to describe the difference between the two strains FSL and SD. But the classifiers have reasonable biological explanations that makes them good candidates for being translational EEG-based classifiers for clinical depression.

ISSN: 1401-5757, UPTEC F15 025 Examinator: Tomas Nyberg

Ämnesgranskare: Alexander Medvedev Handledare: Mia Lindskog

Populärvetenskaplig sammanfattning på svenska

Depression är en allvarlig och vanlig sjukdom. Det är den vanligaste orsaken till handikapp enlight värld- shälsoorganisationen WHO. En komplikation vid behandling av depression är avsaknaden av en tydlig definition och att symptomen för flera olika psykiska sjukdomar överlappar med symptom på depression och det kan därför vara svårt att hitta rätt behandling baserat på symtom. Befintliga antidepressiva läkemedel fungerar för ca 50% av patienterna men har liten eller ingen effekt för andra. Utvärderingen av symptomen hos människor är också i sig subjektiv vilket gör det svårare att på ett tillförlitligt sätt upprepa kliniska diagnoser. Exakta medel för diagnos baserad på inspelningar av mönster i hjärnans aktivitet skulle kunna hjälpa till att ge mer exakta medel för behandling. Denna studie syftar till att beskriva potentiella biomarkörer för klinisk depression baserat på EEG-inspelningar från råttor.

Projektet beskriver signalbehandlingsparametrar för elektroencefalografi-signaler (EEG) med en sig- nifikant skillnad i signalerna från en djurmodell för klinisk depression (FSL) och i signalerna från en icke- deprimerade djurmodell (SD). Signalen från den deprimerade modellen var starkare i gamma-bandet (30 ´ 80 Hz) än den icke-deprimerade modellen under vaket tillstånd och var starkare i delta-bandet (1.5 ´ 4 Hz) under sömn.

Den data som används i detta projekt är begränsad till inspelningar från endast två stammar, en icke- deprimerad modell och en deprimerad modell. Så i bästa fall denna studie kan bara föreslå potentiella biomarkörer eftersom det inte finns några uppgifter att redovisa andra egenskaper som är gemensamma för andra tillsånd som också kan påverka hjärnans aktivitet på ett likartat sätt. Studen kan heller inte urskilja stamskillnader som inte är relaterade till depression men fortfarande skiljer stammarna åt.

Denna studie använde tidigare inspelade data från ett annat projekt för vilket djuromsorg och ex- periment genomfördes i enlighet med protokoll som lämnats in till och godkänts av den lokala etiska kommittén i Stockholm Norra, Sverige.

För att skilja på signalparametrar (såsom signalstyrka) under olika sömntillstånd togs en metod fram för automatiserad klassificering av sömntillstånd. Automatiserad klassificering av sömntillstånd är inte nytt men befintliga metoder använder vanligtvis signaler från både hjärnaktivitet och muskelaktivitet för att skilja vaket tillstånd från djup REM-sömn vars hjärnaktivitet påminner om den under vaket tillstånd.

En annan parametrar som skiljde de två djurmodellerna åt var deras kronotyp. Kronotyp anger benägenheten att sova under en viss tid på dygnet. Den deprimerade modellen hade en dygnsrytm som var skiftad en timme framåt jämfört med den icke-deprimerade modellen.

Studien utgick från 8 råttor i varje grupp och undersökte deras EEG främst under ett dygn (24 timmar) från varje råtta. Båda grupperna sov totalt cirka 50% av de 24 timmarna, men den deprimerade sov totalt lite mer än den icke-deprimerade. Under natten sov de omkring 70% av tiden och omkring 30% av tiden under dagen. Men den deprimerade modellen sov något mindre under natten och något mer under dagen. Men det var också en markant skillnad i stabiliteten hos sömnmönster under dagen.

Under dagen hade den icke-deprimerade modellen ett stabilt mönster av mestadels vaket som avbryts av flera korta men regeulbundna period av sömn. Den deprimerade modellen sov däremot inte bara mer utan sömnperioderna var oregelbundna och betydligt längre.

Signalstyrka kan mätas på flera sätt. Två vanliga metoder är MAD och RMS. MAD utmärker sig genom att inte vara känslig för extremvärden. Kvoten RMS / MAD är ett stabilt mått som lätt kan testas på en annan signal. Kvoten var 1.36 för den deprimerade modellen och 1.41 för FSL. En liten men signifikant skillnad som räckte för att skilja grupperna åt.

En viktig del av arbetet med detta projekt var användningen av interaktiv visualisering av tids- frekvensrepresentationer för att orientera sig i datan och kunna tolka resultaten.

Det gjordes även ett försök att studera effekterna på depressionsmarkörerna av en ketaminbehandling men det gav inget eftersom det var för få djur och sannolikt för höga doser.

Resultaten innebär att EEG-signaler från de två djurmodellerna är olika i flera avseenden som kan tas fram genom signalbehandling av signalen för att få ut parameterskattningar. Det är möjligt att en del av dessa parametrar kan översättas till biomarkörer för klinisk depression hos människor. Eftersom metoden av EEG är icke-invasiv och EEG utrustning är lätt tillgänglig i klinisk praxis är det möjligt att söka efter sådana biomarkörer hos människor också i ett framtida projekt.

Master thesis

Engineering Physics at Uppsala University Thursday 11^th June, 2015 12:32

Finding potential electroencephalography parameters for identifying clinical depression

Johan Gustafsson

Contents

Glossary 5

1 Introduction 6

1.1 The EEG signal . . . 6

1.2 Animal models . . . 8

1.3 Interactive time-frequency analysis - Freq . . . 8

1.4 Purpose . . . 9

2 Method 9 2.1 Data material . . . 9

2.2 Notations and common equations . . . 11

2.3 Workflow . . . 12

2.3.1 Data management . . . 13

2.4 Preprocessing . . . 14

2.4.1 Artefact removal . . . 14

2.4.2 Power normalization . . . 15

2.4.3 Loglognormal spectrogram . . . 15

2.4.4 Spectrogram decomposition . . . 17

2.4.5 Sleep scoring . . . 18

2.4.6 Subset segmentation . . . 19

2.5 EEG parameters . . . 19

2.6 Strain classification . . . 23

2.6.1 Parameter decomposition . . . 23

2.6.2 Strain classifier . . . 24

2.6.3 Classifier quality . . . 25

2.7 Ketamine effects . . . 25

3 Results 26 3.1 Notation of results . . . 26

3.2 Decomposition . . . 26

3.3 Sleep scoring . . . 27

3.4 EEG parameters . . . 30

3.5 Strain classifiers . . . 31

3.6 Ketamine effects . . . 32

4 Discussion 33

4.1 Freq . . . 33

4.2 Preprocessing . . . 34

4.3 Decomposition . . . 34

4.4 Sleep . . . 34

4.5 Strain separation . . . 34

4.6 Conclusion . . . 35

4.7 Future work . . . 35

5 Acknowledgments 35 6 References 35 Appendix 38 A Time-frequency plots of recorded data . . . 38

A.1 Sprague Dawley . . . 38

A.2 Flinders sensitive line . . . 38

Glossary

API

Application programming interface, An API defines a set of routines for building software appli- cations.

biomarker

A trait used for identification and/or diagnostication.

clinical depression

Clinical depression is a severe form of depression. In this text the term depression refers to clinical depression.

delta

EEG activity in the frequency range 1.5 ´ 4 Hz. See table 3.

EEG

Electroencephalography, measures brain activity through electrodes.

EMG

Electromyography, measures muscle activity through electrodes.

FSL

Flinders sensitive line, a strain of rats used as a model depressed behaviours.

gamma

EEG activity in the frequency range 30 ´ 80 Hz. See table 3.

iEEG

Intracranial electroencephalography, measuring brain activity through electrodes placed inside the brain.

OpenGL

The open graphics library is an industry standard API for hardware accelerated rendering of 2D and 3D graphics.

qEEG

Quantitative electroencephalography, the study of brain activity through processing information from many electrodes simultaneously.

RAM

Random access memory, The RAM is the computer memory used to run active programs and keep their data. The RAM puts a limit to how much data the computer can have readily accessible within a few nanoseconds. The size is typically around a couple of GB.

SD

Sprague dawley, a strain of normal lab rats used as a model for non-depressed behaviour.

translational

Applicable on different species, notably both humans and rats.

1 Introduction

This project describes signal processing parameters of electroencephalography (electroencephalography (EEG)) signals with a significant difference between the signals from the animal model of clinical depres- sion and the non-depressed animal model. The signal from the depressed model had a stronger power in gamma (30 ´ 80 Hz) than the non-depressed model during awake and it had a stronger power in delta (1.5 ´ 4 Hz) during sleep.

The dataset used in this project is limited to recordings from only two strains, a non-depressed model and a depressed model. So at best this study can only suggest potential biomarkers as there is no data to account for other properties that are common to other conditions that could also affect the brain activity in a similar way nor to account for strain differences that are unrelated to depression but still separate the strains.

This study used previously recorded data from another project for which animal care and experi- mentation were performed in compliance with protocols submitted to and, approved by the local ethics committee of Stockholm North, Sweden.

Depression is a severe and common disease, that represent the leading cause of disability world-wide [1]. A complication in treating depression is the lack of a clear definition, and that symptoms for several different psychiatric conditions overlap with the symptoms of depression and it can therefore be hard to find the correct treatment based on symptoms. Existing antidepressants drugs work for about 50%

of patients but have little to no effect for others [2, 3]. Evaluation of symptoms in humans are also inherently subjective which makes it harder to reliably repeat clinical assessments. Accurate means of diagnosis based on recordings of patterns in brain activity could aid in providing more accurate means of treatment. This study aims to describe potential biomarkers of clinical depression in EEG recordings from rats.

This project was carried out in the Lindskog Laboratory [4] at the department of Neuroscience at Karolinska Institutet.

1.1 The EEG signal

An EEG signal is created by measuring the electric potential between two electrodes placed either outside the skull (usually what is referred to as EEG) or inside the skull or deep within the brai, (usually referred to as intracranial electroencephalography (iEEG). The signal recorded is the sum of the electrical activity of the individual neurons of the brain. The activation of a neuron causes ion currents across the neuron cell membrane, which create a small bipolar electric field. Figure 1 shows this field next to a firing neuron.

The signal measured outside the skull is the accumulated effect of many neurons firing synchronously.

EEG rhythms are typically studied within a set of frequency bands named by letters from the greek alphabet, see table 3. Each band is notable for being associated to a specific type of brain activity and are translational, meaning that the same bands occur for the same type of different activity in different mammals regardless of brain size. The first EEG was recorded in 1875 by Richard Caton in

Figure 1: The effect of a firing cell on the electric potential in the vicinity of a cell. A reference electrode is placed far away from the cell. Each spike graph shows the normalized electric potential measured at that location where the deepest red color is 1000 times stronger than the deepes blue color. From a distance a firing cell can be modelled by a dipole. The EEG captures the accumulated effect of many cells firing synchronously. The overlayed spike graphs spans 5 ms each. The underlying picture of the cell spans 650 ˆ 650 µm. Image from of E. W. Schomburg, California Institute of Technology, USA.

England [5] and Lemere published the first EEG findings related to depression in 1936 [6]. Lemere found correlations with increased alpha power and increased alpha power is to date still considered a hallmark of depression. In 1973 d’Elia narrowed this further to show increased alpha activity near the forehead (prefrontal cortex) in patients diagnosed with clinical depression when the patient is awake with their eyes closed [7]. This has however not been clear enough to be used as a classifier for individuals while separating the signal from other diseases. This also means that it cannot alone account for comorbidity [8].

EEG patterns are largely conserved across mice, rats, dogs, non-human primates and humans [9], with the exception of the alpha band that is represented by different frequencies in differet species. [10, 9]. The conservation of frequency bands suggests that biomarkers developed for rats will be translational, i.e that the results from this study will also be applicable on human trials in subsequent studies.

An EEG is often recorded with multiple electrodes, giving different signals between different pairs of electrodes on the head. The varying intensities between the signals can be overlayed on an image of a head to create a brain map. Computationally intensive algorithms for studying the vast amount of data generated by a collection of such signals is referred to as quantitative electroencephalography (qEEG).

This study has also employed computationally intensive algorithms, but to study an EEG from just two electrodes. This report does not use the term qEEG as there has been some dispute as to whether an analysis of a 2-electrode EEG should be called qEEG [11].

Some established biomarkers in EEG for clinical depression require long recordings and compares the average power in main frequency bands. For example, sleep patterns are disturbed during depression which shows up clearly in a long recording of an EEG. In addition qEEG studies in patients with old-age depression found increased slow wave activity [12]. Differences in qEEG indicators were found even between unipolar and bipolar depressive disorders [13]. Abnormal qEEG indicators have been found to consistently predict therapeutic response [14, 15]. Prefrontal theta cordance could become an objective marker of change of depressive symptoms, independent of patients’ compliance and symptom dissimulation, more precise than objective and self-rated depression rating scales [16].

However, if there are distinct features that could be detected during a few minutes during wake it will be implementable in the primary care, where depression is most of the diagnosed.

The focus in this study is to develop and study the quality of biomarkers that can be used on shorter recordings during non-sleep. A biomarker that requires a shorter and less involved recording session is better.

Recent development has indicated that an increase in glutamate causes symptoms similar to depres- sion. Glutamate is the main neurotranmisttor in the brain, making up about 80% of the brains electrical

1 2 3 4 5 6

Days

2 10 100

Hz

CWT of EEG (Non-depressed model)

1 2 3 4 5 6

Days

2 10 100

Hz

CWT of EEG (Depressed model)

Figure 2: Time-frequency plot of EEG data, darker shades represent more activity. The non-depressed model (top) clearly has a pattern of EEG activity that matches the daily rhythm, in contrast to the depressed model (bottom) which has a less clear pattern.

activity. This theory is partly funded on the fact that ketamine, a glutamate receptor antagonist, has a fast-acting anti-depressant effect. Thus, it should be possible to detect variations in glutamate levels through an EEG [17]. This report could be used as a prestudy for a forward translational project to segment depression based on EEG [18].

A related application of EEG for diagnosis is the commercial company Mentis Cura who claims to reliably detect Alzheimer’s disease in an early stage from a 5 minute EEG recording [19]. Their method is based on statistical pattern recognition (SPR) on the EEG signals but it cannot alone account for comorbidity [20].

1.2 Animal models

The Flinders sensitive line (FSL) is a rat strain that we will use as a model for depression. The FSL strain was not intentially created as a model of depression. The original intent was to produce a strain that was resistant to a drug for blocking the effects of a specific enzyme (anticholinesterase). The selective breeding program did instead create progressively increased sensitivity to another toxin (DFP) [21]. The FSL rat displays several traits that resemble depression in humans; they do not sleep well, they explore less, they show increased despair, they are bad at learning and remembering, they are restless, and they are bad at controlling stress compared to standard laboratory rats. In addition to this they react positively to anti-depressant medication [22].

Sprague-Dawley (SD) is rat strain commonly used as multipurpose standard laboratory rat in medical research. It is an albino (white fur) rat characterized by being calm and easy to handle. The same strain has been used since 1925.

1.3 Interactive time-frequency analysis - Freq

The approach taken in this study builds on studying the large datasets with software for visually assessing features of interest in the signal and then trying to develop algorithms that capture said features and describe it in a statistical manner. The human eye is good at assessing patterns in images. The software

Freq leverage this ability by making it easy to switch between different representations of the data, allowing the detection of different features in different representations and different perspectives. Being able to compute the spectrogram of a signal with 4¨10⁸samples, zoom into details, change the perspective to look at variations over time in specific bands or which frequencies that might correlate, and interact with the time-frequency resolution was essential to develop the features of interest presented in this study. This would not be possible with Matlab plots alone as there is not enough memory available to compute a full continuous wavelet transform with 40 scales per octave of a signal with 400 million samples (uncompressed this would require a 500 GB image). And even if there was enough memory it would not be possible to navigate it interactively due to the sheer size of the image matrix. Freq solves this by quickly computing the image iteratively as one interact with the data while providing prompt access to study the corresponding waveform of an event and filtering out frequencies of no interest. Figure 2 shows such a rendering of a continuous wavelet transform.

While Freq takes a manual approach to data analysis it was used in conjunction with Matlab to develop numeric parameters that can be evaluated for each of the features of interest. The scripts computing parameter estimates could then be validated by plotting the values on top of the images in Freq to see if they match the predictions of the human eye, and similarly to aid in understanding when and what the parameters and classifiers actually detected.

1.4 Purpose

The hypothesis here is that a simple cheap EEG recording over a small period of time can be used to detect signs of depression. More specifically;

1. This study will develop one or more classifiers that can separate the strains Sprague Dawley from Flinders sensitive line and find out how long recordings that are necessary for the classifier to give correct predictions. Each strain is inbred and the individuals within the group should thus to be homogenuous. From this it is assumed that there is no variation in how severe the depression is within each strain.

2. The classifer will then be applied to EEG data recorded after Ketamine treatment to evaluate if the prediction of the classifier is affected by Ketamine treatment. Ketamine is a proposed anti-depressant so if the classifier weighs more towards the non-depressed model after Ketamine treatment it is a sign that Ketamine works as an anti-depressant. Ketamine has shown promising effects in humans were patients themselves report a rapid anti-depressant effect, but the effects also decay rapidly. The approach proposed here will study what happens to the EEG after a Ketamine treatment and compare it to the baseline of a non-depressed EEG.

2 Method

Several parameters of the recorded EEG data were studied to see how well they describe a separation between the strains, how long recordings that are required to observe a significant difference, and if any difference is affected by Ketamine administration.

We assume that each strain is homogeneous and will not focus on individual differences between rats within the same strain. The Flinders sensitive line (FSL) rats are inbred which makes their DNA very homogenous, and we assume that their expressed traits therefore also are sufficiently homogenous.

This study has studied two strains of rats, depressive and non-depressive, to identify signal processing classifiers that separate the two.

The result of these various preprocessing steps were then used to evaluate the parameters for building strain classifiers which were then used to evaluate treatment effects of ketamine.

The signal will be separated into segments representing different types of activity so that it is possible to study how features vary during different states.

2.1 Data material

The EEG recordings used in this study were single channel recordings between two epidural electrodes, that is, the electrodes were placed on the dura mater above parietal and the prefrontal cortex. The signal

Hz

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dB

-30 -20 -10 0

Signal strength

EEG power (SD) EEG power (FSL) Filter

Hz

1 10 100

dB

-30 -20 -10 0

EEG power scaling by 1/fⁿ

Filter^-1 × EEG Scaling by 1/f^0.42 Scaling by 1/f^1.13

Figure 3: Power spectral density in baseline recordings, the bottom plot illustrates that the EEG power strongly resembles the amplitude response of the recording device. Note that while the signal strength at 100 Hz is an order of magnitude weaker there is still significant signal content remaining after filtering by the recording device. The spectral power was estimated by taking the median of the square of each frequency bin in a spectrogram produced with a Hamming window with a length of 8192 samples and 50%

overlap (window lengths of 256 or 65536 produced virtually identical PSDs). The left plot illustrates that the common mean of both strains is a good estimator of the mean of the separate strains.

from the electrodes were then wirelessly transmitted with a telemetry system to the recording computer where the signal was stored.

The healthy strain was represented by 8 animals from Sprague dawley (SD), and the depressed strain was represented by 8 animals from FSL. The 16 recordings span 9.2 days with a gap of missing data from day number 7. The telemetry system used was a wireless transmitter called F40-EET by Data Sciences International (DSI). The wireless transmitter applied a bandpass filter from 0.1 Hz to 100 Hz before digitizing and then encoded the signal with a varying sample rate around 250 Hz [23]. The recording computer then resamples this data to a pulse code modulation with a sample rate of 500 Hz. Some data is missing due to amplitude reaching out-of-bounds values (from artefacts) or transmission errors. The rats were freely moving with cage mates in their home cages during the recordings.

The prior project used two different levels of Ketamine treatment with 10 mg/kg and 30 mg/kg, compared to a vehicle of 0 mg/kg. The doses were randomly administered and each animal had a dose in the beginning of day 3 and day 6, counting from the beginning of the experiment. Each day starts with 12 hours of lights on to induce a subjective night followed by 12 hours of ligths out inducing a subjective day (the rats are nocturnal, awake when it’s dark and asleep when it’s light).

While the recording device has a specified bandwidth of 1-50 Hz [23] it delivers signal content up to 140 Hz. However, the varying sample rate of the F40-EET is at most 250 Hz and mostly closer to 200 Hz [24]. This means that signal content above 125 Hz should be purely resampling artefacts, and similar artefacts from temporal aliasing are probably present below 125 Hz as well. Figure 3 show a power spectrum of the recorded data. The power of electrical activity in a brain scales with 1{fⁿ, where f is the frequency and n P r1, 2s whose exact value depends on various factors [10].

When the signal is normalized by scaling with fⁿ and compared to the amplitude response of the recording device it is seen that frequencies above 50 Hz are attenuated as expected by the recording device which means that the power of the original signal in the brain follows the same curve 1{fⁿ to at least 100 Hz. Low frequencies below 2 Hz also follow the expected fⁿ curve whereas the theta band (4-10 Hz) is comparatively stronger than the other bands with a peak at 6-7 Hz.

The power of different frequencies in the EEG recording (that is, the square of the Fourier amplitude) is inversely related to temporal frequency f . In these recordings power below 7 Hz followed an estimated scaling by 1{f^0.55 and power above 10 Hz followed an estimated scaling by 1{f^1.20. See figure 3. The Appendix contains listings of detailed time-frequency plots of the studied data.

2.2 Notations and common equations

A vector of recorded single-channel EEG data is denoted y “ tyiu where yiP <. Such a vector may span a short segment of a longer recording, the whole week or a concatenation of several shorter segments.

The different recordings used are denoted Ri and the complete dataset from one recording is denoted y_Ri. Matrices and sets are denoted with capital letters X. A subset function y “ Γpy_Riq produces a smaller vector by selecting one or multiple concatenated segments from the full recording, a commonly used subset function is the baseline function Γ_baseline that selects the full second 24 hour day of the recording. Other examples include functions that only select sleep or awake segments. EEG parameter functions f “ f pyq “ tfipyqu extract some real vector-valued parameter from an EEG segment. A parameter may also be a scalar f “ f pyq.

The sum over j in integer steps from a to b is denoted

j“b

ř

j“a

y_i,j or

b´ař

j“a

y_i,j, or just ř

j

y_i,j if j has a natural range such as the number of elements in y. Or justř y_iif the index is i and no other variable is being summed over. The mean value of a vector y is denoted "meaniy_i" and typically the character µ is used to represent a mean value. Similarily the standard deviation is denoted "std_iy_i"" and represented by the character σ. The median is denoted "median_i y_i". "mad_i y_i" is the median absolute deviation and equation references look like this: (1).

madi yi“ medianipyi´ medianj yjq (1) Regression quality on scalar response: R² and ¯R²

Three measures that will be used repeatedly to evaluate the quality of regressions are R², ¯R² and κ. R² can be interpreted as how much of the variance in a response variable that is explained by explanatory variables. In a linear regression β “ arg minβ||Xβ ´ Y || the response variable is Y , the explanatory variables are X, β is the parameter estimates and ||y|| is some norm of y (for the Euclidean norm (or L², or ||y||2) this regression is a least squares fit). The same concept of response variables, explanatory variables and parameter estimates applies to generalized linear regressions such as a binomial or multinomial regression (or arbitrary regressions for that matter) although the interactions between the variables may be more complex. ¯R² (adjusted R²) takes into account that more explanatory variables (more columns in X) should always explain more of the variance in the response variable Y .

yi “ response variable ˆ

y_i “ predicted response variable p “ number of predictor variables y “ meanty¯ iu

R²” 1 ´

řpyi´ ˆyiq²

řpyi´ ¯yq² (2)

R¯²“ 1 ´ p1 ´ R²q n_y´ 1 ny´ p ´ 1

“ R²´ p1 ´ R²q p

ny´ p ´ 1 (3)

Binomial and multinomial regression

Binomial regression is a generalized linear model for classifying data into one of two groups based on their distance to a hyperplane. The hyperplane is defined by a scalar product between a β vector and explanatory variables βx (where x is assumed to include an intercept which means that one of the values in x is 1 so that β can capture the mean). The hyperplane is found by maximizing the separation in a dataset with known classes and then subsequently used when classifying samples of unknown class.

The result is a probability for each possible class, i.e a value in the range r0, 1s. Binomial regression is used for two classes and multinomial regression is a generalization for more than two classes which uses

the distances to multiple hyperplanes β_k, one for each class k P K. Based on the hyperplane distances δ_k“ β_kx multinomial regression calculate probabilites for each class using the softmax function (4).

softmaxpk, δq “ e^δk řK

i e^δi (4)

The softmax function is approximately 1 when k is the largest element in δ and 0 otherwise but makes a smooth transition between classes. The multinomial (or binomial) probability of class k is thus calculated by (5).

Pr`Y “ k | x˘ “ e^βkx řK

i e^βix (5)

Note that the sum of probabilities of all classes is 1 (6).

ÿ

kPK

Pr`Y “ k | x˘ “ 1 (6)

The MATLAB function mnrfit with default arguments were used to calculate β given a matrix explanatory variables X and known classes Y .

Regression quality on nominal response: Cohen’s κ and adjusted Cohen’s κ

Cohen’s κ (kappa) is a measure of agreement between two nominal classifications (also called interex- pert agreement). Such as classifying according to the most likely response in a binominal/multinomial regression. It takes into account the amount of identical classifications Prpaq, as well as the possibility of agreement by chance Prpeq. The two classifications C1 and C2 are typically a true response variable Y and an estimated response variable ˆY .

κpC1, C₂q “Prpaq ´ Prpeq 1 ´ Prpeq Prpaq “ PrpC1“ C2q Prpeq “ ÿ

cPC1YC2

PrpC1“ cq ¨ PrpC2“ cq

(7)

Where C1Y C2 refers to all possible classifications in C1 or C2. Adjusted cohens kappa ¯κ (8) is a novel measure that applies the idea of adjusted R² to Cohen’s kappa to account for overfitting with too many explanatory variables p compared to the number of samples n_y.

κpC¯ 1, C₂q “ 1 ´ p1 ´ κ²pC1, C₂qq ny´ 1

n_y´ p ´ 1 (8)

2.3 Workflow

The software Freq, developed by the author, was used to navigate and visualize the signals and the software Matlab was used for developing signal processing algorithms.

The named frequency bands listed in table 3 were not assumed in the methods employed in this study.

Rather the spectrograms were analyzed with signal decomposition techniques such as non-negative matrix factorization (NMF), principal component analysis (PCA) and independent component analysis (ICA) to find prominent frequency bands of that describe the EEG recording. The named frequency bands are used though when referring to frequency ranges in the discussions of figures.

The EEG data was first preprocessed to produce segments and data representations that were easier to work with (section 2.4). Candidates for the various parameters were primarily identified through visual inspection of time-frequency plots of the signals using Freq.

Each parameter is defined by a vector-valued function detailed in section 2.5. The parameters were then used as strain classifiers and their qualities as such were evaluated (section 2.6). Finally the effects of Ketamine treatment was evaluated by studing the effects on the strain classifiers (section 2.7). The workflow of the method used in this study is summarized in figure 4.

Preprocessing

EEG recording y_Ri

Artefact removal

Power normalization

Loglognormal spectrogram

Spectrogram decomposition

Sleep scoring

Subset segmentation

EEG parameters

EEG recording y_Ri

Features of interest

Parameters f pyq

Strain classification

Parameter decomposition

Strain classifier

Classifier quality Q_p, Q_κ

Ketamine effects

Figure 4: Workflow for building classifiers, assessing their quality and examine Ketamine effects. Each box is described in sebsequent sections below.

2.3.1 Data management

When studying the signal in MATLAB the data had to be analyzed in smaller subsets. Both in order to run small iterations fast during development, but also to not exhaust the available working memory of the computer. Intermediate copies kept by by algorithms (even built-in MATLAB functions) can easily require an order of 10 times the size of the input, or more if the complexity of the algorithm is non-linear.

During development it was of interest to store intermediate results if they took a long time to compute, but this was not feasible to do for more than a few copies of results of the whole dataset. Each of the steps outlined in figure 4 rhoughly corresponds to such intermediate results.

An important side effect of this project were the improvements to Freq for keeping its memory foot- print low even when studying such large datasets. Instead of keeping the whole file unpacked in memory Freq only loads comparatively small chunks of the source data when computing the corresponding sec- tion of the time-frequency plots. The plot was then resampled to viewing resolution where a pixel value was aggregated from a whole area of the actual time-frequency plot by using a max filter. Overall this provided a nice tradeoff between caching and memory footprint that ended up being faster at churning through data than the previous version of Freq due to improved memory access patterns, so that the overhead of reloading data from disk were hidden by being parallelized. It was effective enough that these datasets could be studied interactively through Freq on a tablet (iPad 2 Mini). Through Freq it was possible to study the whole dataset using both its waveform trace, spectrograms and continuous wavelet transforms when looking for patterns of interest.

Time [s]

-10 -5 0 5 10

Signal [mV]

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

0.8 Artefact: constant value

input constant value result

Time [s]

-10 -5 0 5 10

Signal [mV]

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

0.8 Artefact: high amplitude

input high amplitude result threshold

Time [s]

-10 -5 0 5 10

Signal [mV]

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

0.8 Indirect artefact: short segment

input short segment result

Time [h]

0 night 12 day 24

Artefacts [%]

×10¹⁰

0 1 2 3 4 5 6 7 8

9 Artefacts

SD FSL

Figure 5: Cleaning input data from 3 different types of artefacts and removing transient effects from edges to removed data. The last plot shows the amount of samples in artefacts per hour during the 24h baseline in all 16 animals.

2.4 Preprocessing

The raw data contained artefacts that interferred with the measurements. The different recordings were scaled differently. The spectrogram was decomposed into simpler (fewer) frequency bins, similar to the named frequency ranges listed in the Introduction.

2.4.1 Artefact removal

To avoid false detections originating in non-neurological sources the data was cleaned from artefacts before analysis. There are multiple potential sources for artefacts in a EEG recording [25]. In EEG recordings from humans a common source is eye movement or blinking (ocular artifacts) and other facial muscles that interfere with EEG electrodes placed on the front of the skull. Other muscles such as neck muscles may also interfere. It was previously believed that electric signals from muscular activity (electromyography (EMG)) was confined to high frequencies (above 20 or 40 Hz) but it has since been shown that such signals have a much wider frequency range from 0 Hz to more than 200 Hz. Yet another source of artefacts may be interference from electronic equipment, as for example the 50 Hz or 60 Hz power supply signals.

In these recordings the artefacts should be limitted to movement artefacts and transmission artefacts.

Movement artefacts occur when movements interfere with the wire from the electrode to the transmitter in the abdominal space. Transmission artefacts occur when the rat is in a position where the signal to the receiver in the bottom of the cage is to weak to be decoted. The transmitter sens the signal through a wireless signal encoded in a digital PWM signal which if obstructed by electromagnetic interference would yield a clearly obstructed output signal.

Streaks of constant values were removed. Some segments of the data contained long streaks of constant values (not necessarily only zeros) which are unnatural and is assumed to be a result of a transmission error. When more than 7 values in a row had the exact same numerical value (to available numerical precision) the entire segment of constants was discarded as an artefact. It did appear that 5 equal values in a row could happen by chance (by manual inspection).

Extremely high amplitude was discarded as artefacts with a threshold |yi| ą 20median|y|. The median of the absolute value is a stable measure of scale, unsensitive to outliers. It equals the measure mad if medianpyq “ 0. Such large amplitudes may be a natural effect of an actual voltage gap between the electrodes but it is probably not an EEG signal (a voltage gap originating from synchronously firing neurons).

Some data segments were also missing in the original recording, most likely due to movement during awake that interferred with the transmission. Remaining data contained short bursts of data within segments of otherwise discarded data. Short segments are unreliable for analysis so segments shorter than 20 seconds were also discarded as artefacts.

Transient edges to artefacts were smoothed with a sinusoidal ramp-in and ramp-out window over the course of 10 seconds before and after a discarded segment. Artefacts were encoded as Not-A-Number just like missing data so such segments may be seen in these figures as gaps in trace plots of the waveform or time-frequency plots. Some examples from the artefact removal process of each artefact type is illustrated by figure 5. Note that there are more artefacts per hour during the subjective day. On average 2% of all samples where discarded as artefacts. The EEG parameter Artefacts (section 2.5) examines the amount of artefacts in different subsets of the data.

2.4.2 Power normalization

EEG recordings from FSL rats had a significantly stronger power measured in dB than the recordings from SD (p “ 0.012), see figure 6. The signal power would affect all parameters developed from the data so each signal was normalized by its total median absolute deviation. The mean power (i.e the standard deviation of the signal) was not used because the signal strength was not normally distributed (non-gaussian). This applies both to the signal as a whole as well as all frequency bands independently, see figure 7.

The effects of varying signal power was removed by normalizing the signal to mad “ 1.0, giving a resulting unit proportional to volt [„V]. This assumes that the overall power of a strain is not a biomarker in itself but merely an artefact of the recording setup. Figure 6 shows violin plots with the estimated distributions of the power and mean of the raw data before normalization. By all practical means the mean is identical to 0.0 in all animals as the numerical precision of the recorded data is of the same magnitude. The value closest to 0 except 0 in the data is approximately 3e ´ 7 “ 3 ¨ 10^´7.

Variations in head sizes may cause the corresponding recording to vary in not only amplitude but also frequency range as different frequencies are not attenuated the same way when propagating through the brain and skull. This observation makes variables related to merely comparing frequency power between the strains less reliable.

2.4.3 Loglognormal spectrogram

The loglognormal spectrogram S_m,i is the logarithm of a power spectrogram resampled to a logarithmic frequency range. It is normalized for each frequency bin m with an offset and scale calculated from the loglog spectrogram of all concatenated baseline recordings.

SD FSL

Ampltidue [uV]

-7e-07 -6e-07 -5e-07 -4e-07 -3e-07 -2e-07 -1e-07 0 1e-07

2e-07 EEG mean, p=0.7

Mean, µ Median

SD FSL

Power [dB]

-8 -6 -4 -2 0 2 4

6 EEG power (dB), p=0.012

Mean, µ Median

Figure 6: EEG signal mean voltage and standard deviation. Note that the difference between the weakest and strongest recording is about a factor of 10 (from -5 to +5 dB).

Power distribution

Fourier amplitude

-2σ -σ µ σ 2σ

Hz

0 20 40 60 80

Fourier amplitude squared

-2σ -σ µ σ 2σ

Hz

0 20 40 60 80

Log of Fourier amplitude

-2σ -σ µ σ 2σ

Hz

0 20 40 60 80

Figure 7: Power histograms with the Fourier amplitude compared to the logarithm of the Fourier ampli- tude.

Fktxu “ 1 n_x

n_x´1

ÿ

j“0

xj¨ e^´i2π

k nxj

fk“ k{fs

gm“ exp ˆ

ln 0.4 `ln 120 ´ ln 0.4

40 ´ 1 pm ´ 1q

˙

K_m“

!

k : g_m“ min

m¹ |log fk´ log gm¹| ) K¹“ tKm: |Km| ą 0u

G_mtxu “ mean_kPKm¹ |Fktxu|² Hipxq “ xihi

Y_m,ityu “ 10 ln10pGmtHpyiwo : iwo`wquq Sm,ityu “ Ym,ityu ´ meani¹Ym,i¹ty¹u

std_i1Y_m,i1ty¹u

(9)

Where Fktxu is the discrete fourier transform of x, fs is the sample rate of x, fk is the linearally distributed frequencies of the fourier spectra (fk gives the linear center of each bin), gm is the loga- rithmically redistributed frequencies (gmgives the logaritmic center of each bin). Gm is the resampled fourier transform with squared amplitudes. The resampling is done by taking the mean of a nearest mapping where Km denotes the set of fk bins that goes into a gm bin. H is a window function with a window vector hi(omitted). Yk,ityu is the spectrogram of y and yiwo : iwo`w denotes the subset of the vector y spanning from element yiwo to (but not including) element y<sub>iwo`w</sub>. y¹ is a training baseline segment, concatenated from stratified samples of all baseline recordings.

The logaritmic frequency axis ends up with 22 bins from the 61 bins in the linear frequency axis between 1 Hz and 120 Hz based on a window of length 256 samples and a sample rate of 500 Hz.

2.4.4 Spectrogram decomposition

By visual inspection of a spectrogram visualization it becomes clear that the timevarying spectral content of this dataset mostly falls into a limited set of components with few intermediate situations (see Figure 2).

Either high frequencies are higher than the average (as in awake and REM sleep) or low frequencies are higher than the average (as during non-REM sleep) but it rarely happens for instance that both high and low frequencies are simultaneously higher than average, or are simultaneusly on the average level.

The data thus seems to not be normally distributed. This is also supported by two separate peaks (roughly corresponding to awake/rem and nrem) on the the probability density estimate of the first PCA component listed in figure 9. The independent clusters in the first (strongest) component means that a shift in measures such as mean or std should rather be interpreted as a weakening/growth of a cluster as oposed to a shift in the numeric values. The segmentation based on sleep states does not have this problem.

To separate the spectral content in different types of EEG activity we run an independent component analysis on the time-dependent spectral content of the EEG to score the signal into segments of different activity. As noted in the previous paragraph all frequency components of the signal were non-gaussian which is a requirement for ICA analysis, and an anti-requirement for finding good components through the related principal component analysis. The signal has to be whitened first though, which is accomplished by subtracting the median and dividing by the median absolute deviation, MAD[26] which is a robust measure of scale. Yielding a median=0 and MAD=1. This is analog to whitening a gaussian dataset by transforming the data to a z-distribution (with mean=0 and standard deviation=1).

ICA assumes that components can be linearly combined which is at most approximately true, but it is close enough to be useful albeit not optimal [27].

In order to do an efficient decomposition the spectrogram data has to be normalized first. The spectrogram this analysis is based on uses a hamming window with a width of 256 samples and 50%

overlap. Natural events are typically logaritmically distributed so it makes sense to resample the linearly

distributed Fourier bins to a logarithmic distribution over Hz and express the power of all fourier ampli- tudes in dB which also has a logaritmic scale. This loglog spectrogram is then normalized to compensate for the naturally large differences in strength between high and low frequencies. The normalization is performed by subtracting the mean over all time steps and dividing by the corresponding standard de- viation on each frequency bin independently. The resulting spectrogram is then for no apparent reason smoothed with a very arbitrary box filter that probably does not help covering 4 frequency bins and 200 time steps. This yields what is here refered to as the loglognormal spectrogram. The normalization coefficients (offset and scale for each bin) are saved so that the loglognormal spectrogram can then be computed for an arbitrary segment of EEG data and yield comparable frequency information.

The decomposition performed here also acts as a filter that keeps prevalent properties but discards rare events and noise. How much of the data that is kept after the decomposition can be measured by the sum of eigenvalues after decomposition divided by the sum of eigenvalues before decomposition. This coefficient is the same value as R² to measure how much of a response variable that is explained by a linear regression. A lot of data is also discarded before that when resampling the linear frequency axis of the fourier transform to a logaritmic representation with lower resolution (22 bins compared to 256 bins).

78% of the eigenvalues of the spectrogram are retained with just one PCA component. In addition to evaluate the PCA decomposition the spectrogram was also decomposed using independent component analysis (ICA) and non-negative matrix factorization. With fewer components the results tend to be less sensitive to overfitting as well as easier to interpret.

NMF is a matrix decomposition approach which decomposes a non-negative matrix into two low-rank non-negative matrices [1]. It has previously been successfully applied in the mining of biological data [28].

2.4.5 Sleep scoring

An experienced researcher performed a manual sleep scoring of the 24 hour baseline of the dataset by looking at the waveform of EEG and EMG data where it is possible to recognize and distinguish the shapes of the EEG during deep sleep, rapid eye-movement (REM) sleep and wake activity. This dataset is used as a training dataset for providing spectrogram based sleep scoring for the remaining data.

A period of an EEG recording can be largely divided into one of two states; sleep and awake. The patterns between awake and sleep are rather different. The classification can be done by a computer algorithm, or manually. In this case the dataset included a manual sleep scoring of a baseline recording.

Multinomial regression was used with the manual sleep scoring as a training dataset.

The sleep was categorized into REM (rapid eye movement) sleep and non-REM sleep as the EEG signal differs significantly between those two states. REM sleep is in some aspects more akin to awake than non-REM sleep although REM sleep only happes after being asleep in non-REM sleep for a while.

Figure 2 shows the first 6 consecutive days of two rats to illustrate sleep patterns in a time-frequency plot. The time-frequency plot is a continuous morlet wavelet transform with 40 scales per octave. Each 24 hour day starts with a subjective night as the light is turned on. Theta activity is stronger during non-REM sleep [25] and the periods that are stronger in low frequencies show up as dark patterns in the image. It is possible to count the days as 6 periods in the non-depressed models whereas it is hard to do the same on the depressed model. The differences between the images reflect different sleep patterns as the daily rhythm is different between the strains. The non-depressed model to the left has a regular sleep pattern whereas the depressed model on the right has a more irregular sleep pattern.

This example is illustrative but not representative as there are non-depressed models with less regular rhythms and depressed models with less irregular rhythms. Corresponding images from all 16 rats are listed in Appendix A.

The manual sleep scoring used as a target response variable was done on a resolution of 10 second epochs. The spectrogram based sleep scoring was performed through multinomial regression from the quartiles and median of each frequency bin corresponding to each sleep state in the loglognormal spec- trogram. The MATLAB functions MNRFIT and MNRVAL were used to build the regression β. The spectrogram used had a window length of 256 samples, which corresponds to about 0.5 seconds. The multinomial state scores was then filtered by smoothing with a box filter, weak scores were discarded and the resulting classification was then taken according to the highest multinomial score, or from the nearest timestep that was not discarded. The coefficients of the smoothing window length as well as

the thresholds for discarding samples were optimized for producing a high Cohen’s kappa on half of the recordings for which the manual sleep scoring had been performed. The result section lists the coefficients and the value of Cohen’s kappa on the reamining recordings used as a validation dataset.

The spectrogram based sleep scoring was then used to perform segmentation of the dataset and build subsets for further analysis by concatenating all samples classified as the same sleep state. The function isawakepyⁱq P t0, 1u is used in the EEG parameters below and takes the value 1 of the corresponding spectrogram segment around sample yi was scored as awake, and takes the value 0 otherwise. Conversely isasleeppyⁱq “ 1 ´ isawakepyiq, the non-REM and REM states are not separated by these functions as they are both non-awake states. sleep score^jpyiq P r0, 1s is the multinomial score for state j P tawake, non-rem, remu after smoothing but before discarding scores below the threshold.

2.4.6 Subset segmentation

Table 1: Subsets used for evaluating variables and comparing treatment effects.

Name Description

24h This baseline started 22 hours after the last human contact when the recording was started.

night The first 12 hours of 24h. night refers to the subjective night and since rats are nocturnal animals this is the pe- riod where the lights are turned on (when the animals are typically asleep).

day awake Awake during the last 12 hours of 24h. The subjective day is defined by the period where the lights were turned off.

ketamine night 1 Night starting 2 hours before Ketamine administration.

ketamine day 1 awake Awake during the day starting 10 hours after Ketamine administration.

ketamine night 3 Night starting 46 hours after Ketamine administration.

ketamine day 3 awake Awake during the day starting 58 hours after Ketamine administration.

In order to evaluate variables during sleep only or awake only the data was categorized, or scored, into segments of sleep and non-sleep. The set of signals from all animals was divided into shorter subsets to study how variables differ between strains and over time. The subsets used are listed in table 1.

2.5 EEG parameters

Previously known biomarkers for clinical depression such as changes in alpha power and irregular sleep patterns were verified to validate if the FSL rats seems to function as a model for clinical depression.

This is important to verify as there is then is a higher chance that the results will be translational, i.e that similar or identical results would potentially be found in both humans and rats. And successful treatment of rats that eliminate biomarkers for clinical depression might also have a higher chance of being successful on humans.

The potential features of interest were mostly found through interactive visualization in Freq. Pa- rameters are listed in no particular order. ny refers to the number of elements in the vector y. i “ x h refers to the index in y at x hours into the vector. So for a samplerate fsthe element yx h this refer to element number i “ 3600xfs.

Artefacts

A trait of depression is being less active. As movement may cause artefacts in the recording measuring the amount of artefacts can be used as a coarse proxy of measuring movement and thus measuring differences between the strains. Figure 5 illustrates some examples of segments detected as artefacts and figure 8 illustrates that the amount of artefacts vary between the animals. As artefacts are encoded as

SD FSL

Artefacts [%]

0 2 4 6 8 10 12

Artefacts day, p=0.6

Mean, µ Median

SD FSL

Artefacts [%]

0 2 4 6 8 10 12

Artefacts night, p=0.3

Mean, µ Median

Figure 8: Amount of data discarded as artefacts during day and night respectively.

N aN -values by the preprocessing step this parameter can just check for the occurance of N aN -values in a segment y.

f pyq “ 1 ny

ÿ

i

isnanpyⁱq (10)

Chronotype awake

At which time of the day they are most likely to be awake. This is computed for the 24 hour baseline as the weighted mean of all awake periods centered around noon during the subjective day.

f pyq “ 1 ny

i“30 h

ÿ

i“6 h

i ¨ isawakepymodpi,24 hqq (11)

Chronotype for the day and night subsets does not have the offset:

f pyq “ 1 n_y

i“12 h

ÿ

i“0 h

i ¨ isawakepyiq (12)

This parameter does not have a shortest significant length so the quality measures does not apply.

Sleep

The sleep parameter measures the total amount of sleep in a segment y, measured as a fraction of the segment length.

f pyq “ 1 ny

ÿ

isasleeppyiq (13)

Sleep slope

The sleep slope parameter measures how the sleep scoring varies across a subset segment y. It performs a linear regression onto the sleep scoring.

taj, b_ju “ arg min

a¹_j,b¹_j

ÿ

i

`a<sup>1</sup><sub>j</sub>i ` b¹_j´ sleep scorejpyiq˘2

f pyq “ ta_awake, b_awake, a_non-rem, b_non-rem, a_rem, b_remu

(14)

This parameter only makes sense to compute for long segments, such as a whole day, a whole night, or the entire 24 hour baseline.

Subset segment length

A subset of only awake (or only sleep) consists of many concatenated shorter segments. This parameter is used to studiy if the typical length of these segments vary between strains. Each element y_i in the vector comes from one such concatenated segment and spyiq gives the length of that segment. f pyq gives the average segment length.

f pyq “ 1 ny

ÿspyiq (15)

Subset length

This parameter is used to study if the length of a subset vary between strains.

f pyq “ ny (16)

This parameter only makes sense to compute for a 24 hour period.

MAD

The total signal is normalized to have RMS-power equal to 1 but a segment or a subset may have a different MAD-scale. MAD (median absolute deviation) is a robust measure of scale that is unaffected by outliers. This parameter is used to study if the signal has a different scale in subsets from different strains.

f pyq “ mediani|yi´ medianj y_j| (17)

RMS

The total signal is normalized to have RM S-power equal to 1 but a segment or a subset may have a different RM S-power. This parameter is used to study if the power of different subsets vary between strains.

f pyq “ d

1 n_y´ 1

ÿy_i² (18)

Power spectrum

This parameter describes the mean of squared fourier amplitudes resampled to a logaritmic frequency distribution fmwith 40 frequency bins between 0.4 Hz and 120 Hz. The fourier amplitues are calculated from a spectrogram with a Hamming window of length of w “ 256 and o “ 50% overlap.

Fktxu “ 1 n_x

n_x´1

ÿ

j“0

xj¨ e^´i2π

k nxj

fk “ k{fs

gm“ exp ˆ

logp0.4q `logp120q ´ logp0.4q

40 ´ 1 pm ´ 1q

˙

K_m“

!

k : g_m“ min

m¹ |log fk´ log gm¹| ) K¹“ tKm: |Km| ą 0u

G_mtxu “ mean_kPKm¹ |Fktxu|² Hipxq “ xihi

Y_m,ityu “ GmtHpyiwo : iwo`wqu f_m¹pyq “ meani Y_m,ityu f_m²pyq “ stdi Ym,ityu

f pyq “ rf¹f²s

(19)

Where Fktxu is the discrete fourier transform of x, fs is the sample rate of x, fk is the linearally distributed frequencies of the fourier spectra (fk gives the linear center of each bin), gm is the loga- rithmically redistributed frequencies (gmgives the logaritmic center of each bin). Gm is the resampled fourier transform with squared amplitudes. The resampling is done by taking the mean of a nearest mapping where Km denotes the set of fk bins that goes into a gm bin. H is a window function with a window vector h_i(omitted). Y_k,ityu is the spectrogram of y and yiwo : iwo`w denotes the subset of the vector y spanning from element y<sub>iwo</sub> to (but not including) element y<sub>iwo`w</sub>. f²captures the variability of the power spectrum and is used to take heteroscedasticity into account to see if it can be used as an explanatory variable.

Spectrum dB

This parameter describes the power spectrum in a logaritmic scale. The logaritmic scale is likely to be a better explanatory variable as natural events (such as an EEG) are typically logaritmically distributed, both in power and frequency. The binomial regression 2.6.2 applies an affine transformation to the parameter values but that will only be efficient if the parameter values are approximately normally distributed. The logaritm of the power is more normally distributed than the power measured in squared fourier amplitudes, see figure 7.

g_mpyq “ Parameter: power spectrum fmpyq

fspectrum dbpyq “ 10 log₁₀gmpyq (20)

Alpha power

Others have reported a variation during clinical depression in the alpha power between the left and right prefrontal cortex. This experiment only used one electrode on the prefrontal cortex so it was not able to detect any shift in alpha power between the left and right side. The second best thing to examine is to see if any variation in power shows up at all when comparing the strains.

This parameter catches the mean value of the alpha band in the power spectrogram.

gmpyq “ Parameter: power spectrum fmpyq α “ tm : 8 ă gmă 12u

f pyq “ ÿ

mPα

gmpyq

(21)