Chapter 3 Abstract Mathematical Foundations of … · nals, is best characterized by electroencephalography (EEG). The ability of EEG to report changes in cortical physiology for

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Clinical Advances in Neurophysiology Clinical Advances in Neurophysiology

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AbstractThe bioelectric function of the cerebral cortex is best

characterized by an electroencephalogram (EEG). Nu-merous mathematical analytical tools have recently been developed to objectively study processes that are impos-sible to assess at a glance. The use of these tools is called quantified EEG (qEEG). We analyze some of the main tools used in numerical analysis. First, we describe the analog-to-digital conversion process and the Nyquist theorem. The fundamentals of spectral analysis in the fre-quency domain are then discussed, starting with the Fou-rier theorem. Given the limitations of this approach, two tools for analyzing time-frequency series, i.e., the power spectral density and wavelet transform, are introduced. Finally, we address some of the methods to assess the in-teraction between pairs of time series. The mathematical convolution process, the main tool to assess the combina-tion of channels, and the auto(cross)-correlation are de-scribed. Although not based on this principle, the Pear-son correlation coefficient is also shown to be a measure of correlation. Finally, after the Fourier transformation of the auto(cross)-correlation, a useful synchronization measure and coherence were obtained. We speculate that the discrepancy between the theoretical interest in these tools and their limited use in clinical practice can be par-tially explained by the limited knowledge and confidence in their use by clinical neurophysiologists. Therefore, the basics of these tools need to be elucidated such that they can be used by a growing percentage of neurophysiolo-gists in their daily practice.

Chapter 3

Mathematical Foundations of Quantified Electroencephalography

Lorena Vega-Zelaya1,2, Oscar Garnés-Camarena1,2, Ancor Sanz-García2, Guillermo J Ortega2 and Jesús Pastor1,2

1Clinical Neurophysiology, Hospital Universitario de La Princesa, Spain2Foundation for Biomedical Research, Hospital Univer-sitario de La Princesa, Spain

*Corresponding Author: J Pastor, Clinical Neurophysi-ology, Hospital Universitario La Princesa, C/Diego de León 62, 28006, Madrid, Spain, Tel: +34 91 520 23 87; Fax: +34 91-4013582; Email: [email protected]

First Published February 10, 2016

Copyright: © 2016 J Pastor et al.

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source.



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KeywordsAuto-spectrum; Coherence; Cross-spectrum; Fourier

transform; qEEG; Periodogram; Wavelet transform

IntroductionThe primary function of the cerebral cortex, i.e., the

exchange of information by generating bioelectrical sig-nals, is best characterized by electroencephalography (EEG). The ability of EEG to report changes in cortical physiology for long periods with optimum temporal reso-lution is largely due to the development of mathematical analysis tools for bioelectric signals, commonly known by the acronym qEEG (quantified EEG). qEEG is used to in-troduce elements of objectivity into the analysis of EEG records, making it less dependent on the subjectivity of the observer. qEEG provides access to physiological or pathological conditions that are impossible to evaluate with naked eye, such as the phenomena of correlation or spectral composition.

Recently, qEEG has also been successfully applied in intensive care units (ICU). Neurocritical patients often experience altered levels of consciousness, and clinical examination results are insufficient. The large volume of scientific work that has focused on EEG monitoring (m-EEG) in critically ill patients [1-4] has highlighted the high incidence of epileptic seizures and status, which, being non-convulsive, go unnoticed in a routine clinical

exam. Accumulating evidence (summarized in a consen-sus document published recently by the American Society of Clinical Neurophysiology [5]) supports the use of m-EEG as a diagnostic tool to monitor and titrate therapy with antiepileptic drugs as well as for prognostic assess-ment. The utility of qEEG and its ability to display results that are easily analyzed by the clinician are clearly demon-strated in these studies.

However, qEEG is not only useful for critically ill pa-tient but also to all other areas that require EEG. In fact, qEEG is being applied very intensively in the diagnosis of dementia [6-9], attention deficit hyperactivity disorder [10,11], psychiatric disorders (like schizophrenia, obses-sive-compulsive disorder or depression) [12-17] and of course, epilepsy. However, the type of analysis used for epilepsy may slightly differ in the location of sources or spectral characteristics [18-20]. Moreover, qEEG is also used to study sleep [21].

In this review, we intend to provide a brief but useful summary of some of the most fundamental methods used in qEEG, especially those derived from spectral analysis, because the important clinical results that can be obtained through its use (and the possible misdiagnosis) depend on the correct application of mathematical basis of qEEG. The proper understanding of these methods is essential for the application of mathematical tools with confidence in clinical practice.



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Analog-To-Digital ConversionDigital signal processing necessarily involves the

conversion of analog to digital signals. An analog signal is continuous over a time interval. Voltage signals can be represented as V(t), indicating that this signal is a func-tion of time. These signals are converted into a series of numbers that can be stored, represented and mathemati-cally analyzed.

Modern equipments in clinical neurophysiology (CNP) consist of cheap analog-to-digital converters (ADC) with high sampling rate multiplexers. Briefly, this process samples a registered potential value defined dur-ing a very short time interval (sampling period, or Δt, usually measured in milliseconds or even microseconds s- 10-3 to10-6 s). Thus, the continuous signal is converted into a sequence of discrete measured voltage values at constant intervals of time (Figure 1). Therefore, the notation be-comes V(n), n = 0, 1, 2... Therefore, a continuous function is mathematically changed to a discrete number series. The inverse of the sampling period is called sampling fre-quency, denoted ν = 1/Δt, and is measured in Hertz (Hz).

Importantly, the mathematical transformations of our records will consequently be a discretized, not continuous series, representing the mathematical peculiarities of the tools to be used and the possibility of errors introduced by the ADC itself.

Figure 1: Analog-to-digital conversion. Effect of different sampling rates on an analog signal (top) and thedistortion of the morphology by decreasing the sampling rate. Time is measured in seconds. Each sampled point is represented by a small red dot, whereas the reconstruction of the sig-nal (i.e., the straight line joining two consecutive points)

is represented by a green line.



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Although we will not discuss the Nyquist theorem in detail, we will briefly address it herein [22].Specifically, the theorem states that to maintain all the original infor-mation of the analog signal (which is frequency or band-limited), the sampling rate must be atleast twice the maxi-mum frequency contained in the signal (νmax). Suppose we want to determine a signal whose smaller symmetrical element (∆tmin) has a duration of 1 ms (10-3 s). Therefore, because ∆tmin

-1=νmax, the frequency will be νmax=103 Hz. According the theorem, ν≥2νmax, and consequently, ν≥ 2x103 Hz. This finding is more intuitively demonstrated in Figure 1. At least three time points are required to char-acterize the smallest element: one at the beginning (tinit), another at the maximum value (t1) and another at the end (tend), such that tint<t1<tend. If the difference between both ends is 1 ms, (∆textrem=tend-tinit), the period between either end and the midpoint is one half; thus, ∆textrem/2 = 10-3/2. Logically, to preserve these points, our ADC must be able to sample values at a frequency of (10-3/2)-1, which it is 2 kHz, as shown above. This sampling rate is important because it clearly differs from thesampling frequency for scalp EEG, for which the highest frequency is 60 Hz (im-plying a Nyquist rate of 120 Hz) for an intracranial record-ing aimed to register fast ripples [23-25] to 500 Hz (which is a sampling rate of at least 1 kHz).

Signal Analysis in the Frequency Do-main

Neurophysiological recordings in general and, in particular EEG, are always obtained as time-dependent functions, i.e., in the time domain. This somewhat cryp-tic expression refers to the mathematical formulation of a function for which the domain is the set of values that can take the function. In our case, the values are the instants of time between the origin of the record (tonset) and the end (tend), tonset ≤ t ≤ tend. Therefore, the domain is explicitly stated in parentheses as the dependent variable function, e.g.,V(t).

However, the time domain analysis does not allow us to access much neurophysiological information, such as the most important frequencies in the signal. Therefore, the analysis is moved to the frequency domain.

Before beginning, we will define a periodic function. This is any function that acquires the same value when a defined interval of time, called period (T, s), is reached, which is explicitly defined as f(t)=f(t+nT); n=1,2…

The function value is equal at the following period t + T, t + 2T, t + 3T, etc. The trigonometric functions sine, co-sine and their multiple combinations are typical examples of such functions.



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Figure 2: Fourier’s theorem. Reconstruction of the linear function y = 3/5x defined between 0 and 5 by increasing the terms of the Fourier series (equation 1). A) n = 1 term, B) n = 5 terms, C) n = 20 terms. For each section, the graphs represent the following: a) Function to be set (f(t)), b) Representation of the coefficients as a frequency func-tion (power spectrum); c) resulting function d) the ob-

tained sum of the different spectral components.

Fourier’s theorem states that any periodic and con-tinuous function f(t) (of period T) that complies with spe-cific mathematical conditions -that we will not describe here- can be written as an infinite sum of weighted sine and cosine functions (see [26] for a detailed description). This result is profound, especially because this approach can be used to describe functions that do not resemble the sine and cosine shapes (Figure 2). The Fourier’s theorem can be formally stated as follows:

0

1 1

2 2( ) cos( ) sin( )2 n n

n n

a nt ntf t a bT Tπ π∞ ∞

= =

= + +∑ ∑ Equation 1

Generally, it is called angular frequency to ω=2πν=2π⁄T, which is measured in radians/s (rad/s). We utilize the inverse relationship between ν and T in this ex-pression.

In this equation, two types of coefficients, an and bn, can be defined as follows (a0 is an when n = 0):

/2 /2 /2

0/2 /2 /2

2 2 2 2 2( )cos( ) ; ( ) ; ( )sin( )T T T

n nT T T

nt nta f t dt a f t dt b f t dtT T T T T

π π− − −

= = =∫ ∫ ∫

Equation 2This general expression can represent different func-

tions because the values of the coefficients will differ. In-deed, each function f(t) will have a set of coefficients that



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are proper and determine its spectrum. To represent the dependence of the function on each discrete frequency (i.e., each n = 1, 2, ...), the coefficients of a frequency are combined as the square of the absolute value of their sum, i.e., for each n, we calculate |an+bn |2 (figure 2).

However, Equation 1 is not very useful in practical terms. To be usable, we must find a formula with man-ageable results. To this end, we combined coefficients to define a new coefficient, called cn.

2n n

na ibC −

= Equation 3

This expression introduces a new symbol, i, which is called the imaginary term and is defined as = − . This number does not belong to the body of real numbers ( ) but instead is part of the complex number domain ( ), designed precisely to determine the root of negative numbers. Within the theory of complex numbers, the De Moivre theorem [27] is extraordinarily important for our purposes:

cos( ) sin( )i te t i tω ω ω= +

Combining this theorem with the definitions of an and bn (Equation 2) and with Equation 3, the following expression is obtained:

/2

/2

1 ( )T

i tn

T

c f t e dtT

ω−

−

= ∫Using this definition, equation 1 can be stated as fol-

lows (this proof is simple and omitted here):

( ) i ntnn

f t c e ω∞

=−∞=∑ Equation 4

The great advantage of this expression in equation 1 is that it is a symmetric series whose midpoint is zero. If we change the period to infinity, i.e., T→∞, and remove it from denominator (a rigorous process that can be shown to be valid), we can call F{f(t)}=cn to obtain the continuous Fourier transform (CFT).

{ ( )} ( ) ( ) i tF f t F i f t e dtωω∞ −

−∞= = ∫ Equation 5

The infinite series is then changed to an improper in-tegral to obtain the following:

1 1( ) { ( )} ( )2

i tf t F F i F i e dωω ω ωπ

∞−

−∞

= = ∫ Equation 6

which is the continuous inverse Fourier transform (ICFT). Both equations 5 and 6 are a pair of Fourier trans-



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forms. Consequently, applying operator F to the func-tion f(t), which is defined in the time domain, will give its transform, F{f(t)}=F(iω), which is the equivalent function in the frequency domain. Note that F, which represents an operator (i.e., a mathematical operation applied to a function or vector to obtain another), is not the same as F, which is the transformed function. Furthermore, the in-verse function, i.e.,F-1, allows us to find the original func-tion from the transformed one.

However, the transform of a function is a complex, not real function. Therefore, for x(t) series, the transform is { ( )} ( )F x t X ω= ∈ . Thus, to obtain a real function, we must multiply by its complex conjugate (see [28]) as fol-lows:

*( ) ( )x

X XSN

ω ω=

Equation 7

where N is the number of array elements used to cal-culate the transform. Consequently, the units obtained are the squared, i.e., to the power of two, units of the original function, called the power spectrum. The resulting func-tion is a periodogram, Sx. Therefore, we do not confuse the Fourier transformation of a time series with the peri-odogram. This form of analysis is useful to provide valu-able information about the composition of the original signal frequency, which allows us to identify the dominant

frequencies and better estimate the relationship between bands. Figure 3 shows an example of the use of frequency analysis. Figure 3A shows a periodogram obtained from an EEG (F3-C3) during an absence seizure in a patient with idiopathic generalized epilepsy and the same patient after the administration of midazolam (Figure 3B).The im-portance of these findings for the mathematical analysis of neurophysiological signals is difficult to over estimate [29-35].Undoubtedly, this calculation can be considered the main tool for the numerical analysis of time series.

However, this type of analysis suffers from various problems, such as fluctuations caused by inconsistencies in the statistical estimation of the periodogram [21] or the fact that this analysis applies mainly to stationary signals (whose probability density distribution does not change over time [22]), which is not the case for most of the EEG signals. Therefore, to extend the power of spectral analysis signals to non-stationary or short-term signals, the analy-sis of time-frequency has been introduced, which consists of presenting functions that depend on both the frequen-cy and time G(ω,t).

Numerous techniques are available for time-frequen-cy analysis (for a detailed description, see [21]), but a de-tailed analysis of these techniques is beyond the scope of this work. However, we will focus on two that are most commonly used: the power spectral density (PSD) and wavelet transform (WT).



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Figure 3: Periodogram of EEG recordings. A) Left: Fron-tal differential register for an absence seizure (left). Right: periodogram with a spectral peak at 6.4 Hz indicating the main frequency of ictal activity. B) Registration (left) and periodogram (right) from the same patient following the administration of midazolam. Note that the theta peak has disappeared, whereas a maximum appears in the delta band. The overall structure of both periodograms is simi-lar, although during the crisis, several beta peaks disap-pear with medication. Note that neither the gain of the

records nor the periodograms is equal.

The PSD (see Figure 4A) represents the change in the frequency composition of a signal over time. It is es-timated using spectral analysis methods, such as classical nonparametric methods (fast Fourier transform), mod-

el-based methods (autoregressive or moving average) or time-frequency methods, such as the short Fourier trans-form or wavelet transform [29,30,36-39]. In this type of graphic, one axis (horizontal in the example shown) rep-resents the frequency, whereas the other (vertical) axis represents time. The recording is analyzed in fixed time windows (between 8 and 32 s, usually) from every peri-odogram. The spectral composition for each frequency and time is represented by a color code: red indicates the maximum power, and blue indicates the minimum power for the example shown (codes may vary in other cases). This analytical tool is useful, especially for neurophysi-ologists, allowing the identification of complex spectral changes over time at a single glance.

The other widely used tool is the wavelet analysis (WT). The general expression for a WT is as follows [40]:

*1( , ) ( ) ( )t bW a b f t dtaa

ψ∞

−∞

−= ∫

Equation 8where Ψ* represents the type of wavelet used for the

transformation and the asterisk indicates its complex con-jugate (see [27]). In addition, two parameters are used to define the transformation, such as the scale factor, a (which is > 0), and b, the positional parameter. The main property of this transformation is that it is linear and does not depend on the translation or magnification.

The form or type of wavelet varies [40], the most fre-



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quent forms arethe Haar, Daubeuchies, Morlet and Mexi-can hat forms.

Figure 4: A) time-frequency analysis by PSD. Records (right) and the PSD (left) of two different channels ob-tained during a electrocorticography (ECoG) are shown. Two recordings for each channel obtained in different time windows are shown. The change in the dynamics of the system is well reflected in the spectral composition of the signal. See text for details. B) Time-frequency analysis by WT. The spectral composition is shown for a wide fre-

quency range (y axis) of a typical spike-like element.

To evaluate the properties of equation 8 in the fre-quency domain, the terms W(a, b) and f(t) must be trans-formed and used in equation 5. Subsequently, a convo-lution is performed (see below) with the signal for each wavelet scale (changing the value of a in the equation 8). For subsequent scales, different values are color-coded ac-cording to their amplitude. Thus, by changing the scale, it can be repeatedly applied to the same wavelet signal to cover large frequency ranges.

Therefore, this transformation results in is a more flexible representation of the time-frequency relationship, allowing the use of temporal analysis variable windows [21,41,42], such that large windows allow a finer frequen-cy resolution, whereas smaller time windows are used to increase the resolution of high frequencies. This flexibil-ity, which covers both the lower and higher frequencies, makes the WT an ideal tool for analyzing paroxysmal EEG patterns, such as ictal and interictal discharges. Therefore, a multi-resolution analysis method avoids fixed frequency classical Fourier analysis and similar methods [32] and al-lows the analysis of non-stationary signals, which in prin-ciple is prohibited in classical Fourier analysis.

The proper selection of the type of wavelet and the number of levels to be tested is very important for the de-composition of the signal. For a suitable choice, the domi-nant frequency of the signal must be considered.



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Measures of Interaction Between Two Signals

Before addressing the interaction between signal pairs, we must briefly introduce a mathematical operation called convolution. It is a method to combine two signals, one of which is shifted in time by a defined amount. For signals x(t) and y(t), the convolution (⊗ ) is defined as fol-lows:

0

( ) ( ) ( ) ( )t

x t y t x y t dτ τ τ⊗ = −∫In essence, a signal (x) is multiplied by the offset from

the other version (i.e., y(t-τ)), and this produces a unique value that is the area (i.e., the defined integral) under the resulting function of the product.

However, the convolution calculation is often com-plex, and it is consequently transformed into frequency domain to facilitate calculation. Using equation 5, the transform of the convolution of two time series can be shown to be the product of their transforms (X(ω) and Y(ω))(this proof is simple and omitted here [22]):

{ ( ) ( )} ( ) ( )F x t y t X Yω ω⊗ =

We can now analyze the measures that will determine the relationship between two time series. The correla-

tion determines the dependence between two samples. This measurement can be performed on a single series, a process known as autocorrelation, or between two differ-ent time series, which is called cross-correlation. In both cases, the convolution is essentially calculated as shown above [22]:

( ) ( ) ( )xyC x t y t dtτ τ∞

−∞= +∫ Equation 9

Equation 9 is the expression for the cross-correlation, whereas the expression for autocorrelation is Cxx(τ).

These measures not only identify and quantify a re-lationship between two time signals but also indicate the time point of maximum (or minimum) delay. This meth-od provides information on the directionality of the inter-action and the time required for the transmission of that relationship. Another useful tool for assessing the rela-tionship between two time series is Pearson’s correlation [43], which is defined as follows:

( )( )x y

x y

x y dxdyµ µρ

σ σ

∞

−∞− −

= ∫ ∫ Equation 10

where µx and µY are the mean values of both series, and σx and σY are the standard deviations.

This measure indicates the degree of linear depend-ence between the two series. However, it does not provide



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exactly the same information as the correlation measure-ments based on convolution. In fact, it is not possible to correlate to the same signal (which would be ρxx) because the value will always be a maximum (ρ = 1) in this case. Furthermore, considering the entire signal, a time value, τ, is lacking. Therefore, the spread of information between the two series cannot be determined.

Figure 5 shows an example of the information pro-vided by these tools for two time series that are related to differing degrees, varying from two equal sine series (5A) to two completely random series (Figure 5C).

The Fourier transform of equation 9 provides an in-teresting aspect that links correlation measures to the spectral component of a signal pair. In fact, the following is easily demonstrated for this transformation [22]:

*{ } ( ) ( )xyF C X Yω ω=

In other words, the Fourier transform of the cross-correlation results in the cross-spectrum (Sxy), which is the product of the Fourier transform of both functions, wherein the asterisk indicates the complex conjugate. In the case of autocorrelation, transformation gives the pow-er spectrum or periodogram. This fact is very important because it demonstrates that the power spectrum of a time series represents the same information as its auto-correla-tion.

Finally, we will analyze the ultimate tool of analysis, which is obtained directly from the above and is called coherence (C). It is defined as follows[22]:

2( )

( )( ) ( )

xy

xx yy

SC

S Sω

ωω ω

=

Therefore, coherence is the ratio of the square mod-ulus of the cross-spectrum for two time series x and y divided (or rather,normalized) by auto-spectra (or peri-odograms, namely Sxx and Syy). This measure is advanta-geous because it allows us to determine the synchroniza-tion between the signals for a particular frequency band (therefore, its dependence on the angular frequency is explicitly stated), instead of calculating the degree of syn-chronization for the entire band, such as for the Pearson correlation. This is very important because thecoherence for a particular band of the EEG spectrum is often high, but this high coherence is not observed in the other bands. Figure 6 shows a real example of the use of coherence dur-ing a seizure. We can observe an increase in the coherence for theta band during the post-ictal state compared with the pre-ictal state. This fact allowed us to identify a real ictal state (discarding non-epileptic psychogenic seizure). The patient was then studied using deep electrodes in or-der to ultimately provide an intervention.



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Figure 5: Relationship between time series. A) Equal sine series (absence of random noise), ρ = 1. B) Mixing sig-nals of the sine function more random noise, ρ = 0.328. C) Two signals consist of purely random noise ρ = 0.012. The third row of each section displays the presentation cross-correlation function (in A would be in reality autocorrela-tion). Note that the correlation is maximized in the same time intervals in A and B, although its size is smaller in B.

However, C shows no evident correlation.

Figure 6: Utility of coherence in clinical practice. A) Re-cording of a generalized crisis (blue bar) with anomalous behavior that may appear to be a non-epileptic psycho-genic seizure. The record is full of artifacts by muscle ac-tivity. B) Analysis by bands of pre-ictal of time (red bar) and post-ictal (yellow bar) periods. The black line in each graph indicates the average value of coherence, whereas



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the green dashed lines show the levels of ± 2.5 standard deviation. The horizontal red line shows the level of sig-nificance. The red arrow indicates the increase in coher-ence approximately 5 Hz after the episode, an effect that was not present before the episode (in fact, the remain-ing bands are not affected). C) Average coherence for each scalp electrode (shown as proportional to the diameter of each electrode) and for each frequency band before and after the episode. In the theta band corresponding to the graphic, the average coherence in the electrodes of the

right temporo-parietal region is increased.

ConclusionThe various tools for time series analysis in general

and for neurophysiological EEG recordings in particular have experienced extraordinary development during the last quarter century. Not only have the field applications of these tools increased, but also several new techniques that promise to further expand this range, such as techniques that address the functional connectivity of EEG [44-48] or synchronization measures [49-51], have also been de-scribed.

However, a significant gap exists between the theoret-ical and scientific development of the field and its clinical application. In fact, these tools (other than spectral meas-urements and correlation) are not routinely used in clini-cal practice, despite the considerable increase in publica-

tions describing their use. We attribute this lack of use to several reasons: i) the lack of interest of commercial firms in the generation of easily programmable and accessible applications to implement these tools; ii) the mathemati-cal complexity of some of these tools, which sometimes requires MATLAB®, R or other languages and hinders their implementation by clinicians; and iii) mainly, the general lack of confidence of neurophysiologists when us-ing these tools. The latter is likely the strongest limitation to the clinical use of qEEG. For a neurophysiologist faced a diagnostic problem, knowledge and confidence in what he/she is doing is essential, given the clinical, ethical and legal implications of his act. Therefore, a trusted method of analysis, such as conventional EEG analysis, is prefer-able to ensure solid and informed knowledge of treatment and diagnosis.

However, scientific articles demonstrate the diagnos-tic and prognostic utility of qEEG [5,9,11,16,17], which encourages its implementation. Therefore, we believe a deeper understanding of these mathematical tools for an increasingly large group of neurophysiologists is essential to begin using quantified analyses for clinical practice. Thus, we will soon achieve a quantitative change in the diagnostic and prognostic value of EEG, a greater inter-observer coincidence and an undeniable increase in the scientific basis of the theory of EEG, which is the main knowledge gap in this scientific field [52]. Ultimately, these measures will increase our understanding of the brain.



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AcknowledgmentsThis work was financed by the Ministerio de Sanidad

FIS PI12/02839 and was partially supported by FEDER (FondsEuropéen de /Développement/.Économique/ et /Régional) and PIP 11420100100261 CONICET (Consejo Nacional de Investigaciones /Científicas y Técnicas).

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